Thank  you  for  taking  the  time  to  download  our  Elementary  Math  Digital  Kit.    Enclosed  in  this  kit  are  the  following  materials:    

-­‐ The  Effective  Mathematics  Classroom:  Research-­‐based  Principles  and  Practices    

-­‐ Blackline  Master  Worksheets  § Venn  Diagram  § Number  Lines  § Clocks  and  Calendar  § Thermometer  and  Dot  Paper  

 We  hope  you  save  time  with  these  mathematics  resources!    -­‐-­‐Sadlier    

�

PR

OF

ES

SIO

NA

L D

EV

EL

OP

ME

NT

SE

RIE

S

he students best prepared to meet today’s rigorous mathematics standards will share a set of essential skills. They will think analytically; they will

gather, organize, and interpret data with confidence; and they will makeinformed decisions based on well-considered hypotheses. They will also bestrong problem solvers, relying on a sound understanding of mathematicalconcepts and practices. In addition, they will be articulate communicators, able to use direct and symbolic language to create and interpret explanations(California State Board of Education, 1998; National Council of Teachers ofMathematics, 1989). Effective classrooms provide the support and encour-agement necessary to develop these essential skills.

T

Teachers who seek to equip students with these toolsstrive to create and maintain effective classrooms—classrooms that contribute to cognitive, affective, and effective learning. The best classrooms share anumber of hallmarks. Here we examine several of thesehallmarks and offer strategies for implementing them.

CLASSROOM ENVIRONMENT

Although the building blocks of effective learning arestrongly interconnected, as the environment forlearning the classroom is the logical place to begin ourexamination. The interpretation of a classroom ofstudents as a collection of individuals has in recentyears shifted toward that of the classroom as onemathematical community (NCTM, 1991); onehallmark of the effective classroom is the support it

offers to such a community. Physically, the environ-ment can reinforce knowledge and encourage clearthinking and creativity. Socially and intellectually, it can support new learning, encourage risk taking,build confidence, and provide resources for exploration.

PHYSICAL ENVIRONMENT

In an effective classroom, teaching begins even beforea word is spoken. Attractive visual displays sparkinterest in students with a broad range of learningstyles. Professionally produced displays can helpreinforce concepts, strengthen vocabulary, andemphasize correct procedure. Teacher-made materialsdo much the same, but can be more tailored tospecific student needs. And displays made by studentsnot only showcase their work, but give them at once

The Effective K–6 MathematicsClassroom: Research-Based

Principles and Practicesby

Marie Cooper, I.H.M., Ed.D.Associate ProfessorImmaculata College

�

Volume 5 Sadlier-Oxford

Professional Development Series

SADLIER-OXFORD

�

a sense of ownership and pride. The process ofcreating materials for display leads students toreflect on their learning, which prompts them toorganize and bolster it. Graphs and charts thatdescribe data of special interest to students makethe applications of mathematics real andpersonal.Resource bookshelp studentsconnectmathematics totheir world; theguided use ofthese books helpsstudents betterunderstand howprofessionals findthe information they need. Manipulatives formodeling mathematical ideas give studentsaccess to tools that can clarify, extend, and enrich their own thinking.

A table in the classroom may serve as the focusfor students’ discussion, the creation of groupprojects, and peer reinforcement. “Private areas”in the room may support quiet journal entries orproblem/story construction (Ross, 1998). Whilemany classrooms may not have the space orfurniture for such simultaneous separatearrangements, in most classrooms desks may be rearranged to facilitate distinct uses.

SOCIAL ENVIRONMENT

The nature of interaction between students andteacher provides a second important support to learning and to fostering a community oflearning. The promotion of higher-orderthinking skills—essential if students are todevelop as independent learners—is largely afunction of an effective social environment in the classroom. Teachers seeking to develophigher-order thinking skills in students create an accepting, encouraging atmosphere whilemaintaining high, but not unreasonable,expectations. They model effective strategies and guide students to reflect on and respectfully

critique their own and others’ strategies. Thismethodology encourages depth and strength oflearning (Hogan & Tudge, 1999; Tudge &Caruso, 1988; Webb, 1991).

Some skills and processes lend themselves topractice and reinforcement within groups.

