Mathematics Reform and Learning Differences

Hammill Institute on Disabilities

Mathematics Reform and Learning DifferencesAuthor(s): Betsey GrobeckerSource: Learning Disability Quarterly, Vol. 22, No. 1 (Winter, 1999), pp. 43-58Published by: Sage Publications, Inc.Stable URL: http://www.jstor.org/stable/1511151 .

Accessed: 18/06/2014 19:41

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Sage Publications, Inc. and Hammill Institute on Disabilities are collaborating with JSTOR to digitize,preserve and extend access to Learning Disability Quarterly.

http://www.jstor.org

This content downloaded from 62.122.72.104 on Wed, 18 Jun 2014 19:41:43 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=sage

http://www.jstor.org/stable/1511151?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


MATHEMATICS REFORM

AND LEARNING DIFFERENCES

Betsey Grobecker

Abstract. Reform in mathematics as advocated by the National Council of Teachers of Mathematics (NCTM) has been met with much criticism by special educators. Specifically, researchers are arguing that NCTM reports have neglected the needs of students with learning differences (LD) by failing to attend to the remedial techniques that have demonstrated effectiveness with this population. However, rather than to dismiss the recommendations made, some of the teaching methodologies have been incorporated in traditional, explicitly based instructional programs. This practice is unfortunate because constructivist theory related to the development of logical-mathematical structures of thought holds a completely different philosophical premise than the information-processing perspective. This article will examine constructivist beliefs regarding the evolution of logical-mathematical thought structures so that its potential in meeting the needs of children with LD can be more honestly evaluated.

BETSEY GROBECKER, Ed.D., is an Assistant Professor, Aubur University.

It should not be necessary to stress that many good studies were conducted and reported during the days of statistics' paradigmatic primacy. However, . . . there was an emphasis on experimentation form and statistical significance rather than on a deep understanding of thinking, teaching, and learning. Indeed, a focus on such understanding was rare. (Schoenfeld, 1994, p. 701)

Schoenfeld (1994) saw mathematics education as "in the midst of a phenomenal growth spurt" (p. 706) because it once again acknowledges children's logical orders as the object of legitimate inquiry (e.g., Piagetian inquiries). This emphasis differs from current practice in which the replication of explicitly taught procedures and strategies, isolated from the logical orders of the adult, is transmitted to children in directed forms of instruction. According to Schoenfeld, the new perspective represents a major paradigmatic change whose time has come.

In an effort to de-emphasize explicitly taught skills and strategies while encouraging the expansion of children's logical thinking, the National Council of Teachers of Mathematics (NCTM) (1989, 1991, 1995) has published three reports advocating reform in mathematics curricula, instruction, and assessment. However, the council's recommendations in the first two documents have been criticized as not having ade- quately attended to effective teaching methodologies for children with learning differences (LD) (Giordano, 1993; Hofmeister, 1993; Hutchinson, 1993; Rivera, 1993). Because of this exclusion, the reforms were considered to be of potential harm to children who cannot readily conform to grade-level curriculum in mathematics (e.g., Hofmeister, 1993).

Although lacking in sensitivity to children with LD, some researchers contended that constructivist techniques defined in the NCTM standards could enhance and facilitate mathematics learning (Giordano, 1993;

Volume 22, Winter 1999 43



Hutchinson, 1993; Mercer, Harris, & Miller, 1993; Mercer, Jordan, & Miller, 1994; Rivera, 1993). However, the abandonment of information-processing principles of learning that include direct instruction is not being advocated. Rather, researchers believe that incorporat- ing "techniques" of constructivism into our existing model of learning would best enhance current instructional practices.

In this article, I will argue that the practice of adding fragmented pieces of constructivist techniques to the existing information-processing paradigm that so pow- erfully steers our research greatly distorts the tenets of constructivism. To more accurately represent the science of the constructivist paradigm, the assumptions embedded in the information-processing paradigm that are accepted as truth must be called into question. Science is changing, and we must also change our willingness to question paradigm assumptions and accept and work through the disruption that follows such questioning (Kuhn, 1962). Only through such a process will we be better able to define research methodology that more accurately represents the tenets of the NCTM reform and to assess the effectiveness of such tenets for children who encounter difficulties in mathematics.

This article examines differences between information-processing and constructivist paradigms with regard to (a) the nature of knowledge; (b) the interaction of the learner with the curriculum; and (c) the assessment process. To facilitate a deeper understanding of the constructivist position and its potential for children with LD, protocols will be presented from a lesson with two sixth-grade children receiving remedial intervention in mathematics, due to a discrepancy between mathematics achievement as measured on standardized testing and IQ. Jeff receives support instruction to the regular sixth-grade curriculum whereas Sarah's mathematics instruction replaces the sixth-grade curriculum. Sarah is working on multiplication with two-digit multipliers and long division with one-digit divisors. Both are taught using a traditional mathematics approach and receive help in mathematics daily.

NATURE OF KNOWLEDGE

Information Processing Traditionally, the field of learning differences has

operated under the assumptions held by the mechanistic, Cartesian world view that complex [mental] phe- nomena can be reduced to basic building blocks. Further, mechanisms can be defined to describe how these building blocks interact (Capra, 1982). When complex mental functions are isolated for examination and are conceived of as separately existing building blocks,

learning and assessment will mirror these assumptions. Specifically, tasks are understood as separately existing skills that must be mastered prior to moving to higher- level skills. To teach these skills, explicit, teacher- directed instruction is deemed necessary.

Studies examining the effectiveness of explicit, teacher-directive practices provide what appears to be convincing evidence that such pedagogy facilitates the learning process. For example, in a review of literature that examined mathematics performance in children with no learning difficulties (NLD), Brophy and Good (1986) found that instruction is maximized when students move quickly through a curriculum that breaks knowledge down into small steps where frustration and confusion are minimized. Practice of new learning is additionally necessary, so that mastery, indicated by "consistently smooth and correct responses" (p. 361), can occur. Finally, students need to be taught to integrate new learning with other concepts and skills so they can efficiently apply this learning in problem-solving situations.

With regard to classroom management and routines, students perform best in classrooms with much instruction/supervision by teachers rather than working alone. Structured lessons using techniques such as advance organizers, outlining content, signaling tran- sitions between lesson parts, summarizing subparts as lessons move along, and reviewing main ideas at the end, further enhance instruction. Finally, praise and symbolic rewards facilitate learning, particularly in the lower grades. However, Brophy and Good (1986) stated that these conclusions are limited in their generalization because they are characteristic of traditional classrooms only. Further, these data are reflective of instructional practices of classrooms mainly in the United States during the 1970s. (See Confrey, 1986, for a discussion of these instructional practices.)

Mathematics research related to children with LD has its roots in the information-processing paradigm, which continues to dominate mathematics research. Specifically, the operations of addition, subtraction, multiplication, and division are thought of as hierarchically ordered linear skills that are acquired by transmission from a more knowledgeable other (e.g., Brophy, 1996). (See Baroody, 1987, for further explanation.) Hofmeister (1993) summarized research discussing instructional implications in mathematics problem-solving consistent with the strategies defined by Brophy and Good (1986) and Brophy (1986). Specifically, effective problem-solving methodology includes (a) teaching domain-specific problem-solving skills; (b) using systematic instruction to teach the generalization of problem-solving; (c) investing considerable time in explicitly taught strategies and

Learning Disability Quarterly 44



practice to develop practical problem-solving skills; and (d) integrating the teaching of other content in a given domain such as knowledge of computations to teach problem-solving strategies.

