Editor's Comment:

The articles in this special series on curriculum make significant contributions, first, by proposing that the curriculum employed for all students, including those with learning disabilities, should emphasize thinking, problem solving, and reasoning, a point few would disagree with, given the complexities of life in this postindustrial-information age. The authors also provide some evidence that students with learn-ing disabilities can learn and apply sophisticated concepts, rules, and strategies. Fur-thermore, the authors describe one instructional process that emphasizes higher order thinking, that is, "sameness analysis," This process facilitates the integration of con-cepts, rules, strategies, schema, systems, heuristics, and algorithms. The authors believe that sameness analysis fosters a holistic understanding of a content area, Over the last several years this journal has contained numerous articles that cogently argued for a more holistic approach to education. Sameness analysis is one that lends itself to the holistic approach. Finally, the series of articles quite appropriate-ly addresses the importance of efficient teaching. Some recent research indicates that students with learning disabilities may be receiving less instruction than their nonhandicapped peers—this despite the fact that they have problems in learning, Efficient teaching takes on increased importance with the recent call (by some) for increased or total integration of students with learning disabilities into general classrooms. — JLW

Curricular Interventions for Teaching Higher Order Thinking to All Students:

Introduction to the Special Series

Douglas Carnine

The composition of American class-rooms is becoming increasingly di-

verse. The Regular Education Initiative, calling for the placement of more stu-dents with handicaps in general educa-tion classrooms, was but one impetus for this transformation. Another major fac-tor is the changing demographics in U.S. schools: Minority enrollment is close to or exceeding 50% in five states and the District of Columbia (GEM Project, 1990).

Educators could argue that, in the face of these changes, maintaining current levels of achievement in general educa-tion classrooms would be a reasonable goal. The American public would surely reject such an argument, however, be-cause achievement levels are so low. For example, more than 80% of eighth-grade-age students in the United States cannot correctly solve modestly difficult

problems from their eighth-grade math basal (Anrig & LaPointe, 1989). This 80% failure rate provides surprising con-firmation of an hypothesis made by Ysseldyke (in press): Eighty percent of U.S. students could be considered to have a learning disability by applying one of the various identification procedures currently in use.

Clearly, massive educational reforms are necessary if students with learning disabilities and at-risk students are to suc-ceed in school. Those reforms should in-clude increasing academic-engaged time for students, staff development for teachers, appropriate curricular materials, administrative support, and restructur-ing. Yet, the prognosis for implementing reforms is bleak."Meeting the challenges inherent in such an immense endeavor re-quires that the proponents of restructur-ing now move beyond selling their good

ideas, and begin the much more difficult work of policy design and implementa-tion" (McDonnell, 1989, p. iv).

The challenge of reform has been made even more formidable with the call for a new curriculum that teaches higher order thinking. "Although it is not new to include thinking, problem solving, and reasoning in someone's school curricu-lum, it is new to include it in everyone's curriculum" (Resnick, 1987, p. 7). Special education teachers react to these new goals with puzzlement and frustration. Their students struggle to survive the traditional curriculum.

The present series of articles responds to the call for higher order thinking by explaining and illustrating how curricu-lum can be designed for a full spectrum of students. In fact, the optimal way to organize curriculum to accommodate atypical learners may also be highly ad-vantageous for teaching higher order thinking to general education students (though this probably seems paradoxi-cal—that a curriculum could meet the needs of atypical learners as well as those of mainstream students).

This series of articles explains and il-lustrates the implications of that asser-tion. To be convincing, these explana-tions have to adequately sample the range of disciplines and thinking skills. To that end, the articles deal with social science, natural science, spelling, mathematics, composition and comprehension, reason-ing skills, and problem solving. Most of the authors of the articles have carried out research having to do with curricular issues; many also have been involved in designing and implementing curricular materials.

The series, however, does not dwell on many of the essential aspects of reform that would have to occur for all students to learn higher order thinking skills. The scope is fairly narrow, limited to cur-ricular issues and, to a lesser extent, in-struction. Still, the articles respond to the public's demands for the teaching of higher order thinking by arguing for higher academic expectations in the con-text of practical, empirically tested cur-ricular interventions. The importance of high expectations cannot be overempha-sized. As long as educators believe that a learning disability makes a student in-capable of higher order thinking, they

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will not search out and implement effec-tive interventions. High expectations and effective interventions are essential ingre-dients of reform. Either one, alone, is insufficient.

It is important to acknowledge that these articles do not present definitive data in support of the suggested new directions for curricular reform. The data are adequate, though, for highlight-ing the importance of curriculum, as both a cause of student failure and a target for reform. The data in Table 1, on teaching higher order thinking to spe-cial education students, though limited, should encourage educators to engage in serious debate about teaching higher order thinking to a broader spectrum of students and about the role of curricular materials in reaching this goal.