Research hasshown thatstudents ingrades twothrough fivewho work ingroups scorebetter on high-levelachievement

test questions that require elaboration thanstudents who work alone, although no differenceexists in the results of low-level questions (e.g.,Sharan, 1980). The most dramatic improvementcomes with the effective combination of directinstruction and peer collaboration. Directinstruction offers the opportunity for theorganized presentation of new concepts andskills, while peer collaboration offers teachers the opportunity to interact with individuals andgroups as they implement newly learned skills.It is in the implementation phase thatquestioning has its greatest cognitive effect(Fishbein, Eckart, Lauver, VanLeeuwen, &Langmeyer, 1990). In addition, having toexplain one’s reasoning and negotiate with a partner compels students to clarify theirthinking (Phelps & Damon, 1989). Studentslearn much from each other by giving andreceiving help, by recognizing andresolving differences in theirthinking, and by internalizingprocesses and strategies (Webb & Faravar, 1999).

THE EFFECTIVE MATHEMATICS CLASSROOM

IS CHARACTERIZED BY:

• an engaging, creative, supportive environment;• a student-centered, problem-solving approach;• well-planned, carefully sequenced, multimodal learning activities;• connections to home, community, and student interests; and• a variety of formative and cumulative assessment tools.

Manipulatives give students access to tools that can clarify, extend,

and enrich their own thinking.

2

Professional Development Series

SADLIER-OXFORD

�

3

Group work is not automatically successful,however. Success takes carefulplanning, patience, and thepreparation of the learningenvironment. From the outset,respect for others, goodlistening skills, andcourteous questioningskills must be explicitlystated, modeled, anddemanded. Problems oractivities assigned forgroup work should benonroutine and toocomplex for any studentto solve alone, yetaccessible to groupstrategies (Lacampagne,1993). A good example isa story problem with more information than isneeded, or lacking information that students can infer. When this approach is successfullyemployed, peer learning promotes confidenceand a sense of security that permits the risktaking behind stating new or divergent ideas.

Individual accountability is just as important asgroup interaction; their coexistence is consistentwith the work of Piaget and Vygotsky on thedevelopment of understanding and autonomy inyoung children (Webb & Faravar, 1999). Bothenvironments are essential to the building of asense of competence and self-esteem.

STUDENT-CENTERED,

PROBLEM-SOLVING APPROACH

Meeting today’s mathematics standards demandsa balance of instructional tradition and the bestof current research. Although an extensive bodyof educational literature urges teachers to viewstudents as builders of their own learning,students do not and can not build knowledge by themselves. New knowledge must be builtonto existing frameworks. Teachers, therefore,need clear knowledge of the mathematical

content to be learned, students’prior understanding, and the

strategies needed to takestudents forward. A skilledinstructor helps “scaffold”learning, selecting tasksdesigned to reinforce priorlearning, linking it to new

learning, and applying it to both familiar andnovel situations.

Scaffolding is advancedwhen a teacher intro-

duces new concepts with manipulatives andinvestigative activities,and guides students withdetailed examples (Glaser,1991). The teacher further

scaffolds new learning by leading students tosupply missing terms, operations, reasons, andsolutions in practice examples; by approachingeach concept from a variety of perspectives andactivities to accommodate diverse learning styles(Dunn & Dunn, 1978); and by integratinginstruction both with the larger curriculum and students’ culture (Banks, 1994).

National and state mathematics standardsdemand that every student becomes a confidentand versatile problem solver. Unfortunately,many students view mathematics and problemsolving as the routine application of rules andalgorithms to achieve a closed answer (Franke &Carey, 1997). Such students must be encouragedto give more attention to the reflective processthat helps integrate new knowledge and to theapplication of problem solving beyond theclassroom. Students who see problem solving as avaluable, creative activity build a truer, strongerconcept of mathematics (NCTM, 1989). Thecultivation of higher-order thinking skills,modeling of effective problem-solvingtechniques, and explicit instruction in thevocabulary of mathematical problem solving all

Students learn much from each other by giving and receiving help, by recognizing and resolving

differences in their own thinking, and by internalizingprocesses and strategies.

Professional Development Series

SADLIER-OXFORD

�

4

help students develop problem-solving pro-ficiency (Nicholl, 1996; Richardson & Monroe,1989). Beginning students are best introducedto problem solving by acting out problems,drawing pictures, checking for extra information,reasoning logically, making lists and tables,using simple graphs, and writing numbersentences. Rooting problem solving in learners’everyday interests enlivens both the problemsand students, and makes the activity moreaccessible (Barnett, Sowder, & Vos, 1980). It should be kept in mind, however, that acurriculum based entirely on daily problems isunrealistic and may do more to hinder thetransfer of concepts and skills than help(Sierpinska, 1995).