Currently, much attention is being directed to fos- tering higher cognitive functioning in mathematics, which is inclusive of big ideas, linkage between operations, depth of understanding, and problem-solving (Carnine, 1993). Big ideas (Carnine, 1993, 1997; Dixon, 1994; Dixon & Carnine, 1994; Kameenui & Carnine, 1998) are defined as "those concepts, principles, or heuristics that facilitate the most efficient and broadest acquisition of knowledge ... keys that unlock a content area for a broad range of diverse learners" (Kameenui & Carnine, 1998, p. 8). According to Carnine and his colleagues, big ideas, or the underly- ing problem schemata, should be emphasized rather than overloading students with vocabulary words and formulae. Big ideas are found in the four basic operations (addition, subtraction, multiplication and division), place value, fractions, estimation, probability, volume and area, and word problem-solving. The importance of identifying and teaching problem schemata in the problem-solving process has gained considerable support (see Jitendra & Xin, 1997, for a literature review).

Linkages between operations, a second focus related to higher cognitive functioning, become more clear and comprehensible through the use of big ideas while enhancing the possibility for generalization. Such a knowledge linkage results in a knowledge base that is rich and hierarchically organized. Depth of understanding, the third focus, can be further facilitated by emphasizing declarative knowledge (what), procedural knowledge (how), and conditional knowledge (when). Finally, problem-solving, the last of the four focuses defined by Carnine, is dependent on deeply understood big ideas that are seen as interconnected. Research examining these four focus areas presents what appears to be convincing evidence that they lead to significant progress in mathematics for children with LD (e.g., Bley & Thornton, 1995; Jitendra & Xin, 1997; Kameenui & Carnine, 1998). Constructivism

What is most unique about constructivist theory is the belief that specific, hierarchically ordered skills that are isolated from the complex, logical orders of the adult are not the building blocks of mathematical knowledge. Rather, the four operations of addition, subtraction, multiplication, and division are first and foremost forms and structures of mental activity (Piaget, 1965, 1987a, 1987b) that are "entirely built up from the coordinations of action-schemes and from the ensuing, coherent, deductive modes of reasoning ...

(Sinclair, 1990, p. 27). The coordination of action schemes is a biological process, fueled by the self-regulated activity of anticipating possibilities to solve meaningful problems and altering those actions as they are acted upon and evaluated.

Real-life activities embedded in cultural practices serve as stimuli to generate mathematics problem-solving (Saxe, 1988), while the exercise of this logic in reality enables the truth of one's actions to be validated in a transforming reality (Sinclair, 1990). Informal knowledge, defined as applied, real-life circumstantial knowledge that is drawn upon in the context of familiar, real-life situations (Mack, 1990), has been referred to as "partitioning" (Mack, 1990; Lamon, 1996; Pothier & Sawada, 1983). Lamon (1996) defined partitioning as an operation that generates quantity. As an intuitive, experienced-based activity, it anchors the construction of rational number to a child's informal knowledge about fair sharing.

Unitizing is a complementary cognitive process to partitioning, whose purpose is to conceptualize the amount of a given commodity or share (i.e., "size chunks" of the parts that are subsumed as a whole) before, during, or after the sharing process. Young children do not act on object quantities using reversible structures of operational thought due to a lack of coordination between the parts to each other and to the whole and the simultaneous ability to deduce composite unit structures (i.e., unitizing) from the elements contained within those parts. Specifically, when children initially practice counting, they reflect upon each element that precedes a count as a quantity independent of what is currently under consideration rather than considering the amount as inclusive of all elements that precede it. Over time, this activity expands, thereby enabling the abstraction of successive subgroups and, later, reversible structures of operational logic. The evolution of operational structures enables each composite unit to be included in each successive composite unit, such that the groups are compared successively (e.g., Chapman, 1988; Kamii, 1985, 1994; Piaget, 1965). Conserving addition is thus associative because it conserves the sum of all parts as well as the elements within each of the parts (Piaget, Coll, & Marti, 1987). Specifically, differences between parts result only from the displacement of elements that are considered in the process of changes in distribution.

For example, Sarah and Jeff are given a series of problems that relate to a florist preparing for a wedding. Sarah's responses to the following problem are indicative of additive thought structures. The problem states, "Every time the flowers are taken from the garden, new gardens are created that are bigger




than the last one. For every daisy that he picks [12], six seeds are added. Now, how many seeds will he need to replace the daisies that he picked for your wreath?"

E- "What are you doing Sarah?" (She is writing number problems.)

S- ".. . I'm just taking, I added 6 and 6 together to make 12. And 12 plus 6 more was 18. And 18 plus 6 more was 24 and 24 plus 6 is 30. And 30 plus 6 is 36. And 36 plus 6 is 42. And I still got a little more to do."

As children evaluate self-generated possibilities in tasks such as sharing, dealing, magnifying, and creating multiple sets of equal groups, their structures evolve into an elementary form of distributive reasoning referred to as a "splitting structure" (Confrey, 1994; Confrey & Smith, 1995; Steffe, 1992). In a splitting structure, children have awareness of a contained unit by creating a unit of units that is iterated (i.e., distrib- uted) across groups. This repeated action is preserved by reinitializing (i.e., the process of treating the product of a splitting structure as a basis for the reap- plication of that process) where the origin is always 1. The unit is, therefore, the invariant relationship between a predecessor and a successor in a sequence that is formed by the repeated action. The early stages of the evolution of this structure are additive because the unit iterated is not simultaneously coordinated with (i.e., inclusive of) the unit(s) that it precedes or follows. Thus, children reflect upon the whole as separate parts added together rather than simultaneously anticipating the inclusive relationship between the elements, parts, and whole.

In the previous problem, Sarah repeats a unit of 6, 12 times, in which the unit to be iterated was defined in the problem. However, in the problem to follow that allows the opportunity to define an iterated unit, Sarah demonstrates behaviors indicative of transitional multiplicative structures by defining a splitting structure. This problem followed a similar problem (later provided) in which Sarah determines how many daisies there would be if one third of the 36 flowers were daisies. In that problem, she is initially unable to define a unit to be iterated. Further, in that problem, as well as the one detailed below, both Sarah and Jeff first halve the total amount. The splitting structure of two is a powerful split (Confrey, 1994; Lamon, 1996) that children intu- itively prefer before its relationship to equality is fully understood (Pothier & Sawada, 1983).

The problem detailed below states, "The remaining flowers will be tulips. One fourth of the tulips will be large and the remaining flowers will be small. They can be any color. How many large and small tulips will she have?" Sarah subtracts 12 from 36 for the answer of 24. In the meantime, Jeff tries a series of addition problems

(12+12=24; 12+19=31; 12+24=36). After the last problem, I stop and ask how many tulips are needed to which Jeff responds that he doesn't know. Sarah explains to Jeff, "I took 36 and I took away 12 ... because we already used 12 of 36 flowers and it leaves 24." When I ask Jeff why he thought he has 12+24=36, he states, "Instead of going backwards and taking away I went forward and I added." On the second part of the problem, Sarah's solution differs significantly from that of Jeff.