INSTRUCTIONAL ORGANIZATION

Although this series of articles is con-cerned primarily with curriculum, curricu-lum is not independent of instruction. The variation in methods for organizing instruction to improve learning is strik-ing. On one extreme are child-centered approaches, in which instruction is or-ganized to meet the needs of each in-dividual child. A major concern with this approach is efficiency. If, for example, a teacher divides his or her attention among 30 fourth graders across a 4'/2-hour school day, each student receives less than 10 minutes of individual atten-tion to cover a half-dozen subjects.

The middle of the spectrum for orga-nizing instruction might be called children-centered. Such instructional approaches would include peer tutoring, reciprocal teaching, and cooperative learning. These approaches focus on more than one stu-dent, and the students themselves have considerable responsibility for teaching each other.

At the other end of the spectrum are teacher-centered approaches. Teacher-centered approaches can be sensitive to the needs of the individual student, as in the Socratic, interactive questioning model. A more prosaic form of interac-tive teaching is found in direct instruc-tion, active teaching, and so forth. At the extreme of the continuum of approaches is the lecture-only method, in which the teacher expounds for an entire class

period. The lecture-only method is prev-alent in postsecondary settings, but also is found in many secondary classrooms. There is no interactive teaching, only a few questions from the students who are confident enough to ask. In the present article, the teacher-centered approach means interactive teaching, not a lecture-only method.

The empirical support for these various ways of organizing instruction varies considerably; a review of that research is beyond the scope of this article. One point is quite relevant, though: These variations are not mutually exclusive and, in fact, probably could and should occur every day in every classroom. A child-centered perspective enhances in-

struction by encouraging teachers to at-tend to each child's interests; for exam-ple, by identifying books for reading, topics for writing, and other projects that are of interest to the child.

Children-centered activities foster both academic and social competencies. Al-though various children-centered ap-proaches, such as reciprocal teaching and group process work, are employed in some of the curricular approaches de-scribed in the present series of articles, the primary approach is teacher-centered, interactive instruction. Interactive in-struction seems to be the best suited to introducing and explaining new, complex content, particularly for difficult-to-teach students (e.g., White, 1988).

TABLE 1 Research on Closing the Gap Between Special Education and General Education Students

Reasoning 1. On a variety of measures of argument construction and critiquing, high school students

with mild handicaps in a higher-order-thinking intervention scored as high as or higher than high school students in an honors English class and college students enrolled in a teacher certification program (Grossen & Carnine, 1990b).

2. In constructing arguments, high school students with learning disabilities in a higher-order-thinking intervention scored significantly higher than college students enrolled in a teacher certification program and scored at the same level as general education high school students and college students enrolled in a logic course. In critiquing arguments, the students with learning disabilities scored at the same level as the general education high school students and the college students enrolled in a teacher certification program. All of these groups had scores significantly lower than those of the college students enrolled in a logic course (Collins & Carnine, 1988).

Understanding Science Concepts 1. High school students with learning disabilities were mainstreamed for a higher-order-

thinking intervention in science. On a chemistry test that required applying concepts such as bonding, equilibrium, energy of activation, atomic structure, and organic compounds, the students' scores did not differ significantly from control students' in an advanced place-ment chemistry course (Hofmeister, Engelmann, & Carnine, 1989).

2. Middle school students with learning disabilities were mainstreamed for a higher-order-thinking intervention in science. On a test of misconceptions in earth science, the students showed better conceptual understanding than Harvard graduates interviewed in Schnep's 1987 film, A Private Universe (Muthukrishna, Carnine, Grossen, & Miller, 1990).

Problem Solving 1. On a test of problem solving in health promotion, high school students with mild handi-

caps in a higher-order-thinking intervention scored significantly higher than nonhandicapped students who had completed a traditional high school health class (Woodward, Carnine, & Gersten, 1988).

2. Middle school students with learning disabilities were mainstreamed for a higher-order-thinking intervention in science. On a test of earth science problem solving, the students scored significantly higher than nonhandicapped students who received tradi-tional science instruction (Woodward & Noell, this series).

3. High school special education students were mainstreamed for a higher-order-thinking intervention in math. On a test of problem solving requiring the use of ratios and propor-tions, the students scored as well as nonhandicapped high school students who received traditional math instruction (Moore & Carnine, 1989).

4. Middle school students with mild handicaps were mainstreamed for a higher-order-thinking intervention in earth science. Most of the students with handicaps scored higher than the nonhandicapped control students in problem solving involving earth science content (Niedelman, this series).

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CURRICULUM-THE CONVENTIONAL PERSPECTIVE

Instructional approaches, though cru-cial to the implementation of a curricu-lum, are not central to the articles in this series. The focus is on how the curricu-lum shapes the teaching of new, complex content. How can academic content be analyzed and communicated via curricu-lar materials to promote understanding, transfer, and retention among all stu-dents? A first step in answering this ques-tion is to briefly review two influences on current thinking about curriculum.