The development and application in the primarygrades of an explicit standard heuristic model, or pattern, for problem solving, and theextension and mastery of the heuristic model in the upper grades, affords learners a lifelongproblem-solving tool. With a basic heuristicmodel established, students can add strategiesalong the way, gathering tools suited to a broadvariety of situations.

As students build, practice, and strengthen their problem-solving heuristic model, one laststep can greatly improve their performance andself-regulating skills. If students consistentlytest the reasonableness of their solutions, theybecome more aware of the accuracy and precisionof their answers and strengthen their grip on theprocess. A number of activities built into theproblem-solving process can promote this skill.Teachers can urge students to test their solutionsdirectly and to defend them; to write a summaryparagraph, adding a cross-disciplinary dimen-sion; or to search for alternate solutions, changingconditions, or extending the original problem(Krulik & Rudnick, 1994). All of these measurespromote the independent application of strategies.

Formulas still have their place: Many real appli-cations depend upon them. They are, however,best reserved for the later grades, when studentsunderstand their appropriate use. The complexityof problems increases over the grades, as studentsgrow more proficient. From the beginning,students should be encouraged to write problemsrooted in their own interests, lives, and cultures.Problem solving can and should be linked toeach major topic in the learning sequence, withspecial emphasis given to the application andinterpretation of statistics, critical thinking, and algebraic methods.

ACTIVITIES

Lessons that proceed from a concrete presenta-tion to a symbolic representation promote thedevelopment of deep, interconnected, and easilyaccessible conceptual understanding (Berman &Friederwitzer, 1983; Bohan & Shawaker, 1994).The effectiveness of moving from the concrete tothe symbolic, or abstract, is enhanced when theplan and purpose(s) of procedures are explainedat the outset, and when the lesson is concludedwith discussion and reflection (Brown & Palincsar,1989). The concrete phase of the lesson mayinvolve direct instruction, guided investigation,or group exploration using manipulatives or aform of technology. Immediate application of

Kindergarten,Early First

GradeListen

Imagine what is

happening as the

problem is read.

DrawPicture the

situation.

ThinkPlan what to do.

WriteWork through

the plan.

CheckDoes the answer

make sense?

First and Second Grade

ReadImagine what is

happening.

DrawDraw what

you imagined.

ThinkPlan what to do.

WriteWork through

the plan.

CheckDoes the answer

make sense?

Third throughEighth Grade

ImagineCreate a mental

picture of what is

happening.

NameList the fact(s) and

question(s).

ThinkChoose and

outline a plan.

ComputeWork through

the plan.

CheckIs the answer

reasonable?

A standard heuristic model for problem solvingaffords learners a lifelong problem-solving tool.

Professional Development Series

SADLIER-OXFORD

�

newly acquired concepts in a small group settingoffers the opportunity to reinforce, practice, anddevelop cognitive-triggering discussions (NCTM,1989; Tourniaire, 1986). A holistic approach toassessment strengthens the lessons and providestimely adjustment of instructional strategies.

Lessons about and/or using technology adddimension to the learning of mathematicalconcepts. Lessons about technology guidestudents in the discriminating and effective use of computers, calculators, and the Internetas tools for problem solving; for gathering,organizing, and analyzing information; and for streamlining and checking their own work(Bruck & Crump, 1995). Lessons usingtechnology provide opportunities for patternanalysis, practice in choice of appropriateoperations, and development of creativeproblems; they can also serve as strong supportfor peer interaction, stimulating discussion, and the explanation of strategies and applications(Cox, 1992).

CONNECTIONS TO HOME, COMMUNITY,

AND CULTURE

Students bring to every classroom—even thekindergarten classroom—a wealth of knowledgeand a variety of skills. Their conceptual andprocedural understanding are deeply rooted in

their cultures and practical realities. Obviously,the most immediate and significant influence tostudents’ identity and view of the world isfamily. It is not surprising, then, that familyinvolvement is a consistent hallmark ofsuccessful schools (Henderson & Berla, 1994).Family involvement boosts academic achieve-ment and positive attitude in students and isespecially helpful to at-risks students (Carey,1998; Campbell, 1994; Sears & Medearis, 1992).Similarly, the judicious incorporation ofstudents’ backgrounds and interests intoinstruction increases students’ interest, and,therefore, the impact of that instruction.

