vol 30 issue 1.pdf

Journal for Research in Mathematics Education1999, Vol. 30, No. 1, 2

Relationships Between Research andthe NCTM Standards: An Introduction

Deborah SchifterChair of the NCTM Research Advisory Committee

Once again, the fundamentals of mathematics education are the subject of anational discussion—indeed, of sometimes heated debate—much of it crystallizedby the appearance, in 1989, of the NCTM Standards. In the news media and on theinternet, at PTA meetings and school board hearings, one now routinely encountersexpressions of enthusiasm or concern about the reforms suggested in that document.

Willy-nilly, these discussions inevitably implicate claims about what researchhas or has not shown, should or should not probe, but these claims are not nec-essarily consistent with either what has been done or even what is possible. Yet,among all the contesting voices, scarcely heard are those of the men and womenwho actually conduct mathematics education research.

The reasons for this are no doubt varied, but one, in particular, we, as practi-tioners, can do something about: Mathematics education researchers are unusedto addressing audiences outside the field. Tightly focused on exchanges withpeers, we share assumptions, language, references, goals, and concerns thatmake our discussions opaque to outsiders. These understandings are frequentlyso taken for granted by us that we may not always recognize how they compli-cate efforts to communicate with nonspecialists.

As a step toward redressing this situation, the NCTM Research AdvisoryCommittee asked James Hiebert to produce, for JRME, an abridged version ofhis chapter, “Relationships Between Research and the NCTM Standards,” pre-pared for a volume to be published by NCTM. Although admittedly crafted toreach a larger public, Hiebert’s arguments and insights are nonetheless of inter-est to his colleagues. But there are two other, overriding reasons for placing thispaper in a venue designed to serve the research community:

1. The article is a model for researchers who wish to reach an interested,non–research-oriented public. Especially instructive is the use Hiebert makes ofexamples and analogies intended to help others understand the role that researchcan play in shaping educational policy.

2. The article is a resource that can form the basis of discussion with policymakers, journalists, school administrators, teachers, and concerned parents.

A more complete development of Hiebert’s ideas will be found in an anthologyof articles to be published later this year as part of a project directed by JeremyKilpatrick. The intended audience for this volume is not confined to the researchcommunity; its authors will set out the theory and research findings relevant to thePrinciples and Standards for School Mathematics, to be published in the year 2000.

This material may not be copied or distributed electronically or in any other format without written permission from NCTM. Copyright © 1999 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

Relationships Between Researchand the NCTM Standards

James Hiebert, University of Delaware

The current debates about the future of mathematics education often lead to confusion aboutthe role that research should play in settling disputes. On the one hand, researchers are calledupon to resolve issues that really are about values and priorities, and, on the other hand,research is ignored when empirical evidence is essential. When research is appropriatelysolicited, expectations often overestimate, or underestimate, what research can provide. Inthis article, by distinguishing between values and research problems and by calibrating appro-priate expectations for research, I address the role that research can and should play in shap-ing standards. Research contributions to the current debates are illustrated with brief sum-maries of some findings that are relevant to the standards set by the NCTM.

Key Words: All levels; Policy issues; Reform in mathematics education; Research issues;Review of research; Teaching practice

What is the relationship between what is known from research in mathematicseducation and what is expressed in the NCTM Standards?1 Can we say, forexample, that research supports the Standards? These questions have becomeincreasingly important as debates about reform reach fever pitch. They are fairquestions, even though they do not have simple answers. The answers are notsimple because (a) standards, in any field, are rarely based solely on research, sothe connection between research and standards is never straightforward; and (b)research in mathematics education does not shine equally brightly on all aspectsof the NCTM Standards, so we cannot provide blanket statements.

Journal for Research in Mathematics Education1999, Vol. 30, No. 1, 3–19

Preparation of the more complete version of this article, to be published in an editedbook that will provide a research companion to the NCTM Standards 2000 Initiative’sPrinciples and Standards for School Mathematics, was supported by the Project on theFoundations for School Mathematics funded, in part, by the National Science Foundation(Grant ESI 9727890). Some of the ideas contained in the article can be traced to discus-sions during a working conference, held in Atlanta in March 1998, organized by JeremyKilpatrick and supported by the Project. Thanks to Deborah Schifter, Chair of the ResearchAdvisory Committee (RAC) of the National Council of Teachers of Mathematics, mem-bers of the RAC, and Judy Sowder, for their comments on an earlier draft of the article.

1 The phrase “NCTM Standards,” or just “Standards” (capitalized), will be used for the NationalCouncil of Teachers of Mathematics recommendations for K–12 curriculum, teaching, and assess-ment contained in the initial three-volume set (Curriculum and Evaluation Standards for SchoolMathematics [1989], Professional Standards for Teaching Mathematics [1991], and AssessmentStandards for School Mathematics [1995]) and in the revised volume Principles and Standards forSchool Mathematics (draft, 1998), all published in Reston, VA, by the NCTM.


My goal in this article is to clarify what we can expect from research and thento review, briefly, what we can say, from research, about the Standards. The con-clusion will be that, where relevant research exists, the Standards are consistentwith the evidence. Said another way, the Standards do not violate the relevantfindings from research on teaching and learning mathematics. But knowing theshort answer is not enough. In matters as complex as connecting educationalstandards with research, it is as important to understand the process throughwhich such a conclusion can be reached as the conclusion itself.

WHAT SHOULD WE EXPECT FROM RESEARCH?

How nice it would be if one could look at the research evidence and decidewhether the Standards are right or wrong. This would make decisions simple andbring an end to the debates about the direction of mathematics education in theUnited States. Is this impossible? After all, can’t those in other professions makesuch clear connections? Actually, they can’t. Standards and research rarely havea clear relationship. To understand the reason, we need to consider some of thelimitations of research.

Some Things We Should Not Expect From Research

Standards are not determined by research. Standards in mathematics educa-tion, like those in other fields, are statements about priorities and goals. In educa-tion, they are value judgments about what our students should know and be ableto do. They are chosen through a complex process that is fed by societal expecta-tions, past practice, research information, and visions of the professionals in thefield. The process is similar to the one that operates in selecting standards in otherprofessional fields. Research can influence the nature of the standards that areadopted, but, in the end, research is not the sole basis for selection of the stan-dards. Standards, ultimately, are statements about what is most valued.2

Our society is now in the midst of a crucial debate about goals and values. Itis important that the debate continue until a consensus is reached about our goalsfor students. Research can inform the debate, but the reverse is equally true—theselection of standards signals what research is most relevant. If the paper-and-pencil computation of square roots is omitted from the standards, for example,then research that shows one method of teaching written computation of squareroots is more effective than another probably will be ignored. If ability to inventprocedures to solve new problems is emphasized in the standards, then theresearch on students’ creative problem solving is of great interest.

One of the current dilemmas revolves around written computation skills. Thedebate has not yet developed a clear consensus about their importance. This lack

4 Research and Standards

2 See NCTM’s “Statement of Beliefs” (posted on their website, www.nctm.org) for a descriptionof basic values that underlie the Standards.

of consensus is understandable given the rapid changes in mathematical compe-tencies that are important in the workplace and the increasing availability ofcomputational technologies. But the current uncertainty has implications forinterpreting research. For those who believe that high efficiency with writtencomputation still is essential, the research showing Teaching Method A producesgreater efficiency than Method B is significant; for those who believe that mod-erate efficiency with written computation is sufficient, such research is, at most,of moderate interest. Debates about what the research says will not settle theissue; only debates about values and priorities will be decisive. Until the valueissue is settled, it will be difficult to find common ground for examining theresearch.

What is “best” cannot be proven by research. There is increasing pressure toprove, scientifically, what are the best curricular and pedagogical decisions inmathematics. Should we teach in this way or that way? Should we use this text-book series or that textbook series? Scientific research is looked to for the solu-tion because, after all, science has taken us to Mars with the Pathfinder and hashealed painful backs with incredibly precise arthroscopic surgeries.

Looking to scientific research is a good thing; the more reliable informationwe have, the better will be our decisions. But, in every field, science has its lim-its. Consider the requirements for a healthy lifestyle. Standards are proposed byhealth professionals for living a healthy life—diet, exercise, and so on. Butmedical research does not prove that these standards are the best ones. Is meatgood for you or not? Is it better to use butter or margarine? Should we haveexactly seven servings of fruits and vegetables every day, or would six beenough? These simple sounding questions do not have simple answers. Thereare too many factors that influence the outcomes: how much exercise we get,how much we weigh, our genetic make-ups and past histories, our metabolicrates, and so on. It would be impossible to control all these factors to prove thata certain diet is best.

We have a similar situation in education. Most outcomes are influenced bymore factors than we can identify, let alone control. Does this mean thatresearch is a waste of time? Not at all. Just because researchers cannot provewhether a particular decision is the best one does not mean that research is irrel-evant. In complex environments, such as our bodies and school classrooms,there is a special relationship between research and decision-making. Decisionsoften are based on probability estimates, and research data help us estimate thelikelihood of success. The clearer the results, the more confident we are that weare making good decisions. We make decisions with levels of confidence, notwith certainty.

Here is a simple example. Is it better for students to use calculators or not touse calculators in elementary school? This is a simple enough question and onethat is receiving heated debate. Shouldn’t we be able to prove whether childrenshould use calculators, one way or another? Suppose we try. First, we need to

5James Hiebert

decide what we mean by better and how to measure this construct. Does bettermean that students, at the end, understand mathematics more deeply, solve chal-lenging problems more effectively, execute written computation proceduresmore quickly, like mathematics more? Deciding what better means is not a triv-ial task. It requires being clear about values and priorities. Suppose, for the sakeof argument, that we mean “execute written computation procedures more accu-rately and quickly.” Many people would guess that, if this is the valued outcome,the no-calculator classroom would be the best.

How could we test this hypothesis? How would we set up a fair comparisonbetween the calculator and the no-calculator treatments? A reasonable approachwould be to develop, with our desired learning goal in mind, the best instruc-tional program we could think of with the calculator and the best program with-out the calculator. Using this approach would mean that students in the two pro-grams probably would be completing different tasks and engaging in differentactivities, because different activities are possible with and without the calcula-tor. But now we have a problem because we will not know what caused the dif-ferences in students’ learning. Was it the calculator, the other differencesbetween the instructional programs, or the interactions? Maybe we could solvethis problem by keeping the instructional programs identical; just plop the cal-culators into one set of classrooms and not the others. But into which instruc-tional program should the calculators be plopped—the one designed to maximizethe benefits of the calculator or the one designed to function without calculators?Neither choice is good, because the omitted program would not get a fair test.Maybe we should split the differences. But then we have an instructional pro-gram that no one would intentionally design.

Does this research design problem mean that all the studies on using calcu-lators, and there have been many, are uninterpretable? No. But it does meanthat no single study will prove, once and for all, whether we should use cal-culators. The best way to draw conclusions regarding issues like this is toreview the many studies that have been done under a variety of conditions andlook for patterns in the results. Perhaps studies in the early grades show onekind of pattern and studies in the later grades another pattern. Or, perhapsstudies using the calculators in one way show one pattern of results and stud-ies using the calculators in another way show another pattern. As it happens,this kind of review of calculator use has been done and a partial and tentativeanswer is available (Hembree & Dessart, 1986). The results indicate thatusing calculators, along with common pencil-and-paper activities, does notharm students’ skill development and supports increased problem-solvingskills and better attitudes toward mathematics. This finding does not mean, bythe way, that this is what will be found in every classroom, but it does indi-cate two things: (a) A decision to use calculators wisely during mathematicsinstruction can be made with some confidence; and (b) when calculators areblamed for damaging students’ mathematical competence, it would be usefulto check the full instructional program—the problem is likely to be a poor use


of calculators, or a feature of instruction unrelated to calculators, and not thecalculators themselves.3

If researchers cannot prove that one course of action is the best one, it followsthat researchers cannot prescribe a curriculum and a pedagogical approach for allstudents and for all time. Decisions about curriculum and pedagogy are alwaystentative, made with some level of confidence, a level that changes over timewith new information and changing conditions. Research can, and should, play acritical role in helping educators make informed decisions and set the levels ofconfidence, but we cannot look to research for clear prescriptions.

Research cannot imagine new ideas. Improving the learning opportunities forstudents depends, in part, on coming up with new ideas—new ways of teaching,new curriculum materials, new ways of organizing schools. Generating newideas depends on the creative acts of the human mind. Research, by itself, is nosubstitute. Of course, the research process can place people in position to seethings in a new way and imagine new possibilities, but it is the individual’s inter-pretation, not the research evidence alone, that generates the new ideas.

Suppose we wanted to develop a better method for teaching fractions. We couldbegin by reviewing the research evidence from previous experiments on teachingfractions. We might be able to tell which methods have worked best, but to imag-ine an even more effective approach we would need to use other things we knowabout students’ learning, about classroom processes, about mathematics, and soon. New ideas might be triggered by reading previous research and conductingstudies ourselves, but forming the new ideas requires human creativity.

It is important to remember that the research data tell us something only aboutthe teaching methods or curriculum materials that have been tested. Often, class-room experiments compare a new method with a traditional or “control” method.When the results favor the new method, investigators are tempted to claim thatthe new method should be adopted. But the power of the results is only as greatas the control method against which the new method was compared. It may betrue that, of the two, the new method is more effective, but there may be a thirdmethod that is even more effective.

A good example of this situation can be found in past descriptions of how expertteachers differ from novices (Good, Grouws, & Beckerman, 1978; Leinhardt,1986). Experts were found to teach quickly paced lessons, cover more problems,and ask more recall than explanation questions. Does this result mean that weshould train all teachers to teach in this way? If the two alternatives included in thesestudies were the only options, maybe so. But, suppose the goals change from a focus

7James Hiebert

3 Many of the claims that calculators undermine students’ mathematics learning seem to be prompt-ed by anecdotes and stories of calculators used poorly. Some of these claims, such as those made byDavid Gelernter in his New York Post column (1998), have attracted a good deal of public attention.If these critiques promote a debate about the goals of mathematics education, they could be useful.But, the argument that methods or materials should be eliminated if they can be used poorly is notpersuasive, even when supported by anecdotes; very little would remain in the classroom.Systematically collected data, from large numbers of trials, are much more informative.

on efficient execution of written computation procedures to a balance between abroader set of skills and conceptual understanding, and suppose that there are otherapproaches, developed more recently, that help students achieve these goals evenbetter? Then we need to consider seriously these alternative approaches.

Some Things We Can Expect From Research

Before summarizing what can be learned from research about the effects of dif-ferent instructional approaches, we must continue calibrating our expectations.After all, research is not filled just with limitations; it holds enormous potential.

Research can influence the nature of standards. Although research cannot be thebasis for making the final decision about standards, mathematics education is filledwith examples of ways that research can influence the nature of standards. In theearly 1900s, mathematics was viewed as a valuable subject because learning math-ematics was believed to exercise the mind, and the mind, like a muscle, needed exer-cise to become strong. E. L. Thorndike (1922; Thorndike & Woodworth, 1901)warned educators that the idea of mind as muscle was a poor analogy. Students’minds did not appear to become stronger from studying mathematics (they did notbecome smarter in other areas); they simply learned mathematics. Standards todayrarely prescribe mathematical activity in order to exercise the mind. Thorndike’sresearch encouraged a move away from these kinds of standards.

Research on learning also can have the opposite effect—it can document newpossibilities and draw attention toward new standards. Research on young chil-dren’s ability to solve simple arithmetic story problems before instruction pro-vides one example (Carpenter, Moser, & Romberg, 1982). Standards increasing-ly emphasize young students’ inventions of arithmetic procedures because, inpart, we know they are capable of such inventions.

Research in the subject itself also can shape the kinds of standards that areselected. For example, research and development within mathematics has openedup vast new areas of study, such as coding theory and combinatorics. Relatedtopics in discrete mathematics are now found in the elementary and secondarycurricula and are identified in the NCTM Standards.

Research influences the nature of standards only when the implications ofresearch are valued. Mathematical inventions by students are not included in theStandards simply because students are capable of inventing; they are includedbecause an additional value judgment has been made—that invention is animportant mathematical process. Topics in discrete mathematics are included notjust because they are there but because a judgment has been made about theirimportance in the field of mathematics.

Research can document the current situation. Research can provide informa-tion about how we are doing at the moment—how we are teaching, what cur-riculum materials we are using, and how students are learning. Although this isan obvious role for research, it often is underutilized. Take the case of California(Stigler, 1998). In 1995, faced with falling mathematics achievement scores, the


state superintendent of public instruction appointed a task force to study the sit-uation and propose solutions. Why, if California’s curriculum frameworks hadreceived so much acclaim, were students’ achievement scores so low?Discussion at the task force meetings soon turned to the frameworks. Were theyto blame? Some members thought so; some members defended them.

Lost in those early debates in California was the fact that no information wasavailable on the extent to which the frameworks were influencing mathematicsinstruction in the state’s classrooms. Without knowing what was happening inclassrooms, how could the effectiveness of the frameworks be assessed? This storyis not meant to single out California; few, if any, states regularly collect informa-tion on what is happening inside classrooms. The absence of data collection isunfortunate because without information about the current situation, we makeunwitting mistakes and produce the pendulum swings often evident in education.

Research can document the effectiveness of new ideas. In addition to usingresearch to apply the brakes, research also can be used to step on the accelerator.Research can document what students can learn under what kinds of conditions.Research can show that students can reach certain goals and that some kinds ofinstruction are especially effective in helping them get there. For example, givenappropriate instruction, students at particular ages can learn more about proba-bility (Jones, Thornton, Langrall, Johnson, & Tarr, 1997) or engage in moredeductive reasoning (Fawcett, 1938; King, 1973; Yerushalmy, Chazan, &Gordon, 1987) than they do now. Research of this kind can help to verify thatimprovements in particular areas are feasible, that specific visions of the profes-sionals in the field are reasonable.

By the same token, research also can show that new ideas are untenable. Visionsof what is possible for students might be endorsed enthusiastically by experts butprove to be misinformed and unrealistic. What is crucial is that carefully collectedempirical data be used to distinguish between the new ideas that can be imple-mented effectively and those that can’t. Without such information, we can engagein debates, like those of the California task force, that have no resolution.

An increasingly common debate is illustrated by this excerpt from the April26, 1998, edition of the Riverside Press-Enterprise newspaper:

High failure rates and concerns that students are not learning the math skills theyneed has prompted a third of Inland area high schools trying a new college-prep pro-gram to drop it. Riverside’s Poly High School discontinued College PreparatoryMathematics [CPM] in June after only 27 percent of the Algebra I students earned aC or better. One semester after scrapping the program, the passing rate went up to 42percent. (Sharma, 1998)

As the story continues, it becomes clear that there is no consensus among thelocal stakeholders about whether or not CPM is a failure nor about why it is hav-ing the reported effect. Many opinions are expressed, such as that NCTM-inspired programs like this are doomed to fail, but there are no clear conclusions.Of course, there can be no clear conclusions because no information was col-

9James Hiebert

lected systematically about what was going on in classrooms. We do not knowhow the program was being implemented, so there is no way to evaluate its effec-tiveness.4 Unfortunately, many of the claims and counter-claims about theeffects of new programs are based on these kinds of stories, without the benefitof real information.

Research can suggest explanations for successes and failures. Researchers canprobe beneath the surface and collect information to help us understand the situ-ation and prevent us from making mistakes and engaging in fruitless debates.Consider a recent report by investigators of the QUASAR project, a large-scaleeffort to improve the mathematics education programs of inner-city middleschools. In some QUASAR schools, students’ achievement was not rising asexpected. It would have been easy to conclude that the reform programs were noteffective for some students. But the investigators took a second look, comparingschools in which students’ achievement was increasing with schools in which itwas not (Parke & Smith, 1998). What they found were major differences in thestaffing situations in the two kinds of schools. In the less successful schools, therate of teacher and principal turnover was very high. This turnover resulted in arelatively weak implementation plan and fewer and more superficial changes inclassroom instruction. So, it would be a mistake to conclude that the school’sprogram itself was ineffective; instead, one can conclude only that a weak imple-mentation was ineffective and that this can occur when staff do not have the timeto learn new practices.

WHAT CAN WE LEARN FROM RESEARCH?

The guidelines for what we can expect from research help to interpret theresearch findings that are relevant for the NCTM Standards. The following obser-vations summarize briefly what we know from applying our research machin-ery—taking advantage of what it can do and accounting for its limitations.5


4 Beyond the absence of information about classroom practice, there are other missing elementsin this story, elements that are needed to interpret the “facts.” For example, what does it mean forthe passing grades a teacher assigns to move from 27% to 42%? Are students learning more? Maybethey are, or maybe they are being tested on easier material.

5 Summarizing briefly a large body of research is not an easy task. One is faced with an immedi-ate problem: Which studies should be consulted? One option would be to include only reports of tra-ditional scientific experiments. A team of researchers made this decision in their March 1998 reportto the California State Board of Education: “Review of High Quality Experimental MathematicsResearch,” was prepared by R. C. Dixon, D. W. Carnine, D.-S. Lee, J. Wallin, The National Centerto Improve the Tools of Educators, and D. Chard. The basic issue is how one measures high-quali-ty research. A number of helpful discussions of this thorny question are already available. See, forexample, the presentations in Part V (“Evaluation of Research in Mathematics Education”) inMathematics Education as a Research Domain: A Search for Identity, edited by A. Sierpinska andJ. Kilpatrick (1998), including chapters by F. K. Lester and D. V. Lambdin (“The Ship of Theseusand Other Metaphors for Thinking About What We Value in Mathematics Education Research”)and by G. Hanna (“Evaluating Research Papers in Mathematics Education”); see also Kilpatrick, J.(1993). Beyond face value: Assessing research in mathematics education. In G. Nissen & M.

The Current State of Mathematics Teaching and Learning

What is the current state of classroom teaching? It may surprise some peopleto learn that we have a quite consistent, predictable way of teaching mathemat-ics in the United States and that we have used the same basic methods for near-ly a century (Fey, 1979; Hoetker & Ahlbrand, 1969; Stake & Easley, 1978;Stigler & Hiebert, 1997; Stodolsky, 1988; Weiss, 1978). Here is an often citedaccount from a researcher’s observations of mathematics lessons:

First, answers were given for the previous day’s assignment. A brief explanation,sometimes none at all, was given of the new material, and problems were assignedfor the next day. The remainder of the class was devoted to students working inde-pendently on the homework while the teacher moved about the room answeringquestions. The most noticeable thing about math classes was the repetition of thisroutine. (Welch, 1978, p. 6)

Readers may recognize their own school mathematics experience in this descrip-tion; many people do.

The same method of teaching persists, even in the face of pressures to change.After a decade of mathematics reform in the 1960s, the Conference Board of theMathematical Sciences (1975) found that “Teachers are essentially teaching thesame way they were taught in school” (p. 77). And, in the midst of currentreforms, the average classroom shows little change (Dixon et al., 1998; Stigler &Hiebert, 1997).

Most characteristic of traditional mathematics teaching is the emphasis on teach-ing procedures, especially computation procedures. Little attention is given tohelping students develop conceptual ideas, or even to connecting the proceduresthey are learning with the concepts that show why they work. In the lessons includ-ed in the video study of the Third International Mathematics and Science Study(TIMSS), for 78% of the topics covered during the eighth-grade U.S. lessons, pro-cedures and ideas were only demonstrated or stated, not explained or developed.And 96% of the time that students were doing seatwork they were practicing pro-cedures they had been shown how to do (Stigler & Hiebert, 1997).

Coupled with this information on teaching practices, the TIMSS data alsoshow that the traditional U.S. curriculum is relatively repetitive, unfocused, andundemanding (Schmidt, McKnight, & Raizen, 1996; Silver, 1998). Comparedwith the curricula in other countries, the U.S. curriculum provides few opportu-nities for students to solve challenging problems and to engage in mathematicalreasoning, communicating, conjecturing, justifying, and proving. Much of the

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Blomhøj (Eds.), Criteria for scientific quality and relevance in the didactics of mathematics (pp. 15-34). Roskilde, Denmark: Danish Research Council for the Humanities. Three criteria that were keptin mind for this summary of research were (a) possesses educational significance and scientific merit,(b) is directed toward understanding teaching and learning in classrooms, and (c) converges towarda conclusion using a variety of methodologies. In addition, most of the studies were conducted in theUnited States. Many studies that fit the criteria have been conducted in other countries, but there isalways the question of whether something that works well in one culture can be imported into anoth-er culture.

curriculum deals with calculating and defining, and much of this activity is car-ried out in a rather simplistic way.

What are students learning from traditional instruction? On the basis of themost recent National Assessment of Educational Progress (NAEP)6, we knowthat almost all students learn to add, subtract, multiply, and divide whole num-bers, and the majority learn to do very simple arithmetic with fractions, decimals,and percents. For example, in eighth grade, 91% of students added three-digitnumbers with regrouping, 80% completed a long-division problem, 83% round-ed a decimal number to the nearest whole number, and 58% found the percent-age of a number (Kouba & Wearne, in press; Wearne & Kouba, in press).

We also know, however, that students’ knowledge and skills are very fragileand apparently are learned without much depth or conceptual understanding.This problem becomes evident when we study performance on related items thatrequire students to extend these skills, reason about them, or explain why theywork. For example, only 35% of eighth graders identified how many pieces wereleft if 65 pieces of candy were divided equally among 15 bags with each bag hav-ing as many as possible (Kouba & Wearne, in press). Multistep problems posean even greater challenge. For example, 8% of eighth graders solved a multistepproblem on planning a trip that required adding miles, finding distance frommiles per gallon, and calculating a fractional part of the trip (Wearne & Kouba,in press).

Conclusions. The data confirm one of the most reliable findings from researchon teaching and learning: Students learn what they have an opportunity to learn.In most classrooms, students have more opportunities to learn simple calculationprocedures, terms, and definitions than to learn more complex procedures andwhy they work or to engage in mathematical processes other than calculation andmemorization. Achievement data indicate that is what they are learning: simplecalculation procedures, terms, and definitions. They are not learning what theyhave few opportunities to learn—how to adjust procedures to solve new prob-lems or how to engage in other mathematical processes.

These achievement data indicate that the traditional teaching approaches aredeficient and can be improved. It is curious that the current debate about thefuture of mathematics education in this country often is treated as a comparisonbetween the traditional “proven” approaches and the new “experimental”approaches (Schoenfeld, 1994). Arguments against change sometimes claim thatit is poor policy, and even unethical, to implement unproven new programs. LeeHochberg, a reporter for Oregon Public Broadcasting, recently had this to sayduring a story on reform-minded mathematics teaching for the PBS NewsHourwith Jim Lehrer: “Although there never was any scientific research conducted on


6 NAEP is the best source of information on the achievement of U.S. students because the itemsare matched specifically to the U.S. curriculum, and the sampling design ensures a large and repre-sentative sample of students.

the effectiveness of this style of teaching, the NCTM hoped that it would betterprepare American students for the modern adult workplace” (May 11, 1998).Expressing a similar sentiment, a parent in Bloomfield Hills, Michigan, removedher son from a reform mathematics program because “I like going with what Iknow is proven. I just don’t want to take the chance” (Bondi, 1998).

The commendable part of these arguments is that they claim to promoteresearch-based decision making. That part certainly is appropriate and, in fact, isthe reason for this article. But, presuming that traditional approaches haveproven to be successful is ignoring the largest database we have. The evidenceindicates that the traditional curriculum and instructional methods in the UnitedStates are not serving our students well. The long-running experiment we havebeen conducting with traditional methods shows serious deficiencies, and weshould attend carefully to the research findings that are accumulating regardingalternative programs.

How Effective Are the New Programs?

What are the new teaching methods? Summarizing the alternative methods ofteaching mathematics that are being developed around the country is nearlyimpossible because there are so many programs. Even if we examine only thosethat have been inspired by the Standards and those that are trying to translate therecommendations into practice, it is difficult to lump them into one description.It is possible, however, to focus on one area of the curriculum in which consid-erable work has been done in designing and testing alternative instructional pro-grams—primary-grade arithmetic (Carpenter, Fennema, Peterson, Chiang, &Loef, 1989; Cobb et al., 1991; Fennema et al., 1996; Fuson & Briars, 1990;Hiebert & Wearne, 1992, 1993, 1996; Hiebert et al., 1997; Kamii, 1985, 1989;Villaseñor & Kepner, 1993; Wood & Sellers, 1996). Because many of the inves-tigators are engaged in independent research programs, there are differences inthe alternative instructional programs that are implemented in classrooms. But,there also is a convergence toward some significant similarities, and it is thisconvergence that is of particular interest.

The features that characterize many of the alternative programs in primary-grade arithmetic include the following:

• Build directly on students’ entry knowledge and skills. Many students enterschool being able to count and solve simple arithmetic problems. Alternativeprograms take advantage of this ability by gradually increasing the range ofproblem types and the sizes of the numbers.

• Provide opportunities for both invention and practice. Classroom activityoften revolves around solving problems that require some creative work bythe students and some practice of already learned skills. For example, secondgraders may have been subtracting numbers like 345 – 127 and then are askedto work out their own methods for subtracting 403 – 265 (a problem with a 0in the subtrahend).

13James Hiebert

• Focus on the analysis of (multiple) methods. Classroom discussion usuallycenters on the methods for solving problems, methods that have been present-ed by the students or the teacher. Methods are compared for similarities anddifferences, advantages and disadvantages.

• Ask students to provide explanations. Students are expected to present solutionsto problems, to describe the methods they use, and to explain why they work.

There are research reports of alternative instructional programs in other areasthat share these features. These include, for example, the comprehensive prob-lem-solving program for middle school students commonly referred to as the“Jasper Project” (Cognition and Technology Group at Vanderbilt [CTGV],1997) as well as smaller scale research programs on students’ learning of com-mon fractions (Behr, Wachsmuth, Post, & Lesh, 1984; Mack, 1990), decimalfractions (Wearne & Hiebert, 1988, 1989), percents (Moss & Case, in press), andcalculus (Heid, 1988; Palmiter, 1991).