One obvious influence is psychology. Only the viewpoint that most influences curriculum development—the motiva-tional-developmental perspective — will be touched upon. By presenting develop-mentally appropriate experiences and motivating students, teachers assume that students will be able to learn what-ever is presented. Such motivation is thought to result from activities that match the interests and developmental level of the students and to stimulate learning through choice or discovery. The following quotes from a recent issue of Educational Leadership illustrate this philosophy:

"Children learn letter names and the sounds they represent as a part of the purposeful reading and writing they do, not as a set of meaningless fragments of information" (Strick-land, 1990, p. 23).

"The trick is not to have a multitude of tasks for students to do but to know how to 'read' students so that we can motivate them to em-brace the learning situation. . . . When stu-dents have an investment in what they are doing, they are motivated, involved, and dedi-cated. That is when learning takes place" (Lovitt, 1990, p. 44).

Prevalent in cognitive psychology is the constructivist perspective, which also supports a discovery orientation; how-ever, a review of this research points out that the quality of the curriculum design is paramount —not whether instruction occurs through discovery or explicit teach-ing (Grossen & Carnine, 1990a). The specifics of how to organize and present the content of a discipline must be care-fully considered if students are to succeed at higher order thinking. The articles in

this series illustrate why curricular ac-tivities based primarily on developmental and motivational considerations will be inadequate for many low-achieving students.

A second influence on educators' at-titudes toward curriculum is the text-book, which often defines what and how content is to be presented. "Textbooks dominate instruction in elementary and secondary schools" (Farr, Tulley, & Ray-ford, 1984, p. 59). As the primary tool of the teacher, textbooks are not par-ticularly sensitive to the needs of atypical learners. Tulley and Farr (1985) pointed out that state adoption committees, in an effort to provide a standardized state cur-riculum, have in fact produced a "uni-formity of curriculum." Because publish-ing companies construct their textbooks according to the requirements of the large adoption states (California, Texas, and Florida), textbooks from various publishers are quite similar. Such an ap-proach assumes that all students "require or benefit from the same instructional goals and sequences," but to the extent that curriculum uniformity is achieved, "the ability to meet the diverse needs of students is reduced" (Tulley & Farr, 1985, p. 1).

This uniformity might not be so dam-aging, except that members of textbook adoption committees seldom consider pedagogy or research in selecting text-books (Courtland, Farr, Harris, Tarr, & Treece, 1983; Powell, 1985). As is pointed out in several articles in the series, text-books are very careful to be comprehen-sive in their coverage of topics, but they are seemingly indifferent to the concep-tual coherence of the content and the pedagogical effectiveness of activities that are recommended therein. Hurd (cited in Rothman, 1988) described biology text-books as the world's most beautifully il-lustrated dictionaries. In mathematics, a relatively large percentage of the topics receive only brief coverage (Porter, 1989). On the average, teachers devoted less than 30 minutes in instructional time across the entire year to 70% of the topics that they covered (e.g., telling time might receive 25 minutes during all of first grade). This practice, "teaching for exposure," has become commonplace in American classrooms, largely due to the fact that the practice parallels the rec-

ommendations for topic coverage in mathematics textbooks, which are trying to cover many topics. In reviewing read-ing basals, Durkin (1978-1979) also found pervasive teaching for exposure, which she called "mentioning."

These two influences—the motiva-tional-developmental orientation and textbooks that teach for exposure—have the effect of diminishing the importance of curricular materials. Many primary teachers do not consider textbook activ-ities to be motivating or developmentally appropriate. Intermediate and secondary teachers recognize that textbooks do not promote higher order thinking in the content areas. The present series of ar-ticles does not recommend revising inter-mediate and secondary textbooks accord-ing to the motivational-developmental perspective. Rather, an alternative psy-chological perspective is offered for making curricular materials effective tools for teaching higher order thinking.

THE FOUNDATION OF THINKING - NOTING SAMENESSES

Higher order thinking entails the inte-gration of concepts, rules, strategies, schemas, systems, heuristics, algorithms, and so forth. A discussion of higher order thinking could define these con-structs and their interrelationships. An alternative is to deemphasize descriptive terms and focus on the process that underlies the constructs and the implica-tions of that process for designing cur-riculum. The present series of articles argues that the process that underlies concepts, rules, strategies, and so forth is noting samenesses. Every concept, rule, strategy, and so forth is defined by a fundamental sameness. For example, when one learns a concept, such as "big," one extracts a sameness in quality from several events, then applies it to new, ap-propriate events, such as big houses, big animals, big people. Because the later ar-ticles illustrate how to organize content around important samenesses as a means of teaching a broader spectrum of stu-dents, the process of noting samenesses is crucial to the entire series. This pro-cess will be discussed and illustrated, to foreshadow the upcoming articles.

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William James (cited in Campbell, 1986) pointed out the importance of sameness in his 1890 work, The Prin-ciples of Psychology. "We do not care whether there be any real sameness in things or not, or whether the mind be true or false in its assumptions of it. Our principle only lays it down that the mind makes continual use of the notion of sameness, and if deprived of it, would have a different structure from what it has" (p. 60).