S- "Can we use the tulips? It will be easier. Now how many tulips do we have? Oh, 24."

E- "That's an important question. So Sarah, what are we doing here?" (eff is working on a number problem without the tulips.)

S- "I'm counting 24 flowers out . . . This one should be more easier 'cause you did that one up there. I know more." (Sarah is referring to the problem later detailed about daisies.)

Initially, Sarah appears to be distributing the flowers into groups of four, but then she adds a fifth group. After laying them out, I ask how she will figure what one fourth is.

S- "We're going to have to halve them first." E- "Did it help us before with the daisies to halve

them first?" S- "Yes." E- "Did it? Ok, I see you're putting them into two

equal groups. That tells us what one half is, but do we want one half?"

S- "I'm halving them into 1, 2, 3, 4, 5, 6; 1, 2, 3, 4, 5, 6; 1, 2, 3, 4, 5, 6."

E- "Ok, so what do you have now?" S- "I have, ok, we have 24 and we're going to take

them and half them into one fourth." E- "Ok, well, first we put them into groups of half.

But we didn't really want groups of half. What did we want? We wanted to put them into?"

S- "Four." E- "Four equal groups. So we really didn't have to

do the halving." As children act on and evaluate their thinking activ-

ity in meaningful learning situations, structures expand to such a degree that they are reorganized into second-degree, nested relationships and are referred to as multiplicative structures. In contrast to additive thought structures where composite unit groups are compared successively across one level of abstraction, the distributive property relates multiples of the same composite units as nested orders such that all of the composite units are considered simultaneously. Over time, these structures again reorganize onto higher-order levels, enabling the learner to coordinate and differentiate the complex relationships




inherent in ratio and proportion. Thus, a student progresses from reflecting on a whole number as a single, composite entity (i.e., addition and subtraction) to a whole that is composed of multiple entities nested within each other (see Chapman, 1988; Clark & Kamii, 1996; Kamii, 1985, 1994; Lamon, 1996; Piaget, 1965, 1987a, 1987b; Steffe, 1988, 1992, 1994; Vergnaud, 1982, 1983, 1988, for more specific characteristics of these logical-mathematical structures of thought).

Jeff's responses to the number of seeds to be planted in the garden are indicative of multiplicative thought structures. Because he is consistently using more sophis- ticated reasoning throughout the lesson, I increase the problem difficulty: "For every daisy that is picked, 6 seeds are added. For every large tulip that is picked, 4 seeds are planted, and for every small tulip that is picked, 2 seeds are planted. How many seeds will the gardener need to replace all of the flowers he picked?"

J- "How many flowers are there?" E- "We had 6 large tulips, 18 small tulips, and we

had 12 daisies" . . . J- "Six times 4 is 24. Twenty-four seeds that are

tulips will be planted." E- "Ok, but he wants to know what will he need to

replace all of the flowers he picked." J- "Ok, there's 24." (He correctly multiplies in the

remaining two problems and then adds for the grand total.) "You're going to ask me how I came out with the answer ... That takes 6 flowers but if you lost 6, the seeds it takes is 4. So take 6 x 4 that's 24. There was 12 daisies. For each daisy there was 6 seeds in its place. Twelve times 6 that's 72. Then I took 18 flowers times 2 is 36. You add them altogether, you have 132."

Jeff responds to the tulip problem by simultaneously considering the nested relationships inherent in the problem, thus not being dependent on the flowers as his derives his solution.

E- ".. . Jeff, why don't you show us what you did?" J- "I wrote 4 into 24 and got 6." E- (To Sarah) "Why do you think Jeff did this num-

ber problem?" S- "Because if you take 24 and divide 4." E- "Divide it into four equal groups." S- "Four equal groups and you end up with 6 in

each group." Both students add groups of 6 to determine how

many large and small tulips the woman needs in all, although Jeff's ability to anticipate the nested relationships is noted in his justification, "Ok, 6 x 4, 24/4 = 6." Compare this reasoning to Sarah's justification, "Because it's a lot easier to count by 6s." Because she is ordering the relationships across one level of abstraction, her attention is directed to successive addition of

the splitting unit. Thus, Sarah requires many opportunities to reflect on her actions at the level she is on before her structures reorganize on a higher-order level. This transformation in mental structuring activity is necessary to abstract nested composite unit structures.

While Jeff is abstracting nested composite unit structures from the object quantities, the two-step problem to follow brings to light problems to be overcome while reflecting upon his thinking within this level prior to coordinating the higher-order, nested relationships of ratio and proportion. In the first step of a problem where Jeff determines the number of flower rings each worker would make if there were 8 workers and a total of 160 rings, he has minimal difficulty. Specifically, he comes up with the number problem 8 x 20 = 160 as a result of some trial and error and my questioning of one of his initial solutions (2 x 80 = 160).

E- "All right, tell me what you did here." J- "I took, 'cause there was 8 workers. I thought 9,

8 times what number. I know it had to be a double number would make 160. So the first one I guessed, I took 20 because it's right after 10. Well, 8 x 0 is 0, 8 x 2 is 16. When I multi- plied, it came out to 160."

E- "Ok, is there a different operation that you could have used other than multiplication so that you wouldn't have to guess that number here? Given the information I gave you, is there a different way to look at this?" (He then wrote 160/8 = 20.)

When I ask how many flower rings each worker would make over five days, he needs to have the problem repeated. Even with attempts to facilitate his problem-solving by going over the relevant facts, Jeff is unsure of how to coordinate the multiple nested hier- archies that exist as a result of introducing the second part of the problem. Also explicitly clear is Jeff's inability to reflect upon the quotient as a ratio unit (i.e., 4/1=20/5 in which both the numerator and the denom- inator are the result of a split that comprises a single invariant unit). Specifically, although Jeff acts on the divisor as a splitting structure, the quotient is derived by experimenting with quantities that can be iterated five times until the correct total is arrived at. Thus, the quotient is the product of how many parts, as independent units, are necessary to comprise the whole in which the whole, although nested into second-degree orders, is not yet coordinated as a ratio unit.

E- "How are you doing Jeff?" J- "Confused ... Each time I do this, I come up a

lot short. If I count by 5s in 4 days, they make 20. If I count by 2s I only get 10 in 5 days ... I counted by 3s, I get to 15 in 5 days ... I counted by, that's all."




E- "You got 2s, 3s, 5s, what are you missing?" J- "Fours." (He used his fingers to count.) "Four, 8,

12, 16, 20. Ok." E- "So how much will they have to make a day?" J- "Four." E- "Ok. Instead of guessing by counting by groups,

what operation could you have used?" J- "I could have divided 20 into 160."

E- "Twenty into 160? Why 20 into 160?" J- "You would have had a little bit of trouble divid-

ing 20 into 160." E- "Well, you're starting with a total of how many

wreaths for each worker?" J- "Twenty." E- "And what do you want to know about that 20?" J- "Oh, divide it by 5, you get 4."




Thus, logical-mathematical thought consists of the evolution of increasingly complex forms and structures of mental activity that evolve over time through the coordinations and differentiations of children's thinking activity. The child's logical orders vary significantly from those of the adult such that the complexity of adult logic cannot be assimilated by the child. In the section to follow, the implications of teaching techniques relative to how each position defines mathematical knowledge are discussed. Differences between the two theoretical positions in the section just discussed, as well as the two to follow, are summarized in Table 1.