Regular communication of goals and objectivesbetween the home and school is vital. Families,both English speaking and those where Englishis not the first language, lend greater support to instructional activities when they are aware of the purpose and connection of the activities to the larger picture of students’ achievement.Developing and sharing students’ portfolios with their families strengthens families’ under-standing of instructional strategies and theircommitment to supporting them. It also fostersstudents’ pride in their efforts.

Students are shaped by cultures beyond thefamily as well. Interests in sports, hobbies, andtheir communities can promote learning in anumber of ways. Story problems based on localculture or events add meaning and relevance to learning. The display of statistical represen-tations of students’ concerns and interests

promotes the integration of mathematicswith other school subjects (Ross, 1998).

For older students, especially, thesepractices can lead to the awarenessof deceptive statistics and the care that must be taken in statistical

interpretation.

Lessons using technology provide opportunities forpattern analysis, practice in choice of appropriate

operations, and development of creative problems.

5

Professional Development Series

SADLIER-OXFORD

�

6

STANDARDS-DRIVEN ASSESSMENT

The broad range of backgrounds, abilities, andlearning styles that students bring to the math-ematics classroom requires a variety of teachingand assessment strategies. The rapid growth ofthe Internet makes available a wide variety ofassessment strategies that are designed to guideinstruction and monitor student performance inalignment with the mathematics standards.Though traditional paper-and-pencil tests provideuseful information, they do not give the fullpicture of students’ learning (Stenmark, 1991),nor do they provide the immediate feedbackneeded for a quick shift in instructional strategies.

The Mathematical Sciences Education Board(1991) urges teachers to use assessment strategiesthat are aligned with mathematicalunderstanding, appropriate to instruction, andaccountable in their use. If the key purpose ofassessment is to help teacher and student improvethe learning process, then it must be flexible andmultimodal (Cramer, 1996).

The first step in assessment, especially in veryyoung students, is observing their actions andinteractions with others in small groups.Unobtrusive observation provides information on group dynamics and individual student’sprogress, and can, therefore, help the teacherguide the process (Elkind, 1998; Webb, 1992).Anecdotal records, including observations andinterviews, can be used to document progress and identify specific learning needs (Bruck &Crump, 1995).

Student presentation of problem solutions,whether of routine practice examples or the morechallenging nonroutine problems, has been aclassroom standard for generations. Encouragingstudents to think aloud as they work throughsuch presentations often yields many benefits,opening a window for the teacher on studentthinking and helping support and clarify studentunderstanding (Muth, 1993). Explanation is a

powerful tool in small groups (Vetter, 1992).Used in the full group, such assessment gives allstudents access to diverse thought and problem-solving patterns. Questions directed by theteacher to students encourage them to assumeresponsibility for both listening and evaluatingcarefully (Mewborn & Huberty, 1999). Suchassessment gives students the opportunity toexperience mathematics as mathematicians do,honing one another’s clarity of thought throughquestions, corrections, and further explanations.

Insistence on explanation is key to students’success in mathematics. Simple recitation ofprocedural steps is not enough to promotecognitive growth, though it will reinforce theefficient use of algorithms and rules. It is theeffort to provide the reasons for the steps, theunderlying logic, that promotes deepermathematical thinking.

As an instructional strategy, questioning shapesand scaffolds students’ efforts at understandingand leads students to reflect on how they learn, so that they can learn more effectively (Martino & Maher, 1994). Students’ explanations andresponses to well-framed questions provide the ideal immediacy and flexibility for theteacher and student to decide “what to do next”(Cramer, 1996). The skilled questioner helpsstudents focus on concepts at hand, integrate oldand new learning, reassess beliefs and understand-ings, explore new ideas, and transfer knowledgeto other areas of life and learning (Ross, 1998).

Responses to well-constructed questions requireconsidered thought and careful formulation, butstudies indicate that the average instructor waitsabout one second after asking a question beforeexpecting a response. Waiting three to fiveseconds has been shown to increase the length of responses, the willingness to participate, theevidence and diversity of thinking, the likelihoodof supporting statements, and even an increase instudents’ willingness to ask questions of theirown (Hunkins, 1995).