More recent research, contrasting the performance of experts and novices, underscores the importance of organizing information around critical samenesses. Feltovich (1981) described the expert's knowledge structure as more intercon-nected and hierarchical than the novice's. Experts seem to organize this hierarchy of knowledge around explanatory or causal relationships (Bromage & Mayer, 1981; Chi, Feltovich, & Glaser, 1981). In mathematics, important samenesses have to do with the solution method (Schoen-feld & Herrmann, 1982), while in science, the important samenesses are underlying laws or principles (Silver, 1981). Such hierarchical organizations of knowledge seem to be more important in expert problem solving than a repertoire of metacognitive strategies, which usually are assumed to be needed for higher order thinking.

The central role that noting samenesses plays can also be illustrated through the accomplishments of inventors and scien-tists. For example, Thomas Edison ada-mantly opposed the prevailing notion about how electricity should be used in homes —to feed a single enormous elec-tric arc light. He hypothesized a sameness between how water was distributed to many points in a home via a network of pipes and how electricity could be dis-tributed via a network of wires. He was instrumental in turning this vision into reality.

The ability of Katherine Payne (1989), a biologist, to detect a sameness between vibrations from an organ and those from elephants led her to discover how ele-phants communicate:

Some capacity beyond memory and the five senses seems to inform elephants, silently and from a distance, of the whereabouts and ac-tivities of other elephants.

I stumbled on a possible clue to these mys-

264

teries during a visit to the Metro Washington Park Zoo in Portland, Oregon, in May, 1984. While observing three Asian elephant mothers and their new calves, I repeatedly noticed a palpable throbbing in the air like distant thunder, yet all around me was silent.

Only later did a thought occur to me: As a young choir girl in Ithaca, New York, I used to stand next to the largest, deepest organ pipe in the church. When the organ blasted out the bass line in a Bach chorale, the whole chapel would throb, just as the elephant room did at the zoo. Suppose the elephants, like the organ pipe, were the source of the throbbing? Suppose elephants communicate with one another by means of calls too low-pitched for human beings to hear? (p. 266)

Obviously, a process as basic as noting samenesses does not always lead to prob-lem-solving insights such as those of Edi-son and Payne. Nobel-prize-winner Gerald Edelman, who directs the Neurosciences Institute at Rockefeller University, be-lieves that the noting of samenesses accounts for the moment-to-moment functioning of the brain. The central pro-cedures in Edelman's (1987) scheme of brain functioning are categorization and recategorization—in perception, in rec-ognition, and in memory (Rosenfield, 1988). Categorization and recategoriza-tion, which depend on the learner's capacity to note samenesses, are viewed as the overriding activity of the brain, serving as the basic mechanism for the various brain functions.

The implication of Edelman's (1987) emphasis on samenesses is quite different from those drawn from the localization-of-function theories that strongly in-fluenced special education in the past. Rather than trying to identify strengths and weaknesses at various brain loca-tions, educators can analyze how and why students learn particular samenesses, some of which are desirable from the educator's perspective, some of which are not.

THE FOUNDATION OF MISCONCEPTIONS - NOTING SAMENESSES

The simple fact is that the process is quite indiscriminate, continuing until a seemingly functional sameness is noted. For the young child, the learning of what might be called "plausible" sameness re-sults in all men being called "da da," even the mail carrier. For the preschooler,

who has learned that an object retains its name regardless of its orientation in space, the letter b is still called "b," even after it is rotated in space to look like this: d. Misconceptions are also evident at the other end of the schooling con-tinuum, as demonstrated by Harvard University graduates at commencement (Schneps, 1987). At some time in their lives, they all had this "same" experience: The closer they came to a fire or other heat source, the warmer they got. When asked why summers are warmer, they an-swered that it is because the earth is closer to the sun in the summer. The students erroneously assumed that the warmer summer temperatures have the same cause as warmer temperatures from a fire—proximity to the heat source. The learning of inappropriate samenesses (or misconceptions) is anticipated in Edel-man's work. "But neither can one predict what constitutes information for an organism. The brain must try as many combinations of incoming stimuli as possible, and then select those combina-tions that will help the organism relate to its environment" (Rosenfield, 1988, p. 149).

A final example of naive samenesses will be discussed in a little more detail. These naive samenesses are all applied to zeros in elementary math problems. The origin of the naive sameness and its ap-plication are described in Table 2. In all three examples, students learn how to work problems that do not contain zeros. When the students apply those same pro-cedures to problems containing zeros, many make mistakes based on their mis-conceptions. In the first example in Table 2 (place value), the initial problems are all the same in this way: The students write a digit for each stated digit.

Stated digits: 2 million, 3 hundred 47 thousand, 8 hundred 62

Written digits: 2 3 47 8 62

In 2,007,862, no digit is stated for hun-dred thousands or ten thousands; so the student reponds in this way:

Stated digits: 2 million, 7 thousand, 8 hundred 62

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TABLE 2 Origins and Applications of Naive Samenesses Involving Zeros

Origin 1. Place Value

For each stated digit, the student writes a digit. The teacher says, "Write 2, 347, 562." The student writes these digits: 2, 347, 562.