INTERACTION OF THE LEARNER WITH THE CURRICULUM

Information Processing The learner is generally considered to be naive due to

his or her lack of awareness of the skills, strategies, and routines necessary for task performance as well as the use of self-regulated behavior (Engelmann, Carnine, & Steely, 1991; Miller & Mercer, 1997). By presenting stimuli to learners (e.g., explanation, example, rein- forcement) changes occur in them (Engelmann et al., 1991). These changes consist of learning ideas, relationships, or concepts that the teacher is attempting to convey. "Whatever learning is induced will be a function of the teacher's efforts. When instruction is viewed in this manner, the nature of what the teacher does to communicate concepts and relationships becomes extremely important. The learner is trying to extract qualities from what the teacher says" (p. 294). Because the learner "extracts" thinking qualities from the teacher, learning is greatly facilitated by the replication of sameness both in the strategies students use and in the content of the curriculum that students interact with (Carnine, 1993, 1997). Knowledge that is externally well organized and created by conspicuous strategies designed to teach big ideas also serves as a means to facilitate student memory (Kameenui & Carnine, 1998).

Explicitly taught self-regulation skills additionally help students better determine what a given task is asking of them and to select the appropriate problem- solving sequence. Specifically, strategies are practiced for the purpose of making the student increasingly responsible for recruiting and applying the strategies effectively. As these skills improve, the learner can better monitor, evaluate, and revise strategies while learning (Case, Harris, & Graham, 1992; Dixon, 1994; Englert, Tarrant, & Mariage, 1992; Hutchinson, 1993; Jitendra & Hoff, 1996; Mercer et al., 1994; Miller & Mercer, 1997). Students who lack awareness of the skills, strategies, and resources necessary to perform

tasks, and who fail to use self-regulatory mechanisms to complete tasks, have trouble with mathematics (Miller & Mercer, 1997).

To enhance the process of developing higher cognitive functioning in mathematics, it is imperative that teachers find opportunities to dialogue with students with special attention to developing problem schemata (i.e., big ideas). The quality of this dialogue should be prescriptive and will vary in content and nature depending on the specific needs of students. General techniques incorporated in this dialogue include (a) questioning; (b) providing corrective feedback; (c) encouraging; (d) reflecting with the student; (e) setting goals; (f) discussing a rationale for learning new declarative or procedural knowledge; and (g) discussing transfer (Mercer et al., 1994). Due to their prescriptive nature, questions posed to children tend to inform the "more knowing other" how well the child understands the logical orders of the presenter so that procedural errors can be corrected. Scaffolding from a more knowing peer who guides the students with procedures to solve algorithms is also useful to some students (Hutchinson, 1993). In peer-tutoring situations, high- ability students in mathematics were found to be more effective tutors for students with LD than average- ability students (Fuchs et al., 1996).

In a review of recent research of mathematical word problem-solving instruction for students with mild disabilities (Jitendra & Xin, 1997), instructional techniques delivered by the teacher or researcher were cate- gorized as representational techniques, strategy-training procedures, and task variations. The representational techniques included pictorial (e.g., diagramming), verbal (e.g., analogies and metaphors) or physical (e.g., manipulatives) aids. Strategy-training procedures refer to "specific heuristics" that promote particular task performance (p. 423) and are both cognitive and metacognitive. Finally, task variations consisted of the manipulation of word problem-solving tasks such that the easier tasks were taught before the more difficult ones.

Jitendra and Xin (1997) found that representational techniques are effective only when students are assist- ed with identifying relationships among key components of the problem. They further commented that in considering the use of manipulatives to aid instruction, the teacher needs to balance effectiveness with efficiency "because the challenge for students with disabilities and at-risk students is to catch up with their normally achieving peers" (p. 435)-a contention Carnine (1991) would agree with. Although the use of manipulatives is in question, there is evidence that intervention that combines cognitive and metacognitive self-regulation procedures appears to better assist




students' learning (e.g., Case et al., 1992; Jitendra & Hoff, 1996). However, Jitendra and Xin (1997) stated that their summary findings of studies reviewed are limited due to the lack of studies to support the effectiveness of various instructional strategies. Further, in most of the studies, the instructional procedures were implemented by the researcher and not the teacher.

What is additionally missing in the above study review is the use of real-life problem situations to give students a purpose for writing algorithms. Such practice is contrary to the recommendation that algorithmic-driven instructional materials must be replaced with instruction that "springs from authentic and purposeful contexts" (Mercer et al., 1994, p. 292). In other words, there must be an authentic problem that needs to be solved first and then an equation to solve the problem (Englert et al., 1992). The neglect of these studies to work from an authentic base reinforces the lack of attention to the quality of students' organizing activity while perceiving the replication of fragmented "pieces of information" from the logical orders of the more knowing other as primary to the "acquisition" of mathematical skills.

Constructivism As discussed, the empowering source of logical-

mathematical knowledge has as its foundation children's self-regulated, biologically based structures of organizing activity. Logical knowledge evolves by its coordination (assimilation) and differentiation (accommodation) of forms and structures of organizing activity as puzzlements are resolved. Thus, understanding the learner's organizing activity is primary in the learning process. In fact, children's construction of mathematical logic is advanced by studying their assimilatory operations, goals, and intentions, the quality of the activity generated to achieve such aims, and the results of their activity (Steffe, 1994). Language and objects serve only as tools to constrain and to guide children's thinking when these mediums assist with the process of encouraging reflections relative to what children's forms and structures of organizing activity can support (e.g., von Glasersfeld, 1990). Ultimately, it is children's available mental operations that impose constraints both on what can be placed in the "zones of potential construction" and on situations that can be presented to them (Steffe, 1992, p. 261).

The pedagogical challenge becomes one of facilitat- ing children's "reflective abstractions from, and progressive mathematization of, their initially situated activity" (Cobb, Yackel, & Wood, 1992, p. 23). To do so, the teacher, together with the students, makes sense of mathematics. Thus, the teacher engages in continuous cycles of negotiations among students'

understandings of the lesson, their view of students' thinking and learning, and their knowledge of mathematics. How a lesson should proceed during instruction depends on what is occurring in that moment (Bauersfeld, 1995; Sherin, 1997; Sherin, Mendez, & Louis, 1997). A critical element of successful negotiations is questioning that encourages the activity of thinking and the elicitation of the coordination and differentiation of thought patterns that typify developmental levels sensitive to children (Inhelder, Sinclair, & Bovet, 1974). This technique has been referred to as "graded learning loops" (see Pascual- Leone, 1976, for more description of this technique).

Working with peers in small groups to solve mathematics problems followed by a teacher-facilitated discussion of the strategies used by individual groups is also effective in creating perturbations necessary to expand structures of thought (Yackel, Cobb, & Wood, 1991). While in small groups, the students derive solutions individually followed by an explanation of this solution to their partner. The class discussion that follows may stir further conflict. Because puzzlement is so necessary to anticipating possibilities, students who are homogeneous in their logical constructions work together better in pairs than students for whom differences are greater (Yackel et al., 1991).