What are students learning from alternative programs? Because the goals ofthe alternative programs are somewhat different from those of traditional pro-grams, comparing students’ achievement in the two kinds of programs must bedone carefully. The following conclusions pertain mostly to elementary schoolstudents’ learning of arithmetic, for which the teaching methods in the alterna-tive programs show considerable similarity.

• Instructional programs that emphasize conceptual development, with the goalof developing students’ understanding, can facilitate significant mathematicslearning without sacrificing skill proficiency.

It should come as no surprise that instruction can be designed to promote deep-er conceptual understanding. If students have more opportunity to constructmathematical understandings, they will construct them more often and moredeeply. The question is, at what cost? Will they fail to master other knowledgeor skills that we value? The results show that well-designed and implementedinstructional programs can facilitate both conceptual understanding and proce-dural skill (Carpenter et al., 1989; Cobb et al., 1991; CTGV, 1997; Hiebert &Wearne, 1993, 1996; Hiebert et al., 1997; Kamii, 1985, 1989; Knapp, Shields, &Turnbull, 1992; Mack, 1990; Moss & Case, in press; Wearne & Hiebert, 1988;Wood & Sellers, 1996).

• Students learn new concepts and skills while they are solving problems.

The traditional approach to solving problems in U.S. classrooms is to teach aprocedure and then assign students problems on which they are to practice theprocedure. Problems are viewed as applications of already learned procedures.The alternative instructional programs take a different view. The theory on whichthese programs are based says that students can acquire skills while they devel-op them to solve problems. In fact, the development of the skill, itself, can betreated as a problem for students to solve. Evidence for students’ conceptual andprocedural learning in these programs is presented in the reports cited above; a


summary of these findings is presented in Hiebert et al., 1996.

• If students over-practice procedures before they understand them, they havemore difficulty making sense of them later.

A long-running debate has been whether students should practice proceduresfirst and then try to understand them or should understand the procedures beforepracticing them. The best evidence suggests that if students have memorized pro-cedures and practiced them a lot, it is difficult for them to go back and under-stand them later (Brownell & Chazal, 1935; Mack, 1990; Resnick & Omanson,1987; Wearne & Hiebert, 1988).

Explaining the Lack of Implementation

If it is true that instructional programs can be designed to facilitate more ambi-tious learning goals for students, why don’t we see them more often? Why do weread stories of failed programs, like the story carried in the Riverside Press-Enterprise (Sharma, 1998)? One possibility is that the alternative programs,which show great promise in research settings, are not implemented effectivelywhen adopted by schools and districts. One reason for this situation is simple butunder-appreciated: It is difficult to change the way we teach. The new, moreambitious instructional programs require teachers to make substantial changes.This change doesn’t happen automatically; it requires learning. And learning forteachers, just as for students, requires an opportunity to learn. But most teachershave relatively few opportunities to learn new methods of teaching (Cohen &Hill, 1998; Lord, 1994; O’Day & Smith, 1993; Weiss, 1994).

Research on teacher learning shows that fruitful opportunities to learn newteaching methods share several core features: (a) ongoing (measured in years)collaboration of teachers for purposes of planning with (b) the explicit goal ofimproving students’ achievement of clear learning goals, (c) anchored by atten-tion to students’ thinking, the curriculum, and pedagogy, with (d) access to alter-native ideas and methods and opportunities to observe these in action and toreflect on the reasons for their effectiveness (CTGV, 1997; Cohen & Hill, 1998;Elmore, Peterson, & McCarthey, 1996; Fennema et al., 1996; Franke, Carpenter,Fennema, Ansell, & Behrend, in press; Little, 1982, 1993; Schifter & Fosnot,1993; Stein, Silver, & Smith, in press; Stigler & Hiebert, 1997; Swafford, Jones,& Thornton, 1997). Because most classroom teachers in the United States do notyet have learning opportunities of this kind, it is not surprising that promisingalternative methods are not widely implemented.

CONCLUSIONS

The Standards proposed by NCTM are, in many ways, more ambitious thanthose of traditional programs. On the basis of beliefs about what students shouldknow and be able to do, the Standards include conceptual understanding and theuse of key mathematical processes as well as skill proficiency. The best evidence

15James Hiebert

we have indicates that most traditional programs do not provide students withmany opportunities to achieve these additional goals and, not surprisingly, moststudents do not achieve them. Alternative programs can be designed to providethese opportunities, and, when the programs have been implemented with fideli-ty for reasonable lengths of time, students have learned more and learned moredeeply than in traditional programs. Although the primary evidence comes fromelementary school, especially the primary grades, there is no inconsistent evi-dence. That is, there are no programs at any level that share the core instruction-al features, have been implemented as intended for reasonable lengths of time,and show that students perform more poorly than their traditionally taught peers.

But this is not the end of the story. Alternative programs, consistent with theNCTM Standards, often require considerable learning by the teacher. Withoutnew opportunities to learn, teachers must either stick with their traditionalapproaches or add on a feature or two of the new programs (e.g., small-groupactivity) while retaining their same goals and lesson designs. On the basis of theavailable evidence, it is reasonable to presume that it is these practices that oftenare critiqued as not producing higher achievement.

What we have learned from research now brings us back to an issue of values.We now know that we can design curriculum and pedagogy to help studentsmeet the ambitious learning goals outlined by the NCTM Standards. The ques-tion is whether we value these goals enough to invest in opportunities for teach-ers to learn to teach in the ways they require.

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Author

James Hiebert, H. Rodney Sharp Professor of Education, University of Delaware, School ofEducation, Newark, DE 19716; [email protected]

19James Hiebert

Gender Differences in First-GradeMathematics Strategy Use:

Parent and Teacher Contributions

Martha Carr, Donna L. Jessup, and Diana FullerUniversity of Georgia

In this study we examined how parents and teachers influence the development of gender dif-ferences in mathematics strategy use in the 1st grade. Children were interviewed about their strat-egy use, their metacognitive knowledge about specific strategies, and their perceptions ofparents’ and teachers’ attitudes toward various strategies. Parents and teachers completedquestionnaires about the types of strategy and metacognitive instruction they provided. Previousresults (Carr & Jessup, 1997) were replicated with boys correctly using retrieval during the 1stgrade more than girls and girls correctly using overt strategies more than boys. Boys were influ-enced by the belief that adults like strategies indicating ability and by teacher instruction on retrievalstrategies. Girls’ strategy use was not related to perceived adult beliefs or actions.

Keywords: Addition, subtraction; Children’s strategies; Gender issues; Metacognition; Parents’role; Social factors; Teaching (role, style, methods)

Girls choose to take advanced mathematics courses less frequently than boysdo, and girls are less successful than boys on mathematics achievement tests(American Association of University Women, 1992). These differences occurdespite apparent equality in mathematics skills as measured by classroom grades(Friedman, 1989) and even despite superior performance by girls in some areasof mathematics, for example, calculation (Marshall, 1984). One explanation forgirls’ failure to pursue advanced mathematics courses is that early-developingdifferences in cognitive styles lead to differences in later mathematics achieve-ment (Fennema & Peterson, 1985). There is evidence for differences in girls’ andboys’ styles of doing mathematics. Carr and Jessup (1997) found that to solvecomputational tasks in first grade girls more frequently use manipulatives tocount-on or count-all, whereas boys retrieve basic mathematics facts from mem-ory more often than girls. Fennema and her colleagues (Fennema, Carpenter,Jacobs, Franke, & Levi, 1998) also found gender differences in strategy use: Inearly elementary school, girls tended to use counting procedures modeled with


The writing of this article was supported by Grant DBS-9122032 from the NationalScience Foundation. The authors would like to thank the teachers of Oglethorpe ElementarySchool, Lilburn Elementary School, Lawrenceville Elementary School, Montecello ElementarySchool, and Morgan County Primary School for their help and patience.

Correspondence concerning this article should be addressed to Dr. Martha Carr, EducationalPsychology Department, University of Georgia, Athens, GA 30602.


21Martha Carr, Donna L. Jessup, and Diana Fuller

manipulatives and boys tended to use invented derived facts. Metacognitiveknowledge about mathematics strategies has been shown to support the use ofmodeling with manipulatives for girls, but not boys, at the beginning of the firstgrade (Carr & Jessup, 1997).

Another explanation offered for gender differences in advanced mathematicscourse-taking is that girls have poorer motivation in mathematics. Girls are con-sistently found to be less confident than boys in their mathematics skills (e.g.,Boekaerts, Seegers, & Vermeer, 1995), are less likely to believe that they will besuccessful on mathematics tasks (e.g., Ewers & Wood, 1993), and are more like-ly to hold a learned-helpless self-concept with respect to mathematics (e.g.,Eccles, Wigfield, Harold, & Blumenfeld, 1993). These differences in motivationare believed to result in girls’ being more comfortable and accomplished ineveryday classroom situations but performing poorly in novel situations such astaking achievement tests (Kimball, 1989). Thus, girls’ lack of confidence andlow expectations for success may lead to differences in their strategic approach-es to mathematics. Researchers to date, however, have failed to determine howemerging differences in motivation are related to emerging differences in math-ematics performance.

Those offering a third explanation assume that mathematics skills and beliefsabout mathematics develop in response to environmental and social demands(e.g., Carraher, Carraher, & Schliemann, 1985). According to this explanation,girls and boys are influenced to develop different skills, knowledge, and motiva-tion as a result of their interactions with peers, parents, and teachers. For exam-ple, boys’ increased access to neighborhood activities and resources results in alarger range in mathematics achievement scores for boys and may explain boys’superior performance on achievement tests (Entwisle, Alexander, & Olson,1994). Similarly, the different messages girls and boys receive about themselvesas mathematicians influence their attitudes and approaches to mathematics (e.g.,Parsons, Adler, & Kaczala, 1982). In the present study we examined gender dif-ferences in mathematics with the assumption that parents’ and teachers’ beliefsand reported actions are related to gender differences in children’s strategy useand metacognitive knowledge about strategies.

The Role of Parents

Parents’ beliefs about their children influence the children’s learning of skillsand knowledge (Jacobs & Weisz, 1994). For example, parents support problemsolving by transforming problems into more manageable tasks and by scaffold-ing children’s attempts at problem solving (Parent, Moss, & Gosselin, 1993;Sigel, 1982). Parents also provide direct instruction on strategies (Carr, Kurtz,Schneider, Turner, & Borkowski, 1989). If parents believe that their daughtersare less able in mathematics and will have more difficulty with mathematics thantheir sons (Parsons et al., 1982), they may encourage their daughters’ use ofcounting procedures using manipulatives. Boys, if they are perceived to be more

22 Parent and Teacher Influences

capable, may be given more latitude in their strategy use or encouraged to usemore advanced strategies that are more risky in that they increase the chances offailure. Parents’ attitudes and beliefs, therefore, may be transmitted throughinstruction or comments to children.

Differences in children’s mathematical behaviors, in turn, influence parents’expectations. As an example, parents’ expectations for boys increased after theboys showed improved performance on the California Achievement Test(Alexander & Entwisle, 1988). There is some evidence that it is the response ofadults to a child’s performance, instead of to the gender of the child, that isresponsible for differences in mathematics achievement (Pedersen, Elmore, &Bleyer, 1986). Thus, children’s developing mathematics skills are dependent notonly on their prior performance but also on adults’ reactions to their prior per-formance (Entwisle & Hayduk, 1988). As a part of this study we wanted toexamine how parents, in their instruction of their children, respond to their chil-dren’s early mathematics activities.

The Role of Teachers

We know that teachers tailor strategy recommendations to the grade level ofthe child. For example, teachers more commonly recommend the use of manip-ulatives to children in the earlier grades and the use of derived facts in latergrades (Moely et al., 1989). Although teachers encourage the use of strategiesinvolving manipulatives, they often attempt to shape children’s strategy use bysuppressing other forms of modeling, such as counting on fingers, with sugges-tions to use more “mature” strategies, such as mental calculation (Moely et al.,1986). If teachers believe that boys and girls have different abilities and if thisbelief influences their instruction on strategy use1, they may be directly promot-ing gender differences in the use of strategies. For example, teachers are morelikely to attribute boys’ performance in mathematics to ability and to believe thatboys are more likely to enjoy mathematics, are more competitive, and are morelogical and independent than girls (Fennema, Peterson, Carpenter, & Lubinski,1990). These perceptions may translate into differences in beliefs about appro-priate strategy use by boys and girls, leading to differences in instruction.

Conversely, teachers’ expectations and instruction may be influenced by indi-vidual children’s characteristics and activities. Preschool girls, in contrast toboys, are more likely to be responsive to teachers, to attend more to teachers’instructions, and to seek out reassurance and guidance from teachers (Fennema& Peterson, 1986). In addition, girls and boys seek out different forms of infor-mation from teachers (Fennema & Peterson, 1986). Teachers may interpret chil-dren’s actions to mean that children are less or more capable in mathematics andrespond to these actions accordingly. Early-emerging strategy use, therefore,

1 The term strategy use is used here to refer both to particular strategies used to solve problems andto the metacognitive regulation of strategy selection.


should influence teachers’ instruction on strategies and on the metacognitiveinformation children can use to regulate strategies.

The purpose of this study was to examine how the development of gender dif-ferences in mathematics strategy use in the first grade is occasioned by parents’and teachers’ reported actions and children’s perceptions of adults’ beliefs. Toexamine the relationship between intentional, direct instruction and children’sstrategy use, we assessed parents’ and teachers’ instruction and beliefs throughreported instruction on strategies and on metacognitive aspects of strategy use.Parents’ and teachers’ beliefs were also assessed via children’s perceptions ofthese beliefs. Our goal was to determine whether and how adults’ instruction isperceived by children and to determine whether children’s perceptions of adults’beliefs about strategies influenced their strategy use and their metacognitive reg-ulation of strategy use.

We believed that children’s strategy use and related metacognitive informationabout strategy use at the beginning of the school year would influence teachers’and parents’ instruction by midyear. Parents’ and teachers’ instruction, in turn,was believed to influence children’s perceptions of adults’ beliefs, children’sstrategy use, and children’s strategy-related metacognitive knowledge. Further,we believed that children would react to their perceptions of parents’ and teach-ers’ beliefs toward strategy use.

METHOD

Participants

A total of 92 children, 4 (2 girls, 2 boys) from each of 23 first-grade class-rooms, were recruited to participate in the project. Their teachers randomlyselected the children to participate in the project (usually by picking names outof a hat). Letters requesting permission for the children to participate were sentto the parents of the selected children; the letters also described the expectedparental involvement in the project. Thirteen parents either did not return the per-mission form or refused to participate, and 13 children were selected as replace-ments, using the same process. Data for some children are incomplete because 4children moved before the end of the school year. The average age of the chil-dren when they started the project was 6 years, 5 months. Children from low,middle, and upper middle socioeconomic statuses took part in the study; 21% ofthe sample were African American, 74% were Caucasian, and 4% were fromother groups (primarily Asian).

The schools were located in northern and central Georgia. Two schools (10classrooms) were in the suburbs of Atlanta and three schools (13 classrooms)were in small towns in Georgia. The teachers had been teaching for an averageof 12.5 years in total and an average of 8.92 years at first-grade level. Teachers’experience ranged between 1 year and 26 years. All teachers were female andcertified to teach first grade. One teacher held a PhD, 10 teachers held MA orMEd degrees, and 12 teachers held BA or BS degrees.


The instruction in the classrooms varied within and across schools. All teach-ers used manipulatives in the instruction of mathematics. Three teachers used achild-centered approach to mathematics instruction. Five teachers usedMathematics Their Way (Baratta-Lorton, 1988) as the primary form of instruc-tion, and the remaining 15 teachers used textbook-based instruction combinedwith instruction on the use of manipulatives.

Design

The children were interviewed individually outside of the classroom settingthree times (October, January, and April) while they progressed through the firstgrade. The interview spacing allowed us to observe changes in children’s strate-gy use and made data collection manageable. Interviews were videotaped so thatboth verbal and nonverbal responses could be automatically recorded, leavingthe investigator free to concentrate on interacting with the child. Only one of thetwo investigators worked with the children at any one time. The interviews yield-ed information about children’s strategy use, metacognitive knowledge aboutspecific strategies, and perceived parents’ and teachers’ beliefs about strategies.In January teachers completed a questionnaire about each child. They were askedabout the types of mathematical strategies and metacognitive information aboutthe regulation of strategies they taught their students. Similarly, a third investi-gator conducted phone interviews with parents asking about the types of mathe-matical strategies and related metacognitive information about regulating strate-gies that parents presented in the home.

Procedure and Materials

Children’s strategy use. Following the procedure used in a previous study(Carr & Jessup, 1997), we told children that they would be solving 10 additionand 10 subtraction computation tasks. The children were told to do their best and,if they wished to do so, to use the available counters provided for their use. Theinvestigator also told the children that they would be asked questions about howthey computed the answers. A low table was used so that the counting behaviorswould be apparent to the investigator and could be videotaped.

The 10 addition computational tasks (e.g., 3 + 4 = ) were randomly presentedfirst, followed by the random presentation of the 10 subtraction computationaltasks. The same computational tasks were used in each interview so that indi-vidual differences could be observed in children’s responses to the same tasksover repeated sessions. Addition computation tasks were presented first so thatchildren would experience some success before they moved on to the more dif-ficult subtraction computation tasks. The computation tasks included 10 single-digit tasks, 10 tasks with one double-digit number, and 4 tasks with two double-digit numbers. The tasks were presented on small cards, and the investigator stat-ed the task when the card was presented. After completing each computationtask, the children were asked how they had solved that task. Children’s respons-


es were checked against observed activities. This technique is a commonly usedprocedure and has been shown to be a valid measure of actual strategy use(Siegler, 1989).

Strategies were categorized as being overt, covert, or retrieval by two raterswho determined the number of computation tasks on which each child correctlyused each strategy. Categorization of strategies into more precise descriptions ofstrategy use (e.g., categorizing overt strategies as counting-all or counting-onthrough direct modeling) was limited by the small number of times the childrenused some strategies and by the focus of the study, which was on the socialobservability of strategies. If a child made no attempt to solve a computationtask, the task could not be categorized. Interrater reliability for strategy catego-rization was α = .95.

Strategies in which children used the counters or counted on fingers were clas-sified as overt strategies because these strategies were visible to others and,therefore, were open to comment by parents and teachers. For example, count-ing-on and counting-all strategies directly modeled with physical objects asdescribed by Carpenter and Moser (1984) were categorized as overt strategies.

Strategies were classified as covert when no manipulatives were used andwhen the child described counting numbers in his or her head. Counting-onstrategies that were not directly modeled using manipulatives or fingers wereplaced in this category, together with derived-facts strategies in which childrenused basic math facts to calculate answers. Both of these strategies were catego-rized as covert because they were not observable but required some calculation.When children used covert strategies but were unable to describe the strategyused, the strategy was identified either by asking the child to replicate the pro-cedure on his or her fingers or by asking which number he or she started to countfrom (e.g., for the computation task 8 + 3, the child was asked whether he or shestarted to count from 1, 8, or 3).

A strategy was categorized as retrieval if the child described pulling the infor-mation from memory or said that the information just “popped” into his or herhead. Retrieval was distinguished from covert-strategy use by examining theamount of time it took a child to solve a computation task. A child who reportedretrieval and who was quick in his or her response was judged by the raters tohave used retrieval. If the initial response from the child was vague, the investi-gator probed by asking the child, “Did you count numbers in your head?” or “Didthe answer just pop into your head and you didn’t have to think about it at all?”Guessing was not categorized separately because children rarely reported guess-ing and because their incorrectly retrieved answers were typically close to thecorrect number so that they could have pulled a close but incorrect response frommemory.

Children’s metacognition. We assessed children’s metacognitive knowledgeabout the regulation of mathematics strategies during the time the children com-pleted the addition and subtraction computation tasks. When the children initial-ly used a strategy, they were asked for a rationale for the use of that strategy: “I


noticed that you just used ______ when you figured out that answer. Why did youget the answer that way?” Next, the children were asked about the different sit-uations in which they would use the strategy: “When do you use ______ to fig-ure out math problems? When don’t you use _____ to figure out math problems?”If a child was not clear in his or her responses, the investigator working with thatchild said, “Tell me more about that.” This request helped prompt the child togive a more complete response. After the children had completed all the compu-tation tasks, they were asked about the strategies that they had not used. To avoidthe problem of children altering their strategy use in response to the investiga-tor’s questions, our questioning about unused strategies occurred after the com-pletion of the entire set in order. As an example, the children were asked, “Whenyou were doing all the problems, I noticed that you didn’t use ______ . Why not?Are there any times that you would use that way? When would (or wouldn’t) thatbe a good way to use?”

Metacognitive knowledge was assessed during solution of the computationtasks instead of after completion of an entire set of computations to assure thatthe children connected the questions with the appropriate computation tasks. Onepoint was scored for each metacognitive response for both the “why” and“when” questions. If the same response was given for “when” and “when not,”only 1 point was scored. Possible responses included comments about the speedor ease of use of a strategy, the capacity of a strategy to help in learning or devel-opment (e.g., “First kids learn with fingers, then they count in their head”), theusefulness of the strategy for difficult or easy computational tasks (e.g., “Justknowing is used for easy problems”), the existence of physical or visual cues,and the reliability or effectiveness of the strategy. Scores for the metacognitivequestions were summed. A score of 0 meant that a child made no metacognitivecomments, and the maximum score of 9 meant that a child made an appropriatemetacognitive response to each of the three metacognitive questions (why, when,and when not) for each of the three strategy categories. Children were not givencredit for responses that did not appear to the raters to be accurate or correctassessments of why and when strategies should be used.

The metacognitive scores were summed for several reasons. It was believedthat metacognitive knowledge about one strategy could influence the use notonly of that strategy but also of other strategies. For example, a child might claimthat retrieval is used for easier computational tasks and overt strategies are usedfor harder computational tasks. Thus, the knowledge that strategies are used ondifferent computational tasks on the basis of the difficulty of the computationaltask is not metacognitive knowledge unique to one strategy but is knowledge thataffects all strategy use. In addition, scores for individual response categories ofmetacognitive knowledge were not examined independently because the limitednumber of responses made it impossible to examine the effect of differentresponse categories on strategy use and because aggregation improved reliabili-ty (Rushton, Brainerd, & Pressley, 1983). Interrater reliability for scoring onmetacognitive knowledge was α = .86.


Children’s perceptions of parents’ and teachers’ beliefs. The children werefirst asked about their perceptions of their teachers’ beliefs about the appropri-ateness of different strategies using three cartoon pictures as visual aids in eachof three paired comparisons. Each cartoon picture showed a child using retrieval(a child with a light bulb above his or her head in a bubble), a covert strategy (achild counting numbers in a bubble above his or her head), or an overt strategy(a child holding up his or her fingers and counting, with numbers). Covert count-ing was compared to overt counting, retrieval was compared to overt counting,and covert counting was compared to retrieval. The investigator asked the chil-dren, for example, “Which does your teacher think is better, to count on your fin-gers (points to one picture) or to figure something out in your head (points to theother picture)? Why does your teacher think that?” Children’s perceptions oftheir teachers’ beliefs about strategies were scored as metacognitive beliefs if thechildren referred to information teachers might provide students to regulate strat-egy use, including information about the ease of the computational task, the util-ity of the strategy, the reliability of the strategy, or the accuracy of the strategy.Children’s perceptions of their teachers’ beliefs about abilities, labeled abilitybeliefs, were based on the children’s perceptions of teachers’ comments that astrategy made the child look smart.

Scores on children’s perceptions regarding their teachers’ ability beliefs andmetacognitive beliefs were summed across the three strategy comparisons to cre-ate separate scores for perceived ability beliefs and perceived metacognitivebeliefs. The sums were intended to provide indicators of the overall strength ofchildren’s perceptions of their teachers’ ability and metacognitive beliefs. Ascore of 0 for ability meant that for none of the three comparisons had the childremarked that his or her teacher preferred strategies because they indicated abil-ity. A maximum score of 3 meant that for each of the three comparisons a child’steacher was perceived to prefer a strategy because it indicated high ability.Typically, children perceived that teachers held ability beliefs when the compar-ison included retrieval. The same scoring technique was used for perceivedmetacognitive beliefs. The possible score range for both the perceived abilitybeliefs and perceived metacognitive beliefs was 0 to 3.

Next, children’s perceptions of their parents’ beliefs about strategy use wereexamined. The procedure was the same as for teachers’ beliefs; the same ques-tions and pictures were presented again, but this time the children were askedabout their parents. Scored responses included perceptions of parents’ beliefsabout ability (e.g., they say I am smart if I use this) and about metacognitiveprocesses that might help children regulate the use of strategies (e.g., because itwill help you learn). Perceived parental beliefs were scored and the scores weresummed in the same way as for the perceived teachers’ beliefs. A score of 0 indi-cated no perceptions of beliefs about ability or metacognitive strategies. A max-imum score of 3 indicated that the child attributed metacognitive or abilitybeliefs to his or her parents for each of the three comparisons. Interrater reliabil-


ities for the perceived parent and teacher beliefs were α = .93 and α = .98,respectively.

Parent-Strategy Questionnaire. In January parents were interviewed over thetelephone about the type and quantity of the home instruction on mathematicalstrategies; we used data from those interviews as indicators of the quantity ofmetacognitive information on strategies taught in the home. The questionnairewas originally developed by Kurtz, Schneider, Borkowski, Carr, and Rellinger(1990) and for the purposes of this study was modified to focus on mathematicsinstruction. A telephone interview rather than a written questionnaire was usedin most cases to assure complete responses to the questions and to provide oppor-tunities for the interviewer to explain or restate misunderstood questions. If theparent could not be contacted by telephone or did not own a telephone, the ques-tionnaire was sent home with the child. Of the 92 parents who returned permis-sion slips only 3 could not be contacted or did not return the questionnaire.

At the beginning of the interview we explained that the purpose of the ques-tions was to learn about how mathematics was taught in the home. Parents wereasked about how they dealt with their children’s difficulties with addition andsubtraction computational tasks and whether they taught any mathematics strate-gies to their child in the home. Because the focus of this study was on the influ-ence of instruction on children’s use of retrieval, overt, and covert mathematicsstrategies and the metacognitive information used to regulate the use of mathe-matics strategies, parents’ responses were categorized into one of four possiblecategories: instruction on overt strategies, instruction on covert strategies,instruction on retrieval strategies, or the provision of metacognitive informationabout strategy use. Interrater reliability for response categorization for theParent-Strategy Questionnaire was α = .81. The questionnaire and scoring cate-gories are presented in Appendix A. Some questions were open-ended and par-ents were allowed to report more than one instance of instruction per category;hence there is no maximum number of points awarded. Scores for the four cate-gories ranged from –1 to 6 for overt-strategy instruction, 0 to 2 for covert-strat-egy instruction, 0 to 4 for retrieval-strategy instruction, and 0 to 4 for instructionon metacognitive information about strategy use. (A score of –1 was given forovert-strategy instruction if using manipulatives was discouraged.)

Teacher-Strategy Questionnaire. This questionnaire was completed in Januaryby each teacher with help available from the investigator. The teacher question-naire was structured in the same way as the parent questionnaire, and the samecategories were used for responses. The first nine questions were identical tothose on the parent questionnaire and were scored in the same way. Questions10, 11, 12, and 13, listed in Appendix B, were about teachers’ instruction relat-ed to metacognitive information that would allow children to appropriately reg-ulate the use of the strategies. Interrater reliability for response categorization forthe Teacher-Strategy Questionnaire was α = .86. Some questions were open-ended and teachers were allowed to report more than one instance of instruction


per category; hence there is no maximum number of points awarded. The actualranges of scores for the four categories were 3 to 11 for overt-strategy instruc-tion, 0 to 4 for covert-strategy instruction, 0 to 6 for retrieval-strategy instruction,and 0 to 8 for instruction on metacognitive information about strategy use.

RESULTS

The data were examined first for gender differences in overt-strategy use, covert-strategy use, and retrieval. Means and standard deviations for correct and incorrectstrategy use, for children’s metacognitive knowledge about strategies, and for chil-dren’s perceptions of parents’ and teachers’ ability beliefs and metacognitivebeliefs about strategies for girls and for boys are presented in Table 1.

The first analysis was a repeated measures ANOVA. Correct overt-strategyuse scores as measured in October, January, and April were the dependent mea-sures, and gender was the independent measure. The gender by time interactionwas significant, F (2, 172) = 8.02, p < .01, with girls increasing their overt-strat-egy use over the school year more than boys.

The second analysis was a repeated measures ANOVA in which correct-retrieval scores as measured in October, January, and April were the dependentvariables and gender was the independent variable. The main effects for genderand time were significant with boys consistently using retrieval correctly morethan girls and all children increasing their correct use of retrieval across the year,F (1, 86) = 7.58, p < .01 and F (2, 172) = 70.83, p < .01, respectively.

In the third repeated measures ANOVA we examined correct covert-strategyuse as measured at the three times with gender as the independent measure. Nogender effects were found for covert-strategy use. Across the school year, how-ever, children improved their abilities to correctly use covert strategies, F (2,172) = 14.75, p < .01.

As can be seen, the gender differences in strategy use found by Carr and Jessup(1997) were replicated here in that boys increased their use of retrieval across theschool year more than girls. In contrast, girls increased their use of overt strate-gies more than boys.

Predictors of Parents’ and Teachers’ Reported Instruction

Do children’s early abilities to select and use strategies predict parents’ andteachers’ cognitive and metacognitive instruction on strategies later in the schoolyear? Researchers suggest that adults are influenced by their perceptions of chil-dren’s skills (e.g., Pedersen et al., 1986). If that is the case, then initial individ-ual differences in strategy use and knowledge about the regulation of strategiesby boys and girls as measured in October should predict teacher and parentinstruction as reported in January. Means and standard deviations for parents’and teachers’ instruction on strategies and related metacognitive knowledge asreported in January are presented in Table 2.