Professional Development Series

SADLIER-OXFORD

�

7

The use of journals and mathematical storywriting by students offer alternate or additionalopportunities for learners to reorganize and reflect on their thinking (Gagne & White, 1978),thereby expanding and enriching the backgroundknowledge they bring to the task (Leonard, 1999).Students who understand their own learning willbe more likely to continue to learn successfullythroughout their lives.

CONCLUSION

The teacher seeking to promote excellence inmathematics instruction supports achievement inhis/her students with a balance of traditionalmethods, proven elements of current research, and a commitment to continued growth andlearning in both content and instructionalstrategy (NCTM, 1991). Teachers are challengedto know students as learners and to model goodmathematical thinking. The American Federationof Teachers adds to this challenge the call toexercise skill and judgment in the choice,development, use, and interpretation of assessments (1990).

Today’s teachers face many great challenges,but they are also aided by powerful supportstructures. The choice of a research-based, carefully sequencedmathematics textbookmeeting today’srigorous national,

state, and local standards, along with strongteacher-support features can save time and anxiety,leaving teachers free to practice creativity andmaximize the teaching-learning process. Pro-fessional journals provide access to currentresearch, innovative implementation strategies,and the experience of other instructors. Directcommunication with other mathematics teachersreinforces efforts, supports explorations, andprovides access to the shared expertise of theteaching community. Sharing within a school,school district, or professional association hasalways been fruitful; now the Internet providesaccess to a much broader community. Internetsites designed by and for teachers featureactivities, lesson plans, supplementary materials,and suggestions for assessment. Opportunitiesexist for standards-specific online assessment andongoing monitoring of student performance. Mostsites offer the opportunity to “talk” with otherteachers, sharing goals and concerns and gainingthe breadth and objectivity that is provided by

the thinking of many otherprofessionals. In addition, the

Internet provides rewardingopportunities for enrichingstaff development andtraining by providingteachers with the latestresearch and teachingstrategies that reflect the research.

The use of journals offers additional opportunities for learners to reorganize and reflect on their thinking. Students who understand their own learning will

most likely continue to learn successfully throughout their lives.

Copyright ©2001 by William H. Sadlier, Inc. All rights reserved.

REFERENCESAmerican Federation of Teachers, National Council on Measurement in Education, &

National Education Association (1990). Standards for Teacher Competence inEducational Assessment of Students. Washington, DC.

Banks, J.A. (1994). “Transforming the Mainstream Curriculum.” EducationalLeadership, 51 (8), 4-8.

Barnett, J.C., Sowder, S., & Vos, K.E. (1980). “Textbook Problems: Supplementing andUnderstanding Them.” In Krulik, S. (Ed.), Problem Solving in School Mathematics:1980 Yearbook, Reston, VA: National Council of Teachers of Mathematics.

Berman, B. & Friederwitzer, F.J. (1983). “Teaching Fractions without Numbers.” SchoolScience and Mathematics, 83 (1), 77-82.

Bohan, H.J. & Shawaker, P.B. (1994). “Using Manipulatives Effectively: A Drive Downa Rounding Road.” Arithmetic Teacher, 41 (5), 246-248.

Brown, A.L. & Palincsar, A. (1989). “Guided, Cooperative Learning and IndividualKnowledge Acquisition.” In Resnick, L.B. (Ed.), Knowing, Learning and Instruction:Essays in Honor of Robert Glaser. Mahwah, NJ: Lawrence Erlbaum Associates.

Bruck, K.S. & Crump, I. (1995). “What Works in Math: Give Us a Better Way”;“Math Tools á la MTV”; “Hands-on Geometry—Just Plane Fun.” Learning, 23 (4),63-69, 71.

California State Board of Education (1998). The California Mathematics AcademicContent Standards, Prepublication Edition. http://www.cde.ca.gov/board/k12math standards.html,

http://www.cde.ca.gov/board/intro.html.

Campbell, P.B. (1994). Programs to Encourage Girls in Math and Science: Some Researchand Evaluation Results. Groton, MA: Campbell-Kibler Associates.

Carey, L.M. (1998). “Parents as Math Partners: A Successful Urban Story.” TeachingChildren Mathematics, 4 (6), 314-319.

Cox, Margaret J. (1992) “The Computer in the Science Curriculum.” InternationalJournal of Educational Research, 17 (1), 19-35.