2. Subtraction 714

^ 2 - 2 7 8

5 7 4

Each time the student renames, she rewrites the top digit to the left as a number that is 1 less. In the example above, 5 is rewritten as 4, and 8 is re-written as 7.

3. Division

231 3T693

Each time the student divides, he writes a digit in the answer. In the example above, the answers for the division prob-lems are 2, 3, and 1.

Application

For each stated digit, the student writes a digit. The teacher says, "Write 2, 007, 862." The student writes 27,862. No digit is stated for hundred thousands or for 10 thousands, so nothing is written.

Jo12 - 2 7 8

5 7 4

Each time the student renames, she rewrites the top digit to the left as a number that is 1 less. In the example above, the student does not think 0 can be rewritten as a number that is 1 less, so the student skips over to 8 and rewrites it as 7.

2 8 3)624

Each time the student divides, he writes a digit in the answer. In the example above, he did not divide 3 into 2, so he did not write anything above the 2. He divided 3 into 24, so he wrote 8 above the 4 in 24.

Written digits: 2 7 8 62

This misconception would not cause mistakes if the digits were stated in this way:

2 millions, 0 hundred thousands, 0 ten thousands, 7 thousands, 8 hundreds, 62

2 0 0 7 8 62

The second example in Table 2 is renaming. The renaming examples are all the same in this way: Borrow from the digit to the left. Here are several examples:

3V8 3V8 3 ^ 8 3 ^ 8 3°^8 - 6 9 - 6 9 - 6 9 - 6 9 - 6 9

However, in the next example, the stu-dents assume that they cannot borrow from zero, so they borrow from the digit next to the zero, the 3:

V8 - 6 9

The human mind is an incredible de-vice for noting samenesses. It formulates sameness based on memory and input from the senses but is often insensitive to intentions, even teachers' best inten-tions. This point is difficult for adults to appreciate, because they are so knowl-edgeable about the content of the school curriculum. As a reminder of how adults search for samenesses and must rely on what is presented during instruction, you are about to learn something new: damon. You start with a damon and add damon. Here are some examples:

9 and you add 1. The new damon is 10. 10 and you add 2. The new damon is 12. 3 and you add 4. The new damon is 7. 3 and you add 6. The new damon is 9. The next two are for you to figure out: 4 and you add 11. 9 and you add 7.

Did you answer 15 and 16, respective-ly? If so, you learned an obvious same-ness, but still a misconception. The cor-rect answers are 3 and 4. This damon example is intended to remind you how it feels to be left hanging—a familiar

feeling for too many students. (You can read more about damon later in the article.)

One implication of the sameness anal-ysis is that, as students are constantly seeking out samenesses, they sometimes fail, as was illustrated with writing num-bers with zeros, renaming with zeros, and explaining why the earth is warmer in the summer. This is the darker side of the sameness analysis —explaining students' failure.

DESIGNING CURRICULUM AROUND IMPORTANT SAMENESSES

The bright side can lead to success. If sameness is the psychological key for organizing curriculum, the content itself must be the lock. The mechanism that allows the key and the lock to function is the organization of content in ways that highlight important samenesses. This operationalizes what Flavell (1971) called building "cognitive structures":

The really central and essential meaning of "cognitive structure" ought to be a set of cognitive items that are somehow interrelated to constitute an organized whole or totality; to apply the term "structure" correctly, it ap-pears that there must be, at minimum, an ensemble of two or more elements together with one or more relationships interlinking these elements, (p. 443)

As implied by the math problems with zeros and the interviews with the Har-vard graduates about why summer is warmer, building appropriate cognitive structures is not easy. First, competing, inappropriate structures develop through the mind's unceasing search for same-nesses. Second, the more crucial same-nesses within a discipline are not nec-essarily obvious and are seldom given prominence through developmentally appropriate, motivating experiences or through conventional textbooks.

Consider a topic as seemingly simple, yet hotly contested, as beginning reading. A phonics curriculum highlights impor-tant samenesses by selecting words in which the same letter always represents the same sound, such as the short a in

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man, sat, dad. Selecting words in this way restricts the types of stories that can be composed for early first grade. These restrictions, which appear to inhibit meaning, interest, real-life relevance, and so forth, offend educators concerned with motivation. The issue appears to be this: Do the complexities of beginning reading for the 6-year-old justify re-strictions on what he or she reads in school? In other words, does preserving phonic samenesses justify the use of highly constrained stories in very early reading instruction? It seems so, accord-ing to the latest research review man-dated by Congress, Beginning to Read: Thinking and Learning About Print (Adams, 1990). Interpreting the research on beginning reading is not the point. The issue is, For what content and to what ex-tent should curricular materials be organized around important samenesses?