Table 1 Means and Standard Deviations (in Parentheses) for Strategy Use, Metacognitive Knowledge, andPerceived Beliefs

October January AprilCorrecta Incorrectb Correct Incorrect Correct Incorrect

Overt-strategy useGirls 3.30 2.43 7.68 3.30 7.09 2.05

(3.83) (3.28) (4.96) (3.23) (4.67) (2.18)Boys 3.41 2.59 4.23 2.00 3.53 1.75

(3.93) (3.19) (4.46) (2.77) (3.35) (2.42)Retrieval

Girls 2.23 2.09 3.23 0.75 4.96 0.68(2.24) (3.92) (2.16) (1.70) (3.31) (1.75)

Boys 3.50 3.77 5.02 2.93 7.00 2.30(3.47) (5.90) (3.88) (4.47) (3.88) (3.59)

Covert-strategy useGirls 2.18 1.41 3.07 1.05 3.91 1.02

(3.04) (3.20) (3.49) (1.86) (3.38) (1.55)Boys 2.43 0.68 3.71 1.77 3.89 1.21

(2.89) (1.07) (2.86) (2.20) (2.69) (1.46)Metacognitive knowledgec

Girls 4.73 7.43 8.36(4.16) (5.00) (5.71)

Boys 5.50 7.39 7.14(4.45) (5.07) (5.20)

Perceived parents’ ability beliefsd

Girls 0.14 0.16 0.05(0.41) (0.53) (0.30)

Boys 0.25 0.25 0.32 (0.58) (0.62) (0.74)

Perceived teachers’ ability beliefsGirls 0.05 0.11 0.11

(0.30) (0.39) (0.44)Boys 0.09 0.24 0.21

(0.43) (0.61) (0.68)Perceived parents’ metacognitive beliefs

Girls 0.57 0.75 1.09(0.90) (0.94) (1.18)

Boys 0.89 0.68 0.82(1.08) (1.05) (1.17)

Perceived teachers’ metacognitive beliefsGirls 0.55 0.91 0.96

(0.85) (1.10) (1.08)Boys 0.84 0.80 1.00

(1.01) (1.11) (1.10)

aCorrect strategy-use for a given strategy is the number of problems that a child correctly used thatstrategy for out of all the problems given. The maximum possible score for each strategy is 20.

bIncorrect strategy-use for a given strategy is the number of problems on which the child unsuc-cessfully attempted to use that strategy, out of all the problems given.

cThe maximum score of 9 for metacognitive knowledge would show that children gave a metacog-nitive rationale for the questions (why, when, and when not to use a strategy) for each of the threestrategies (overt, covert, or retrieval).

dThe maximum score for perceived parents’ and teachers’ metacognitive beliefs and ability beliefswas 3. A score of 3 for perceived teachers’ metacognitive beliefs would show that for each of thethree paired comparisons the children responded with a perceived teachers’ metacognitive belief.


Table 2 Means and Standard Deviations (in Parentheses) for Parents’ and Teachers’ Reported Strategyand Metacognitive Instruction

Parent Teacher

Overt-strategy instruction Girls 3.86 6.35

(1.46) (1.69)Boys 4.12 5.60

(1.03) (1.53)Instruction on retrieval

Girls 2.09 3.09(1.21) (1.90)

Boys 1.97 3.23(1.23) (1.84)

Covert-strategy instruction Girls 1.23 1.68

(0.81) (1.46)Boys 1.42 1.95

(0.71) (1.93)Metacognitive instruction

Girls 0.93 3.60(0.83) (1.56)

Boys 1.00 3.79(0.89) (1.58)

Note. Several of the questions for parents and teachers were open-ended questions, allowing for mul-tiple examples of the instruction on overt, covert, and retrieval strategies and on metacognitiveknowledge. As a result it is impossible to provide a maximum possible score because the scoredepends on parents’ and teachers’ responses.

It should be noted that parents and teachers had fewer opportunities to report instruction of covertstrategies in contrast to retrieval strategies, overt strategies, or instruction on metacognition. Thus,variations in means may be due to this limitation instead of actual differences in adult instructionon covert strategies.

In the regression analyses shown in Table 3, gender by strategy use interac-tions were of particular interest because we wanted to see whether parents andteachers interpreted strategy use differently for girls and boys. For example, par-ents may respond to boys’ use of retrieval by providing further instruction onretrieval, but they may not respond in the same way to girls’ use of retrieval. Inthis case, the significant gender by use of retrieval interaction would indicate thatgender differences in strategy use emerge from differential responses to girls’and boys’ use of the same strategy. Interaction terms were dropped from theequations if they were found to be nonsignificant. For these and all the otherregression analyses done for this study, separate analyses were run for retrieval-,overt-, and covert-strategy use to avoid problems with dependence among thepredictor variables.

As can be seen in Table 3, only children’s correct use of covert strategies asmeasured in October significantly predicted parents’ reports that they instructedchildren to use these strategies. Neither gender nor metacognitive knowledgepredicted instruction on strategies in any of the analyses. The gender by strategyuse interactions were also nonsignificant predictors of parent instruction. So,with the exception of children’s use of covert strategies, there is little relation-


ship between children’s early metacognitive knowledge or strategy use and par-ents’ reported instruction.

Children’s early strategy use and gender were better predictors of teachers’instruction than of parents’ instruction. As can be seen in Table 3, children’s useof retrieval in October predicted teachers’ reported instruction on the use ofmetacognitive knowledge and retrieval strategies. Girls were more likely thanboys to receive metacognitive instruction and instruction on retrieval. The gen-der by retrieval interaction was also a significant predictor of teacher instructionthat provided metacognitive information about strategies and instruction onretrieval. To interpret the significant interactions, we calculated coefficients toprovide separate weightings for boys and girls. In regard to the gender byretrieval interaction predicting the instruction that provided metacognitive infor-mation about strategies, boys (coeff = .40) were more likely than girls (coeff =.01) to receive instruction that provided metacognitive information about strate-gies only if they correctly used retrieval. In regard to the gender by retrieval

Table 3Regression Analyses of Strategy Use and Metacognitive Knowledge as Predictors of Adults’Reported Instruction

Parents TeachersInstruction type and child variables β p value β p value

Metacognitive instruction (January)October measures of

Gender 0.05 .64 0.40 .01 Metacognition 0.11 .29 0.11 .30 Correct retrieval 0.08 .48 0.40 .00 Gender by retrieval interaction -0.39 .01

F (4, 84) = 3.16, p = .02October measures of

Gender 0.03 .77 0.03 .75 Metacognition 0.14 .21 0.13 .24 Correct overt-strategy use –0.08 .46 –0.08 .45

October measures ofGender 0.03 .75 0.04 .74 Metacognition 0.12 .32 0.02 .89 Correct covert-strategy use –0.01 .99 0.25 .03

Instruction on retrieval strategies (January)Gender 0.00 .99 0.32 .03 Oct. metacognition 0.02 .84 0.19 .06 Oct. correct use of retrieval –0.12 .27 0.32 .01 Oct. gender by retrieval interaction –0.43 .01

F (4, 84) = 3.31, p = .01Instruction on overt strategies (January)

Gender –0.06 .57 0.16 .15 Oct. metacognition –0.15 .19 0.01 .95 Oct. correct use of overt strategies 0.11 .32 –0.01 .90

Instruction on covert strategies (January)Gender –0.13 .21 –0.05 .61 Oct. metacognition –0.08 .48 0.11 .36 Oct. correct use of covert strategies 0.31 .01 0.02 .87

F (3, 84) = 3.15, p = .03

Note. All β and p values rounded to closest hundredth.


interaction predicting instruction on retrieval, for boys the relationship betweencorrectly used retrieval and teachers’ reports of instruction on retrieval was pos-itive (coeff = .32). The relationship was negative for girls (coeff = –.11). Boys,but not girls, who correctly used retrieval were more likely to receive instructionin the use of retrieval from teachers. Girls received instruction that providedmetacognitive information about strategy use and retrieval strategies regardlessof their abilities to successfully use retrieval strategies.

These data indicate that teachers respond differentially in their instruction tothe correct use of retrieval. Gender differences emerged in how teachers respond-ed to girls’ and boys’ strategy use in that whereas all girls received moremetacognitive and retrieval instruction, only boys who already retrieved wellreceived instruction that provided metacognitive information about strategy useand retrieval strategies. We believe that teachers may have been trying to teachthose girls who were not spontaneously using retrieval to use retrieval via directinstruction on this strategy and to use metacognitive information about strategiesvia direct instruction that provided metacognitive information about strategy use.Teachers might have been reinforcing the use of retrieval strategies in boys, how-ever, only if boys were already capable of using them.

It is unclear why teachers responded to children’s use of retrieval but not totheir use of overt strategies. Although overt strategies, particularly the use ofcounters, were commonly used in these first-grade classrooms and were the basisof some of the instructional programs, teachers may view overt strategies asbasic skills instead of as viable options for problem solving. Thus, teachers mayspend little time emphasizing overt strategies beyond simple instructions at thebeginning of the year. Instead, teachers may focus on and reinforce what theyconsider to be more “advanced” mathematics, such as retrieval or covert strate-gies. This practice may be a reflection of teachers’ understanding of the devel-opment of mathematics skills as moving from overt and concrete to covert andabstract representations of numbers (Carpenter & Moser, 1984).

Parents appeared to be unaffected by their children’s early emerging strategyuse, perhaps because parents were not aware of early strategy use and, as a result,did not consider it when making decisions about how to instruct their children onstrategy use and how to provide metacognitive information about strategy use. Itremains to be seen whether parents’ instruction will be influenced by children’sactions later in the children’s academic careers.

Parents’ and Teachers’ Instruction and Children’s Strategy Use andMetacognition

How well do parents’ and teachers’ reported instruction on strategy use and themetacognitive knowledge necessary to use strategies predict children’s com-mand of metacognitive information about strategy use and their strategy use? Ifgender differences in strategy use are influenced by parents and teachers, thenreported instruction should predict gender differences in strategy use. To exam-ine this hypothesis, parents’ and teachers’ reported instruction as measured in


January was used to predict children’s strategy use and metacognitive informa-tion about strategy use in January and April.

Parents’ and teachers’ instruction were first regressed on children’s strategyuse and children’s metacognitive knowledge as measured in January and then onchildren’s metacognitive information about strategy use and their strategy use asmeasured in April. In each case, separate analyses were performed for the parentand teacher data. For these equations, interaction terms including a gender byparents’ strategy instruction, a gender by parents’ instruction on metacognitiveknowledge, a gender by teachers’ strategy instruction, and a gender by teachers’instruction on metacognitive knowledge were tested to examine whether parents’and teachers’ instruction interacted with gender. Initial analyses were run withboth interaction terms (e.g., parents’ metacognitive instruction by gender andparents’ strategy instruction by gender) and nonsignificant interaction termswere dropped from the analyses. In Table 4 we present the significant regressionanalyses for the parent and teacher data.

Table 4Results of Regression Analyses on Adults’ Instruction (Jan.) as Predictor of Children’s StrategyUse and Metacognitive Knowledge

January AprilCorrect strategy use and instruction β p value β p value

ParentsOvert strategies

Gender 0.34 .00 0.70 .00Parents’ metacognitive instruction –0.04 .73 0.20 .15Parents’ instruction on overt strategies 0.07 .47 –0.07 .47Gender by parents’ meta. instruction interaction –0.50 .01

F (3, 83) = 3.81, p = .01 F (4, 78) = 6.42, p = .00Retrieval strategies

Gender –0.29 .01 –0.28 .01 Parents’ metacognitive instruction –0.04 .68 0.08 .45 Parents’ instruction on retrieval –0.13 .20 –0.12 .25

F (3, 84) = 3.25, p = .03 F (3, 80) = 3.15, p = .03Covert strategies

Gender –0.04 .73 –0.23 .15 Parents’ metacognitive instruction 0.11 .32 –0.16 .27 Parents’ covert-strategy instruction 0.19 .08 0.30 .01Gender by parents’ metacog. instr. interaction 0.43 .02

F (4, 79) = 3.47, p = .01Teachers

Overt StrategiesGender 0.34 .00 1.33 .00Teachers’ metacognitive instruction –0.17 .10 –0.13 .21Teachers’ instruction on overt strategies 0.01 .95 0.25 .11Gender by teachers’ overt instr. interaction –1.07 .01

F (3, 83) = 4.58, p = .01 F (4, 80) = 5.79, p = .00Retrieval strategies

Gender 0.06 .74 0.14 .49 Teachers’ metacognitive instruction 0.28 .01 0.19 .09Teachers’ instruction on retrieval 0.29 .05 0.31 .04Gender by teachers’ retrieval instr. interaction –0.45 .04 -0.52 .02

F (4, 84) = 6.19, p = .00 F (4, 82) = 4.50, p = .00



Parental variables. As can be seen in Table 4, gender was the only significantpredictor of correct overt-strategy use in January. The parents’ metacognitiveinstruction by gender interaction term and the gender term were both significantpredictors of correct overt-strategy use in April. In January and April, girls weremore likely to correctly use overt strategies. An examination of the coefficientscalculated to interpret the significant interaction term indicated that the more par-ents reported providing information about strategy regulation to their daughters,the less likely their daughters were to correctly use overt strategies in April (coeff =–.30). Boys, in contrast, were more likely to correctly use overt strategies inApril if their parents had provided instruction on metacognitive informationabout strategy use (coeff = .20).

When the effect of parents’ instruction on both strategies and metacognitiveinformation about strategy use on correct use of retrieval (as measured in Januaryand April) was examined, only gender was a significant predictor. Boys weremore likely to correctly use retrieval than girls. Parents’ instruction did not pre-dict children’s correct use of retrieval.

There were no significant predictors of correct covert-strategy use in January.In April, however, the parents’ instruction on covert strategies term and the gen-der by parent instruction on metacognitive knowledge about strategies interac-tion term were significant. Children instructed by their parents to use covertstrategies were more likely to be capable of correctly using these strategies. Theinteraction indicated that for girls the relationship between parents’ metacogni-tive instruction and the correct use of covert strategies was positive in April(coeff = .27). For boys, the more metacognitive instruction provided by parents,the less likely boys were to correctly use covert strategies (coeff = –.16).

Teacher variables. As can be seen in Table 4, only gender significantly pre-dicted the correct use of overt strategies for the January data. In April the correctuse of overt strategies was predicted by gender and the gender by teacher instruc-tion on overt strategies interaction was significant. In both January and April,girls were more likely to correctly use overt strategies. However, the interactionindicated that the more teachers taught girls to use overt strategies, the less like-ly girls were to correctly use overt strategies in April (coeff = –.82). In contrast,if teachers reported teaching boys to use overt strategies, the boys were betterable to use these strategies correctly in April (coeff = .25). No significant pre-dictors of correct covert-strategy use were found.

Teachers’ reported instruction on retrieval and on metacognitive informationabout strategy use and the gender by teacher instruction on retrieval strategiesinteraction term were all significant predictors of the correct use of retrieval inJanuary. Teachers who taught their students to use retrieval strategies and whoprovided metacognitive information about strategy use were more likely to havestudents who correctly used retrieval. The significant interaction indicated thatfor boys the relationship between the instruction on retrieval and the correct useof retrieval was positive (coeff = .29). This relationship was negative for girls


(coeff = –.16). In regard to the effect of teachers on the use of retrieval in April,teachers’ instruction on retrieval strategies and the gender by teacher instructionof retrieval strategies interaction term were significant predictors of the correctuse of retrieval. Teachers who instructed their children to use retrieval strategieswere more likely to have children who did so. However, the significant interac-tion indicated that this relationship is qualified in that the relationship betweenteachers’ instruction on retrieval and the correct use of retrieval was positive forboys (coeff = .31) and negative for girls (coeff = –.21).

Fennema and Peterson (1986) found that girls and boys benefited in differentways from their interactions with teachers. We also found this difference whenwe examined the roles of teachers in children’s strategy use. In this case, boysshowed improvement in their use of retrieval- and overt-strategy use as a func-tion of teachers’ strategy instruction, particularly in the April data. Girls, in con-trast, did not appear to benefit as much as boys and, in fact, were hurt in theirstrategy use by their interactions with their teachers. There are several possibleexplanations for these differences. One is that teachers spend more time workingwith boys on their mathematics (Leinhardt, Seewald, & Engel, 1979) and thatboys’ strategy use may have improved as a result of this attention. Fennema andPeterson (1986) suggested a second possibility: Girls and boys benefit from dif-ferent types of instructional interactions with their teachers. In this case, theinstructional interactions appear to benefit boys more than girls in first-grademathematics. A third explanation is that teachers’ instruction on overt strategiesmay follow failure by girls in the use of overt strategies and failure by boys inthe use of retrieval. Redirecting boys to use overt strategies may have promotedthe correct use of these strategies for boys, but girls would have been redirectedto use the same strategies they were already using unsuccessfully.

Parents’ instruction predicted children’s strategy use only after gender differ-ences were established—there were no significant predictors either of Januarystrategy use or of the use of metacognitive information about strategy use. ByApril, among children whose parents had provided metacognitive instruction,boys were more likely than girls to be capable of correct overt-strategy use.Parents’ instruction on covert strategies, particularly for girls, predicted the cor-rect use of covert strategies.

As with teachers’ instruction, parental instruction on these strategies may haveoccurred as a response to perceived deficiencies in children’s strategy use. Forexample, boys who were not particularly successful in their use of retrieval mayhave been redirected to use overt strategies through metacognitive instructionwith the purpose of improving their mathematics skills. Similarly, by the end ofthe school year parents may have felt it necessary to encourage girls to abandonovert strategies in favor of covert strategies. As was found with the teachers’data, parents’ instruction on overt strategies to girls was negatively related to thecorrect use of these strategies. As was true with teachers, this relation may be amismatch in parents’ instructional styles relative to students’ genders, or parentsmay have been attempting to reinforce and reteach strategies that some girls had


not successfully mastered. Thus, parents’ contributions to strategy use are pri-marily in the alteration of established patterns in boys and girls in the second halfof the school year, with boys benefiting in their use of overt strategies and girlsbenefiting in their use of covert strategies.

Parents’ and Teachers’ Instruction and Perceived Parents’ and Teachers’Beliefs

Are parents’ and teachers’ instruction related to strategy use and metacogni-tive information about strategy use reflected in children’s perceptions of parents’and teachers’ attitudes toward strategies? Entwisle and Baker (1983) indicatedthat children are aware of parents’ expectations. It is, therefore, reasonable tobelieve that children’s perceptions of adults’ attitudes would correlate withreported adult instruction. To examine this possibility, parents’ and teachers’instruction on metacognitive information about strategy use and parents’ andteachers’ strategy instruction as measured in January were used as predictors ofperceived parents’ and teachers’ ability and metacognitive beliefs as measured inJanuary and April. As with the prior analyses, adults’ instruction by gender inter-action terms were included to determine whether parental instruction predictedperceived parents’ beliefs as a function of gender. There were no predictors ofperceived parents’ and teachers’ metacognitive beliefs.

The analyses included an examination of the effect of parents’ reportedinstruction on children’s perceptions of parents’ ability beliefs. As can be seen inTable 5, in January overt-strategy instruction was highly related to children’sperceptions of adults’ ability beliefs. Gender, parents’ reported overt-strategyinstruction, and the gender by overt-strategy instruction interaction term weresignificant predictors of perceived parents’ ability beliefs. In April the same pat-tern occurred with gender, parents’ instruction on overt strategies, and the gen-der by parent overt-strategy instruction interaction term being significant. In bothJanuary and April, boys were more likely than girls to say that their parents pre-ferred strategies that make one look smart. At both times, children of parents whoinstructed them to use overt strategies were unlikely to perceive that their parentspreferred “smart-looking” strategies, such as retrieval. The significant gender byovert-strategy instruction interactions indicated that this relationship was quali-fied in that parents of boys who provided overt-strategy instruction were unlike-ly to be perceived as preferring strategies that made children look smart (per-ceived ability beliefs) in January and April (coeff = –.63 and –.73, respectively).The coefficients for girls for these interactions (.60 in January and .64 in April),in contrast, indicated an opposite and positive relationship between parents’overt-strategy instruction and girls’ perceptions that parents liked strategies (typ-ically retrieval or covert strategies) because they made them look smart.

The regression analyses presented in Table 5 indicate similar results for teach-ers. In April, boys were more likely than girls to say that their teachers preferredstrategies that made them look smart. Similarly, teachers who instructed children


to use overt strategies were less likely to have children who perceived abilitybeliefs for teachers. The gender by teacher instruction on overt strategies inter-action term was also significant. Boys whose teachers provided instruction onovert strategies were less likely than girls to report that their teachers preferredstrategies because they made them look smart (boys’ coeff = –.57). But for girlsthe perception that teachers like strategies because they indicate ability was pos-itively related to teachers’ instruction on overt strategies (coeff = .69).

Children believe that to be good at mathematics means to be able to solveproblems quickly and effortlessly (Kloosterman, 1996). On the basis of our datawe suggest that these perceived beliefs and the use of strategies such as retrievalseem to be the default values for boys in the absence of adult reinforcement ofovert strategies. Boys were more likely than girls to believe that adults preferredstrategies that indicated ability. This perception, however, was not related toparental instruction of retrieval as would be expected. Instead, parents’ andteachers’ instruction related to overt strategies appeared to suppress boys’ per-ceptions that adults preferred “smart” strategies. Thus, parents who do not make

Table 5Regression Analyses for Adults’ Instruction as Predictors of Children’s Perceptions of AbilityBeliefs

January AprilAbility beliefs and instruction β p value β p value

Perceived parents’ ability beliefsRetrieval instruction

Gender –0.10 .34 –0.22 .04 Parents’ metacognitive instruction –0.12 .25 –0.08 .48 Parents’ instruction on retrieval –0.06 .58 –0.09 .41

Overt-strategy instructionGender –1.22 .00 –1.51 .00Parents’ metacognitive instruction –0.10 .29 –0.10 .33Parents’ overt-strategy instruction –0.63 .00 –0.73 .00Gender by parents’ overt-strat. instr. interaction 1.23 .00 1.37 .00

F (4, 82) = 5.40, p = .00 F (4, 78) = 5.74, p = .00Covert-strategy instruction

Gender –0.11 .30 –0.24 .02Parents’ metacognitive instruction –0.11 .32 –0.05 .64Parents’ covert-strategy instruction –0.06 .56 –0.15 .18

Perceived teachers’ ability beliefsRetrieval instruction

Gender –0.14 .19 –0.08 .45Teachers’ metacognitive instruction 0.07 .55 0.05 .66Teachers’ instruction on retrieval –0.16 .19 0.03 .82

Overt-strategy instructionGender –0.14 .19 –1.14 .01Teachers’ metacognitive instruction 0.02 .85 0.05 .63Teachers’ over-strategy instruction 0.04 .73 –0.57 .00Gender by teachers’ overt-strategy instr. interaction 1.26 .01

F (4, 80) = 3.56, p = .01Covert-strategy instruction

Gender –0.14 .17 –0.08 .48Teachers’ metacognitive instruction 0.01 .90 0.06 .57Teachers’ covert-strategy instruction –0.15 .16 0.10 .39



an effort to teach overt strategies may be implicitly teaching boys to value speedand apparent lack of effort in problem solving.

The opposite pattern occurred for girls relative to teachers’ and parents’instruction. Overt strategies were positively related to reported instruction onovert strategies. It is not clear why girls who receive more overt-strategy instruc-tion would simultaneously perceive that teachers and parents like strategies thatindicate high ability. This finding is particularly puzzling in that perceived abil-ity beliefs were reported primarily when children were asked about retrieval. Inthe future researchers need to examine better how girls interpret adults’ strategyinstruction and adults’ comments about strategies. On the basis of these data wesuggest that girls are receiving two different messages, one supporting the use ofovert strategies and one valuing quick, “smart” strategies such as retrieval.

Perceived Parents’ and Teachers’ Beliefs and Children’s Strategy Use andMetacognitive Knowledge

Finally, what is the relationship between, on the one hand, children’s percep-tions of parents’ and teachers’ beliefs and, on the other hand, children’s strategyuse and metacognitive information about strategy use? We know that children’sawareness of parents’ expectations predicts their performance (Entwisle &Baker, 1983). In this case, children’s perceptions of adults’ beliefs may affect thedevelopment of gender differences in strategy use and related metacognitiveknowledge. We examined this hypothesis by checking whether perceived par-ents’ and teachers’ metacognitive beliefs predicted the development of metacog-nitive knowledge about strategies and strategy use. We also determined whetherperceived parents’ and teachers’ ability beliefs predicted strategy use andmetacognitive knowledge about strategies.

In the following sections describing regression analyses, perceived parents’ability and metacognitive beliefs and gender were used as predictors of retrieval-,overt-, and covert-strategy use and metacognitive knowledge about strategies.Similarly, perceived teachers’ ability and metacognitive beliefs and gender wereused as predictors of retrieval-, overt-, and covert-strategy use and metacognitiveknowledge. Interaction terms of gender by perceived parents’ ability beliefs,gender by perceived parents’ metacognitive beliefs, gender by perceived teach-ers’ ability beliefs, and gender by perceived teachers’ metacognitive beliefs wereincluded to examine whether girls and boys were differently influenced in theirstrategy use and metacognitive knowledge by their perceptions of parents’ andteachers’ beliefs. Children’s strategy use and metacognitive knowledge as mea-sured in January were used as criterion variables for the predictor variables mea-sured in October. Variables measured in January were regressed on children’sstrategy use and their metacognitive knowledge as measured in April.

Parental variables. As can be seen in Table 6, perceived parents’ abilitybeliefs significantly predicted children’s correct use of retrieval strategies inJanuary and April. The gender by perceived parents’ ability beliefs interaction


term was also a significant predictor of correct retrieval in January. The interac-tion indicated that boys were much more likely to correctly use retrieval strate-gies if they perceived their parents to have ability beliefs (coeff = .55). This inter-action was not found for girls (coeff = .11). Gender was a significant predictorof the correct use of retrieval in April but not January, with boys being more like-ly than girls to use retrieval correctly.

Table 6Regression Analyses of Perceived Adults’ Beliefs as Predictors of Children’s Strategy Use andMetacognitive Knowledge

January AprilChild variables/adult beliefs β p value β p value

ParentsRetrieval strategies

Gender –0.15 .14 –0.26 .01Parents’ metacognitive beliefs 0.02 .86 0.02 .86Parents’ ability beliefs 0.55 .00 0.22 .04Gender by parents’ ability beliefs interaction –0.44 .00

F (4, 88) = 6.51, p = .00 F (3, 84) = 3.97, p = .01Overt strategies

Gender 0.36 .00 0.39 .00Parents’ metacognitive beliefs 0.09 .35 0.10 .30Parents’ ability beliefs –0.01 .93 –0.05 .59


Gender –0.05 .61 0.01 .95Parents’ metacognitive beliefs 0.25 .02 –0.09 .41 Parents’ ability beliefs –0.02 .84 0.01 .94

Metacognitive knowledgeGender 0.07 .48 0.11 .28Parents’ metacognitive beliefs 0.38 .00 0.40 .00Parents’ ability beliefs 0.09 .36 0.09 .35

F (3, 89) = 5.18, p = .00 F (3, 84) = 6.29, p = .00Teachers

Retrieval strategiesGender –0.27 .01 –0.26 .01Teachers’ metacognitive beliefs 0.00 .97 0.12 .24Teachers’ ability beliefs 0.17 .36 0.21 .05

F (3, 88) = 3.58, p = .00 F (3, 84) = 3.90, p = .01Overt Strategies

Gender 0.37 .00 0.59 .00Teachers’ metacognitive beliefs 0.12 .23 0.10 .47Teachers’ ability beliefs 0.03 .79 0.01 .88Gender by teachers’ metacog. belief interaction –0.37 .02


Gender –0.05 .63 –0.01 .96Teachers’ metacognitive beliefs 0.13 .21 0.28 .01Teachers’ ability beliefs 0.01 .91 0.05 .64

Metacognitive knowledgeGender 0.06 .53 0.09 .35Teachers’ metacognitive beliefs 0.44 .00 0.41 .00Teachers’ ability beliefs –0.08 .41 0.02 .83

F (3, 88) = 7.10, p = .00 F (3, 84) = 6.06, p = .00



When the predictors of the correct use of overt strategies were examined, onlygender significantly predicted the correct use of overt strategies in January andin April. The only predictor of children’s metacognitive knowledge about strate-gies as measured in January and in April was children’s perception that parentsvalued strategies for metacognitive reasons.

Teacher variables. In looking at the predictors of retrieval, we found that gen-der was a significant predictor of correct retrieval in January and April with boysusing retrieval more than girls. The perception that teachers had beliefs associat-ing ability with strategy in use significantly predicted children’s correct use ofretrieval in April but not in January. In April, children were more likely to cor-rectly use retrieval when they believed teachers preferred strategies, typicallyretrieval, that made students look smart.

In looking at the predictors of overt-strategy use, gender significantly predict-ed the correct use of overt strategies as measured in January and April. Girlswere more likely to use overt strategies. In addition, the gender by perceivedteacher metacognitive beliefs interaction term predicted the correct use of overtstrategies. In April the significant interaction indicated that girls (coeff = –.37)were less likely than boys to correctly use overt strategies if in January they hadperceived their teachers as having metacognitive beliefs related to strategy use.For boys (coeff = .10) there was a slightly positive relationship between per-ceived teacher metacognitive beliefs as measured in January and the correct useof overt strategies as measured in April. Neither perceived ability beliefs nor per-ceived metacognitive beliefs predicted covert-strategy use.

As with the parent data, perceived teachers’ metacognitive beliefs were pre-dicted by children’s metacognitive knowledge in January and April. The morestrategy-related metacognitive knowledge children used, the more likely theywere to report perceptions that parents and teachers held metacognitive beliefs.

Our data indicate that children’s perceptions of adults’ beliefs influenced gen-der differences in strategy use. Children who saw adults as having beliefs abouta relationship between ability and strategy use were more likely to use retrieval,particularly boys who perceived parents to have such beliefs. These data are inline with Carr and Jessup’s (1997) data, which showed that boys were more like-ly to make comments about the need to be competitive in strategy use, especial-ly in the case of retrieval. Furthermore, these data indicate that boys’ preferencefor competitive situations (McClintock & Moskowitz, 1976; Peterson &Fennema, 1985) may be related to the information they gather from adults.