Self-regulation thus exists as a result of the innate, biological need for mental structuring activity to seek aliment to resolve disequilibration and maintains itself as accommodations are made by the expansion and eventual reorganization of structures to neutralize disturbances. Because logical-mathematical structures expand in the process of seeking solutions, new puzzlements will arise that again require greater adaptive behavior to be resolved (Piaget, 1971, 1980, 1985; Steffe, 1994). The more complex the possibilities that are anticipated, the more "intelligent" the activity of the child. If children's current ways of understanding do not give rise to problematic situations to be resolved, there is no need for existing cognitive structures to expand and reorganize. To solve a problem, one must first see the problem (Cobb, Wood, & Yackel, 1991; von Glasersfeld, 1989, 1995). Hatano (1988) argued that familiar types of problems suppress the cognitive incon- gruity necessary to produce puzzlement, thus reducing the need to seek comprehension beyond a defined algorithmic procedure.

I would also add that without differences to reflect on, practice serves the purpose of memorizing dupli- cated procedures of others rather than the expansion and refinement of one's own thinking activity by evaluating and modifying actions on objects. Errors that reflect efforts to work through a problem are of tremendous value because they serve to make teachers




and students aware of thinking patterns and problems to be thought through again and overcome (Bovet, 1981; Duckworth, 1996). Errors also indicate when instruction is reaching beyond the child's "zone of potential construction" (Steffe, 1992). Specifically, when child-ren require external focus on task elements to maintain their mental attention, it signifies a lack of conceptualization of the mathematical task and its principles in relation to what the child is being required to do (Sinclair & Sinclair, 1986). The effect of narrowing the content focus (e.g., big ideas) to direct the student to explicitly defined means to meet a pre- determined goal results in "context- and problem-specific routines and skills rather than insight, self-confi- dence, flexible strategies, and autonomy" (Bauersfeld, 1988, pp. 37-38). If we continue to stress a child in this type of situation, errors will reflect the increasingly fragmented thinking and will serve no constructive purpose. In turn, children's attention to tasks will be decreased as they become increasingly overwhelmed. "[T]he child will become discouraged and lose concen- tration and all spontaneous curiosity. The equilibrium of the child's schemes-be they practical, representa- tive, or conceptual schemes-is likely to be upset at all levels" (Bovet, 1981, p. 6).

When questions, and the errors children make relative to these questions, serve the purpose of understanding children's logical orders as they interact with a task, the hierarchical skill sequence that is deemed so necessary to the construction of mathematical knowledge becomes increasingly less relevant. Specifically, as a teacher worked with the content of children's authentic experiences as a base for algorithmic instruction in a regular classroom, the hierarchical nature of knowledge became less important (Fennema, Franke, Carpenter, & Carey, 1993). Further, much vari- ability in children's thinking replaced the standard procedures dictated by adult logic (Fennema et al., 1993; Schifter, 1997). Schifter (1997) additionally observed that when students applied memorized algorithms whose procedures lacked meaning to them, they stopped reflecting on the number quantities and the nature of the operations used to represent their problems. Such are the dynamics of specific skill instruction; it creates forms of knowledge that tend to be static in nature (Cobb et al., 1992) due to the sepa- ration of learning activity from cognitive activity.

When working with meaningful real-life problems, objects genuine to the situation under discussion should be available to children. Such objects serve as a source of reflection for children as they act on and evaluate their actions on object quantities. If children are shown how to use manipulatives to solve problems and/or if manipulatives are used to overtly direct

children's learning activity in a predestined means- ends manner, we are deprived of the investigative knowledge of how children organize the problem. At the same time, we severely restrict learning by limit- ing children's self-evaluation of their actions necessary to create and resolve puzzlement (Baroody, 1996; Duckworth, 1996; Inhelder et al., 1974). As children's logical structures evolve, they become less dependent on manipulatives due to the complex nature of the composite unit structures that they have constructed to generate possibilities. Further, as children's organizing activity becomes increasingly more nested over time, their ability to engage in mental reflection (with or without objects) over a sustained period of time and to consider the solution of another relative to one's own thinking also increases (Piaget, 1987a, 1987b).

The protocols to follow are examples of the interaction of Jeff and Sarah with the curriculum according to constructivist tenets of learning. The problem discussed was the first one given to Sarah and Jeff. Although I initially encouraged them to work together, as their heterogeneous grouping became more evident, Jeff was given more complex problems to better engage his mental reflections. Particular elements to attend to in the protocols include (a) the use of questions to pro- voke puzzlement and the limitations imposed on creating puzzlement by sharing solutions in a heteroge- nous group; (b) the effect of anticipating solutions, testing hypotheses, and evaluating the consequences of one's actions in relation to self-regulation; (c) the use of student knowing as the source for algorithms; and (d) Sarah's preference for using manipulatives as a tool for reflection. The problem states, "A woman is planning a wedding and she wants to decorate the house with flower rings. She wants a total of 36 flowers on each ring. One third of the flowers have to be daisies. How many daisies will she have? If she wants twice as many blue flowers [daisies] as white and yellow flowers [daisies], how many flowers will she have?" The two questions were presented separately.

Similar to the tulip problem (previously detailed) that followed this problem, Sarah needs to work with the flowers to think about the problem. Contrary to the tulip problem, she has no plan of attack that makes sense to her as she reflects on her solution. Specifically, she initially wants to add or subtract, being unable to conceptualize the nested, second-order composite unit structures.

S- "Is it like adding or taking away?" E- "Well, you're taking one third of 36." S- "Ok, you're not adding." E- "Right, you're not adding. Would we be adding

to 36 if we're taking one third of 36?"




S- "No." E- "Ok, we're taking a part of 36. The thing is what

part of 36? That's the question." (She is stumped.) "Do you think it might be helpful just to count out 36 flowers?"

S- "That's just what I was thinking of doing." Jeff tries unsuccessfully to write fraction problems

and then decides he wants to take half of 36, "Cause if I could find out half of 36, then from there I could find out one third of 36." After taking half of 36, he realizes he is at another dead end.

E- "Look at what you did to get one half. So what are you going to do to get one third?"

J- "Divide 3 into 36?" E- "Think so? Why don't you try it?"

Sarah continues to be perplexed by the problem. Jeff joins her after solving the problem above, although nei- ther can decide exactly how to section the flowers, even with Jeff's algorithm. Sarah's anticipations about the next step depicts her difficulty in abstracting the nested, complex composite unit structures of the problem.

E- "Ok, we have 36 here. Now what did he do with the 36 if he wanted to find one third? Do you know what you did? Jeff, can you show me with the flowers what you did?"

S- "He probably did, maybe add or take 3 of them away?"

E- "Ok, when you divide by 3 what do you do? Do you take 3 away? What does that mean when you divide by 3?"

J- "When you divide, it makes the numbers small- er. It's like simplifying."

Initially they divide the flowers into half. Upon agreeing that they don't have one third, Jeff offers that you have to "divide by 3 and come up with 12." He divides the piles.

E- "What do you think Jeff's doing here, Sarah?" S- "He's dividing." E- "So what is he sectioning the flowers into? Are

you putting them into halves?" J- "Putting them into thirds." E- "Does that make sense that we're putting them

into thirds?" S- "Oh, ok. I get it now. I have to catch up to you." E- "Is this one third? Where's our other third?"