The contribution of a sameness anal-ysis can be illustrated in geometry, wherein students learn equations, first for surface area and later for volume of various figures. Students are typically expected to learn seven formulas to calculate the volume of 7 three-dimen-sional figures:

Rectangular prism: Nw»h = v Wedge: !/2»Nw»h = v Triangular pyramid: 1/6»l»w«h = v Cylinder: 7r»r2«h = v Rectangular pyramid: 1/3«l»w«h = v Cone: !/3»7r»r2»h = v Sphere: 4/3»7r»r3 = v

These equations do not prompt higher order thinking about volume, just the need to memorize formulas. The same-ness analysis reduces the number of for-mulas students must learn from seven to slight variations of a single formula— area of the base times the height (Bxh) — which brings conceptual coherence to exercises involving volume.

Rectangular Prism Wedge Cylinder

B*h B*h B*h

Pyramid Rectangular Triangular Cone Sphere

B^/s-h B-!/3*h B . / s ^ h B . ^ . h

For the regular figures —rectangular prism (box), wedge, cylinder—the volume

is the area of the base times the height (B»h). For figures that come to a point (pyramid with a rectangular base, pyra-mid with a triangular base, and a cone), the volume is not the area of the base times the height, but rather the area of the base times Vi of the height (B« lA*h). The sphere is a special case—the area of the base times 2A of the height (B»2/3«h)— where the base is the area of a circle that passes through the center of the sphere, and the height is the diameter. The same-ness analysis makes explicit the core con-cept that volume equals base times height. This core concept is obscured in math textbooks that present seven different formulas.

The articles in this series briefly review the criticisms of conventional curricular materials and then illustrate how impor-tant samenesses can serve as a framework for organizing the curriculum. The ar-ticles are somewhat unusual in the degree to which they deal with the specifics of a curriculum. This depth is a natural outgrowth of a shift in emphasis from the psychology of the child as an end in itself, to identifying and teaching the core concepts of a discipline. For example, the article on social studies (Kinder & Bur-suck) illustrates how major events in U.S. history can be explained through a prob-lem-solution-effect model, with prob-lems usually stemming from economics, though occasionally from concerns about religious freedom or human rights. Solu-tions are more variable: for example, fighting, moving, inventing, making an accommodation, or tolerating a problem. In the article on science, Woodward and Noell explain causal models in earth science (e.g., based on convection) and in chemistry (e.g., based on equilibrium) that integrate a variety of seemingly unrelated phenomena. Each of the other content articles (mathematics and spell-ing) describes important samenesses that provide a framework around which cur-riculum for higher order thinking can be organized. The articles exemplify Bruner's (1960) claim: "The basic ideas that lie at the heart of all science and mathematics and the basic themes that give form to life and literature are as simple as they are powerful" (p. 12).

It is important to remember that the sophistication of learning garnered by noting important samenesses is not be-

yond the grasp of students with learning disabilities or at-risk students (cf. Table 1). All of the articles in this series (ex-cept the one concerning teacher training by Simmons, Fuchs, & Fuchs) demon-strate that students with special needs can learn and apply sophisticated concepts, rules, and strategies. These findings justify higher expectations for students who typically are not taught higher order thinking skills.

Organizing curriculum around impor-tant samenesses also has implications for the organization of instruction. The quality of learning in children-centered activities such as cooperative learning, reciprocal teaching, and peer tutoring can be constrained by the quality of text-books, as is illustrated in later articles in this series. For example, children-centered activities based on texts that touch upon many topics superficially, as unrelated fragments, might improve student recall of the fragments but not build cognitive structures that lead to higher order think-ing. In contrast, children-centered activ-ities applied to content organized around important samenesses foster higher order thinking. For example, one of the chil-dren-centered approaches used in the social studies intervention (Kinder & Bur-suck) is reciprocal teaching. In the com-position and comprehension intervention (Englert & Mirage), instruction includes a peer work group.

The inference is that children-centered activities are strongly influenced by the curricular content of the activities. Ac-tivities that are like real life, or that lead children to discover, do not necessarily make salient the fundamental samenesses in a discipline. As noted earlier with be-ginning reading, the identification of important samenesses, as in phonics instruction, may even conflict with "meaningfulness."

The conflict between naturally occur-ring organizations of curricular material (e.g., literature in beginning reading) and organizations based on a sameness anal-ysis (phonically constrained stories for beginning reading) accounts for one criti-cism of the sameness analysis. Another criticism is that making important same-nesses explicit in general education class-rooms will "hold back" above-average students. Some recent reviews of research indicate that this is not the case for prob-

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lem solving in computer science (Dalley & Linn, 1985), in learning to design scientific experiments (Ross, 1988), or in logical and analogical reasoning (Grossen & Carnine, 1990a). It might be more ac-curate to say that using a curriculum organized around important samenesses is not as important for less cognitively complex topics with above-average stu-dents. For below-average students, in-cluding students with learning disabilities and at-risk students, making important samenesses explicit is fairly important at all levels of cognitive complexity. These hypothesized relationships are displayed in Figure 1.