Girls’ correct use of overt strategies in April was negatively related to per-ceived teachers’ metacognitive beliefs as measured in January. The negativerelationship that occurs for girls may be explained by teachers’ reinforcingmetacognitive instruction for girls who have been unsuccessful in their use ofovert strategies. The poorer the girls’ performance using overt strategies, themore likely teachers are to provide information about strategies. Teachers, how-ever, do not seem to provide such information for the boys, perhaps because boyswere predominantly using retrieval.


DISCUSSION

These data replicated the gender differences found by Carr and Jessup (1997)with boys correctly using retrieval during the first grade more than girls and girlscorrectly using overt strategies, such as counting on fingers and counters, morethan boys. In this study we extend prior research in that we examined the rela-tionship between the emergence of gender differences in children’s strategy useand (a) instruction by parents and teachers as measured through self-reports and(b) children’s perceptions of parents’ and teachers’ attitudes and preferenceswith regard to strategy use. The primary contribution of parents was throughchildren’s perceptions of parents’ preferences. Teachers’ reported instruction, incontrast, was more strongly related to children’s strategy use than to children’sperceptions of teachers’ attitudes regarding strategy use.

Much of the instruction given on strategy use seems to be either intentionallyor unintentionally beneficial to boys. For example, both teachers and parentsdirected boys to use overt strategies, and by the end of the school year boys werecorrectly doing so. Teachers were also likely to direct boys more than girls to useretrieval strategies in the second half of the school year. Parents had an effect onboys’ strategy use via boys’ perceptions of their parents’ beliefs about strategies.On the basis of these data, taken together, we suggest that boys’ strategy usedevelops in part as a function of adult intervention. Researchers should furtherexamine exactly how adults influence boys’ strategy use and why parents andteachers do not influence girls’ strategy use in the same way.

In several instances girls actually seem to be hurt by their interactions withteachers and parents. Girls did not benefit much from instruction on retrieval orovert strategies. Nor did adults’ providing metacognitive information about theregulation of strategies help girls in their use of overt strategies. Additionally,girls who were given more overt-strategy instruction were more likely to per-ceive parents and teachers as believing that strategies were good when they madethe student look smart. We have discussed some possible reasons for these out-comes in prior paragraphs; however, we believe that it is important to learn moreabout how and why adult-child interactions do not seem to benefit girls.Certainly, parents and teachers have the best intentions in their instructionalinteractions with girls. It will be necessary to determine in future research underwhat circumstances these instructional interactions go awry for girls.

With the exception of the relationship between teachers’ instruction onmetacognitive regulation of strategies and children’s correct use of overt strate-gies in April, we did not find much evidence that girls’ overt-strategy use wasinfluenced by perceived beliefs of parents or teachers or by direct instruction.Nor does girls’ overt-strategy use appear to be influenced by perceptions ofpeers’ attitudes (Carr & Jessup, 1997). It may be that girls possess different men-tal operations and cognitive structures that lead them to use overt strategies.Benbow and Stanley (1980) believed that gender differences in achievement testperformance are due to fundamental differences in cognitive skills. There is lit-


tle evidence in the literature on young children’s mathematical cognitive devel-opment that this belief is true (L. P. Steffe, personal communication, September1995) because no studies have specifically examined early developing genderdifferences in mental operations and cognitive structures.

The results of the study are limited in that we dealt only with parents’ andteachers’ reported instruction and with children’s perceptions of adults’metacognitive and ability beliefs. Actual strategy instruction and adult-childinstructional interactions were not assessed. It would be of particular interest toexamine differences in mothers’ and fathers’ interactions with their sons anddaughters. Mothers and fathers approach strategy use and problem solving in dif-ferent ways (McGillicuddy-DeLisi, 1982). When instructing their children inmathematics, fathers and mothers are likely to use different techniques. In addi-tion, parents may be reacting to children in ways of which they are unaware and,therefore, would not have reported via our interviews and questionnaires.Observation of parent-child interactions in problem-solving situations wouldprovide needed information about what parents really say and do in their inter-actions with their children.

Similarly, it will be necessary to document how teachers and children interactin their use of strategies in the classroom. The research of Fennema and Peterson(1986) indicated that girls and boys elicit different types of feedback from teach-ers. Researchers should examine how teacher-student interactions accompanythe emergence of gender differences in strategy use. Of particular interest wouldbe the balance of power between peers and teachers as predictors of children’sstrategy use.

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APPENDIX A: PARENT QUESTIONNAIRE

1. Do you do math work with your child at home? Do you use any of the follow-ing ways with your child? (a) different ways to count on fingers (scored 1 pointovert); (b) counters or other materials (scored 1 point overt); (c) flash cards(scored 1 point retrieval); (d) memory tricks or techniques (scored 1 pointretrieval); (e) different ways to count in the head (scored 1 point covert); (f) Isthere anything else you do?2. Which way do you teach your child to solve addition or subtraction problemsmost often? (a) I teach them to remember the answers to problems (memorizing)(scored 1 point retrieval); (b) I teach them ways to use counters (scored 1 pointovert); (c) I spend about the same amount of time teaching my child to remem-ber the answers to problems as I do teaching my child different ways to use coun-ters (scored 1 point each for overt & retrieval); (d) We do no work.3. If your child is having problems doing math in school, what do you do to helpyour child? Each response was categorized as indicating the instruction onretrieval strategies (1 point per response), overt strategies (1 point per response),covert strategies (1 point per response), or the instruction on metacognitiveknowledge to regulate strategy use (1 point per response). Parents may providemultiple responses for each strategy or for instruction on multiple strategies. 4. Do you think that your child needs to count on fingers before he or she is ableto figure out problems in his or her head (Yes or No)? If parents responded yes,the response was scored 1 point as being in the overt-strategies category.5. If you try to teach your child a math skill but your child doesn’t seem to under-stand, what do you do? Each response was categorized as indicating instructionon retrieval strategies (1 point per response), overt strategies (1 point perresponse), covert strategies (1 point per response), or instruction on metacogni-tive knowledge to regulate strategy use (1 point per response). Parents may pro-vide multiple responses for each strategy or for instruction on multiple strategies. 6. Children will often count on counters or fingers to help them solve math prob-lems. Are there times that you encourage counting on counters or fingers?“Encouraged the use of manipulatives” was scored 1 point in the overt-strategycategory. This question was asked to determine whether parents were aware ofthe need for children to use counters or fingers to externally represent numbers. 7. Are there times that you discourage counting on counters or fingers?


“Discouraged the use of manipulatives” was scored as –1 point in the overt-strat-egy category. We asked this question to find out whether parents actively dis-couraged or suppressed the use of manipulatives as opposed to encouraging orignoring this type of strategy use.8. Have you taught your child or encouraged your child to count in his or herhead? “Encouraged the use of mental calculation” (1 point was scored forcovert-strategy instruction). 9. Is it important for your child to get the answer from memory? Do you encour-age your child to just know? (One point was scored for retrieval if the parentresponded yes.)

APPENDIX B: ADDITIONAL QUESTIONS ON TEACHERQUESTIONNAIRE

10. Do you specifically instruct this child to check his or her work (circle one:Always, Often, Sometimes, Seldom, Never)? Please write a few examples of whatyou specifically say to this child regarding the importance of checking his or herwork. There were two parts to this question. For the Likert scale, points wereawarded as follows: always = 5, often = 4, sometimes = 3, seldom = 2, and never= 1. For the examples, 1 point was awarded for each unique example of instruc-tion related to metacognitive knowledge about checking techniques. The scoresfor the first and second parts of the question were summed. 11. If this child is having difficulty with a math problem that requires severalsteps, what do you say or do to help him or her? Responses were categorized asmetacognitive if the teacher told children to plan out steps (1 point), thinkthrough problems (1 point), check the work at each stage (1 point), or think aboutwhat is going wrong following a mistake (1 point). One point was awarded foreach different example of metacognitive knowledge and strategy regulation. Ifthe teacher made all the suggestions listed above, this question would be scored4 points. 12. What do you do if this child doesn’t seem to think about math problems andanswers without thinking (is impulsive)? Scored the same as Question 11.13. Imagine that you have explained an arithmetic lesson and this child doesn’tseem to understand. What do you do? Scored the same as Question 11.

Authors

Martha Carr, Professor, Educational Psychology Department, University of Georgia, Athens, GA30602; [email protected]

Donna L. Jessup, School Psychologist, Walker-Spivey School, Fayetteville, NC 28306Diana Fuller, Instructor, Truett-McConnell College, 1201 Bishop Farms Parkway, Watkinsville,

GA 30677

The Dilemma of Transparency:Seeing and Seeing Through

Talk in the Mathematics Classroom

Jill AdlerUniversity of the Witwatersrand, Johannesburg, Gauteng, South Africa

In this article talk is understood to be a resource for mathematical learning in school. As a resourceit needs to be both seen (be visible) to be used and seen through (be invisible) to provide accessto mathematical learning. Lave and Wenger’s (1991) concept of transparency captures this dualfunction of talk as a learning resource in the practice of school mathematics. I argue that the dualfunctions, visibility and invisibility, of talk in mathematics classrooms create dilemmas for teach-ers. An analytic narrative vignette drawn from a secondary mathematics classroom in South Africaillustrates the dilemma of transparency that mathematics teachers can face, particularly if theyare teaching multilingual classes.

Key Words: Bilingual issues; Communication; Language and mathematics; Social and culturalissues; Teacher knowledge

One feature of the changing political landscape in South Africa has been therapid racial integration of state schools. Since 1990, historically “whites only”schools have opened to all South African pupils, creating multilingual1, cultural-ly diverse classrooms. My purpose in this article is to open up discussion of theneed to explore the benefits and constraints of explicit mathematics languageteaching by presenting what can be described as a dilemma of transparency forteachers in multilingual secondary mathematics classrooms.

In this article I draw from a qualitative study of South African secondary math-ematics teachers’ knowledge of their practices in their multilingual classrooms(Adler, 1996b), different aspects of which have been published elsewhere (Adler,1995, 1996a, 1997, 1998). Some English-speaking teachers in the study taught inschools that had recently and rapidly desegregated. In initial interviews in thestudy, they talked about the value and benefit of what I have called “explicitmathematics language teaching” (Adler, 1995). In explicit mathematics language


This article is drawn from my doctoral thesis (Adler, 1996), supervised by ProfessorShirley Pendlebury (University of the Witwatersrand, Faculty of Education), in associationwith Dr. Kathryn Crawford (University of Sydney, Faculty of Education). The article is alsoan elaboration of a paper presented at the 21st Conference of the International Group of thePsychology of Mathematics Education (PME21) in Lahti, Finland, July 1997.

1I use multilingual in the same way as Levine (1993), to mean “classrooms in which pupils bringa range of main languages to the class.”


teaching, language itself, and particularly talk, becomes the object of attention inthe mathematics class and a resource in the teaching and learning processes. Nowthat their classes included pupils whose main2 language was not English, theseteachers realized that they needed to be more explicit about instructions for tasksand more careful in their use of mathematical terms and their expression of ideas.In interviews, the teachers said that they had found, to their surprise, that beingexplicit about mathematical language benefited all pupils in their mathematicsclasses, irrespective of their language histories.

While the wider study progressed, one of the teachers, Helen3, specificallyproblematized the issue of explicit language teaching. For Helen, successfulmathematics learning was related to pupils’ saying what they think in conciseand precise mathematical language. She had tried to develop mathematical lan-guage teaching as part of her practice in her multilingual classroom. When shereflected on her teaching during the study, however, she became aware ofinstances in which her explicit language teaching, in her terms, went on “toolong.” There was too much focus on what and how something was said, and themathematics under consideration was lost. She began to question what explicitmathematics language teaching meant in practice and whether and how it actu-ally helped. Helen’s experiences and reflections provoked questions like “Howdoes one pay attention to appropriate ways of speaking mathematically withoutconflating medium and message?” “How does a mathematics teacher focusattention on the form of speech in class without losing mathematical meaningand conceptual focus?”

I argue here that Lave and Wenger’s (1991) idea that access to a practicerequires its resources to be “transparent” (although this idea is not usuallyapplied to language as a resource or developed in school settings) can be usefuland illuminating when applied to the use of language in schools. I will presentwhat I call a teaching dilemma of transparency. The horns of this dilemma are,on the one side, that explicit mathematics language teaching, in which teachersattend to pupils’ verbal expressions as a public resource for class teaching,appears to be a primary condition for access to mathematics, particularly forpupils whose main language is not the language of instruction. On the other side,however, there is always the possibility in explicit language teaching of focusingtoo much on what is said and how it is said.

How teachers manage this dilemma needs to be addressed. Teachers’ decision-making at critical moments, although always a reflection of both their personalidentities and their teaching contexts, requires the ability to shift focus betweenlanguage per se and the mathematical problem under consideration. The chal-

48 The Dilemma of Transparency

2I use main language in place of what is often referred to as home language, vernacular, or moth-er tongue. By main language I mean the language of greatest day-to-day use and facility for thespeaker. In today’s complex multilingual society, many people speak more than two languages; itmay be that more than one is a main language and it is not appropriate to signal one as the secondlanguage; moreover, mother tongue is not necessarily synonymous with main language.

3This is a pseudonym.

lenge, of course, is to judge when and how such shifts are best for whom and forwhat purpose.

These assertions will be instantiated and illuminated through an analytic nar-rative vignette (Erickson, 1986) based on an episode in Helen’s multilingualGrade 11 trigonometry class together with her reflections on the episode. I beginwith some theoretical and methodological comments and then contextualizeHelen’s teaching in the wider study and in education more generally to enablethe reader to situate the episode, the reflections, and the discussion that followand form the substance of the article.

SOME THEORETICAL AND METHODOLOGICAL COMMENT

The wider study from which this article is drawn is framed by a socioculturaltheory of mind in which consciousness is constituted in and constitutive of activ-ity in social, cultural, and historical contexts. In particular, Lave and Wenger’ssocial practice theory (1991) and Mercer’s sociocultural theory (1995) provideanalytic tools for describing and explaining some teaching dilemmas in multilin-gual mathematics classrooms.

Lave and Wenger (1991) have described becoming knowledgeable about apractice like mathematics as the fashioning of identity in a community of prac-tice. Becoming knowledgeable means becoming a full participant in the practice,which includes learning to talk in the manner of the practice. They argued thatlearning occurs through legitimate peripheral participation in the learning cur-riculum of the community and entails having access to a wide range of ongoingactivity in the practice—access to old-timers, other members, information,resources, and opportunities for participation. Such access hinges on the conceptof transparency.

The significance of artifacts in the full complexity of their relations with the practicecan be more or less transparent to learners. Transparency in its simplest form mayimply that the inner workings of an artifact are available for the learner’s inspec-tion.… Transparency refers to the way in which using artifacts and understandingtheir significance interact to become one learning process. (pp. 102–103)

If an apprentice carpenter, for example, is to become a full participant in thepractice of carpentry, it is not sufficient that he or she learns to use a particularcutting tool—a carpentry resource. He or she also needs to understand how andwhere this tool developed in the practice of carpentry as well as how and for whatpurpose it is used now. Thus, access to artifacts in the community both throughtheir use and through understanding their significance is crucial. Artifacts (whichinclude material tools and technologies) are often treated as givens, as if theirhistories and significance are self-evident. Yet artifacts embody inner workingsthat are tied up with the history and development of the practice and that are hid-den. These inner workings need to be made available.

More pertinent to this article is the way Lave and Wenger (1991) elaboratedtransparency as involving the dual characteristics of invisibility and visibility:

49Jill Adler

invisibility in the form of unproblematic interpretation and integration (of the arti-fact) into activity, and visibility in the form of extended access to information. Thisis not a simple dichotomous distinction, since these two crucial characteristics are ina complex interplay. (p. 102)

Access to a practice relates to the dual visibility and invisibility of itsresources. Lave and Wenger (1991) used the metaphor of a window to clarifytheir concept of transparency. A window’s invisibility is what makes it a win-dow. It is an object through which the outside world becomes visible. However,set in a wall, the window is simultaneously highly visible. In other words, thatone can see through it is precisely what also makes it highly visible. For Laveand Wenger, the “mediating technologies” (p. 103) in a practice, like the car-pentry tool, need to be visible so that they can be noticed and used, and they needto be simultaneously invisible so that attention is focused on the subject matter,the object of attention in the practice (e.g., the cupboard being made by the car-penter).

Managing this duality of visibility and invisibility of resources for mathemat-ics learning in school can create dilemmas for teachers. Pupil discussion of amathematical task illuminates this duality if one understands talk as a resource inthe practice of school mathematics. (See the example provided later in the arti-cle.) Discussion of a task should enable the mathematical learning and so beinvisible4. It is the window through which the mathematics can be seen. At thesame time, the specificity of mathematical discourse inevitably enters such dis-cussion and can require explicit attention; that is, it needs to be visible. Learnersneed to understand the significance of mathematical talk. These are the dualcharacteristics of a transparent resource. It is possible, however, that in the math-ematics class the discussion itself becomes the focus and object of attentioninstead of a means to the mathematics. Then it obscures access to mathematicsby becoming too visible itself. This possibility might well be exaggerated in mul-tilingual situations to which learners bring a number of different main languages.In short, practices that are more or less transparent can enable, obstruct, or evendeny participation and, hence, access to the practice.

Lave and Wenger’s (1991) concept of transparency was developed in contextsof apprenticeship in which there is a situated and continuous movement fromperipheral to full participation in a practice. This movement also implies a situ-ated and continuous shifting between the visibility and invisibility of resourcesin use. Lave and Wenger focused on a learning curriculum, arguing that learningis not necessarily tied to explicit and planned instruction but is tied instead to par-ticipation in the practice. However, the school is a very different context fromthat of an apprenticeship. Lave and Wenger recognized this difference, but bytheir own admission they did not address what, for example, could be different


4Meira’s (1995) analysis of tool use (resources) in mathematics classrooms distinguishes “fields ofinvisibility,” which enable smooth entry into a practice, and “fields of visibility,” which extend infor-mation by making the world visible.

and specific about working with the dual visibility and invisibility of resourcesfor mathematics learning in school5.

As Mercer (1995) has argued, (mathematical) knowledge produced in the con-text of schooling is quite specific and is different from knowledge produced ineveryday contexts. Within the context of schooling he distinguished betweeneducational discourse—the discourse of teaching and learning in the classroom(e.g., ways of asking and answering questions in class)—and educated dis-course—new ways of using language (e.g., in algebra “let x be any number”),“ways with words” (p. 82) that would enable pupils to become active membersof wider communities that use this educated discourse6. Learners can developfamiliarity with and confidence in using new educated and educational discours-es only by using them. Teachers know that pupils participate in class in varyingways. In this sense they all, to some extent, engage in educational discourse.However, they also need opportunities to practice being users of educated dis-courses. Often there is a mismatch between the educational discourse in play (theways in which words are being used in the classroom) and the educated discoursethey are meant to be entering. So, in relation to mathematical discourse, theteacher’s role is to translate what is being said into mathematical discourse tohelp frame discussion, to pose questions, to suggest real-life connections, toprobe arguments, and to ask for evidence. The language practices of the class-room (educational discourse) must “scaffold students’ entry into mathematical[educated] discourse” (p. 82):

[Teachers] have to use educational discourse to organise, energise and maintain alocal mini-community of educated discourse. We can think of each teacher as a dis-course guide and each classroom as a discourse village, a small language outpostfrom which roads lead to larger communities of educated discourse.…Teachers areexpected to help their students develop ways of talking, writing and thinking whichwill enable them to travel on wider intellectual journeys…, but they have to startfrom where learners are, … and help them go back and forth across the bridge fromeveryday discourse into educated discourse. (Mercer, 1995, pp. 83–84)

I argued earlier in this article that as a teaching and learning resource, talkneeds to be both visible and invisible so that it can provide access to schoolmathematics. Mercer’s (1995) argument suggests a mediational role for teacherswhen they assist learners in crossing the bridge between talk as the invisible win-dow through which mathematics can be seen and, in Helen’s terms, more explic-it, visible mathematical language teaching.

From this sociocultural perspective, the teaching and learning of mathematicsin multilingual contexts needs to be understood as three-dimensional. It is notsimply about access to the language of learning (in this case English). It is alsoabout access to the language of mathematics (educated discourse) and access to

51Jill Adler

5See Moschkovich (1996) for an interesting argument for situated and continuous code-switchingpractices in bilingual settings.

6In Mercer’s terms, educated discourse in school mathematics will include the mathematics reg-ister (Halliday, 1978, as cited in Pimm, 1987).

classroom cultural processes (educational discourse). How do teachers managethe tensions in use of formal mathematical language and informal language, onthe one hand, and in the language of instruction that is not the main language ofthe pupils, on the other hand?

During 1992 and 1993 I undertook a qualitative study to find out how mathe-matics teachers in multilingual classrooms manage their complex practices. Aswith all qualitative methods, the sample in this study was small, purposive, andtheoretical (Cohen & Manion, 1989; Rose, 1982). Six secondary mathematicsteachers from the three different multilingual contexts in South Africa wereselected:

1. Two teachers were from recently desegregated historically White stateschools in which English was the dominant language in and around theschool; the teaching staff was White and English-speaking. There wereincreasing numbers of pupils with other main languages; hence, classes in theseschools were multilingual. Helen was one of these teachers.

2. Two teachers were from township-based Black state schools in which neitherteachers nor pupils had English as their main language. In addition, they didnot all share the same main language.

3. Two teachers were from private schools that had predominantly Black pupilswho did not have English as their main language and who brought a range ofmain languages to class. Teachers were predominantly White and English-speaking.

Each of the six teachers was a fully qualified and experienced secondary math-ematics teacher with a personal and professional interest in the study as well asa willingness to participate in the study. Furthermore, in spite of the politicalturbulence at the time, the teachers were able to facilitate access to their schoolsand classrooms. Thus, in addition to being theoretical and purposive, this sam-ple of six teachers was also an opportunity sample (Cohen & Manion, 1989;Rose, 1982).

To investigate teachers’ knowledge, I needed two sources of data. First, it wasnecessary to have teachers talk about their practices. Second, I needed data onactual classroom practices. Hence, interviews with teachers were supplementedwith observations of their classroom practice and with teachers’ reflections ontheir observed classes. The methods used to collect data were (a) an initial semi-structured, in-depth, interactive interview; (b) a report-back session, with the sixteachers interviewed to discuss and partially validate my initial analysis andinterpretation of their interviews; (c) up to 3 hours of observation of at least twolessons on consecutive days (videotaped) in one or two of each teacher’s class-es; (d) reflective interviews with each teacher on the videos of his or her class-room(s); and (e) the teachers’ participation in a series of follow-up workshops(three in all) on issues and aspects of the data that the teachers themselves want-ed to discuss with one another and to pursue. In preparation for these workshops,some of the teachers, including Helen, undertook small action-research projects


to further explore issues that had arisen for them during the research process. All interviews and workshops were audiotaped and transcribed. Analysis of

these transcriptions revealed noticeable presences and silences across differentteachers and their different multilingual contexts (Adler, 1995). Although teach-ers in different contexts emphasized different issues, a common thread across theinterviews and workshops was the expression of tensions and contradictions intheir practices.

The notion of a “teaching dilemma” became the key to unlocking teachers’knowledge of teaching and learning mathematics in complex multilingual set-tings. Teaching dilemmas are discussed in existing literature on teaching (e.g.,Berlak & Berlak, 1981; Lampert, 1985). For the Berlaks, a language of dilem-mas captures

contradictions that are simultaneously in consciousness and society. . . . [Dilemmas]capture not only the dialectic between alternative views, values, beliefs in persons andin society, but also in the dialectic of subject (the acting I) and object (the society andculture that are in us and upon us). (pp. 124–125)

Teachers in different multilingual contexts revealed different teaching dilem-mas when they spoke about their teaching, thus supporting the notion of teach-ing as a contextualized social practice (Adler, 1995). Tensions concerning code-switching7 (using more than one language in class) were emphasized by Blackteachers in township schools (Adler, 1998). Tensions related to mediation wereemphasized by teachers who had tried to create more participatory-inquiryapproaches in their classrooms (Adler, 1997). Helen and other teachers whoseclassrooms rapidly became multilingual faced the inherent tensions in explicitand implicit language practices in their multilingual classrooms and what I haveinterpreted as the dilemma of needing both to see and to see through mathemat-ical language in class.

Of course, what teachers reflect on and talk about is only part of what theyknow. What happens in practice? In particular, how does Helen’s practice illu-minate the dilemma of transparency, her explicit mathematics language teaching,and the need for both visibility and invisibility of talk in her class?

THE CONTEXT

Helen and Her Focus on Explicit Language Teaching

Helen is White and English-speaking8 with 6 years experience as a secondarymathematics teacher. During the workshops she invited the other participatingteachers to struggle with her over whether or not explicit language teaching actu-ally helps, over whether and how working on pupils’ abilities to “talk mathe-

53Jill Adler

7Code-switching is an individual’s (more or less) deliberate alternation between two or more lan-guages for a range of purposes.

8Interestingly, Helen’s mother is French, and she grew up speaking French and English at home.Helen also speaks and understands some Zulu.

matics” is a good thing. In the language of this article, she thus raised the issueof talk as a transparent resource in the mathematics classroom. That the dilem-ma of transparency was particularly strong for Helen is not surprising consider-ing her view of mathematics as language and her view of language as a crucialresource in the practices in her classroom. In short, Helen appeared to share Laveand Wenger’s (1991) notion that becoming knowledgeable means learning totalk or, in Mercer’s terms, learning educated (i.e., mathematical) discourse. Inher initial interview she said that her greatest thrill was when pupils couldexpress themselves, describing their thinking, in mathematical language. Sherepeated this view in her reflective interview: “’Cause if they start to describesomething to me in accurate mathematical language, it does seem to reflect somekind of mastery.”

Through her reflections and her discussions with the other teachers during theworkshops, Helen came to mean by explicit mathematics language teachingmore than the teacher’s making mathematical and classroom discourse explicit.She included teachers’ encouraging and working on pupils’ verbalizations in themathematics classroom with the following:

1. Attention to pronunciation and clarity of instructions. When she discussed oneof her videotaped lessons, Helen said, “One of the issues was linguistic, . . .the sound issue between sides with an s and sizzzze. A lot were hearing sizewhen I was saying sides, and we picked up on that issue.” She pointed out thatthe pronunciation of particular words by pupils or the teacher or both couldbe a problem in a multilingual mathematics classroom. Teachers’ instructionscould be misunderstood. For Helen, clear speech and clear instructions wereimportant; she thought that they could improve clarity for all pupils, not justlearners whose main language was not English.

2. Pupil verbalization (putting things into words) as a tool for thinking9. Helenraised for discussion with the other teachers her view that pupils’ saying whatthey were thinking would help them know the mathematics under consider-ation: “Debbie, who did that very nice summary at the end of the last lesson,has got absolutely no idea at this stage. For me it seemed that if she had donethis great summary the day before, that she should have been able to do that.”

3. Verbalization of mathematical thinking as a display of mathematical knowl-edge. Helen articulated on numerous occasions the point that if pupils couldclearly say what they were thinking, then they knew the mathematics underconsideration: “Now listen to how clearly Rosie verbalises that, … and sheis a successful student. There must be a relationship.”

4. Pupil verbalization as a tool for teaching. The teachers agreed that pupils’ say-ing what they were thinking would, at least, help the teacher to know what


9In sociocultural terms, this is the dialectic between language and thought, in which paraphrasingis associated with personal appropriation of cultural concepts and ideas (i.e., within a community ofpractice) (Leontiev & Luria, 1968).

learners were construing and to respond appropriately. One summed up thisview in the workshop discussion: “Hearing what it is pupils think and artic-ulate can help you [the teacher] see what they understand.”

Clearly, Helen regarded pupils’ verbalization in the mathematics classroom asa resource. That verbalization is a tool for thinking and a display of mathemati-cal knowledge has been recognized by Barnes (1976). In fact, all a teacher hasaccess to are the forms of language students use to display knowledge (Pimm,1996). That pupil verbalization is a tool both for thinking and for teaching meansthat language functions as a psychological tool when students put their mathe-matical ideas into words and as a cultural tool10 for the sharing and joint con-struction of knowledge (Mercer, 1995; Vygotsky, 1978) when the teacher usespupil verbalization as a tool for teaching. Thus, although for Helen the practiceof explicit language teaching entailed being explicit about mathematical dis-course, explicit language teaching was bound up with her view of a strong andcomplex relationship between language and learning.

The School and Class

Helen taught in an historically White state school for girls. This school dera-cialized faster than many similar schools, and at the time of this research study,fewer than 50% of the pupils were White. The school was well equipped. Theclass in which observation and videotaping were carried out was a mixed-abilityclass of 30 pupils. English, Sesotho, and Zulu, all now official languages inSouth Africa, were some of the main languages spoken by pupils in this class.There were also immigrant pupils, one of whom had arrived in the countryrecently from Taiwan and spoke no English. The language of instruction in theschool was English, and all public interaction in Helen’s classes was in English.

Helen’s Approach

Helen’s classes, although largely teacher directed, were also interactive andtask based. Group-based tasks were followed by whole-class, teacher-directedreaction to reports pupils gave. In Mercer’s (1995) terms, Helen’s approachentailed an educational discourse that included situations in which pupils talkedwith one another during their interaction on tasks, reported verbally on thesetasks and interactions, and engaged with Helen in public verbal interactions. Itwas during these public interactions that Helen paid explicit attention to educat-ed discourse.

Helen’s approach and the resulting classroom culture that included pupil-pupildiscussion and verbalization were not surprising in light of her views of mathe-

55Jill Adler

10It is important to note here (see Bernstein, 1993) that language as a cultural tool is a tool for learn-ing. But language itself is a producer of relations of power. This point is also made by Ivic (1989).Although language is a resource in the classroom, it is does not function in any simple, unproblem-atic way.

matics as language as well as her concern that mathematics should be contextu-alized and learning should be meaningful and lasting. Moreover, her approachreflected a significant shift away from the “drill and practice” model dominant inSouth African mathematics classrooms. Helen also held strong views on accessto mathematics for both girls and the racially disadvantaged in South Africa. Itis thus important to note here that Helen engaged with the issues of code-switch-ing and effective mediation. Her overarching concern, however, and thus thefocus of this article, was whether or not explicit mathematical language teachingdoes help students—whether it makes mathematics more accessible.