(Sarah took 3 from a group.) S- "The other third is over here." E- "Well, are 3 equal thirds here?" J- "No." E- "So show me 36 divided into thirds. I don't have

3 equal thirds here." (Jeff and Sarah both count 12 in each group.)

E-"Does that make sense here? Thirty-six divided by 3 is 12?"

J- "I guess so." For the second part of the problem, Jeff quickly

derives the solution without the use of the flowers to help. "The white must be, both will be 3, and then blue must be 6. 'Cause 3+3=6 and 6+6=12." However, Sarah feels the problem is very difficult, so I reduce its complexity.

E- "Why don't we look for 6 blue daisies ... If she wants an equal amount of yellow and white flowers, we're starting with a total of 12 and we have 6 blue flowers. Now if she wants an equal number of yellow and white flowers, how many will she have that are yellow and how many flowers will she have that are white?"

S- "She'll probably have 3 white flowers and 3 yellow flowers."

When asked to write number problems to show their thinking, both Jeff and Sarah have problems. Jeff initially writes 6/3, then erases it. He asks if fractions could be used and I tell him to use what he wants to show his thinking. My attention then turns to Sarah who is struggling. She attempts to explain her thinking verbally without writing a number fact.

S- "Ok, 12 and I took 6 out of it." E- "Ok, so show me your number problem where

you took 6 out of that. What wouldyour number problem be?"

S- "I would have 6 left." E- "Ok, so show me with your number problem

where you took 6 out of that. What would your number problem be?"

S- "I would have 6 left." E- "Ok, but show me how you got 6 left." S- "I took away." E- "Ok, so show me taking away." S- "Take away. I took away 6 out of 12. And that

left 6. And then I took, I have 6 of them to make 12. 'Cause that would half that up and that would make 3."

E- "Ok, so half of 6 is 3. How did you show half of 6 is 3? How would you show that with a number problem?"

S- "I'd probably take away 3 ... I'd take away 3 and that would leave out of 6. I'd take away 3 and that makes 3. So altogether you have 6 blue flowers, 3 white flowers, and 3 yellow flowers." (She writes her number problem vertically as 12-6=6-3=3.)

E- Ok, now you told me you would have half of 6 and you didn't know how to write your number problem to show me half of 6. Instead you did subtraction. Can you rethink that number problem to show me half of 6?"

S- "Half of 6?" (She groans.)




At this point, I ask Jeff to assist, and he explains his completed solution to us, "I did 2x3, got 6 and then 2x6 and got 12." Note that Jeff has two separate solutions rather than acting on the whole as a starting point and breaking down its nested composite unit structures in relation to one, inclusive whole.

E- "Sarah's having trouble where she said half of 6 and she's not sure what number problem she would use to show half of 6. What would you do?"

S- "Half of 6. Three plus 3." J- "Well, you could do... " (Was hesitating.) E- "What about division?" S- "Take away?" J- "No, 2 times, 2 divided by 6 is 3."

Unfortunately, because there has been so much resistance to and/or misunderstanding of constructivist principles in the field of LD, there has been limited research on the type of logical structures that underlie children's problem-solving. In the research that has investigated the quality of composite unit structures abstracted by children with LD (Grobecker, 1997, 1998), these children generally displayed less coordinated structures than their same-aged peers (e.g., using additive thought structures versus multiplicative thought structures). Grobecker (1997, 1998) argued that the structures of logical-mathematical activity in children with LD lack the degree of coordinations in their mental structuring activity necessary to assimilate the same type of higher-order composite unit structures in problems compared to their peers with NLD. Thus, if children cannot "remember" mathematics facts, lack strategies, or show unusual error strategies (e.g., Carnine, 1997; Case et al., 1992; Geary, 1993; Ginsburg, 1997), their organizing activity is simply not yet coordinated enough to reflect on those equations in a way that makes them meaningful. The children depicted in this article, as well as previous research (e.g., Swanson, 1993; Wansart, 1990), demonstrate that children with LD are capable of adapting their thinking when engaged in problem-solving situations appropriate to their logical orders. If there are gaps between skills traditionally taught and the quality of operational structures available to deduce logical relationships, then children with LD are given less opportunity or the means to construct stronger schemata (Baroody & Ginsburg, 1986, 1991).

For example, Sarah's replacement teacher stated that Sarah has difficulty remembering the operational processes of multiplication and division, thus requiring much practice and repetition. I would argue that Sarah is being pushed to exceed in tasks that she is not able to meaningfully reflect upon, thereby creating the

need for sameness in routines such that her learning activity is mechanical and routine. The goal of intel- lectual education, which is learning to master the truth by oneself at the risk of taking the time necessary for meaningful learning activity (Piaget, 1973), is thus abandoned. As a result, Sarah is removed from the powers of her self-regulated activity that logical-mathematical knowledge is embedded in. The division algorithms Sarah is being explicitly taught were never used to solve problems, even with her reflections on Jeff's division algorithm in the tulip problem. Thus, even though the same language was used, Sarah reflected upon the problem differently than Jeff, and Jeff reflected upon the problem differently than I.

Similar dynamics were observed in an investigation of the mathematics performance of slower learners or low-achievers in a mathematics reform classroom (Murphy, 1997). Specifically, in response to the question, "To what extent do you feel that slower learners or low-achievers benefit from an emphasis on alternative solution strategies," the teachers interviewed were divided, their responses ranging from some benefit to minimal benefit. The benefits included increased self-esteem as a result of explaining solutions to the class and a decrease in pressure due to the acceptance of different solutions. Those who felt that the children benefited only minimally stated that the alternatives confused them and that they were better off with only one explicitly taught strategy.

The latter teacher comments could support the contentions of many researchers that some children may best benefit from explicit instruction. However, because of the previously discussed problems related to such a practice, I am suggesting a different alternative. Specifically, we should consider replacement in mathematics as children's errors become increasingly more random and nonconstructive while simultaneously becoming increasingly more reliant on explicit instruction to complete a task. With replacement, the teacher could adjust her questions and the nature of authentic problems posed to the level of children's unique constructions so that they can reflect on their own understandings. Most likely, there would be other students in the class with similar thought structures that they could be paired with to enable healthy conflict rather than the shared frustration observed by Yackel et al. (1991). Specifically, shared frustration was observed when students worked with peers whose logical-mathematical constructions were significantly more advanced than theirs. Had I not altered the problems for Jeff and Sarah and separated them to work on their own solutions, both would have become mentally disengaged; Jeff would be bored and Sarah would be frustrated.




Teachers also need to understand the developmental evolution of children's forms and structures of mental activity consistent with constructivist tenets. I am concerned that the teachers in Murphy's (1997) study believed that explicitly taught strategies would be more helpful than altering the quality of the problems and questions posed to students. Such a position calls into question the teachers' awareness of the importance of working with children's logical orders. Giordano (1993) proposed that special education teachers may support and want to adhere to traditional calculation-based programs, in part, because they may lack experience and expertise with other programs. Teachers have to believe that a particular way of teaching mathematics is going to bring about a change in a student's learning to use a methodology effectively in their classrooms (Allinder, 1996; Fennema et al., 1993; Grant, Peterson, & Shojgreen-Downer, 1996).