The basis for the hypothesized rela-tionships in Figure 1 lies in the findings from the brain research cited earlier — the noting of samenesses. Students who are facile, intuitive learners (i.e., above-average students) note important same-nesses fairly readily. They categorize and recategorize at a rapid rate and in a flex-ible manner, without need for an instruc-tional environment that emphasizes im-portant samenesses and, in effect, warns the learner about misconceptions. With content that is not highly complex, these students can "figure out" important same-nesses without getting seriously misled.

Less capable students benefit when these important samenesses are made ex-plicit. Kail (1984) reviewed several studies showing significantly longer memory search times for individuals with handi-caps. This slower rate could reflect ineffi-cient processes for identifying samenesses. By making important samenesses explicit, the curricular interventions illustrated in this series might compensate for such in-efficient processing. This is a possible ex-planation for why these students benefit from a curriculum designed according to the sameness analysis (White, 1988).

THE NEED FOR EFFICIENT INSTRUCTION

The articles in this special series illus-trate how the sameness analysis leads to a holistic understanding of a content area, rather than fragmented knowledge. That is the overriding purpose of the sameness analysis: to foster coherent schemata of important bodies of knowl-edge. At a simplistic level, the notion is to "teach smart." However, the realities

Great

Importance of Organizing Information Around Important Samenesses

Moderate

Slight

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• B A V

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AA

• S E V

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AA

LOW High Cognitive Complexity

Figure 1. Relationship between importance of organizing information around important samenesses and cognitive complexity for above-average (AA), average (A), below-average (BA), and special educa-tion (SE) students.

of schools force another priority upon educators—"teach efficiently." Sameness analyses can lead to certain efficiencies. For example, when students analyze his-tory with a problem-solution-effect model and are familiar with common causes (economics and human rights) and typical solutions (fighting, moving, in-venting, making an accommodation, or tolerating a problem), the students can more quickly come to understand new historical events. In addition, hierarchi-cally organized information is easier to remember than randomly organized in-formation. Finally, the number of terms and isolated facts required of students can be reduced because they are learn-ing higher order organizations that sub-sume many of these bits and pieces.

However, even greater efficiencies are required for at-risk students and students with learning disabilities. As Haynes and Jenkins (1986) reported, students with learning disabilities in an eclectic resource room program ended up getting no more instructional time than their nonhandi-capped peers in the general classroom. Allington and McGill-Franzen (1989) reported that students with handicaps ac-tually received less instruction than their nonhandicapped peers. In addition, stu-dents with special needs, particularly those from disadvantaged backgrounds, typically received less help from their

parents at home. Because special needs students have

fewer learning opportunities, the time that is available for instruction must be used very efficiently. Students should be engaged as much as possible in academic activities that produce rapid learning. There are two central aspects of efficient teaching: Structure academic activities so that they are as productive as possible, and minimize wasted time. An example of an unproductive activity is teaching students to use manipulatives to divide with two-digit divisors. A lot of time can be wasted in passing out and collecting manipulatives. Moreover, even when the students are actively using manipulatives to work a dozen problems, such as 21)178, the time is not being productively used. The activity consumes too much time in mindless manipulation (breaking one block of 100 cubes and seven blocks of 10 cubes into individual cubes) and in tedious counting (forming groups of 21).

The other central aspect of efficient teaching—reducing wasted time—can be achieved through teaching techniques (frequent questions, constructive feed-back, active monitoring, etc.). These techniques were given prominence by Rosenshine (1976) and continue to re-ceive attention (e.g., Christenson, Yssel-dyke, & Thurlow, 1989). Most of the curricular interventions described in ar-

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tides in the series employed efficient teaching techniques. Whatever is going to be taught, whether subtraction facts or problem solving in science, efficient instruction is very important, particularly for students with learning disabilities. There is too little time to meet all the in-dividual needs of students; efficient in-struction allows more of these needs to be met.

It is important to note that these tech-niques are usually used with traditional textbooks. For a comparatively simple topic, such as subtraction facts, the out-come would be fairly reasonable. The teacher would present facts at a rapid rate, give feedback, monitor students' responses, and so forth. Students could memorize the subtraction facts as un-related pieces of information and could be successful.

Efficient teaching techniques can also improve student scores in more advanced topics. For example, in many science courses, students are tested on terminol-ogy. Efficient teaching techniques can facilitate the rote learning of science vocabulary. The disadvantage is that the students can get a good grade and still come away with little understanding of science.