Helen introduced trigonometry to one of her Grade 10 classes with an outdooractivity in which students investigated the lengths of shadows caused by the sunat different times of the day. This activity was followed by activities in whichgroups of pupils measured and compared the ratios of the lengths of sides of aright-angled triangle having one angle of 40 degrees. Later, when groups report-ed what they had learned, Helen attempted to develop their understanding of con-stant ratios and related these ratios to the programming of trigonometric ratiosinto a calculator.

In the first workshop11 (which occurred after the initial interview in whichHelen expressed her firm commitment to explicit language teaching and after shehad observed and reflected on her video), Helen asked the other teachers to helpher grapple with whether “saying it” actually is indicative of understanding, ofknowing. Helen then followed up her question with her own action research. Sheplanned a double lesson (1 hour) on trigonometry for the same students who wereby this time in Grade 11. She organized the lesson around group discussion of aset of tasks, tape-recorded the discussions of two of the student groups, and invit-ed me to observe and videotape the lesson. She wanted to listen carefully to howpupils engaged in discussion on mathematical tasks and to reflect more system-atically on her assumptions about a strong relationship between language andlearning and about the values of explicit mathematics language teaching.

After Helen had viewed the videotape and listened to the tape-recordings ofthe student groups, she brought her reflections from this action research to thesecond workshop with the other teachers in the research study. The vignettebelow provides insight first into how Helen coped in practice with pupils’ mean-ings and with their mathematical expression and second into reflections on herpractice. Together with some of my own commentary, the vignette illuminatesthe dilemma of transparency. The episode and reflections presented in thevignette are neither typical nor rare (Erickson, 1986). Instead they are instancesthat illustrate and create a space for opening dialogue on an important element ofteachers’ knowledge of their practices in multilingual classrooms—an elementquite apparent in newly deracialized schools in South Africa.


11The initial interviews, classroom observations, and reflective interviews were all completed byNovember 1992. The three workshops with the teachers took place in February, May, and August thefollowing year.

A VIGNETTE—A CLASSROOM EPISODE

The episode described below took place in the first trigonometry lesson ofGrade 11 and was part of Helen’s action research in the year following the ini-tial interviews and videotaping of her teaching trigonometry to her Grade 10class. In this lesson Helen asked pupils in groups of four to discuss whattrigonometry meant to them and then to report back their meanings to the rest ofthe class in a “maximum of 2 minutes per group … using key words and puttingacross [the] main ideas.”

Most of the groups related trigonometry to determining “the size and sides ofthe angles,” stating that “there are six ratios”; most presentations included chalk-board diagrams showing two similar right-angled triangles as shown in Figure 1.

57Jill Adler

θ θ

Figure 1. Similar triangles.

Specifically, two groups’ explanations included the following expressions: “Uh,we said the ratio of two angles is independent to the size of the angle in the othertwo triangles,” and “Therefore, we came to the same thing that the ratio of twosides is independent to the size of the tri, of the angle in two triangles.”

After all the presentations, Helen moved to the front of the class. She drew thestudents’ attention to various aspects of the reports and then focused explicitlyon the students’ expressions quoted above. [Note: . . . indicates a short pause; His Helen; S6 (for example) is Student 6, when the name of the student is notknown; the name of the student is used if it was clearly articulated in the lesson;Ss is a number of students talking at the same time.]

H: Say that to me slowly, the.…S6: (H writes as pupil talks) The ratios of the two sides … is independent to the size of

the angles … in the two triangles….H: Is independent to …?S6: The two tri … is independent, no, the two sides is independent.… H: The ratios of the two sides is independent to?S6: The size of the angles in the two triangles (and H finishes writing).H: Let’s look at that statement carefully. I need some distance. (She moves back from

the board and then reads slowly) “The ratios of the two sides is independent … tothe size of the angle … in the two triangles.” What does that statement mean to, uh,to anyone?

S6: It means that, uh, whether the angles … when you’ve got two triangles, and theangles come up to the same degree, you, uh, it doesn’t matter how long or short thetriangle is, your angles, as long as your angles are equal (inaudible).

H: Now listen to what you said. “How long or short the triangles are?”S6: The length, the length of the triangle. H: Triangle is a shape.Ss: (Mumbling) The length of the sides.H: The length of the sides of the triangle. Okay. You know. Let’s just look at this word

independent. Okay. Now I know when I teach this, I use the word independent andthen you think, “Well that’s a nice fancy word to use. If I just repeat it nicely in theright sentence, then she’ll be very impressed.” But, when you use the word inde-pendent, you’ve got to know what it means. What does it mean? Phindile?

Phin: (Some mumbling) It stands on its own.

Helen first questioned the pupils’ expression of “long or short” triangles, andpupils responded indicating their awareness in this interchange that they wereexpected to be more mathematically precise in what they were saying. She ledthem to say “the length of the sides” of the triangle and then pulled the wordindependent out on its own and attended to its meaning. She then returned tofocus on the sentence in which it was placed:

H: Okay. All right. Is that statement true?Ss: (Some say no; some say yes.)H: Must I put a true or a false at the end of it?Ss: (Some say true; some say false.)H: Okay. Who says it’s true? (S6 raises her hand.) S6 says it’s true ’cause she said it.

(Students laugh.) Okay, who says it’s false? (Students laugh.) What do you think,Phindile?

Phin: I don’t know; I don’t understand the sentence.H: Okay, let’s try and sort out the sentence. “The ratios of two sides”—that’s a true

part of the line, uh, of the sentence. Does that make sense?Ss: Yes.H: Okay. “Ratios of two sides”—we know we always talk about opposite to

hypotenuse or adjacent to opposite or something … we are talking about a ratio andwe are talking about two sides.… “Is independent.” Okay. Wait. The most impor-tant word in the sentence is independent? Right. So one thing is independent ofanother. So maybe if I just change this [to] to of, … we can start. So the ratio isindependent from what? Size of the angle in the two triangles? …

Ss: (Some mumbling of “It’s true.”)H: Who says it’s true? Why? S7: Because, Ma’am, um, I think it means that, no, uh, if if you, if you have, uh, one

big triangle and you have one small triangle and you have the same angle in bothof them, uh, the the size of the angles is equal, then the ratio of the, of the sideswon’t change.

H: Now listen to what you’re saying. You’re saying you’ve got, … you said to me(and H links the italicized words below to related words on the board as she speaks)you’ve got the size of two triangles and then you said that the angle inside them isthe same, okay. So if we want to, is what she said different to what is on the boardat the moment?


Ss: (Some say no, and some say yes.)H: She said to me, “The ratio of the two sides is independent of the size of the trian-

gle, when you’ve got the same angle in all of them.” So it is not true to say that theratios are independent of the size of the angle. The size of the angle is exactly whatmakes the fundamental difference. Because if I’ve got two triangles, these twobeautiful triangles over here, 40, 40 (and she writes 40 [degrees] in the corre-sponding angles of the two similar triangles on the board), and these two over here,20, 20 (and again writes these angle sizes for another set of similar triangles on theboard [see Figure 2]).… Would I get—if I say spoke about … sine here [40-degreeangle in the first triangle] and sine here [20-degree angle in the third triangle]?Okay? Will I get the same answer?

Ss: No.H: No! I’ll get two different answers. So it is not true to say to me it is independent of

the size of the angle—because the angle, if it is 40, makes the difference to 20,right? It’s the size of the angle that makes the difference.… Does that make senseto you?

Ss: No.H: What doesn’t make sense?S2: Ma’am?H: Ja (local word meaning yes)?S2: It makes a difference to what?H: It makes a difference … to.… (Students laugh.) Where was I starting off? … um,

let me start again. …

59Jill Adler

40° 20° 20°40°

Figure 2. Two sets of similar triangles.

Helen then recapped by drawing attention to diagrams on the board, to reiter-ate how two different right-angled triangles each with a 40-degree angle wouldhave the same ratios between their corresponding sides as two different right-angled triangles each with a 20-degree angle. But the two sets of ratios will bedifferent precisely because the angles in the triangle pairs are different. She thenasked the pupil who first articulated the sentence to state, in her own words, whatshe understood.

Both in this lesson and in the other lessons videotaped and observed the pre-vious year, Helen directed pupils’ reporting back. After each group reported, shedirected whole-class, teacher-pupil interaction on what had been presented,focusing attention on problems and reformulating and recapping when neces-sary. It was in this part of the lesson that explicit language teaching was evident.In Mercer’s (1995) terms, it is here that Helen made explicit and intentional

instructional moves to bridge or scaffold what pupils say and conventional math-ematical discourse.

In the episode described above, Helen asked what the statement with “indepen-dent to the size of the angles” meant, inviting rethinking and further elaboration.She tried to engage pupils in making sense of the statement. When S7 expresseda clear explanation, she focused on this explanation, reformulated it, and asked theclass to compare the two versions—what had just been said and what was writtenon the board. She assisted by recapping and stressing that the “angle makes thefundamental difference” only to find that the focus of the mathematical discussionwas lost on the pupils. She therefore reformulated and recapped again, and then,as she reflected, she had “gone on too long.” Helen’s practice had come to includeperiodic focusing of her and her pupils’ attention on how to “speak mathematics,”that is, how to use educated discourse, and she faced a new challenge becauseexplicit language teaching could also cause confusion. I have called this challengethe dilemma of transparency, of talk as a resource in the classroom bearing thedual characteristics of visibility and invisibility.

HELEN’S REFLECTIONS

Helen spoke about explicit language teaching on numerous occasions and invarious ways throughout the study—in her initial interview, in her reflectiveinterview, and in the workshops with the other teachers. However, it was in thesecond workshop, as reflected in the quotes below, that the dilemma of trans-parency, of managing the visibility and invisibility of language as a resource forteaching and learning mathematics in multilingual classrooms, became mostclear.

For the opening of the second workshop, Helen played the video from thepoint at which the student said, “The ratio of the two sides is independent to thesize of the angles in the two triangles” (when Helen was writing what was beingsaid, word for word, on the board for the class to think about). She then said tothe other teachers,

Just after the sentence is written on the board and I ask, “What do you understand bythis statement?” the one child puts forward a perfect explanation. She talks about theangle being the same in both triangles and then she talks about the depth of the tri-angles, or whatever, and I pick up on that … and then this [other] child now does itabsolutely perfectly. So, [those are] two very good expressions of what is going on.And yet when you ask the class, “Is this sentence [sentence she has written verbatimfrom the first student] correct?” there is this complete silence. So the question for meis, even in the minds of those two children who put forward such consistent expla-nations, what’s going on with them … that they cannot … um … pick up incorrect-ness in the sentence?

Helen went on to revisit the question she had raised in the first workshop: “Ifthey can say it, do they know it?” She then posed a central question on verbal-ization that points to the dilemma of transparency:


In retrospect, when I look at that lesson, I went on but much too long (laughter), onand on and on, and I keep saying the same thing and I repeat myself, on and on.…But the thing is then if you have a sense that there is a shared meaning amongst thegroup, can you go with it? Um … when the sentence is completely wrong? … Canyou let it go? Can a teacher use a sense of shared meaning to move on? I think thisis a central question in terms of the verbalization and discussion.

In concluding her presentation for discussion to the workshop, Helen remarkedhow clearly she remembered that episode and the particular moment when, in herattempt to teach mathematical language explicitly, the mathematical focus of thelesson was lost. She remembered being “completely thrown” by S2’s interjec-tion: “Ma’am? … It makes a difference to what?”

DISCUSSION

Helen’s working assumption of a strong relationship between language andthought was seriously challenged when she observed pupils who could expresstheir thinking on one day but could not on the next, who could express clear andcorrect mathematical thinking but could not discern problematic expressions ofothers, and who said things “wrong” but created a sense for Helen that they hadsome grasp of the mathematics they were discussing. She also saw how in herfocus on language teaching and in her attention to the pupils’ use of the termindependent, the pupils lost their focus on the mathematical and trigonometricproblem from which that use arose.

This vignette, presenting an episode in Helen’s class, and her reflections on theepisode reveal the tensions in whole-class interaction when attention is focusedon pupils’ mathematical verbalizations and highlight the dilemma this explicitmathematics language teaching can create for teachers. Through Helen’s actionsand reflections one can see what is known only too well—that some mathemat-ical ideas are difficult for pupils to verbalize precisely and with meaning.

The specific challenges for Helen lay in scaffolding educated discourse and inmoving between talk used for thinking while pupils work on a task and talk usedas a display of knowledge. I have argued that, in sociocultural terms, teachingand learning mathematics entail this moving back and forth. Helen providedopportunity for pupils, among themselves, to elaborate and then share theirmeanings of the term trigonometry. Through her elicitation of pupils’ thinkingshe discovered her students’ confusion, and she moved to clarify the issuethrough a particular scaffolding process. She worked explicitly on pupils’expressions of their mathematical ideas. She asked questions in her attempt tobring into focus the incorrect use of the concept and term independent, and shefinally reformulated and recapped, emphasizing in clear (to her) mathematicallanguage what she saw as most significant in the trigonometry description thathad emerged from the pupils. But this explicit language teaching was a struggle.

Helen’s practice and her knowledge of it help us identify a fundamental peda-gogic tension in the explicit way she dealt with language issues, particularly talk,

61Jill Adler

in her multilingual mathematics class. She harnessed talk as a resource in herclassroom. As a resource in her practice, the transparency of talk (i.e., itsenabling use by learners) is related to both its visibility and its invisibility.Specifically, Helen attended to pupils’ expressions as a shared public resourcefor class teaching. This characteristic of classroom talk is not shared by speechin many other settings (Pimm, 1996). The language itself becomes visible andthe explicit focus of attention. It is no longer the medium of expression, but,instead, it is the message—that to which the pupils now attend.

The classroom episode shows Helen struggling to mediate the scientific con-cepts (Vygotsky, 1986) of constant ratios, dependence, and independence whenthey arise in school trigonometry. She did this mediation in her multilingualclassroom, in which the complex three-dimensional dynamic of access toEnglish, to mathematical discourse, and to classroom cultural processes inter-sects with her educational and political beliefs as well as with her view of math-ematics as language. Helen focused on correct ways of speaking mathematical-ly, thus attempting to provide access to English and to mathematical discourse.These attempts occurred, however, within her classroom culture, within whichlanguage was used simultaneously to explore and to display mathematicalknowledge. And problems emerged.

On reflection, Helen felt that her attempt to enable access to mathematical(educated) discourse brought with it the problem of “going on too long.” Inexplicitly making mathematical language visible, she caused it to becomeopaque, obscuring the mathematical problem. It is in this instance that the dilem-ma of transparency—of whether (and when) to make mathematical languageexplicit or leave it more implicit—can be seen. For Helen, there were both polit-ical and educational dimensions to this dilemma. If she focused on language fortoo long, she would inadvertently obscure the mathematics under consideration.If she left too much implicit, she would then run the risk of losing or alienatingthose who most needed opportunity for access to educated discourse. She won-dered about the possible effects of leaving a shared sense of trigonometric ratiosbut a public display of incorrect mathematical language: “If they don’t say itright, can I let it go?”

Of course, there is a world of difference between “what they are saying iswrong” and “I can’t get at what they are trying to say to me” (Pimm, 1996).Teachers like Helen (including other teachers in the wider study) were concernedabout their verbalizing and having pupils verbalize “correct” mathematical lan-guage, about using language as a shared public resource in the mathematicsclassroom. And although access to educated mathematical discourse is impor-tant, Helen’s classroom illustrates how explicit mathematical language teachingcan initiate a dilemma of transparency.

The fundamental tension between implicit and explicit practices with respectto language issues in multilingual mathematics classrooms is revealed in theepisodes of Helen’s teaching. As I have argued elsewhere (Adler, 1997), thesekinds of issues are present in all classrooms, but they are present in particularly


heightened form in multilingual classrooms. There are no simple answers here,nor is it the purpose of this article to provide answers. Instead, in this article Ipresent a description and analysis of an instance of a teacher grappling with theissue of transparency while she tried to embrace new practices and make mathe-matical knowledge available in her particular multilingual classroom.

CONCLUSION

Through Helen’s experience and her reflections on it, one sees that explicitmathematics language teaching, although beneficial, is not necessarily alwaysappropriate. This kind of explicit teaching can result in a language-related dilem-ma of transparency with its dual characteristics of visibility and invisibility.Helen’s particular questions and reflections, and the discussion they provoked inthe workshops, highlight tensions teachers can experience when they try to initi-ate new and different forms of instruction.

Lave and Wenger’s (1991) notion of transparency can illuminate classroomprocesses. Both visibility and invisibility are part of transparency in the practiceof teaching mathematics. Resources need to be seen to be used. They also needto be invisible to illuminate aspects of practice. For talk to be a resource formathematics learning it needs to be transparent; learners must be able to see itand use it. They must be able to focus on language per se when necessary, butthey must also be able to render it invisible when they are using it as a means forbuilding mathematical knowledge. For school mathematics teachers, it is notsimply a matter of going on too long but of managing and mediating the shift offocus between mathematical language and the mathematical problem (which ofcourse are intertwined). There is no resolution to the dilemma of transparency formathematics teachers; there is only its management through awareness and care-ful instructional moves when making talk visible in moments of practice.

REFERENCES

Adler, J. (1995). Dilemmas and a paradox: Secondary mathematics teachers’ knowledge of theirteaching in multilingual classrooms. Teaching & Teacher Education, 11 (3), 263–274.

Adler, J. (1996a). Lave and Wenger’s social practice theory and teaching and learning school math-ematics. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th conference of the InternationalGroup for the Psychology of Mathematics Education (Vol. 2, pp. 3–10). Valencia, Spain:University of Valencia.

Adler, J. (1996b). Secondary teachers’ knowledge of the dynamics of teaching and learning mathe-matics in multilingual classrooms. Unpublished doctoral dissertation, University of theWitwatersrand, Johannesburg, South Africa.

Adler, J. (1997). A participatory-inquiry approach and the mediation of mathematical knowledge ina multilingual classroom. Educational Studies in Mathematics, 33, 235–258.

Adler, J. (1998). A language of teaching dilemmas: Unlocking the complex multilingual secondarymathematics classroom. For the Learning of Mathematics, 18 (1), 24–33.

Barnes, D. (1976). From communication to curriculum. Harmondsworth, England: Penguin.Berlak, A., & Berlak, H. (1981). Dilemmas of schooling: Teaching and social change. London:

Methuen.

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Bernstein, B. (1993). Foreword. In H. Daniels (Ed.), Charting the agenda: Educational activity afterVygotsky (pp. xii–xxiii). London: Routledge.

Cohen, L., & Manion, L. (1989). Research methods in education (3rd ed.). London: Routledge.Erickson, F. (1986). Qualitative methods in research on teaching. In M. C. Wittrock (Ed.), Handbook

on research on teaching (3rd ed., pp. 119–161). New York: Macmillan.Ivic, I. (1989). Profiles of educators: Lev S. Vygotsky (1896–1934). Prospects: Quarterly Review of

Comparative Education, XIX, 427–435.Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems in practice.

Harvard Educational Review, 55, 178–194.Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge,

England: Cambridge University Press.Leontiev, A. N., & Luria, A. R. (1968). The psychological ideas of L. S. Vygotskii. In B. B. Wolman

(Ed.), Historical roots of contemporary psychology (pp. 338–367). New York: Harper & Row.Levine, J. (1993). Learning English as an additional language in multilingual classrooms. In H.

Daniels (Ed.), Charting the agenda: Educational activity after Vygotsky (pp. 190–215). London:Routledge.

Meira, L. (1995). Mediation by tools in the mathematics classroom. In L. Meira & D. Carraher (Eds.),Proceedings of the 19th international conference for the Psychology of Mathematics Education(Vol. 1, pp. 102–111). Recife, Brazil: Universidade Federal de Pernambuco.

Mercer, N. (1995). The guided construction of knowledge: Talk among teachers and learners.Clevedon, England: Multilingual Matters.

Moschkovich, J. (1996). Learning math in two languages. In L. Puig & A. Gutiérrez (Eds.),Proceedings of the 20th conference of the International Group for the Psychology of MathematicsEducation (Vol. 4, pp. 27–34). Valencia, Spain: University of Valencia.

Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. London:Routledge.

Pimm, D. (1996). Modern times: The symbolic surfaces of language, mathematics and art. In L. Puig& A. Gutiérrez (Eds.), Proceedings of the 20th conference of the International Group for thePsychology of Mathematics Education (Vol. 1, pp. 35–50). Valencia, Spain: University ofValencia.

Rose, G. (1982). Deciphering sociological research. London: Macmillan. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes.

London: Harvard University Press.Vygotsky, L. S. (1986). Thought and language (Rev. ed., A. Kozulin, Ed.). Cambridge, MA: M.I.T.

Press.

Author Note

Jill Adler, Professor of Mathematics Education Development, Mathematics Department, University ofthe Witwatersrand, Private Bag 3, P O Wits, 2050, South Africa; [email protected]


Motivation for Achievement in Mathematics:

Findings, Generalizations, andCriticisms of the Research

James A. Middleton, Arizona State UniversityPhotini A. Spanias, Arizona State University

In this review we examine recent research in the area of motivation in mathematics educationand discuss findings from research perspectives in this domain. We note consistencies acrossresearch perspectives that suggest a set of generalizable conclusions about the contextual fac-tors, cognitive processes, and benefits of interventions that affect students’ and teachers’ moti-vational attitudes. Criticisms are leveled concerning the lack of theoretical guidance driving theconduct and interpretation of the majority of studies in the field. Few researchers have attemptedto extend current theories of motivation in ways that are consistent with the current research onlearning and classroom discourse. In particular, researchers interested in studying motivationin the content domain of school mathematics need to examine the relationship that existsbetween mathematics as a socially constructed field and students’ desire to achieve.

Key Words: Achievement; Attitudes; Beliefs; Motivation; Review of research

National assessment data from the 1980s (Carpenter, Corbitt, Kepner,Lindquist, & Reys, 1981; Dossey, Mullis, Lindquist, & Chambers, 1988) haveindicated that American children tend to enjoy mathematics in the primarygrades but that this level of enjoyment tends to fall dramatically when childrenprogress into and through high school. In addition, although students feel thatmathematics is important, the number of students who want to take more math-ematics in school is declining steadily (Dossey et al., 1988). These statistics seemalarming when coupled with the fact that children do not possess the mathemat-ical knowledge that they will need to function smoothly in our increasingly tech-nological society. The problem is considered important enough for the NationalCouncil of Teachers of Mathematics (NCTM) to place the motivational domainsLearning to value mathematics and Becoming confident in one’s own ability astwo of its foremost goals for students as an attempt to change the nature of schoolmathematics (NCTM, 1989).

Our purpose in this review is to describe theoretical orientations guidingresearch in mathematics motivation and to discuss findings in terms of how theyfacilitate or inhibit achievement. First, we discuss definitions of motivation anddistinctions among types of motivation in education. Second, we discuss theo-retical orientations and describe representative research from these orientations.Third, findings from the reviewed studies are drawn into a set of conclusions rep-



resenting what is known regarding students’ motivation in mathematics, howinequities in mathematics education are reflected in students’ motivational pat-terns, and the role of the teacher in enhancing or inhibiting students’ motivation.Last, we raise criticisms regarding the role of theory in informing research andthe lack of adequate conceptualization prevalent in operationally defining moti-vation, achievement, and mathematics as a content domain.

Although pertinent work has been done in motivation outside the domain ofmathematics, in this article we focus on studies in which the participants werestudents in mathematics classes or mathematics teachers. We made this choice inresponse to criticisms that context has been largely ignored in studies of teach-ing and learning (Romberg & Carpenter, 1986). Moreover, there is convincingevidence that student effort and performance can be better explained by task-spe-cific analyses of motivation in mathematics than by general measures of motiva-tion (Seegers & Boekaerts, 1993). By focusing on studies within the mathemat-ics education literature, we hope to draw out conclusions that are sensitive to thecontext of school mathematics. This analysis allows exposition and criticismregarding the limitations of our knowledge about motivation related to mathe-matics as a content domain.

JUST WHAT ARE MOTIVATIONS?

Simply stated, motivations are reasons individuals have for behaving in agiven manner in a given situation. They exist as part of one’s goal structures,one’s beliefs about what is important, and they determine whether or not one willengage in a given pursuit (Ames, 1992). Two distinct types of academic motiva-tion interrelate in most academic settings—intrinsic and extrinsic motivation.Academic intrinsic motivation is the drive or desire of the student to engage inlearning “for its own sake.” Students who are intrinsically motivated engage inacademic tasks because they enjoy them. They feel that learning is importantwith respect to their self-images, and they seek out learning activities for thesheer joy of learning (Middleton, 1992/1993a). Their motivations tend to focuson learning goals such as understanding and mastery of mathematical concepts(Ames & Archer, 1988; Duda & Nicholls, 1992; Dweck, 1986). Students whoare extrinsically motivated engage in academic tasks to obtain rewards (e.g.,good grades, approval) or to avoid punishment (e.g., bad grades, disapproval).These students’ motivations tend to center on such performance goals as obtain-ing favorable judgments of their competence from teachers, parents, and peers oravoiding negative judgments of their competence (Ames, 1992; Ames & Archer,1988; Duda & Nicholls, 1992; Dweck, 1986).

When individuals engage in tasks in which they are motivated intrinsically,they tend to exhibit a number of pedagogically desirable behaviors includingincreased time on task, persistence in the face of failure, more elaborative pro-cessing and monitoring of comprehension, selection of more difficult tasks,greater creativity and risk taking, selection of deeper and more efficient perfor-

66 Motivation in Mathematics

mance and learning strategies, and choice of an activity in the absence of anextrinsic reward (Lepper, 1988). Moreover, intrinsic motivation is related to stu-dents’ perceptions of their competence in mathematics, to whether they are moti-vated by curiosity or by grades, and to whether their orientation toward academ-ic achievement can be characterized as a mastery orientation. Intrinsic motiva-tion in other subject areas seems to be only moderately correlated with these vari-ables (Gottfried, 1985).

Researchers have found that although achievement, ability, and perceivedcompetence each contribute to students’ desire to learn mathematics, intrinsicmotivation is more complex than the additive effects of these domains. Whenstudents see themselves as capable of doing well in mathematics, they tend tovalue mathematics more than students who do not see themselves as capable ofdoing well (Eccles, Wigfield, & Reuman, 1987; Midgley, Feldlaufer, & Eccles,1989), but these expectations of success also influence short-term strategy use(Meece, Wigfield, & Eccles, 1990; Pokay & Blumenfeld, 1990), thereby inhibit-ing or augmenting achievement. It is likely that students must feel comfortablewith mathematics, must be challenged to achieve, and must expect to succeedbefore the development of intrinsic motivation can begin.

The findings of these studies suggest that the decline in positive attitudestoward mathematics can be explained in part as functions of lack of teacher sup-portiveness and classroom environment. These findings, along with results fromnational assessments (Dossey et al., 1988), suggest that motivational patterns arelearned and, what is particularly distressing, that students generally learn to dis-like mathematics and that this dislike becomes an integral part of their mathe-matical self-concepts.

When one looks at the subtle ways in which motivations are formed, modified,and sustained, it becomes clear that there is no such thing as an unmotivatedchild. Children are motivated. Motivations help guide children’s activity; theyprovide a structure for evaluating the outcomes of activity; and they help deter-mine whether or not children will engage in future mathematical activity. Thefollowing discussions describe prominent approaches to investigating and apply-ing motivational theory. We present the main theories, review research, and dis-cuss results in terms of classroom practices that facilitate or inhibit students’developing productive motivational patterns.

THEORETICAL ORIENTATIONS

Behavioral Theories of Motivation

Throughout most of the 20th century, behaviorist theories of motivation dom-inated the literature. In this perspective, motivations are seen as incentives forperforming a given behavior (Spence, 1960). Newer reformulations of these the-ories (McClelland, 1965, cited in Covington, 1984) have focused on the poten-tial conflict between an individual’s perceived necessity for success and per-ceived necessity for avoiding failure.

67James A. Middleton and Photini A. Spanias

Although the declining popularity of behavioral research has led to a decliningnumber of studies in this paradigm, this theoretical orientation has provided power-ful knowledge about student motivation in mathematics. First, research indicates thatsuccess in mathematics is a powerful influence on the motivation to achieve.Students perceive success as reinforcing, and they will engage in mathematics if theyexpect to be successful. In addition, students will not only engage more, they will alsotend to enjoy tasks for which they have a moderately high probability of successmore than tasks for which the probability of success is near chance (Dickinson &Butt, 1989). Although success may not be the only determinant of on-task behavior,it is clearly related to the achievement motivation of children in mathematics.

Second, and more important, an orientation toward achieving success in math-ematics can be built into the mathematics classroom. When students are givenincentives to achieve, the motivation and achievement of entire classes can beraised (Alschuler, 1969). When children are rewarded for choosing a high levelof personal success in mathematics, they tend to enjoy mathematics more andachieve more than when they are not given incentives. Slavin (1984), for exam-ple, recommended the provision of group incentives to motivate students toachieve (i.e., providing a group reward for individual learning). Because thegroup score is rewarded, children are motivated to help others in the group andare pressured to learn well themselves; through this practice, individual account-ability is emphasized. This practice allows students to attribute their successes tothemselves and their failures to the group, thus reducing the individual’s onus forfailure proportionately to the number of students in the group.

Severe limitations are, however, evident in this paradigm, which depends onachievement measures that use either multiple-choice tests or well-defined prob-lems. It is unclear how more realistic problems, ones that provide more avenuesfor failure, would affect the success rate of children. Also unclear is whether suc-cess should be defined as success with a problem as a whole or in the steps nec-essary to solve the problem. The operational definition of success inherent inbehaviorist research, with a focus on discrete observable behaviors, may be toomolecular in scope or too removed from children’s attitudes to be a valid indexof their achievement motivation. Time-on-task is often used as an index of moti-vation (e.g., Dickinson & Butt, 1989). Reliance on time-on-task, however, intro-duces a confounding variable into the research design: The difficulty level of aproblem is related to the time required to solve the problem, independent of moti-vation. In addition, because behaviorist theories have not traditionally been con-cerned with individual differences, they fail to provide information on how stu-dents define success and failure in mathematics.