ASSESSMENT Information Processing

Several changes have been recommended in assessment as a result of efforts by researchers to incorporate principles perceived as constructivist into our current paradigm. For example, Bryant and Rivera (1997) identified five major areas of change in evaluation procedures, which include the use of (a) portfolios, (b) criterion-referenced tests, (c) curriculum-based measurement, (d) calculation-error analysis using structured interviews and checklists/rating scales, and (e) error analysis in word problems. An emphasis on these five areas reflects, to some degree, the NCTM Assessment Standards for School Mathematics (1995), which include (a) comparing each student's performance with specific performance criteria and away from student comparisons; (b) sharing responsibility for performance; and (c) reaching beyond the analysis of right/wrong answers.

Unfortunately, the performance criteria described by Bryant and Rivera (1997) also include attention to rate of performance and the mastering of specific skills and strategies. These learning characteristics may be deemed necessary for efficient learning in children with LD due to the perceived need to catch children with LD up to their peers (e.g., Jitendra & Xin, 1997), as well as the belief that mathematics knowledge consists of acquiring specific skills and procedures as discussed. Examination of these learning characteristics is an alteration of the NCTM Assessment Standards (1995) because attention is focused on students' abilities to effectively regulate and reproduce explicitly taught strategies.

Also, the use of pre-/posttest designs to examine the effectiveness of trained cognitive and metacognitive

strategies (Case et al., 1992; Jitendra & Hoff, 1996) is not in the spirit of the reform. Specifically, such teaching methodologies most often neglect children's logical orders while emphasizing the replicated thinking of another. Although strategy training has demonstrated improved performance in tasks that are similar to those that the child has been instructed in (e.g., Jitendra & Hoff, 1996), generalization of this information in situations that vary significantly from the context in which strategies are taught is question- able. Generalization is the outcome of working with children's reflections that have as their source their dynamic, evolving structures of mental activity.

Because computation and word problem-solving are believed to measure two distinct abilities that can be captured on standardized testing, much credence continues to be given to standardized tests such as the Woodcock-Johnson Tests of Achievement (Revised) (WJTA- R) and the Kaufman Tests of Educational Achievement (KTEA). However, Parmar, Frazita, and Cawley (1996) found that when the classroom performance of fifth- grade students was compared to their performance on the WJTA-R, the two did not match. They argued that much caution is necessary when interpreting the outcomes of these tests because performance on these tests is not consistent with student performance in the classroom. Further, the limitations of standardized measures to make "equitable" judgments that "reflect the ways in which [students'] unique qualities influ- ence how they learn mathematics and how they communicate that knowledge" (NCTM, 1995, p. 15) are debatable (see also Baroody, 1987). Constructivism

The ability to capture the individual complexities of learning is limited by comparisons to a "group mind" (Meltzer & Reid, 1994). Through the process of standardization, psychometric tests become superficial tools because extremely complex and dynamic learning activity acquires a rote, mechanistic nature (Sawada & Caley, 1985). Further, because the adult observes a fragment of behavior removed from children's logical orders, the nature of their deep, transformative, organizing activity is left undiscovered both by the adult and by the children.

When performance on tasks allowing for observations of children's logical constructions of pre- operational, additive, and multiplicative thought structures was compared to performance on the calculation and applied problems of the WJTA-R, dis- crepancies in thinking between the two measures were evident for children both with and without LD (Grobecker, 1997, 1998). Specifically, students with additive thought structures often provided correct answers to multiplication facts and/or word problems on a measure




that is assumed to measure multiplicative reasoning. The think-aloud protocols on the WJTA-R revealed that the thinking of all children consisted of the recall of highly similar strategies and routines to "solve" problems. Further, the children with LD achieved well within one standard deviation of the mean on both calculation and applied problems even though they were delayed in their logical-mathematical constructions compared to their peers with NLD. Thus, although children perform well on what is taught to them, we need to question exactly what it is we are teaching them as well as the manner in which current assessment tools mask the weaknesses of such techniques.

In light of research that is uncovering the highly complex nature of logical-mathematical thought, the isolation of specific skills from their dynamic activity for the purpose of evaluation can be called into question. If mathematics is construed in terms of evolving structures of organizing activity, then what needs attending to is the quality of children's thinking structures when operating on number quantities. Such observations will serve to define appropriate materials and questions to extend children's logical orders. For example, Sarah's logical structures can be best engaged and, thus, expanded by providing real-life problems that have the potential for her to abstract nested, second-order, composite unit structures. Using her reflections as a source for puzzlements that become known as she acts on objects and expresses her logic in various representations that include number facts, her mental structuring activity will reorganize itself into multiplicative structures over time. Working with a peer whose structures are at a similar level will further stimulate the puzzlement necessary for the expansion of logical-mathematical structures.

Jeff, on the other hand, often uses a haphazard, trial-and-error algorithmic approach even though his logical-mathematical structures can support more concise reflections. Such errors typify the effect of teaching algorithms apart from children's logical orders that support the symbols used. Presenting opportunities to refine his problem-solving approach by linking more of his thinking activity with the algorithms written to express such activity would be beneficial (e.g., Mack, 1990, 1993; Post, Cramer, Behr, Lesh, & Harel, 1993).

To expand Jeff's logical-mathematical structures, his thinking needs to be challenged in complex, two-step problems where it is necessary to compare the nested relationships contained within two wholes that are coordinated with each other. Also, opportunities to reflect upon complex, nested composite unit structures within a single whole would appropriately

challenge Jeff's thinking. As Jeff uses his logical activity to solve problems that have meaning to him, his solutions will become more flexible and sophisti- cated and he will come to think of rational numbers as existing as single, invariant entities (i.e., ratios) over time (Mack, 1990; Post et al., 1993). It is essential that the teacher attends to the logical orders of Jeff and Sarah as the starting point for all questions and techniques to follow. If Jeff and Sarah are scaffolded to perform beyond what they can intelligently reflect upon, adaptable learning will decrease while the need for sameness in routines will increase.

CONCLUSION The purpose of this article was to differentiate the

philosophical perspectives of the information- processing and constructivist theories in the area of logical-mathematical knowledge. It was argued that although elements of constructivist theory are begin- ning to be explored with children with LD, they are being significantly distorted by information-processing assumptions about learning that we hold to be true. As a result, new insights that could broaden and extend our understanding of how to help these children have not been given appropriate opportunity for growth and expansion.

To fully appreciate what the constructivist perspective has to offer, we have to reconsider our beliefs about what constitutes cognition and learning and the relationship between these two mental processes. As long as we continue to define mathematical learning as hierarchically ordered specific skills that exist external to the learner and that need to be acquired by transmission from a more knowledgeable other, our tools and methodologies will be supported as valid. But if we are willing to consider logical-mathematical knowledge as biologically based dynamic forms and structures of mental activity that are internally constructed and reorganized at higher levels as children seek solutions to meaningful problems, then our teaching methodologies and assessment tools need significant modification.

A fair and more accurate investigation of the tenets of the mathematics reform consistent with constructivist philosophy demands that we: (a) educate ourselves and teachers on the developmental evolution of logical- mathematical structures of thought and the tenets of constructivist philosophy in general; (b) design methodologies that investigate the nature of logical-mathematical activity in children with LD; (c) test the effectiveness of teaching methodologies consistent with constructivist tenets; and (d) reconsider the value of current assessment tools and the race to "catch children up" to a standardized norm.