For more conceptually demanding ac-tivities, such as learning science concepts (not just terminology) and analyzing word problems, efficient teaching prac-tices may not be sufficient (Moore & Carnine, 1989). In fact, when students are taught to analyze word problems ac-cording to recommendations from tradi-tional textbooks (e.g., read, analyze, plan, and solve), the lack of a strategy based on a sameness analysis can lead to frustration. Darch, Carnine, and Gersten (1984) found that efficient teaching prac-tices, such as frequent assessment with extra instruction, were not beneficial and possibly were harmful. When students in the traditional treatment failed a forma-tive assessment, they received more in-struction in reading, analyzing, planning, and solving. Because this strategy was not specific enough to be of much help, the extra practice problems just brought extra frustration. The students' perfor-mance and attitudes deteriorated. In short, efficient teaching techniques by themselves do not necessarily lead to the acquisition of higher order thinking, just

possibly more efficient learning of lower order skills.

In fact, the lack of efficient teaching techniques might explain, in part, the relative failure of more than 100 cur-riculum development projects funded by the National Science Foundation be-tween 1956 and 1975: "These projects stressed the learning of fundamental con-cepts over facts and the use of discovery methods, student inquiry, and multi-media materials to supplement textbooks" (McDonnell, 1989, p. 40). The child-centered approaches (i.e., discovery and inquiry) might have impeded the learn-ing of fundamental concepts.

If special needs students are to learn higher order thinking skills, new curricu-lar materials must be developed, such as those organized around important same-nesses. Moreover, educators must believe that these students are capable of higher order thinking. Finally, educators must be efficient if these students are to receive sufficient intensive instruction to become proficient. Orchestrating these three re-quirements — curriculum, expectations, and techniques —is extremely difficult, yet absolutely necessary.

OVERVIEW OF THE SERIES

The series is presented in two parts; one appears in this issue and one in the next. The first part covers major content areas:

Social Science by Kinder and Bursuck Natural Science by Woodward and Noell Spelling by Dixon Mathematics by Engelmann, Carnine,

and Steely

The second part covers the tools of higher order thinking and staff develop-ment, and includes a commentary and a conclusion:

Problem Solving and Transfer by Niedel-man

Composition and Comprehension by Englert and Mariage

Reasoning by Grossen Staff Development by Simmons, Fuchs,

and Fuchs Commentary by Jenkins, Stein, and

O'Connor Conclusion by Kameenui

CONCLUSION

It is important to note that the inter-ventions described in the following arti-cles are illustrations of how the curricu-lum can be organized around important samenesses. Many other organizations are possible, and some would quite like-ly be superior. In any case, efficient teaching techniques, cooperative learn-ing, and metacognition are wasted, to some degree, when curricular material presents higher order content in a frag-mented manner that is most amenable to rote learning. The problem of fragmented knowledge is particularly acute for at-risk students and students with learning dis-abilities who are not typically offered alternative, richer organizations by fami-ly or peers. These students require cur-ricular material that is organized to model higher order thinking and is taught efficiently (by the teacher and by other children).

There are several justifications for such an approach. First, there is a reasonable amount of data in support of such an ap-proach, although the results are far from conclusive. Second, the research on ex-pert knowledge underscores the role of hierarchically organized knowledge based on important samenesses. Third, the ap-proach may compensate for a potential processing deficit (Kail, 1984) by mak-ing important samenesses explicit. Fourth, the approach addresses a salient charac-teristic of students with learning dis-abilities by reducing the memory load required by the curriculum. Finally, the approach addresses the practical con-straints of the classroom; it can be implemented in an efficient, practical manner.

Although learning important same-nesses in complex content probably bene-fits above-average students, it is essen-tial for at-risk students and students with learning disabilities, whether the goal is merely to keep them from dropping out of school or, a higher goal, that they thrive in school.

Collaboration on an Integrated Model

Providing such a school environment, where secondary students who are at risk or have learning disabilities learn higher

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order skills, is extremely difficult. In fact, the various curricular interventions de-scribed in this series have always been im-plemented in isolation. The interventions have never been combined in a single classroom to determine the additive effect of focusing on higher order thinking in science, social studies, math, spelling, composition, and so forth. Setting up model classrooms that integrate several of these interventions would be very challenging. For one thing, the students would need to have received reasonable instruction in elementary grades, so that they possessed rudimentary reading, writing, and math skills. Second, a suf-ficient number of students would have to be in school consistently. Finally, teachers would need to be willing to be involved in an intense learning experience, stem-ming from the content of the interven-tions and their implementation.

Note on damon: A simple explanation activates a familiar schema, rendering damon embarrassingly simple. Damon is clock time, involving addition with base 12:

10 and add 1. The new damon is 11. 11 and add 1. The new damon is 12. 12 and add 1. The new damon is 1. 12 and add 2. The new damon is 2. 6 and add 8. What's the new damonl 6 and add 9. What's the new damonl

ABOUT THE AUTHOR

Douglas Carnine is a professor of special educa-tion at the University of Oregon. His interests in-clude curriculum design and educational policy. Ad-dress: Douglas Carnine, University of Oregon, 805 Lincoln St., Eugene, OR 97401.

AUTHOR'S NOTE

Readers who might be interested in exploring the possibility of collaborating in establishing a model classroom may write to Professor Carnine.

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