The most compelling argument against the use of incentives or coercion, how-ever, is the “hidden costs of reward,” well described by Lepper and Greene(1978). Engaging in an intrinsically motivating activity under conditions thatmake obvious the fact that the activity is merely a means to an end will diminishsubsequent intrinsic motivation because the presence of the reward is the prima-ry reason for the student to engage. Consequently, in the absence of the reward


students become less likely to engage in similar tasks in the future. The mostsalient (and most misrepresented) feature of this line of research is not thatrewards necessarily undermine intrinsic motivation but that the expectation oftangible task-contingent rewards tends to weaken the intrinsic desire to learn.When rewards are not expected, intrinsic interest does not seem to be affectedadversely nor do noncontingent rewards seem to have any real effect on subse-quent intrinsic motivation (Deci, 1972; Lepper, Greene, & Nisbett, 1973).Lepper, Keavney, and Drake (1996) even suggested that judicial application ofreward contingencies can be beneficial for developing sufficient skill in a pursuitso that intrinsic motivation can develop. Although this longstanding principlehas recently been contested (Cameron & Pierce, 1994, 1996), a plethora ofresearch suggests that when rewards are used to get someone to engage in someactivity, the probability of subsequent disillusionment with the activity increasessignificantly (Kohn, 1996; Lepper et al., 1996; Ryan & Deci, 1996).

Attribution and Learned Helplessness Theories

Researchers in the 1960s and early 1970s, when they began to examine individ-uals’ perceived reasons for their successes and failures, found that success is not auniversal motivator. Much of an individual’s intention to initiate behavior dependson the value that the consequences of success have for him or her (Atkinson, 1964).Researchers began to focus attention on what factors students perceive to be thecauses of their successes and failures. Attribution theories deal with how the out-comes of an activity are evaluated in relation to the individual’s perception of hisor her own contribution (i.e., ability and effort) and the contribution of the taskdemands (i.e., difficulty, consistency, precedent) (Weiner, 1972).

In mathematics education, attribution theory is the most widely held of the the-oretical orientations discussed in this article, perhaps because (a) attribution the-ories are cognitive, describing the processes by which motivations are acquiredand changed and (b) they are applicable to a remarkable range of domains.Moreover, attribution theories provide a middle ground between competingmodels of motivation such that findings can be discussed in terms of reinforcersand contingencies or in terms of students’ thoughts, plans, and goals.

Attributions and achievement in mathematics. Students in the lower elementarygrades are generally highly motivated to learn mathematics. They believe that theyare competent and that working hard will enable them to succeed. Many first andsecond graders do not distinguish between effort and ability as causes of successin mathematics (Kloosterman, 1993). However, there is considerable evidence thatsome students begin to differentiate ability for different content domains as earlyas kindergarten or first grade (Wigfield et al., 1992). By the middle grades, manystudents begin to perceive mathematics to be a special domain in which smart stu-dents succeed and other students merely “get by” or fail. They begin to believe thatsuccess and failure are attributable to ability and that effort rarely results in a sig-nificant change in their success patterns (Kloosterman & Gorman, 1990).


When students attribute their successes to ability, they tend to succeed; whenthey attribute their failures to lack of ability, they tend to fail. Gender studieshave shown that girls tend not to attribute their successes to ability but do tendto attribute their failures to lack of ability, exactly the attributional style thatleads to failure. For example, Meyer and Fennema (1985) studied the relation-ship between students’ attributions of success in mathematics in the 8th gradeand their subsequent achievement in 11th grade. This study was a departure frommost attribution research, at least as it related to mathematics education, in thatit assessed the relationship between attributions and future success in mathemat-ics instead of current success. The authors found that attribution of success toability was the most consistent correlate of Grade 11 achievement. Conversely,attribution of failure to lack of ability was the most consistent correlate of lackof achievement for both males and females. For girls in particular, when abilitywas controlled for, attributing failure to lack of ability was associated with lowerachievement. However, attributing failure to lack of effort was also a significantpredictor of lack of achievement on computation problems and high-level, con-ceptual mathematics tasks. Boys’ attributions were not as pronounced as girls’for these variables. The authors concluded that attributions may be more impor-tant as predictors of success in mathematics for females than for males.

Kloosterman (1988) studied how seventh graders perceived the role of suc-cesses and failures in influencing their motivational attributions, their mathe-matical self-confidence, and their beliefs about effort as a mediator of mathe-matical ability and failure as an acceptable phase in learning mathematics. Hefound that attributional style (a combined score, scaled in the direction of inter-nal, stable attributions) was the best predictor of mathematical self-confidence.The belief that effort is a mediator of ability and that failure is an acceptablephase in learning mathematics also contributed to students’ self-confidence inmathematics. Although girls, more often than boys, felt that failure was anacceptable phase in learning mathematics, the fact that girls also thought abouttheir failures more than boys did may have contributed to differential effects likethose reported by Meyer and Fennema (1985).

These findings are significant in that when students conceive of ability asamenable to change or augmentation through effort, they tend to expend moreeffort in mathematics and, thus, are better achievers than students who believethat ability is fixed. Because the belief that occasional failure is acceptable inlearning mathematics predicts mathematical self-confidence, the practice ofallowing children to struggle with challenging problems, even in the elementarygrades, is supported. When children who have not experienced difficult problemsin mathematics encounter a problem that cannot be solved in a routine fashion,they may have their confidence shattered unless they believe that occasional mis-takes are a part of learning mathematics.

By the time they reach college, students generally have formed stable attribu-tions regarding their successes in mathematics. Because the attributional patternsof students in mathematics-related majors tend to focus on ability and effort as


the causes for success and lack of effort for failure, females, who tend to attributetheir failures to ability, may be systematically excluded from mathematicsmajors as a result of their prior mathematics education (Amit, 1988; Bassarear,1986). In addition, because students with unstable attributions for the causes offailure in mathematics tend to dislike mathematics greatly (Lehmann, 1986),these students may also be filtered out of mathematics-related majors.

Amit (1988) studied the attributions of university students in five major areasand found that, overall, females tend to attribute their successes in mathematicsto external and unstable causes, whereas males attribute their successes to abili-ty, an internal and stable factor. When attributions of success were analyzed tak-ing academic major into account, however, students tended to attribute theircauses of success and failure the same way regardless of gender. Students choos-ing mathematics as a major tended to attribute their successes to ability and theirfailures to other factors. In fact, as the mathematical requirements for participa-tion in college majors increased, so did the attribution of success to the internalfactor of ability. Students who attribute their failures in mathematics to internalfactors and their successes to external factors are unlikely to choose a collegecurriculum with substantial mathematics content.

Learned helplessness and dealing with failure. An outgrowth of attributiontheory has been the specific attention of researchers to learned helplessness, acondition in which, because of lack of successes and the attribution of failure tolack of ability, individuals begin to view success as unattainable (e.g., Dweck,1986). Unfortunately these beliefs persist as a result of educational environmentsthat (a) place high value on ability and lower value on effort and (b) offer littleopportunity for individuals with diverse learning styles to supplement their abil-ities with sustained effort (Covington, 1984). Because helpless individualsbelieve that success is out of their grasp and attribute failure to internal factors,learned helplessness often becomes perceived as a trait (i.e., stable and unchang-ing) (Dweck, 1986). Helpless individuals tend to show little motivation for chal-lenging tasks, and, in fact, when facing a challenging task, they display lowerachievement than can be attributed to ability.

Although the findings of most studies regarding learned helplessness are dis-heartening, there is some evidence that attributions can be positively influencedthrough classroom instruction. For example, Relich (1984) hypothesized thatwhen students are provided attribution retraining in conjunction with skills train-ing, their feelings of learned helplessness should be reduced and their mathe-matics achievement should be positively affected.

Those providing attribution training attempted to make students aware thatthey were achieving success on increasingly difficult problems as a result of atleast average ability and high effort. Students who received the attribution train-ing displayed superior self-efficacy gains and fewer learned-helplessness char-acteristics compared with students receiving no attribution training.

Relich (1984) then proposed a causal model that contrasted the direct effectsof attribution training with the mediated effects on achievement and learned


helplessness. Results of a path analysis indicated that although the attributiontraining had a moderate direct influence on achievement, stronger paths resultedfrom mediation through self-efficacy. The attribution training also had a directinfluence on reducing learned helplessness; reducing learned helplessness, inturn, had a direct effect on students’ development of self-efficacy. Thus, itseemed reasonable to predict that the attribution training’s effects on achieve-ment were mediated through self-efficacy via reduction of learned helplessness.

Intervention and the role of the teacher. Attribution training has been found tobe effective in helping students develop positive motivational patterns andincrease performance in other content domains as well (Williams, 1993).However, a major difficulty in designing appropriate intervention strategies inthe mathematics classroom is the tendency for teachers’ attributions to paralleland reinforce those of their students. Teachers tend to initiate more concern withboys, prompt boys more, and have more social interaction with boys than withgirls (Fennema & Peterson, 1984, 1985). Thus teachers may unwittingly under-mine their students’ achievement motivation by reinforcing failure-oriented attri-butions, especially for their female students.

For the most successful students, teachers tend to attribute success more toability for boys than for girls, and teachers more often see boys as the most suc-cessful students in the class. When less successful girls fail, teachers tend toattribute their failure to lack of ability, lack of effort, and task difficulty, where-as boys’ failure is more often attributed solely to lack of effort (Fennema,Peterson, Carpenter, & Lubinski, 1990). It seems then that teachers’ attributionsof their students’ successes and failures are reflected in the ways in which theyinteract with boys and girls in their mathematics classes. These differences ininteraction patterns, in turn, tend to contribute to differential gender-related moti-vation and achievement patterns.

Goal Theories: Relating Mathematics to What Is Valued

Goal theorists delve more deeply into the cognitive bases of the reasons peopledo what they do. They are concerned with understanding how people think aboutengaging in meaningful (or meaningless) activity, and they also conduct researchon people’s perceptions, interpretations of academic and social information, andpatterns of self-regulation (Ames & Ames, 1984). Moreover, researchers whoground their work in goal theory often incorporate the generalized findings fromthe attribution literature and attempt to posit how reasons for success and failureare related to what is valued (Ames & Archer, 1988; Dweck & Leggett, 1988).

Duda and Nicholls (1992) suggested that the basic dimensions of goal orienta-tions correspond directly to distinct implicit theories (or beliefs) of how successis achieved in academic work (see also Ames, 1992; Ames & Archer, 1988;Dweck, 1986). An individual with a mastery (or learning goal) orientation val-ues the improvement of skill or knowledge in a given domain and believes thatsuccess depends on working hard, attempting to understand the domain, and col-


laborating with others. An individual with an ego (or performance goal) orienta-tion values establishing “superiority over others” (Duda & Nicholls, 1992, p.290) and believes that success depends on social comparison and assertion ofsuperior ability. A third orientation, work avoidance, is an especially disturbinggoal pattern in which working hard is not valued. An individual with this goalorientation believes success results from, for example, “behaving nicely in class”or other behaviors superfluous to study and academic thoughtfulness. Workavoidance is often developed as a coping method for preserving feelings of ade-quacy by eliminating any threatening or difficult activities so that a legitimatenegative evaluation of one’s ability cannot be made by others (see Covington &Beery, 1976, for example).

The interplay between goal structures and intrinsic motivation. An individu-al’s intrinsic motivation is mediated through the types of goal structures he or shehas created (Meece, Blumenfeld, & Hoyle, 1988). In particular, possession of amastery goal orientation will positively mediate intrinsic motivation such thatone will become more actively involved in a cognitive task. An ego goal orien-tation (i.e., primarily seeking social recognition) has much less effect on one’sdeveloping active cognitive engagement patterns.

Motivational patterns have both generality and specificity. The patterns of goalorientations and beliefs about success listed above seem to be general orientationsthat students, at least by the time they are in high school, apply across differentdomains in their lives. However, feelings of personal satisfaction, relevance, andboredom seem to be created by students with respect to specific tasks (Duda &Nicholls, 1992; Seegers & Boekaerts, 1993). A child may enjoy solving storyproblems in arithmetic and yet feel that her ability is undervalued by her teacheror peers. In such cases, the ego goal of gaining favorable judgments of compe-tence may begin to undermine her intrinsic enjoyment of the task. Both the salien-cy of goals and the strength of her intrinsic orientation toward the task are impor-tant pieces of information the child will use to determine her engagement patterns.

Because of different beliefs about the natures of different academic subjects,even mastery goals can have differential effects on learning. Students who viewmathematics as a fixed body of knowledge tend to develop goals of memorizationof facts and procedures. These students also tend to emphasize determining correctanswers as the primary goal of mathematics learning. Students who view mathe-matics as a process, guided by their own search for knowledge, tend to value con-structing relational understanding of concepts, and consequently they are motivat-ed intrinsically because the knowledge they develop is their own (Underhill, 1988).

Fortunately, the ways in which teachers structure classroom inquiry can great-ly influence students’ views of mathematics and can lead students to developmore powerful conceptual structures in the process (Cobb et al., 1991; Cobb,Wood, Yackel, & Perlwitz, 1992). Students in inquiry-based classrooms are lesslikely to develop ego goals than are students in more traditional classrooms.Moreover, students in inquiry-oriented classrooms are less inclined to believethat conformity to the solutions of the teacher or others leads to success in math-


ematics, and they tend to believe more strongly that the classroom is a placewhere success is defined as attempts to understand mathematics and explain theirthinking to others. These attitudes contribute to increased student performanceon conceptual and nonroutine tasks that persists even in the face of poor instruc-tion later on (Cobb et al., 1991; Cobb et al., 1992).

Goal orientation has been found to be a strong predictor of achievement(Henderson & Landesman, 1993). Students with mastery goals tend to performbetter than those with ego goals regardless of the learning situation.

Students’ goal structures also interact significantly in situations that involveextrinsic rewards. When students are provided with both coherent goals forachievement and an extrinsic reward, they tend to achieve more than students towhom stated goals are not presented (Schunk, 1984). Moreover, when an activi-ty is not intrinsically motivating, dispensing rewards may not be productive aca-demically unless the rewards are coupled with an appropriate goal structure. Itseems likely that when goals have no intrinsic value to the students, some rewardor instruction that exerts social pressure on the student must be tied to the goalsto make achieving them worthwhile (Brown & Walberg, 1993).

Theories of the Self: Personal-Construct Theories

Personal-construct theories are idiographic approaches to examining individualdifferences in human thought (Snow, Corno, & Jackson, 1996). They are based onthe premise that individuals construct knowledge about their worlds and use thisknowledge to predict outcomes of activities (Kelly, 1955). The purpose ofemploying personal-construct approaches in the study of motivation is to describeconstruct systems of individuals in order to uncover the ways they evaluate activ-ities. Usually this description involves some sort of “mapping” of the relation-ships between constructs to ascertain the cognitive structure underlying the moti-vation. Whereas those using other approaches to the study of motivation are typ-ically concerned with the outcomes of motivational processes (e.g., ability attri-butions, achievement, etc.), personal-construct psychologists are interested in theprocesses themselves: They assume that motivation results from rational cogni-tive processes, and they provide a method for understanding these processes.

Owens (1987), for example, used personal-construct theory to describe twoteachers’ attitudes toward mathematics and mathematics teaching. Although theteachers’ conceptions of their mathematics backgrounds were remarkably similarand although they tended to rate themselves as most similar to the person theyconsidered their “best” mathematics teacher, their concepts of what makes a goodmathematics teacher differed markedly. The teacher who felt that more difficultmathematics was enjoyable also felt that inquisitiveness was a desirable trait fora mathematics teacher. The other teacher, who enjoyed mathematics that was eas-ier, rated inquisitiveness least desirable as a trait for a mathematics teacher.

Owens concluded that their constructs about mathematics and mathematicseducation play a powerful role in determining how teachers anticipate their


teaching roles. In addition, it seems reasonable to assume that the teachers’ priormathematics education experiences, especially identification with their mathe-matics teachers, play a pivotal role in determining what aspects of mathematicsare motivating and thus how they approach teaching mathematics.

Lucock (1987) found that children in high-ability mathematics tracks tended tofind mathematics easier, tended to enjoy doing mathematics more, and tended toconsider mathematics to be more useful than did children in lower ability tracks.These findings are hardly surprising. However, when children who enjoyedmathematics were asked to perform routine work (i.e., learning without under-standing), they became disillusioned with mathematics and tended to give up. Inaddition, gender differences were found between the ways in which high-abilityboys and low-ability girls internalized success in mathematics tasks. Lucockfound that high-ability boys tended to fail with confidence; that is, their confi-dence in their abilities was fairly robust in spite of failure. Low-ability girls tend-ed to succeed with diffidence; that is, their insecurity tended to be robust evenwhen they were successful.

Constructing an intrinsic motivation for mathematics. Middleton, Littlefield,and Lehrer (1992) attempted to test a theory of how academic activities come tobe regarded as intrinsically motivating. Their analysis revealed that childrentended to organize their constructs into three general categories: arousal, or thecognitive stimulation afforded by an activity; personal control, or the degree towhich the activity was considered a free choice or of appropriate difficulty; andinterests (a loosely defined category), or the degree to which the students likedthe activity, the importance of the activity, and their ability in performing theactivity. Students, girls in particular, seemed to identify with their teachers inevaluating the motivational value of academic tasks (as was also found byOwens, 1987). In addition, children tended to rate mathematics as less fun asthey progressed from elementary to junior high school. On the basis of the resultsof the study, Middleton et al. developed a model of academic intrinsic motiva-tion. They asserted that when one first encounters an academic activity, she willtend to evaluate the stimulation (challenge, curiosity, fantasy) it provides and thepersonal control (free choice, not too difficult) the activity affords. If her arousaland control requirements are met consistently, she may choose to include theactivity among her interests.

Using this model, one can gain some insight into the reasons that motivationalattitudes seem to be so stable over time. If a student has classified mathematics asan interest, she will tend to engage in mathematics with enthusiasm without hav-ing to evaluate the engagement requirements of the task at hand. If she has clas-sified mathematics as “not an interest,” she will tend to avoid engagement with-out evaluating the task at hand. Thus, once mathematics activities have been clas-sified with respect to interest, little further evaluation takes place. Because onemust continually and consistently evaluate arousal and control to classify an activ-ity, it seems likely that only radical and consistent change of the requirements forengagement in mathematics activities will effect change in motivational patterns.


Other research has indicated that teachers and students can be highly similar inthe ways in which they define intrinsic motivation in their classrooms but thathighly motivated students may tend to focus more on high arousal and less oncontrol when engaged in mathematics activities, whereas less motivated studentsmay tend to focus on low arousal and more on control (Middleton, 1995). Inaddition, teachers seem to have little background knowledge pertaining to howstudents view mathematics activities from a motivational perspective. The teach-ers’ own personal constructs of what makes mathematics intrinsically motivatingplay a pivotal role in determining the types of activities they choose or design fortheir classrooms. Overall, however, teachers who are better able to predict theirstudents’ motivational constructs seem to be better able to fine-tune their instruc-tion to meet the motivational needs of their students.

Middleton (1993b) examined the changes teachers made in their motivationalconstructs after a year of implementing a reform-oriented pilot curriculum thatprovided students with more opportunities to learn, more choices of strategiesand activities, and more challenging problems than a traditional curriculumwould provide; its activities were situated within real-world contexts. The dataindicated that teachers’ beliefs about intrinsic motivation broaden and expandbefore they deepen and differentiate and that carefully designed curricula, cou-pled with strong professional development experiences, can influence a shift inteachers’ attitudes toward providing an atmosphere conducive for the develop-ment of students’ intrinsic motivation. Teachers became more attuned to the con-ceptual complexity and challenge of the mathematics activities, placed lessemphasis on task ease in defining what makes mathematics motivating, andbegan to perceive the importance of personalizing curricula to make the mathe-matics more meaningful for their students.

Results of studies in the personal-constructs paradigm have shown that moti-vations in mathematics education are highly individual, are related to per-ceived ability, and are relatively stable with regard to success and failure.Some of the individual differences in motivations can be explained in relationto students’ identification with their mathematics teachers. Perhaps moreimportant, researchers can begin to outline how academic activities can be tai-lored to students’ individual differences such that intrinsic motivation in math-ematics can be fostered by paying attention to stimulation, control, and inter-est factors.

Researchers in the personal-constructs paradigm, however, have provided onlylimited knowledge of students’ motivational thought processes. The major limi-tation thus far has been that they have made little attempt to explicate the perti-nence of extrinsic motivators to mathematics learning. Further research in thisparadigm is critical to understanding the roles of grades and other incentives ininfluencing student motivation. In particular, because they deal with the process-es by which students evaluate mathematics activities as worthwhile, personal-constructs methodologies seem uniquely useful for discovering why intrinsicmotivation is superior to extrinsic motivation in academic areas.


A second limitation of personal-constructs studies is that they are prone toexperimenter bias in the interpretation of measures of construct organization.Without well-articulated models guiding their interpretation, results of personal-constructs studies are difficult to interpret substantively.

Descriptive Studies

The last approach discussed in this review deals with descriptive studies.Included in this category are studies that have some theoretical orientation but donot fit neatly into any of the categories mentioned previously. For reasons ofclarity and cohesion, we have grouped descriptive studies according to similari-ties in both the variables examined and the motivational patterns discovered.

Mathematics anxiety. Individuals who perceive mathematics as difficult andtheir ability to do mathematics as poor generally avoid mathematics, if possible(Hilton, 1981; Otten & Kuyper, 1988). Such students are termed math anxious.Hoyles (1981), for example, examined the stories told by students about inci-dents (in their mathematics education histories) that they felt reflected significantinfluences on their learning. She was interested in discovering the perceivedcauses of their mathematics anxiety. Students tended to derive satisfaction froma task when they were involved in successful work, and they tended to blametheir dissatisfaction on the teachers. The students seemed to appreciate teacherswho provided a structured, logical progression for students’ work as well as suf-ficient explanation, encouragement, and friendliness (see also Quilter & Harper,1988). Although the sources of satisfaction and dissatisfaction were similar formathematics compared to other subjects, the ways in which students internalizedthese experiences were markedly different. Students were much more concernedwith their own roles in mathematics versus in learning other subjects. They alsotended to have strong feelings about what they were capable of doing, and theytended to internalize these feelings into their self-concepts. The stories Hoylesstudied showed that students’ anxiety, feelings of inadequacy, and shame werecommon in interpreting their bad experiences in mathematics and that studentsgenerally recall more bad experiences in mathematics than in other subjects.Despite these similarities with respect to their recollections of mathematicslearning, pupils differed in the ways in which they could achieve satisfaction. Forsome, challenge added to their satisfaction; others stressed understanding of the“whys” as well as the “hows.” Some were satisfied with just being able to knowwhat to do to solve a problem successfully, and many were quite concerned withthe accuracy of their work and the grades they received.

Nakamura (1988) described motivational differences between high-achievingand underachieving mathematically gifted students. One of the primary results ofher research indicated that gifted children who exhibit high achievement tend toexperience flow (a construct that corresponds to enjoyable engagement in mean-ingful activity) more often and anxiety less often in schoolwork than their lowerachieving counterparts. Higher achievers also tend to spend considerably more


time than low achievers in activities that afford high challenge and require well-developed skills. These activities, according to Nakamura, are those associatedwith the greatest amount of enjoyment for the high achievers. Lower achievers,conversely, tend to avoid challenge. Instead, they choose activities with chal-lenge below their ability level, presumably to avoid the anxiety caused by highlevels of task difficulty. In other words, higher achievers tend to enjoy academ-ic challenge, whereas lower achievers tend to feel overwhelmed by challenge.

In short, when teachers emphasize understanding of mathematical conceptsand provide facilitative classroom environments, students tend to be more recep-tive and less anxious with regard to mathematical activities than when teachersstress rote activities and are perceived to be authoritarian. Students who havegood experiences in mathematics tend to be less math-anxious and less inhibitedin pursuing mathematics-related careers than students who have bad experiences.In mathematics, perhaps because it is viewed as a difficult and important subject,students tend to internalize their experiences into their self-concept more than inother subject areas.

Motivation and underrepresented populations. Rohrkemper and Bershon(1984) examined the efficacy statements minority students used to motivatethemselves to solve mathematics problems correctly. Their findings indicatedthat some children may begin to feel a lack of efficacy in mathematics as earlyas third grade. In addition, a high proportion of students reported negative innerspeech (e.g., “If I don’t get this right, I will maybe fail”) at the outset of problemsolving. These negative self-perceptions with regard to mathematics may under-mine students’ abilities and efforts to persist when faced with difficult problems.

In addition to the attribution literature, many other studies have documentedgender differences in students’ mathematics motivation. The consistent patternthat develops is that females are socialized into viewing mathematics as a maledomain and into perceiving themselves as being less able than males to do math-ematics (Fennema & Sherman, 1976). Males tend to feel more confident in learn-ing mathematics, are more convinced of the usefulness of mathematics, and iden-tify more with mathematics, in general, than females. Gender-role stereotypingdoes not solely affect females with low ability and motivation. Even girls withhigh ability may perceive mathematics as a male domain, or they may defer tothe “dominant male role” because of other social pressures whether or not theyperceive mathematics as a male domain (Jackson & Coutts, 1987).

In the middle grades, students’ motivations toward mathematics tend to crystal-lize into their adult forms. Students who like mathematics tend to report that theystarted liking mathematics at about the seventh grade. Students who dislike math-ematics report that they started disliking mathematics at about the seventh grade.Their reasons for liking or disliking mathematics seem to focus on the transitionfrom elementary to middle school instructional patterns, especially the perceivedsupportiveness of the teacher and new rules for determining success in mathemat-ical tasks (Eccles et al., 1987; Midgley et al., 1989). Girls in particular tend to iden-


tify with their mathematics teachers, and this identification is related to girls’ inter-est in mathematics (Fennema & Peterson, 1985). By the time students get to highschool, interest in mathematics becomes one of the best predictors of students’ per-ceptions of the quality of their mathematical experiences, more so than ability orthe desire to achieve (Schiefele & Csikszentmihalyi, 1995).

The research on gender differences in mathematics seems to paint a consistentpicture. Like the research on students’ attributions, other research on gender dif-ferences has indicated that mathematics is perceived by females as a maledomain. Females tend to defer to males when interacting in mathematics class,even when their abilities would indicate that deference is unwarranted. Girls alsotend to identify with their mathematics teachers more than boys do. In addition,inasmuch as motivational factors seem to predict academic achievement morefor girls than for boys, it is reasonable to assume that girls’ feelings of disinter-est and even anxiety in mathematics contribute to gender-related differences inachievement. What is not known is what factors cause girls to be less motivated.Research regarding these causes is necessary to dispel the myth that girls areinherently less mathematically able than boys (see Secada, 1990).

Intervention studies (e.g., Croom, 1984), however, have shown that appropri-ate instruction, guidance, and continued support can positively influence studentsfrom underrepresented populations to continue studying high school mathemat-ics and can foster improved attitudes toward school and toward mathematics andscience in particular.

CONCLUSIONS

The Current State of Research on Motivation in Mathematics

Although research on motivation may not be in its infancy, it has barelyreached toddlerhood, and, like a toddler, it seems to be going in many directions,frequently getting into trouble. However, some consistencies are evident acrossstudies, and these consistencies represent the current boundaries of our knowl-edge. Drawing together the findings from the studies reviewed in this article, weare beginning to define the body of knowledge pertaining to motivation in math-ematics as it exists today.

First, findings across theoretical orientations indicate that students’ percep-tions of success in mathematics are highly influential in forming their motiva-tional attitudes. Research indicates that the effort a person is willing to expend ona task is determined by the expectation that participation in the task will result insuccessful outcomes, mediated by how much the individual values either partici-pation in the task itself or the extrinsic rewards associated with success in the task(Brophy, 1986). Students need a relatively high degree of success in mathematicsfor engagement in mathematics to be perceived as worthwhile (Alschuler, 1969),and they need to feel that success in mathematics is attributable to their ability andeffort (Fennema & Peterson, 1985). In addition, students’ beliefs about the nature


of mathematics and mathematics learning greatly influence their definitions ofwhat success in mathematics is. Current practice leads students to develop atti-tudes that value speed of computation, following the example of the teacher, andcorrectness of answers over learning and understanding (Kloosterman, 1993).

Moreover, learned helplessness, lack of success, and the perception that failureis due to lack of ability seriously undermine students’ motivation to learn; thesefactors may also affect the ability to process complex mathematical information(Dweck, 1986). Students also seem to require a healthy appreciation for the roleof failure in mathematical problem solving (Kloosterman, 1988). The likelihoodof failure in a task increases the task difficulty, thus increasing the value of suc-cess (e.g., Brophy, 1987). Further, learning appropriate coping strategies for fail-ure is necessary for developing a healthy mathematical self-concept.

Motivations develop when students evaluate the demands of the mathematicaltask (Seegers & Boekaerts, 1993). To allow students to feel successful in mathe-matics without undermining either the value of success or a healthy attitudetoward failure, teachers must structure tasks such that they present an appropriatelevel of challenge and difficulty for students (e.g., Middleton et al., 1992). Thus,mathematics activities must be difficult enough that students are not bored, yettasks must allow for a high degree of success given appropriate effort by the stu-dent. Moreover, students should be encouraged to attribute their successes to acombination of ability and effort and their failures either to insufficient effort (sofailures can be overcome through renewed diligence) or to confusion or relianceon inappropriate strategies (so failures can be overcome with additional prepara-tion). Students must not be given cause to believe that their failures are due to lackof ability for fear of exacerbating their feelings of learned helplessness.