Cramer, S.R. (1996). “Assumptions Central to the Quality Movement in Education.”Clearing House, 69 (6), 360-364.

Dunn, R. & Dunn, K. (1978). Teaching Students through Their Individual LearningStyles: A Practical Approach. Englewood Cliffs, NJ: Prentice Hall.

Earp, N.W. & Tanner, F.W. (1980). “Mathematics and Language.” Arithmetic Teacher,28 (4), 32-34.

Elkind, D. (1998). Educating Young Children in Math, Science, and Technology.Washington, DC: The Forum on Early Childhood Science, Mathematics, andTechnology Education.

Fishbein, H.D., Eckart, T., Lauver, E., VanLeeuwen, R., & Langmeyer, D. (1990).“Learners’ Questions and Comprehension in a Tutoring Setting.” Journal ofEducational Psychology, 82, 163-170.

Franke, M.L. & Carey, D.A. (1997). “Young Children’s Perceptions of Mathematics inProblem-solving Environments.” Journal for Research in Mathematics Education,28 (1), 8-25.

Gagne, R.M. & White, R.T. (1978). “Memory Structures and Learning Outcomes.”Review of Educational Research, 48, 187-222.

Glaser, R. (1991). “The Maturing of the Relationship between the Science of Learningand Cognition and Educational Practice. Learning and Instruction, 1, 129-144.

Henderson, A.T. & Berla, N. (Eds.). (1994). A New Generation of Evidence: The FamilyIs Critical to Students Achievement.” Washington, DC: National Committee forCitizens in Education.

Henningsen, M. & Stein, M.K. (1997). “Mathematical Tasks and Student Cognition:Classroom-based Factors that Support and Inhibit High-level MathematicalThinking and Reasoning.” Journal for Research in Mathematics Education, 28 (5),524-549.

Hiebert, J. (1999). “Relationships between Research and the NCTM Standards.”Journal for Research in Mathematics Education, 30 (1), 3-19.

Hogan, D.M. & Tudge, J.R.H. (1999). “Implications of Vygotskian Theory for PeerLearning.” In O’Donnell, A.M. & King, A. (Eds.). Cognitive Perspectives on PeerLearning. Mahwah, NJ: Lawrence Erlbaum Associates.

Hunkins, F. (1995). Teaching Thinking through Effective Questioning, 2nd ed. Norwood,MA: Christopher-Gordon.

Kjos, R. & Long, K. (1994). “Improving Critical Thinking and Problem Solving in FifthGrade Mathematics.” Saint Xavier University. Manuscript.

Krulik, S. & Rudnick, J.A. (1994). “Reflect...for Better Problem Solving and Reasoning.”Arithmetic Teacher, 41 (6), 334-338.

Lacampagne, C.B. (1993). State of the Art: Transforming Ideas for Teaching and Learning.Washington, DC: Government Printing Office.

Leonard, J. (1999). “When the Task Is Not Just a Task: What One Mathematics TeacherLearned about Facilitating Student Discourse.” Paper presented at the annual meetingof the American Educational Research Association. Montreal, Canada. April 4-8.

Martino, M.M. & Maher, C.A. (1994). “Teacher Questioning to Stimulate Justificationand Generalization in Mathematics.” Paper presented at the annual meeting of theAmerican Educational Research Association. New Orleans, LA. April 4-8.

Mathematical Sciences Education Board (1991). “For Good Measure: Principles andGoals for Mathematics Assessment.” Report of the National Summit on Assessmentof Mathematics. Washington, DC. 16-19.

Mewborn, D.S. & Huberty, P.D. (1999). “Questioning Your Way to the Standards.”Teaching Children Mathematics, 6 (4), 226-227, 243-246.

Muth, D.K. (1993). “The Thinking Aloud Procedure: A Diagnostic Tool for MiddleSchool Mathematics Teachers.” Middle School Journal, 24 (4), 5-9.

National Council of Teachers of Mathematics (NCTM). (1989). Curriculum andEvaluation Standards for School Mathematics. Reston, VA.

National Council of Teachers of Mathematics. (1991). Professional Standards for TeachingMathematics. Reston, VA.

Nicholl, B. (1996). “Developing Minds: Critical Thinking in Grades K-3.” Paper presented at the California Kindergarten Conference. San Francisco, CA.