Second, motivations toward mathematics are developed early, are highly sta-ble over time, and are influenced greatly by teacher actions and attitudes.Students seem to consolidate their motivational attitudes toward mathematics injunior high school (Eccles et al., 1987), and these attitudes in the middle gradespredict the courses taken and mathematics achievement in high school and col-lege (Amit, 1988; Meyer & Fennema, 1985). These motivations are internalizedinto students’ self-concepts, thus affecting how they see themselves with regardto mathematics-related activities. Students with high self-concepts related tomathematics tend to be more focused on the selection and use of specific strate-gies for successful problem solving and are more likely to pursue further studyin mathematics (Meece et al., 1990; Pokay & Blumenfeld, 1990).

The preponderance of students’ recollections of bad experiences (e.g.,Oldfather, 1992) explains in part why students’ liking of mathematics tends todecrease when they get older and why enrollment in higher level mathematicscourses has declined. These students do not see mathematics as being integral totheir academic self-concepts, and they try to avoid the anxiety resulting frominvolvement in mathematical tasks. Because anxious or alienated students areunlikely to have or to develop the motivation to learn mathematics, the teacher


should be patient, encouraging, and supportive of students’ individual learningstyles. Students will feel more comfortable taking risks if they know that theywill not be criticized or humiliated for making mistakes (Brophy, 1987).Students tend to attribute their feelings about mathematics to their identificationwith influential teachers or to their reactions to bad experiences, for which theyblame teachers (Hoyles, 1981; Otten & Kuyper, 1988).

It is unclear, however, what role culture plays in the ways in which motivation-al strategies are implemented in the classroom. Hess and Azuma (1991), for exam-ple, found that Japanese students are expected to be more self-motivated thanAmerican students. In Japan, overt control of tasks by the teacher is minimal, effortis valued over ability, and determinations of interest and success are primarily leftup to the student. In the United States, motivation is still primarily stimulus dri-ven—that is, teachers in the United States are expected to make instruction inter-esting and appealing, and students are less likely to be blamed for inattention if thetopic is personally unappealing. In essence, students are expected to dislike math-ematics and are not provided direction or support when they fulfill this expectation.

Third, providing opportunities for students to develop intrinsic motivation inmathematics is generally superior to providing extrinsic incentives for achieve-ment. To facilitate the development of students’ intrinsic motivation, teachersmust teach knowledge and skills that are worth learning. In other words, studentsmust understand that the mathematics instruction they receive is useful, both inimmediate terms and in preparing them to learn more in the fields of mathemat-ics and in areas in which mathematics can be applied (e.g., physics, business,etc.). Use of ill-structured, real-life problem situations in which the use of math-ematics facilitates uncovering important and interesting knowledge promotesthis understanding. However, utility and importance are not sufficient to devel-op students’ intrinsic motivation.

Students who come to value and enjoy mathematics increase their achieve-ment, their persistence in the face of failure, and their confidence (Gottfried,1985; Lehmann, 1986; Meece et al., 1990; Pokay & Blumenfeld, 1990).Tailoring activities to provide stimulation and student control and matchingactivities with students’ interests increase intrinsic motivation (Middleton,1993b). Providing incentives for success, however, can and does encourage stu-dents to achieve (Alschuler, 1969). Further research regarding interaction ofextrinsic and intrinsic motivation in the context of the classroom is necessarybecause no academic task is free from the influence of either.

Fourth, inequities exist in the ways in which some groups of students in math-ematics classes have been taught to view mathematics. Girls, in particular, maybe influenced through gender-role stereotyping, teacher expectations, and peerpressure to view themselves negatively with respect to mathematics motivation(Fennema & Peterson, 1985; Meyer & Fennema, 1985). Girls, far more thanboys, feel that their failures are due to a lack of ability in mathematics, and thisattributional style may lead them to believe that success in mathematics is unat-


tainable (Benenson & Dweck, 1986). Unfortunately, teachers’ thoughts andbehaviors tend to reinforce learned helplessness in girls, further widening thegender gap in mathematics achievement (Fennema et al., 1990).

Last, and most important, achievement motivation in mathematics, though sta-ble, can be affected through careful instructional design. If students realize thattheir successes are meaningful and result both from their abilities and from a highdegree of effort, they are likely to believe that they can do mathematics if theytry (Relich, 1984). Providing group incentives leads to cooperation and recipro-cal instruction in mathematics problem solving so that all children are givenopportunities to succeed (Slavin, 1984). Creating interesting contexts withinwhich problems are situated stimulates students’ imaginations and illustrates tothem that mathematics is useful in various applications (Bransford et al., 1988).Most important, a supportive, authoritative teacher serving as a model and as afriend gives children the confidence and feelings of self-worth necessary to becomfortable in mathematics (Covington, 1984).

In addition, teachers who are more attuned to bettering their students’ motiva-tional belief systems are better able to adjust their classroom practice to motivatetheir students (Middleton, 1995). This finding would suggest that preservice andin-service programs could profit from detailed examination of the research find-ings in the field of motivation, including the studies reviewed here. Particularattention should be paid to developing strategies for assessing students’ motiva-tional beliefs in the classroom so that teachers’ awareness will be linked to theinstructional sequence. In such a program teachers would be able to use practi-cal knowledge about how students’ beliefs are formed and changed to tailor theirinstruction to better influence their students to take charge of their own learning.

Thus, it seems that there is hope after all. Motivation to achieve in mathemat-ics is not solely a product of mathematics ability nor is it so stable that interven-tion programs cannot be designed to improve it. Instead, achievement motivationin mathematics is highly influenced by instructional practices, and if appropriatepractices are consistent over a long period of time, children can and do learn toenjoy and value mathematics. There is a building body of evidence that indicatesthat the larger, more general goals of schooling can be restructured and rein-vented with a fair degree of success so that the school culture becomes conducivefor student learning and motivation (e.g., Maehr & Anderman, 1993). Theresearch reviewed in this article also provides evidence that classroom practicecan be positively reinvented so that the culture of the classroom can become con-ducive for learning and enjoying mathematics.

Little is known, however, about the socially constructed nature of motivations.What happens in the mathematics classroom when students work together andcreate a shared reality? Do different interpretations of mathematics support amotivating environment for some children but not others? Preliminary findingsindicate that students in cooperative groups perceive the input of others in verydifferent ways and react to the social situation in both positive and negative ways


(Mulryan, 1992). Naturalistic, observational research seems a plausible first stepin examining how individual and social motivational sets become negotiated inmathematics problem-solving situations.

A FEW CRITICISMS

Although the current research on motivation in mathematics education hasprovided profound insights into why students achieve and why they fail, we havesome criticisms pertaining to the lack of theoretical guidance driving the conductof, and implications drawn from, the majority of studies. The research on moti-vational variables in mathematics education has been primarily descriptive andinadequately conceptualized. Often motivation has been thrown “into the pot” toadd a little spice to studies originally focused on other factors—such as mathe-matics achievement.

Particularly evident is the lack of conceptualization of how mathematics moti-vation develops over time. With few exceptions, researchers have neglected toexamine the motivations of students while they change and develop over sever-al years of instruction. If we as mathematics educators are interested in effectingchange in students’ motivational patterns, we need further research regarding theacquisition, consolidation, and maturation of students’ motivations.

In addition, measurement procedures have been primarily atheoretical andpoorly defined. A prime example is the operational definition of motivation asstudent engagement (observed affect, time on task) without the use of comple-mentary measures. Although students’ motivations should influence their engage-ment patterns, engagement itself is not motivation. Engagement can be influencedby a number of factors that distort the actual reasons behind students’ levels oftask involvement—fatigue, for example. At the other end of the spectrum, thoseconducting most motivation studies reviewed in this article have used self-reportmeasures as indices of motivation without actually looking at and listening to chil-dren who are engaged in mathematical activity. The potential biases associatedwith self-report measures of attitude have been clearly delineated (Gall, Borg, &Gall, 1996; Pintrich & Schunk, 1996). When secondary measures of motivationare used, some additional measure should be administered as a validity check.

Moreover, even the theoretically driven studies are limited in their explana-tions of why students are motivated to achieve. Most describe personality corre-lates of motivation, differences or similarities in existing groups, or the correla-tion between motivation and achievement. Few attempts to predict and then testcausal relationships between factors influencing motivation have been made. Tobuild a more extensive body of knowledge about motivational factors in mathe-matics education, mathematics education researchers must attend to theoreticalor model-based research that is designed to ascertain causal and interactive rela-tionships between motivational domains and student achievement (McLeod,Reyes, Fennema, & Surber, 1984). Moreover, through these models they mustbegin to examine the interplay of motivational factors as they exist in the social


and cognitive worlds of the child. Researchers using causal modeling have madea first attempt at large-scale description of the web of factors influencing andaffected by motivational structures. Further research along this line of inquiryholds promise for untangling the causal relationships between motivation andachievement. At the other end of the spectrum, naturalistic studies of studentsengaging in meaningful activity can provide powerful insight into the ways indi-viduals and social groups define motivational constructs, modify these defini-tions that are based on situational variables, and abstract workable goal structuresthat inform future engagement.

But even with the application of appropriate methodologies, nearly all theresearch conducted in the area of mathematics has utilized a model of mathemat-ics instruction that is not conceptually driven. Researchers studying a conceptualmodel of instruction have found that the effects of such instruction on studentmotivation are quite different from the effects of traditional instruction (e.g.,Bransford et al., 1988; Cobb et al., 1992; Middleton, 1993b). In addition, whenstudents who are motivated to learn mathematics concepts in a meaningful wayare forced to work on routine, skills-related mathematics problems, their enjoy-ment of mathematics tends to plummet (Lucock, 1987). Thus, even the positiveresults from studies using more traditional models of mathematics teaching andachievement may be irrelevant or even misleading (Romberg & Carpenter, 1986).

One final criticism is aimed toward the use of theories in motivationalresearch. Although studies may be situated within a theoretical framework, littleattempt has been made to test the adequacy of current theories. Researchers haveused theories to explain behavior, but they have done little to increase the accu-racy, precision, and applicability of these theories. Consequently, very few newtheories or reformulations of existing theories of motivation have been forth-coming. Noticeably absent are approaches that capitalize on research in the cog-nitive science domain. Because they are focused on individual differences, cog-nitive science approaches may prove to be powerful theoretical tools for themotivation researcher, especially in the area of goal theory, by providing theo-retical means for examining volitional decision-making processes (e.g., Corno,1993; Cruz, 1992). A primary goal for future researchers should be the testingand refinement of motivational theories so that their range of applicability can bedelineated and exploited.

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Authors

James A. Middleton, Associate Professor, Department of Elementary Education, Arizona StateUniversity, Box 870911, Tempe, AZ 85287-0911; [email protected]

Photini A. Spanias, Instructor, Department of Elementary Education, Arizona State University, Box870911, Tempe, AZ 85287-0911


Elementary Preservice Teachers’Changing Beliefs and Instructional

Use of Children’s MathematicalThinking

Nancy Nesbitt Vacc and George W. BrightUniversity of North Carolina at Greensboro

In this research, we examined changes in preservice elementary school teachers’ beliefs aboutteaching and learning mathematics and their abilities to provide mathematics instruction that wasbased on children’s thinking. The 34 participants in this study were introduced to Cognitively GuidedInstruction (CGI) as part of a mathematics methods course. Belief-scale scores indicated that sig-nificant changes in their beliefs and perceptions about mathematics instruction occurred acrossthe 2-year sequence of professional course work and student teaching during their undergradu-ate program but that their use of knowledge of children’s mathematical thinking during instruc-tional planning and teaching was limited. Preservice teachers may acknowledge the tenets of CGIand yet be unable to use them in their teaching. The results raise several questions about factorsthat may influence success in planning instruction on the basis of children’s thinking.

Key Words: Children’s strategies; Constructivism; Early childhood, K-4; Pedagogical knowl-edge; Planning, decision making; Preservice teacher education; Teacher beliefs

This study was designed and carried out as an attempt to document the effect ofintroducing preservice elementary school teachers to Cognitively Guided Instruc-tion (CGI) (Carpenter, Fennema, Peterson, & Carey, 1988). CGI’s effectivenessin changing teachers’ beliefs about mathematics instruction and the nature ofmathematics instruction in primary grades is well documented (Fennema et al.,1996; Fennema, Franke, Carpenter, & Carey, 1993; Peterson, Fennema,Carpenter, & Loef, 1989). Teachers prepared in CGI spend more time having theirstudents solve problems, listen more to their students, and are more likely toexpect students to find multiple solution strategies to problems than teachers whoare not prepared in CGI (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989).CGI also results in improved performance by primary-grade students on bothstandardized and problem-solving tests (Carpenter et al., 1989; Fennema,Carpenter, & Peterson, 1989; Peterson et al., 1989). In question is whether simi-lar findings would accrue to the integration of CGI within preservice teacher edu-cation programs.


Preparation of this article was supported in part by National Science Foundation (NSF) GrantMDR-8954679 to the University of Wisconsin (UW). All opinions expressed are those of theauthors and do not necessarily reflect the positions of either NSF or UW.


The research reported in this article was part of a larger project, the PrimaryPreservice Teacher Preparation Project (funded by the National ScienceFoundation) that was designed to begin to investigate the effects of includinginformation about CGI in preservice teacher education programs. The projectwas conducted through the University of Wisconsin and involved preserviceteacher education programs at three sites. The data reported here are from theproject site at the University of North Carolina at Greensboro. Specifically, weexamined (a) preservice teachers’ beliefs about mathematics instruction and (b)mathematics instruction provided by two of these preservice teachers. In ouranalysis of mathematics instruction we focused on how the two preservice teach-ers used knowledge of their students’ mathematical thinking in instruction dur-ing the student-teaching semester.

BACKGROUND

Cognitively Guided Instruction

CGI is an approach to helping “teachers use knowledge from cognitive scienceto make their own instructional decisions” (Carpenter & Fennema, 1991, p. 10).Children’s knowledge and the teacher’s understanding of that knowledge are cen-tral to instructional decision making. Teachers plan instruction using research-based knowledge about children’s mathematical thinking and well-defined tax-onomies of problem types and children’s solution strategies for arithmetic opera-tions (Carpenter & Fennema, 1991; Carpenter & Moser, 1983). Teachers seekspecific information about individual students’ thinking and understanding andthen adjust the level of content to match individual students’ performance levels.

A single model of a “CGI teacher” does not exist. Instead, teachers use CGI in amanner that fits their own teaching styles, knowledge bases, and beliefs, as well asthe needs of their students. Similarities do exist, however, across “CGI classrooms.”For example, students in CGI classrooms spend most of their mathematics instruc-tion time solving various problems by creating their own solutions instead of by fol-lowing a set of procedures provided by an outside source such as the teacher or themathematics textbook. Students also spend a considerable amount of time sharingtheir solution strategies and asking questions of one another and the teacher untilthey have developed an understanding of the problem solutions.

Teachers who use CGI principles when teaching (a) believe that their under-standing of children’s thinking is a critical component of instructional planning,(b) facilitate children’s problem solving and discussions of children’s thinking,(c) listen to their children and question them until the students’ thinking becomesclearer, and (d) are willing and able to make instructional decisions that areappropriate to the mathematical needs of their students (Fennema et al., 1996).As a result, significant positive correlations exist between CGI and students’mathematics problem-solving achievement (Peterson et al., 1989), ability tosolve complex addition and subtraction word problems (Fennema et al., 1989),and level of recall of number facts (Carpenter et al., 1989).

90 Preservice Teachers’ Beliefs and Instruction

Experienced teachers are able to apply the research-based knowledge that theygain while learning about CGI to an already existing set of understandings aboutchildren’s thinking and about their own preferred teaching styles. In contrast, pre-service teachers are likely to have limited knowledge about children’s mathemati-cal thinking and to be in the process of developing preferred styles of teaching;indeed, their teaching styles may be shifting repeatedly while they gain experienceand pedagogical content knowledge (Shulman, 1986). Thus, their preparation inCGI may not be synthesized and applied in a manner similar to that of experiencedteachers, and the extent to which they consider CGI principles in instruction maybe significantly different from that of experienced teachers.

Beliefs About Mathematics Instruction

Teachers’ beliefs about teaching and learning mathematics significantly affect theform and type of instruction they deliver (Clark & Peterson, 1986; Richardson,Anders, Tidwell, & Lloyd, 1991). If teachers’ beliefs are compatible with the under-lying philosophy and materials of a curriculum, there is greater likelihood that thecurriculum will be fully implemented (Hollingsworth, 1989; Richardson, 1990).These findings are supported in the CGI literature. A critical factor in a teacher’s useof CGI principles is his or her beliefs about teaching and learning mathematics(Fennema et al., 1996; Fennema et al., 1993; Peterson et al., 1989). Preparation inCGI helps primary teachers organize and expand their knowledge about children’sthinking while they construct instructional strategies on the basis of what they arelearning about their students’ thinking (Fennema et al., 1996; Fennema et al., 1993).Thus, the processes of learning about research on children’s mathematical thinkingand using that knowledge while interacting with students are associated withchanges in both teachers’ beliefs and the type of instruction they provide their stu-dents. Precisely how these findings apply to preservice teachers is unclear.

Preservice teachers’ general beliefs about teaching are tenacious (Holt-Reynolds, 1992) as are their beliefs about teaching and learning mathematics(Ball, 1989; McDiarmid, 1990). Learning new theories and concepts may havelittle effect in changing preservice teachers’ general beliefs about teaching prac-tices (Calderhead & Robson, 1991; Kagan, 1992). Instead, preservice teachers’beliefs seem to be drawn from previous vivid episodes or events in their lives(Pajares, 1992); their beliefs about teaching and learning appear to be general-izations derived from their own experiences as students (Holt-Reynolds, 1992;Knowles & Holt-Reynolds, 1991). Posner, Strike, Hewson, and Gertzog (1982)suggested that for existing beliefs to be replaced or reorganized, new beliefs needto be intelligible and appear plausible. For example, the framework underlyingthe content presented in mathematics methods courses needs to be consistentwith the framework of the mathematics education program that preservice teach-ers observe and implement during field experiences. If the two frameworks arein conflict, the theories and concepts presented during the mathematics methodscourse may not seem plausible and may be rejected.

91Nancy Nesbitt Vacc and George W. Bright

Schram, Wilcox, Lanier, and Lappan (1988) found that preservice teachers’beliefs about what it means to know mathematics were challenged when concep-tual development, group work, and problem-solving activities were emphasizedduring a mathematics content course. However, emphasizing these componentshad little effect on the preservice teachers’ beliefs about what should be includedin elementary school mathematics education. Schram and Wilcox (1988) con-cluded that instead of changing beliefs, some preservice teachers fit existingbeliefs to their new experiences. These conclusions were supported byMcDiarmid (1990), who indicated that many preservice teachers resisted changeeven when a course was designed specifically to challenge their underlying beliefsabout mathematics education. Despite their experiences in the course, most of thepreservice teachers in his study ended the course still believing that a teacher’srole is to explain the answer instead of to help students develop understanding.

It appears that even full-time teaching during a teacher preparation program maynot be a powerful change agent inasmuch as preservice teachers’ beliefs remainstable across the student-teaching experience (Calderhead & Robson, 1991;McDaniel, 1991; McLaughlin, 1991). Zeichner and Liston (1987) found thatinstead of changing beliefs, preservice teachers became more skillful in expressingand implementing their points of view. These studies, however, focused on beliefsabout teaching and learning, in general. Whether their results generalize to preser-vice teachers’ beliefs about teaching and learning mathematics is unclear.

Brousseau and Freeman (1988) indicated that teacher preparation programsgenerally do not challenge students’ initial beliefs about mathematics education.As a result, preservice teachers may conclude their programs of study withoutexamining their own perspectives about teaching and learning mathematics.Kagan (1992), on the basis of a review of 40 learning-to-teach studies conduct-ed between 1987 and 1991, identified three elements that seem essential forchanging preservice teachers’ beliefs. First, preservice teachers need to haveextended opportunities to interact with and study students. Second, the contentof their university courses needs to be connected to the exigencies of classroomteaching; university courses need to focus on procedural knowledge and practi-cal strategies as well as theory. Third, their field experiences need to includeopportunities to work with classroom teachers who engage in ongoing self-reflection by questioning and reconstructing their own pedagogical beliefs. Asdiscussed later, the first two elements were included in the present study as partof the teacher preparation program. The classroom teachers (i.e., the on-siteteacher educators) who supervised the field experiences of the participants in thisstudy may have engaged in self-reflection practices, but it was not part of the cri-teria for their selection as field-experience supervisors.

Teacher Preparation Program

In our elementary education teacher preparation program, students arerequired to complete 46 semester hours of liberal arts courses that include 6


hours of mathematics course work and to complete a second major that consistsof a minimum of 24 semester hours of course work in one of the arts or sciences.The program incorporates professional development schools (PDSs) that supportsustained experiences in classrooms to help preservice teachers integrate whatthey are learning about teaching (i.e., theoretical frameworks) with what they areobserving, doing, and experiencing in classrooms (i.e., practice). As a result ofthe partnership between the university and PDSs, classroom teachers serve as on-site teacher educators. They meet with university faculty to plan field experi-ences for the preservice teachers and sometimes model instructional activities aspart of the methods courses. They typically are willing for undergraduates to tryout various instructional methods during the field experiences.

Preservice teachers take all their professional courses in cohort groups beginningin the junior year. The sequence of professional course work includes a mathemat-ics methods course taught during the fall semester of the senior year. Preserviceteachers also complete 10 hours per week of internship in the PDSs during bothsemesters of the junior year and the fall semester of the senior year. Full-time stu-dent teaching is completed during the spring semester of the senior year in the sameclassroom in which the senior fall-semester internship is completed.

METHOD

Participants

Thirty-four members of an undergraduate cohort of preservice teachers tookpart in the study. At the beginning of the study, they were commencing their 2-year sequence of professional course work in elementary education.

Only two of the on-site teacher educators working with this cohort were expe-rienced CGI teachers, and both taught at the same PDS. One was a third-gradeteacher and the other taught kindergarten. Thus, only two preservice teachers inthe cohort completed their senior-year field experiences in classrooms of experi-enced CGI teachers.

Because this study was undertaken to document changes in preservice ele-mentary school teachers’ beliefs about teaching and learning mathematics, wewanted to monitor changes in their instruction. Two preservice teachers (Helenand Andrea) were selected as cases for in-depth study because of similarities intheir senior-year field experiences. They completed their senior-year internshipsand student teaching in adjacent third-grade classrooms. Thus, they shared acommon grade-level curriculum, worked with the same school personnel outsidethe classroom (e.g., administrators, resource teachers), and were not in PDSs thatmight have had different school philosophies. Helen’s on-site teacher educatorwas a third-grade teacher with extensive experience with CGI. In contrast, theCGI experience of Andrea’s on-site teacher educator was limited to participationin a 2-hour “awareness” workshop about CGI. Other differences existed betweenthe two preservice teachers: Their junior field experiences were in different


schools and at different grade levels, and Helen’s second major was psychologywhereas Andrea’s was speech communication.

Cohort Leaders

The cohort was led by the first author, who is a faculty member in theDepartment of Curriculum and Instruction. She taught these students’ mathe-matics methods course and conducted their weekly seminars during the threesemesters of internship and student teaching. An experienced classroom teacher,who was a full-time graduate student in the same department, assisted with theleadership of the cohort. These two leaders served as the liaison between the uni-versity and the PDSs and also worked collaboratively with the on-site teachereducators in supervising field experiences and student teaching.

The university faculty member, who was an experienced classroom teacher, hadbeen prepared in CGI through professional development workshops at theUniversity of Wisconsin. Also, she spent a considerable amount of time each yearworking with children and observing in experienced CGI teachers’ classrooms.The graduate student participated in one of the workshops at the University ofWisconsin during the summer prior to the cohort’s mathematics methods course,and she was an observer during the mathematics methods course.

Mathematics Methods Course

The 3-semester-hour mathematics methods course met once a week, for 2 hoursand 50 minutes per session, during the fall semester of the students’ senior year.Course content centered around process learning and the national curriculumreform; problem solving, communicating mathematically, reasoning, and makingmathematical connections were emphasized. Course requirements were designedto provide opportunities for preservice teachers to focus on children’s thinkingand included (a) conducting two case studies (with assessment interviews) of stu-dents in the internship classroom; (b) planning, implementing, and evaluating amathematics lesson in the internship classroom; and (c) carrying out observationsof three mathematics lessons taught by the on-site teacher educator. In conduct-ing their case studies, the preservice teachers had opportunities to focus on thethinking and understanding of individual students. Teaching the mathematics les-son helped the preservice teachers learn to monitor children’s thinking duringwhole-class instruction. During classroom observations the preservice teacherscould focus on different aspects of the teacher’s role as it relates to understandingchildren’s thinking: the amount of wait time used by the on-site teacher educatorwhen she questioned students, the types of questions (i.e., factual, open-ended,restated, and probing) the teacher asked along with the responses of the students,and the instructional procedures the teacher employed during the lesson.

CGI was introduced through a five-session module. We introduced problemtypes for the basic operations and children’s solution strategies (Carpenter,Fennema, & Franke, 1993) during the first four sessions, and knowledge of chil-


dren’s geometrical thinking (Lehrer, Fennema, Carpenter, & Osana, 1992) wasaddressed during the fifth session. In general, to introduce problem types and solu-tion strategies, we presented a mathematics story problem and asked the preserviceteachers to find alternative solutions to the problem. After sharing some solutionstrategies, the preservice teachers viewed videotaped examples of children’s solu-tions to the same problem and discussed how their solutions were similar to or dif-ferent from those of the children. We also focused the discussion on what problemsmight be given next to an individual child, thus encouraging the preservice teach-ers to begin using knowledge of children’s thinking to plan instruction. During thefifth session, in a PDS second-grade classroom, the instructor conducted a demon-stration geometry lesson that focused on the children’s visual, descriptive, and rela-tional thinking about shapes. The discussion that followed this lesson centeredaround the information gained or not gained about the students’ thinking and mod-ifications in the lesson that would have provided additional information about thestudents’ understanding. Immediately following this discussion, the preserviceteachers, individually or in pairs, used the same instructional activities with a stu-dent from another second-grade classroom. The session concluded with the pre-service teachers sharing what they learned or did not learn about their students’geometrical thinking, with possible reasons for their outcomes.

Instrumentation

To assess changes in the preservice teachers’ beliefs about teaching and learn-ing mathematics, we administered the CGI Belief Scale (Peterson et al., 1989)four times: beginning of the professional preparation program (i.e., start of thefall semester of the junior year), beginning of the mathematics methods course(i.e., start of the fall semester of the senior year), beginning of student teaching(i.e., start of the spring semester of the senior year), and end of student teaching(i.e., end of senior year).

The Belief Scale consists of 48 items designed to assess teachers’ beliefs,which are categorized on four subscales: Role of the Learner, RelationshipBetween Skills and Understanding, Sequencing of Topics, and Role of theTeacher. Respondents rate each item using a 5-point Likert scale of stronglyagree, agree, undecided, disagree, or strongly disagree. Each subscale measuresinterrelated but separate constructs. High scores on the Role of Learner subscaleindicate a belief that children, instead of being receivers of knowledge, are ableto construct their own knowledge. High scores on the Relationship BetweenSkills and Understanding subscale indicate the belief that skills should be taughtin relationship to understanding and problem solving rather than in isolation.High scores on the Sequencing of Topics subscale indicate a belief that thesequencing of topics for instruction should be based on children’s natural devel-opment of mathematical ideas rather than on the logical structure of formal math-ematics. High scores on the Role of Teacher subscale indicate a belief that math-ematics instruction should facilitate children’s construction of knowledge ratherthan consist of the teacher’s presentation of knowledge. Peterson et al. (1989)


reported that internal consistency estimates for each subscale ranged from .57 to.86; internal consistency of teachers’ scores on the total belief scale was .93.

In addition to the data from eight on-site formal observations of each prospec-tive teacher (two by each cohort leader and four by the on-site teacher), data forthe more in-depth study of Helen and Andrea included reflective journal entriesduring the mathematics methods course and student teaching, four videotapedmathematics lessons during the student-teaching semester, and three open-endedinterviews. One interview was conducted during the fall semester of the junioryear, one during the fall semester of the senior year, and one at the end of stu-dent teaching. The interviews were planned by the authors in collaboration witha third departmental faculty member who also had been prepared in CGI.Graduate students conducted the first two interviews as part of the requirementsfor a component on ways of knowing in a human development course that wastaught by the third departmental faculty member. The final interview was con-ducted by a doctoral student who was paid to conduct the interviews. To elimi-nate any effect due to an interviewer’s leading the interviewees’ responses in agiven direction, we chose interviewers who were not knowledgeable about CGI.Each interview focused on the teacher’s role in mathematics education (e.g.,How do you figure out what children know in mathematics?); the final interviewalso addressed decisions that each participant had made while teaching a lessonthat the interviewer had observed (e.g., What, if anything, happened during thelesson today that caused you to change your plans for the lesson?).

RESULTS

We present the findings of this study in two parts. Results concerning the preser-vice teachers’ beliefs are presented first followed by the results related to Helen’sand Andrea’s beliefs and their use of CGI-based knowledge in their teaching.

Preservice Teachers’ Beliefs

The 34 preservice teachers’ mean scores on the Belief Scale across the fouradministrations are given in Table 1. Changes in the mean total scores during con-secutive administrations of the Belief Scale were 2.3, 24.7, and 12.2, respective-ly. The preservice teachers’ Beliefs Scale scores changed little during the firstyear of their program, increased significantly during the mathematics methodscourse, and continued to increase significantly across the student-teaching expe-rience. A repeated-measures analysis of variance and follow-up paired t-tests ofscores at adjacent times showed a significant overall time effect (p < .0001) for allfour subscales as determined by Wilks’s Lambda and its associated F-statistic.Subscale means changed little across the first year of the study, but there were sig-nificant increases (p < .0001) in the means of all four subscales during the math-ematics methods course. Also, on average, the preservice teachers’ belief scorescontinued to increase significantly (p < .005) during student teaching, with thegreatest increase occurring in their beliefs about sequencing of topics.



Table 1Mean Scores (and Standard Deviations) on the Belief Scale Across Administrations for the 34Preservice Teachers

Beginning Beginning Beginning of End ofSubscales of program of methods student teaching student teaching

Role of Learner 36.5 35.7 42.2*** 44.9*(5.2) (6.0) (6.7) (5.8)

Skills and Understanding 36.9 37.9 44.7*** 48.0*(4.7) (5.1) (6.6) (7.4)

Sequence of Topics 39.3 40.3 45.3*** 48.7**(4.6) (4.9) (6.1) (5.0)

Role of Teacher 38.5 39.4 46.0*** 48.8*(5.0) (4.8) (6.2) (5.3)

Total beliefs 151.2 153.5 178.2*** 190.4*(14.8) (17.5) (21.7) (20.4)

Note. The maximum score is 60 for each subscale and 240 for the total score.*Change from previous mean significant at p < .005. **Change from previous mean significant at p< .0005. ***Change from previous mean significant at p < .0001.

Helen’s and Andrea’s Beliefs and Their Use of CGI Principles

When we analyzed the data concerning Helen and Andrea, we found differ-ences in (a) the ways their beliefs, as measured by the Belief Scale, changedacross their preparation program and (b) their use of CGI principles during math-ematics instruction.

Belief changes. Helen’s and Andrea’s respective scores on the four adminis-trations of the Belief Scale are presented in Table 2. Although belief scores ofboth participants increased across the 2 years, the changes varied by preserviceteacher. Helen’s overall belief scores about teaching and learning mathematicsincreased substantially across the methods course and continued to increase dur-ing student teaching. In contrast, Andrea’s overall belief scores increased con-siderably across the first three semesters of her preparation program, includingsubstantial increases during the mathematics methods course, but little changeoccurred during the student-teaching experience.

Table 2Helen’s and Andrea’s Subscale and Total Scores on the Belief Scale Across Administrations

Beginning Beginning Beginning of End ofSubscales of program of methods student teaching student teaching

HelenRole of Learner 35 36 48 47Skills and Understanding 37 42 52 58Sequence of Topics 45 45 49 56Role of Teacher 46 47 49 49Total beliefs 163 170 198 210

AndreaRole of Learner 30 32 51 51Skills and Understanding 34 42 48 52Sequence of Topics 33 39 55 52Role of Teacher 31 33 51 49Total beliefs 128 146 205 204

Note. The maximum score is 60 for each subscale and 240 for the total score.


The two preservice teachers differed in their belief-score changes by subscalearea. Their belief scores about the role of the learner during mathematics instruc-tion increased somewhat similarly across the four administrations of the BeliefScale with the greatest change occurring during the mathematics methods courseand no change occurring during student teaching. Their belief scores about math-ematical skills and understanding increased continually across the program, butHelen’s score changed most during the semester when she was enrolled in themathematics methods course, whereas Andrea’s seemed to change most becauseof her experiences during the first year of the program. Helen’s belief scoresabout the sequencing of mathematics topics did not change during the first yearof the program, increased somewhat after the mathematics methods course, andcontinued to increase during student teaching. Andrea’s belief scores in this areaincreased somewhat during the first year and substantially during the mathemat-ics methods course, but the scores decreased somewhat during student teaching.Helen began the program with a belief about the role of the teacher that tendedmore toward a perspective of the teacher’s facilitating student learning, andAndrea began the program with a set of beliefs that tended more toward theteacher’s telling students what they need to know. By the conclusion of thepreparation program, however, Helen and Andrea shared similar perspectivesabout the teacher’s role. Helen’s belief scores related to the role of the teacherremained rather stable across the 2 years. In comparison, Andrea’s belief scoresabout the role of the teacher increased considerably during the mathematicsmethods course but decreased slightly during student teaching.

Helen’s use of CGI principles. Helen’s reflective writings and interviews illus-trate her beliefs across the program of study. Although Helen’s score on the BeliefScale at the beginning of the study indicated a belief that the role of the teacherwas to facilitate learning, she indicated in a written reflection paper at the begin-ning of the mathematics methods course that the teacher’s role was to model prob-lem solutions for students. During the mathematics-methods-course semester, shestated that a teacher should question children to find out what they were thinkingas they solved problems. After Helen had gained instructional experience duringstudent teaching, she questioned the use of worksheets and stated in a journalentry that teachers need to help students understand what they are learning. Thisreflection supports the changes that occurred in her beliefs about skills and under-standing during student teaching. Although the most significant change in Helen’soverall belief scores on the Belief Scale occurred during the mathematics-meth-ods-course semester, her reflective journal entries and comments during inter-views indicate that changes occurred across the entire 2 years:

Children learn mathematics through modeling others. I think children need some guid-ance but then they go use their own “style.” (Journal entry, September, Junior Year)

[Teaching mathematics] means to look at how children are thinking and how theysolve their mathematics problems … making information relevant to what the chil-dren are going to do later in life.… You ask questions to get them to share their pre-

vious knowledge and you observe them and keep records.… Listening to children’sconversation can tell you a lot. (Journal entry, December, Junior Year)

This probably sounds horrible to some people, but I have not really had a chance tolook through the math textbook [for this grade] until today (a teacher workday).…Regular “student pages” are almost identical to what I used in elementary school,boring, fill-in-blank pages.… “Fun stuff” was in the teachers’ guide … fun, useful,and very “CGI-ish.” It was almost as if they were saying, “OK, once your studentssuffer through these boring worksheets, then you can let them … try to understandit. (Journal entry, February, Senior Year)

[The teacher’s role] is to make the children want to learn and to facilitate their learn-ing rather than telling them [what they need to know]. (Interview, April, Senior Year)

Helen appeared to believe in the principles of CGI, and she was fairly suc-cessful in applying some of these during student teaching. She planned andimplemented instruction that was based on problem solving, and she facilitatedstudent understanding and critical thinking through a rather high level of ques-tioning. This finding is illustrated by the following excerpt from a lesson shetaught in February, shortly after she assumed full-time responsibility for the classduring student teaching. The third graders were solving the following problem:“Tyrone wants to buy seven baseball cards. Each card costs 5 cents. How muchmoney does he need?” Prior to this segment in the lesson, one of the students inthe class, Erik, had shared how he had solved the problem by counting out dimesand pennies: “I knew I could get 20 cents out of 4 nickels, so I had ten, twenty,thirty, and then I counted pennies, one, two, three, four, five to equal thirty-five.”(A fictitious name has been assigned to each student.)

Helen: Who can show me a way to do it where you use all the same coins?Alice: I started counting by fives and he wanted … to buy seven baseball cards and

each cost 5 cents, so I got seven nickels and started counting.Helen: Why did you use a nickel?Alice: Because they were 5 cents each and I counted by fives.Helen: How did you know when to stop [counting]?Alice: When I ran out of nickels.Helen: You counted out the seven nickels first?Alice: Yes.Helen: Okay, very good.… Who can tell me what is different between the way Erik did

it and the way Alice did it? Susan: Erik had more coins than Alice.Helen: “Erik had more coins than Alice.” Very good.… Who can tell me something alike

about what they did? They did one thing exactly alike.… Can anyone come up [tothe overhead projector] and write a number sentence for this story problem?

Helen’s use of the teaching strategy of asking questions that facilitated andpromoted children’s critical thinking (e.g., asking students to compare differentsolution strategies) was evident across the student-teaching semester. She alsoasked probing questions to gain further information about students’ solutionstrategies (e.g., “How did you know how many nickels to put down and how


many pennies?”, “Why did you use 20 children?”, and “Would you say that theunits or the longs would give you more of an exact answer? . . . Why the units?”).Yet, there were also occasions during some of Helen’s lessons when she demon-strated behaviors that seem indicative of directed teaching. This fluctuationbetween facilitating and telling was evident during a lesson in mid-April, oneweek prior to the conclusion of her student-teaching experience.

Helen: What do you think this … means, unit of measure?Jose: What you used to go around the book.Helen: What about this word right here? That’s one of your spelling words. Anybody

remember how to say that word?Bobbie: Perimeter.Helen: Perimeter, and that’s what you have been measuring.… That’s the distance

around something.… Who can tell me one unit of measure you used?Erin: Those cubes.Helen: Okay, I’m gonna put “long base 10s” (writes on chart). And how many of those

did it take?Erin: Ten.Helen: So, we would say the perimeter of your book is 10 long base 10s. Did anyone

use the long base 10s and get a different answer?Michael: I got 11.Helen: You got 11 (records number on chart). Who has an idea about why they think

this is different?

After the preceding segment, students discussed reasons for getting differentanswers when they had used the same book and unit of measurement; Helenasked probing questions that resulted in the students’ identification of differencesin how the two students placed the base-ten longs on their respective books.Subsequently, the students discussed units that could be used to measure theperimeter of the room. Prior to the exchange excerpted below, Helen had indi-cated that everyone was going to use a piece of string and had asked how theywould use it to determine the perimeter.Deanna: You could take the strings and go around the room and then take the ruler to see

how long each string was, so you’d know how long the string was to count howlong they are.

Helen: Okay, to see how many inches or feet there are? … Okay, do we need to use theceiling?

Deanna: No.Helen: We can use what?Tien: The floor.Helen: The floor. Anywhere, really; you can use the wall. I think it would be easiest,

well I don’t know. It might be easier to use the wall. Whatever you want touse.… You all came up with some good ways to figure out the perimeter.… I’mgoing to give each two people a string … [and] assign you a wall.… So, if youhad this wall, where are you going to start?… So one partner—I need a volun-teer—will hold it there? So Sandy is going to hold it there and I’m going tobring it around here. How many strings is the wall so far?… Okay, you let goof your end, Sandy, and bring it around the wall. How many strings is that?


Although Deanna’s response indicated a clear understanding of how the room’sperimeter could be measured, it appears Helen believed that she needed todemonstrate the process before the students could proceed on their own. Afterthe measurements for each wall were determined, the lesson was concluded witha rather rich student-centered discussion that focused on determining the perime-ter of the room on the basis of the measurements obtained for each wall.

Helen: How can I tell what the perimeter of the room is?Erin: Add all of those [measurements for each wall] up?Helen: Why should I add all of those together?Erin: ‘Cause that’s how, um, how, um, you know, um how long the walls are.Helen: Let’s see if we can do that. Can we round these up? Those that have a half?

Think that would be okay?Tien: Yeah. Two halves are a whole.Helen: How did you know that?Tien: Half of the string.… Helen: (Interrupts Tien) Okay, this is half of it.Tien: And half and half puts together a whole string.Helen: Okay so if we put this half and this half, we have one whole?Tien: Yes.Helen: So how am I going to remember to do that?Jeremy: Cross the halves out.Helen: Cross the halves out and put a 1 (does this on overhead). Okay, that works for

me. Does everyone understand that?… So our room is thirty-eight and a halfwhat?

Students: Feet. Strings. Yards. (Shared simultaneously)Helen: Strings. Those strings are close to a yard, but we’re just going to say strings.

And what’s that called?Students: Perimeter.

It is interesting to note that during this part of the lesson, which illustrates afocus on children’s thinking, there surfaced occasionally shades of teacher-cen-teredness, illustrated by Helen’s use of the pronoun I (referring to herself) andby her telling students that the unit of measurement is strings.

Andrea’s use of CGI principles. As documented by the reflections presentedbelow, when Andrea began her program of study, she believed that memoriza-tion of facts was the framework for learning mathematics. By the conclusion ofthe mathematics methods course, she indicated that children needed to haveopportunities to discover mathematical concepts through explorations of differ-ent problems and to build on what they already know. She also indicated thatasking students questions was more important than telling them what they needto know. This perspective remained stable across the student-teaching semester.

Children learn mathematics through memorizing the facts and symbols of math.They then take these skills and really learn them through much practice. (Journalentry, September, Junior Year)


[The teacher lets] the children … discover learning instead of getting up there andtelling them what [they] are going to learn. [The teacher lets] them discover throughtheir manipulatives or whatever … the concepts and build on what they already knowinstead of just … doing everything for them and telling them everything. [The teacherfigures out what children know by letting] them have a chance to talk,… write, and… show you what they already can do. (Interview, December, Junior Year)

I am realizing that my students perform better when given story problems than whenthey see the same problems written out in number-sentence formation. (Journalentry, March, Senior Year)

During an interview near the end of student teaching, Andrea stated that themost important thing she had learned about teaching mathematics was the impor-tance of questioning and trying to find out what students were thinking. However,she did not seem to realize that for the purpose of making informed instructionaldecisions teachers have to interpret students’ responses to understand what theyknow. Instead, she appeared to focus more on whether the students’ answersmatched the responses she was expecting. In a journal entry that she wrote fol-lowing a lesson she had taught during her third week of full-time student teach-ing, Andrea reported that her “best” questions were those designed to get studentsto show their processes. The transcript of this lesson, however, showed that shemoved away from students who gave wrong answers and followed up only withstudents who gave correct answers. At one point in this lesson, she asked herthird-grade students, “How are the ways [that certain students solved the problem]different? … Could you tell anything about what the person was thinking?”Although responses to these high-level questions might have provided her withvaluable information to use in planning future instruction, Andrea gave the stu-dents only a few seconds to think about or respond to the questions. She did notprobe the two responses that she accepted, thus missing an opportunity to gainmore in-depth knowledge of those students’ understanding. This type of compar-ative questioning did not occur again during any of her videotaped lessons.Instead, she appeared to pursue only correct thinking or thinking that was alignedwith a predetermined procedure that she wanted the students to learn. This prac-tice is illustrated with the following segment from a March lesson, during whichshe seems to have had in mind a predetermined procedure for writing a fraction.

Andrea: What fraction of your M&Ms is purple?Rashida: Well, I only had 14 M&Ms and only one was purple, so I had one fourteenth.Andrea: One fourteenth. How did you get your 14? How did you know what went on the

bottom?Rashida: Because it’s the number that you had, the denominator.Andrea: It’s the denominator and that is what again?Rashida: The number of things that you have.Andrea: Okay, and how did you all come up with your top number?Latasha: You see how many you got in the purples?Andrea: Of the purples. Okay.… Now I would like to know the fraction of pinks and yel-

lows.


Kim: Five fifteenths.Andrea: Now how did you get five fifteenths?Kim: Because I counted both of these as 3 plus 2 equals 5 and then I counted these

[everything except the pinks and yellows] and that was 15.Andrea: Okay, when we do fractions, do we subtract the part … the yellows and pinks

from the overall group number? How do we get that bottom number? Wheredoes our number come from? We need to decide how we get that first.

Terry: From the amount, the number of everything.Andrea: The number of every single one or just.…Terry: The number of every one in the group that you’re using.Andrea: Okay, every one in the group that we’re using. So, how many M&Ms do you

have altogether, including every one of every color?Kim: Twenty-four.Andrea: Twenty-four. So what is the fraction of the yellow and the pink if you’re using

all 24?… Okay, where do we put that?Kim: At the top.

Andrea appeared uninterested in hearing about interpretations that the studentsmight be developing, and there was no discussion of the quantities that any of thefractions might represent. When Andrea asked at the end of the lesson what thestudents had learned about fractions, one responded, “The top number is thenumerator and that’s the one that we chose stuff to put in, and the bottom num-ber is the denominator and that shows all the things that are together.” The partsof the symbol seemed to have been learned as isolated from one another; studentsdid not seem to have had a chance to make sense of fractions as quantities. Thislesson, which seems to be representative of Andrea’s instruction during the lat-ter part of student teaching, illustrates that her focus during mathematics instruc-tion became more directed toward procedure building with the teacher being theultimate authority on what procedures were to be learned.

In late April, when asked to describe CGI, Andrea stated, “It is more of letting[students] find [the mathematics] through knowing the appropriate or best ques-tions to ask.” Yet, although she appeared to believe in the importance of askingquestions to determine what students were thinking, her own questions seemedto be quite controlling; she appeared to want students to learn one predeterminedway of solving problems. Any changes she wanted to make in their thinkingappeared to be those that aligned students’ thinking with her own.

DISCUSSION

Preservice Teachers’ Beliefs

In general, the preservice teachers in this study appeared to change their beliefsto a more constructivist orientation about the learning of mathematics duringtheir teacher-preparation program. Furthermore, it seems reasonable to concludethat preservice teachers are able to develop views of instruction that are differentfrom telling.


The greatest change in the preservice teachers’ beliefs, as measured on theBelief Scale, occurred during the semester in which the mathematics methodscourse was taught. This phenomenon indicates that dealing explicitly with math-ematics pedagogy and a research-based model of children’s mathematical under-standing may influence preservice teachers’ thinking about teaching and learn-ing mathematics. The preservice teachers’ beliefs continued to change fairly sig-nificantly during the student-teaching semester to reflect greater concern aboutthe mathematical understanding of the students and greater awareness of theneed to help students make sense of mathematics. However, the extent to whichthe preservice teachers changed beliefs guiding their own teaching varied, asexemplified by Helen’s and Andrea’s instruction.

Helen’s Beliefs and Use of CGI Principles

Overall, Helen appeared to believe that children could solve problems withoutinstruction and that the knowledge she gained from listening to children talkingabout their thinking could help her make informed decisions. Yet, her use of thisinformation in planning instruction was unclear. Unlike most of the preserviceteachers in this study, Helen was supervised during her senior year by an on-siteteacher educator who was experienced in using CGI. As a result, she had seenCGI principles being incorporated in regular mathematics instruction before sheassumed full responsibility for planning and implementing instruction. Also, theon-site teacher educator encouraged Helen to gather information about students’thinking and to use that information to adapt instruction.

During student teaching Helen involved students in various problem-solvingactivities that extended beyond the basic arithmetic problem types. She also pro-vided students with various challenging problems, and she encouraged them toshare their different solution strategies. Thus, Helen established a learning envi-ronment that provided her with numerous opportunities to assess students’ think-ing and understanding during each lesson. She also indicated that she used stu-dents’ journal entries as a form of assessment.

On the basis of the levels of mathematics instruction defined by Fennema et al.(1996), Helen’s mathematics instruction at the conclusion of the preparation pro-gram would be categorized as Level 3: “Provides opportunities for children tosolve problems and share their thinking. Beginning to elicit and attend to whatchildren share but doesn’t use what is shared to make instructional decisions” (p.412). And on the basis of the Fennema et al. (1996) levels of cognitively guidedbeliefs, her beliefs about teaching and learning mathematics also would be cate-gorized as Level 3: “Believes that children can solve problems without instruc-tion. Believes only in a limited way that his or her students’ thinking should beused to make instructional decisions” (p. 413).

The instruction Helen provided appeared to be consistent with her beliefs.Students spent most of their time during mathematics solving problems and shar-ing their solution strategies. Usually two to four problems were completed dur-


ing each lesson, and students were encouraged to look for connections, to reason,and to communicate mathematically. She generally assigned the same problemsto the entire class with students working independently, although they could con-sult with others in their group of four if they wished. Helen also attended to whatthe children said or wrote in their journals for the purpose of understanding howthey solved the problems, but she did not seem to use this knowledge in planningsubsequent instruction as would be expected of a Level 4 teacher. As with theLevel 3 teachers in Fennema et al.’s (1996) study, “understanding children’sthinking appeared to be an end in itself rather than a means by which to planinstruction” (p. 418). Helen did not deviate from her planned lessons on the basisof what students said or did during an activity, and her instructional planningseemed to be directed mainly by curriculum objectives. Indeed, Helen stated dur-ing an interview near the end of student teaching that she planned instruction onthe basis of the North Carolina Standard Course of Study, without acknowledg-ing any role for her own understanding of the children’s thinking. Helen appliedsome of the principles of CGI (e.g., she encouraged students to create their ownproblem solutions and she asked high-level questions), but she missed opportu-nities to follow up on students’ thinking; she failed to fully interpret the sense ofwhat the students said in response to her questions.

Andrea’s Beliefs and Use of CGI Principles

When Andrea began student teaching, she appeared to believe that children areable to find their own solutions to problems and that their sharing of solutionstrategies provides helpful information for planning instruction. Although thesebeliefs seemed fairly stable throughout Andrea’s student-teaching experience, asevidenced by her belief scores and journal entries, such beliefs were not evidentin her instruction. Indeed, the relationship between her beliefs and instructionseemed to become more divergent while she gained teaching experience. Forexample, she asked questions at the beginning of the student-teaching semesterthat appeared to be attempts to challenge students’ reasoning skills by gettingthem to compare solution strategies. However, she abandoned this level of ques-tioning during the last half of the semester. Also, throughout her teaching, sheincreasingly guided children to accept particular solution strategies that she hadidentified, independent of what the students were thinking.

In a written evaluation of one of Andrea’s lessons, the on-site teacher educa-tor indicated that Andrea needed to “use every opportunity to model and rein-force any new activity being introduced.” Further, Andrea was not encouragedby the on-site teacher educator to investigate in detail what children were think-ing. Instead, as Andrea indicated in an interview near the end of student teach-ing, she taught “CGI-type” lessons “only on days when a lesson was going to bevideotaped.… The rest of the time [we went] by the textbook.” This choiceseemed to be encouraged by the on-site teacher educator. Andrea seemed willingto ask students questions and to create an environment in which children could


give wrong answers without being embarrassed; she encouraged students toexplain their reasoning for problem solutions. We cannot know, of course,whether this is a sufficient base of knowledge and beliefs from which she canprogress to develop into a CGI teacher.

Using the levels defined by Fennema et al. (1996), we considered Andrea’sbeliefs at the conclusion of the preparation program to be in transition betweenLevel 2 (i.e., “Struggling with the beliefs that children can solve problems with-out instruction and should use their own strategies” [p. 413]) and Level 3. Wecategorized her level of cognitively guided instruction (Fennema et al., 1996) asLevel 2: “Provides limited opportunities for children to engage in problem solv-ing or to share their thinking. Elicits or attends to children’s thinking or useswhat they share in a very limited way” (p. 412). Andrea occasionally plannedlessons that involved problem solving, and during these lessons she had studentsfind and then share their solutions. However, although she often appeared to belistening to and accepting a student’s thinking, she did not seem to actively tryto understand how the student solved the problem. She did not ask probing ques-tions to gain clarity, and she often ignored answers that were incorrect.

Case Comparisons and Contrasts

It is not clear why these two preservice teachers differed in their use of CGI prin-ciples during student teaching, nor is it clear whether either preservice teacher’sprogress might be representative of the progress preservice teachers, in general,could make. Did having an on-site teacher educator who was an experienced CGIteacher affect Helen’s progress? If so, would Andrea have attained instructionalLevel 3 (Fennema et al., 1996) had her on-site teacher educator been more knowl-edgeable of CGI? Is becoming a Level 3 teacher a realistic expectation for preser-vice teachers even though some experienced teachers do not achieve this level after4 years of CGI experience (Fennema et al., 1996)? Did Helen’s more constructivistset of beliefs at the beginning of her preparation program affect the extent to whichshe was able to use what she had learned about CGI, compared with Andrea’sbeliefs and level of use? Did the second major (i.e., psychology for Helen andspeech and communication for Andrea) affect the background knowledge that eachpreservice teacher brought to the teaching experience?

It is encouraging that both Helen and Andrea came to believe that children’smathematical thinking is important and that instruction needs to be based on prob-lem solving. At differing points during their full-time student teaching, eachdemonstrated competencies in encouraging students to find their own solutions toproblems, and they asked questions that encouraged critical thinking and reason-ing skills. Perhaps this is all that can be expected of preservice teachers; ability toapply knowledge gained from listening to children’s solution strategies may not bea realistic expectation for the 2-year sequence of professional course work duringan undergraduate teacher preparation program. At the same time that preserviceteachers are being prepared in CGI, they also are gaining baseline knowledge about


teaching and learning in general and are identifying basic elements such as theirown preferred teaching style; these two factors generally have been addressed byexperienced teachers prior to learning about CGI. Developing beginning compe-tencies in the area of mathematics education while achieving in-depth knowledgeof children’s mathematical thinking and applying that to one’s instructional plan-ning may be too much to expect of novice teachers. Indeed, attaining instructionalLevel 2 (Fennema et al., 1996) as a student teacher may be commendable.

CONCLUSIONS

Students in primary grades can develop fragile mathematics knowledge that pro-duces correct answers in some contexts, but the knowledge may not transfer toother contexts. Similarly, preservice teachers can develop fragile knowledge aboutteaching that in some contexts may produce behavior consistent with CGI princi-ples, but this behavior may not transfer to all teaching contexts. As we found in thisstudy, preservice teachers may acknowledge the tenets of CGI and yet be unableto use them in their teaching, perhaps in part because of their lack of teaching expe-rience. Unlike the inservice teachers in previous CGI studies, the preservice teach-ers in this study were establishing a knowledge base about children’s thinking andlearning and were beginning to develop competencies as mathematics teachers atthe same time that they were attempting to construct instructional strategies on thebasis of what they were learning about their students’ understanding.

There is also a concern about the extent to which the preservice teachers’ lev-els of mathematical understanding may have affected their use of CGI principlesduring mathematics lessons. For example, Andrea’s focus in the excerpt from herlesson on fractions may reflect her own lack of understanding about fractions asquantities as well as her expectation for use of a predetermined procedure for writ-ing the fraction. Because we have no data about the fractions instruction thatHelen provided during her field experiences, it is unclear how her teaching of thistopic might have changed our perception of her use of the CGI framework. Datacollected during the mathematics methods course indicated that she viewed themultiplication of fractions as producing larger answers and division of fractionsas producing smaller answers. There was no opportunity to observe how this mis-understanding might have affected her instruction during student teaching.

Beliefs related to the use of CGI principles appear to be manifested by eachteacher in the ways that instruction is carried out in the classroom. Considerable per-sonal reflection on one’s beliefs and behavior would seem to be necessary for oneto develop coherent pedagogy; short, reflective journal entries may not provide ade-quate opportunity for reflection. Other contexts for reflection (e.g., debriefings afterclassroom observations by an outsider, meetings with peers to discuss the progressof using CGI principles) may be necessary. It is not clear whether preservice teachereducation programs can structurally accommodate these needed “reflection events.”

The results of this study seem to be counter to the previous research findingthat preservice teachers’ beliefs are resistant to change (Calderhead & Robson,


1991; Holt-Reynolds, 1992; Kagan, 1992; McDiarmid, 1990; Schram et al.,1988; Zeichner & Liston, 1987). On the basis of their beliefs scores, interviewstatements, and reflective journal entries, we conclude that, on average, the pre-service teachers in this study changed their beliefs to a more constructivist ori-entation as a result of their teacher preparation program. On the basis of the find-ings of this study, we believe that CGI may provide preservice teachers with thefoundation for an intelligible and plausible alternative approach to teaching andlearning mathematics while at the same time offering them opportunities to linktheir new set of beliefs to previous conceptions (Posner et al., 1982). Using thetaxonomy of problem types and solution strategies as a guide for planninginstruction, listening to how children solve problems, and exploring children’sgeometrical thinking may have provided the preservice teachers in this studywith the reinforcement needed to support changes in their beliefs.

The results of this study support Kagan’s (1992) conclusions that extensivefield experiences and linkages between theory and practice are essential elementsfor changing preservice teachers’ beliefs. The belief changes that occurred forthe preservice teachers in this study may be attributed, in part, to the numerousopportunities they had to interact with and study students during field experi-ences (i.e., 10 hours per week of internship across three semesters prior to full-time student teaching). Also, the case studies, assessment interviews, and obser-vational activities that the preservice teachers completed during the mathematicsmethods course may have influenced belief changes because of the opportunityeach provided for connecting the content of the methods course to the exigenciesof classroom teaching. Data on Kagan’s second factor (i.e., working with class-room teachers engaged in ongoing self-reflection) were not examined as part ofthis study, although reflection is an expected part of a PDS teacher’s role.

Overall, the 34 preservice teachers in this study changed beliefs and perceptionsabout mathematics instruction across the 2-year sequence of professional coursework during their undergraduate program. We cannot be certain, however, whetherthe changes were fundamental or superficial. We also cannot be certain of theeffects that different factors had on their changing beliefs. The data indicate the pos-sibility that intensity of experience and a focus on children’s thinking in the mathe-matics methods course may be keys for helping preservice teachers change theirviews. Programs of minimal duration and programs that provide limited field expe-riences and give minimal attention to focusing on the needs of children may not beas successful in facilitating changes in preservice teachers’ perceptions. Preparationin the use of CGI principles through a 5-week module during the mathematics meth-ods course seemed to affect preservice teachers’ beliefs about teaching and learningmathematics, and their beliefs continued to change significantly during studentteaching. Yet, findings from the more in-depth study of Helen and Andrea indicatethat the extent to which they are able to incorporate these beliefs in their instructionvaries. Perhaps the amount of time (i.e., five sessions) spent on CGI during themathematics methods course may have been insufficient for these novice teachers.Another factor, however, appears to be the amount of consistency that exists among


the philosophical perspectives of the teacher educators with whom preservice teach-ers work. Helen and Andrea completed the same mathematics methods course,completed student teaching at the same school and at the same grade level, followedthe same school mathematics curriculum, and were supervised by the same univer-sity teacher educators. The major difference in their student-teaching experienceswas that Helen’s on-site teacher educator was an experienced CGI teacher andAndrea’s on-site teacher educator had limited knowledge of CGI. Thus, althoughwe believe that providing preservice teachers with a robust research-based model ofchildren’s thinking during a mathematics methods course changes their beliefsabout teaching and learning mathematics, their abilities to incorporate these beliefsduring student teaching may depend on the support preservice teachers receive fromthe classroom teacher who supervises their student-teaching experiences. In Helen’scase, the mathematics methods course, university teacher educators, and on-siteteacher educators held consistent philosophical perspectives. Andrea did not expe-rience this level of coherence and thus may have been placed in the awkward posi-tion of believing in one approach to teaching and learning mathematics and havingto follow a different approach because of the environment to which she wasassigned. It appears that if preservice teachers are to internalize coherent applica-tions to teaching and learning mathematics, the environment in which they studentteach and the support they receive need to be consistent with the principles beingadvocated in their professional preparation program.

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Authors

Nancy Nesbitt Vacc, Associate Professor, Department of Curriculum and Instruction, Curry Building,University of North Carolina at Greensboro, Greensboro, NC 27402-6171; [email protected]

George W. Bright, Professor, Department of Curriculum and Instruction, University of NorthCarolina at Greensboro, Greensboro, NC 27402-6171; [email protected]


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