Panchyshyn, R., Enright, B., & Monroe, E.E. (1986). “Word Frequency in MathematicsProblems, Grades 1-8.” New Horizons, Journal of Education, 27, 91-97.

Phelps, E. & Damon, W. (1989). “Problem Solving with Equals: Peer Collaboration as aContext for Learning Mathematics and Spatial Concepts.” Journal of EducationalPsychology, 81 (4), 639-646.

Richardson, M.V. & Monroe, E.E. (1989). “Helping Young Children Solve WordProblems through Children’s Literature.” School Science and Mathematics, 89 (6), 515-518.

Ross, Elinor Parry. (1998). Pathways to Thinking: Strategies for Developing IndependentLearners K-8. Norwood, MA: Christopher-Gordon.

Schoenfeld, A.H. (1992). In Grouws, D.A. (Ed.). Handbook of Research on MathematicsTeaching and Learning. New York: Macmillan, 334-371.

Sears, N.C. & Medearis, L. (1992). “Natural Math: A Progress Report onImplementation of a Family Involvement.” Paper presented at the meeting of theRocky Mountain Research Association. Stillwater, OK.

Sharan, S. (1980). “Cooperative Learning in Small Groups: Recent Methods and Effectson Achievement, Attitudes, and Ethnic Relations.” Review of Educational Research,50, 241-271.

Sierpinska, A. (1995). “Mathematics: ‘In Context,’ ‘Pure,’ or ‘with Applications?’” Forthe Learning of Mathematics, 15 (1), 2-15.

Stenmark, J.K. (1991). Mathematics Assessment: Myths, Models, Good Questions, andPractical Suggestions. Reston, VA: National Council of Teachers of Mathematics.

Tourniaire, F. (1986). “Proportions in Elementary School.” Educational Studies inMathematics, 17 (4), 401-412.

Tudge, J. & Caruso, D. (1988). “Cooperative Problem Solving in the Classroom:Enhancing Young Children’s Cognitive Development.” Young Children, 44 (1), 46-52.

Vetter, R.K. (1992). “The Learning Connection: Talk-through.”Arithmetic Teacher,40 (3), 168.

Webb, N.M. (1991). “Task-oriented Verbal Interaction and Mathematics Learning inSmall Groups.” Journal for Research in Mathematics Education, 22 (5), 366-389.

Webb, N.M. & Faravar, S. (1999). “Developing Productive Group Interaction inMiddle School Mathematics.” In O’Donnell, A.M. & King, A. (Eds.). CognitivePerspectives on Peer Learning. Mahwah, N.J.: Lawrence Erlbaum Associates, 117-149.

Sadlier-OxfordA Division of William H. Sadlier, Inc.

1-800-221-5175www.sadlier-oxford.com

Code # 9645-4

Cop

yrig

ht ©

Will

iam

H.S

adlie

r, In

c.P

erm

issi

on to

dup

licat

e gr

ante

d to

use

rs o

f Pro

gres

s in

Mat

hem

atic

s.

Venn Diagrams; Operation Wheels T71

T72 Number Lines

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800 900 1000

0 10 20 30 40 50 60 70 80 90 100

Cop

yrig

ht ©

Will

iam

H.S

adlie

r, In

c.P

erm

issi

on to

dup

licat

e gr

ante

d to

use

rs o

f Pro

gres

s in

Mat

hem

atic

s.

T76 Clocks; Calendar

Cop

yrig

ht ©

Will

iam

H.S

adlie

r, In

c.P

erm

issi

on to

dup

licat

e gr

ante

d to

use

rs o

f Pro

gres

s in

Mat

hem

atic

s.

121

2 3

4

56

7

8910

1112

1

2 3

4

56

7

8910

11

0

10

20

30

40

50

60

70

80

90

100

F°�

0

10

20

30

40

50

60

70

80

90

100

F°�

0

10

20

30

40

50

60

70

80

90

100

F°�

-20-10

0102030 0

10

20

30

40

50

-10

-20

-30

405060708090

100110120

F°� C°�

-20-10

0102030 0

10

20

30

40

50

-10

-20

-30

405060708090

100110120

F°� C°�

Cop

yrig

ht ©

Will

iam

H.S

adlie

r, In

c.P

erm

issi

on to

dup

licat

e gr

ante

d to

use

rs o

f Pro

gres

s in

Mat

hem

atic

s.

Thermometers; Dot Paper T77