REFERENCES Allinder, R. (1996). An examination of the relationship

between teacher efficacy and curriculum-based measurement and student achievement. Remedial and Special Education, 16, 247-254.

Baroody, A. J. (1987). Children's mathematical thinking. New York: Teacher's College.

Baroody, A. J. (1996). An investigative approach to the mathematics instruction of children classified as learning disabled. In D. K. Reid, W. P. Hresko, & H. L. Swanson (Eds.), Cognition approaches to learning disabilities (3rd ed., pp. 545-615). Austin, TX: PRO-ED.

Baroody, A. J., & Ginsburg, H. P. (1986). The relationship between initial meaningful and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 75-112). Hillsdale, NJ: Erlbaum.

Baroody, A. J., & Ginsburg, H. P. (1991). A cognitive approach to assessing the mathematical difficulties of children labeled as learning disabled. In H. L. Swanson (Ed.), Handbook on the assessment of learning disabilities: Theory, research, and practice (pp. 177- 228). Austin, TX: PRO-ED.

Bauersfeld, H. (1988). Interaction, construction, and knowledge: Alternative perspectives for mathematics education. In D. A. Grouws, R. J. Cooney, & D. Jones (Eds.), Perspectives on research on effective mathematics teaching (pp. 27-46). Reston, VA: National Council of Teachers of Mathematics.

Bauersfeld, H. (1995). The structuring of the structures: Development and function of mathematizing as a social practice. In L. P. Steffe & J. Gale (Eds.), Constructivism in education (pp. 137-158). Hillsdale, NJ: Erlbaum.

Bley, N. S., & Thornton, C. A. (1995). Teaching mathematics to students with learning disabilities. Austin, TX: PRO-ED.

Bovet, M. C. (1981). Learning research within Piagetian lines. In P. Axelrod & D. Elkind (Eds.), Topics in learning and learning disabilities (Vol. 1) (pp. 1-9). Rockville, MD: Aspen Systems Corporation.

Brophy, A. J. (1986). Teaching and learning mathematics: Where research should be going. Journal for Research in Mathematics Education, 17, 323-346.

Brophy, J., & Good, T. L. (1986). Teacher behavior and student achievement. In M. C. Wittrock (Ed.), Handbook of research on teaching, 3rd ed. (pp. 328-375). New York: MacMillian.

Bryant, B. R., & Rivera, D. P. (1997). Educational assessment of mathematics skills and abilities. Journal of Learning Disabilities, 30, 57-68.

Capra, F. (1982). The turning point: Science, society and the rising culture. New York: Bantam Books.

Carnine, D. (1991). Curricular interventions for teaching higher order thinking to all students: Introduction to the special series. Journal of Learning Disabilities, 24, 261-269.

Carnine, D. W. (1993, October). Effective teaching for higher cognitive functioning. Educational Technology, 29-33.

Carnine, C. W. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141.

Case, L. P., Harris, K. R., & Graham, S. (1992). Improving the mathematical problem-solving skills of students with learning disabilities: Self-regulated strategy development. The Journal of Special Education, 26, 1-19.

Chapman, M. (1988). Constructive evolution: Origins and development of Piaget's thought. New York: Cambridge University Press.

Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5. Journal for Research in Mathematics Education, 27, 41-51.

Cobb, P., Wood, T., & Yackel, E. (1991). Analogies from the philosophy and sociology of science for understanding classroom life. Science Education, 75, 23-44.

Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2-33.

Confrey, J. (1986). A critique of teacher effectiveness research in mathematics education. Journal for Research in Mathematics Education, 17, 347-360.

Confrey, J. (1994). Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 293-332). Albany: SUNY Press.

Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26, 66-86.

Dixon, B. (1994). Research guidelines for selecting mathematics curriculum. Effective School Practices, 13, 47-55.

Dixon, R., & Carnine, D. (1994). Ideologies, practices, and their implications for special education. The Journal of Special Education, 28, 356-367.

Duckworth, E. (1996). The having of wonderful ideas and other essays on teaching and learning. New York: Teachers College Press.

Engelmann, S., Carnine, D., & Steely, D. G. (1991). Making connections in mathematics. Journal of Learning Disabilities, 24, 292-303.

Englert, C. S., Tarrant, K. L., & Mariage, T. V. (1992). Defining and redefining instructional practice in special education: Perspectives on good teaching. Teacher Education and Special Education, 15, 62-86.

Fennema, E., Franke, M. L., Carpenter, T. P., & Carey, D. A. (1993). Using children's mathematical knowledge in instruction. American Educational Research Journal, 30, 555-583.

Fuchs, L. S., Fuchs, D., Karns, K., Hamlett, C. L., Dutka, S., & Katzaroff, M. (1996). The relation between student ability and the quality and effectiveness of explanations. American Educational Research Journal, 33, 631-664.

Geary, D. C. (1993). Mathematical disabilities: Cognitive, neu- ropsychological, and genetic components. Psychological Bulletin, 114, 345-362.

Ginsburg, H. P. (1997). Mathematics learning disabilities: A view from developmental psychology. Journal of Learning Disabilities, 30, 20-33.

Giordano, G. (1993). Fourth invited response: The NCTM standards: A consideration of the benefits. Remedial and Special Education, 14, 28-32.

Grant, S. G., Peterson, P. L., & Shojgreen-Downer, A. (1996). Learning to teach mathematics in the context of systematic reform. American Educational Research Journal, 33, 509-541.

Grobecker, B. (1997). Partitioning and unitizing in children with learning differences. Learning Disability Quarterly, 20, 317-335.

Grobecker, B. (1998). The evolution of proportional structures in children with and without learning differences. Poster session presented at the 1998 American Educational Research Association, San Diego, CA.

Hatano, G. (1988). Social and motivational basis for mathematical understanding. In G. B. Saxe & M. Gearhart (Eds.), Children's mathematics (pp. 55-70). San Francisco: Jossey-Bass.

Hofmeister, A. M. (1993). Elitism and reform in school mathematics. Remedial and Special Education, 14, 8-13.

Hutchinson, N. L. (1993). Second invited response: Students with disabilities and mathematics education reform-Let the dialogue begin. Remedial and Special Education, 14, 20-23.




Inhelder, B., Sinclair, H., & Bovet, M. (1974). Learning and the development of cognition. London: Routledge & Kegan Paul.

Jitendra, A. K., & Hoff, D. (1996). The effects of schema-based instruction on the mathematical word-problem-solving performance of students with learning disabilities. Journal of Learning Disabilities, 29, 422-431.

Jitendra, A., & Xin, Y. P. (1997). Mathematics word-problem- solving instruction for students with mild disabilities and students at risk for math failure: A research synthesis. The Journal of Special Education, 30, 412-438.

Kamii, C. (1985). Young children reinvent arithmetic: Implications of Piaget's theory. New York: Teachers College Press.

Kamii, C. (1994). Young children continue to reinvent arithmetic: Implications of Piaget's theory. New York: Teachers College Press.

Kameenui, E. J., & Carnine, E. W. (1998). Effective teaching strategies that accommodate diverse learners. Upper Saddle River, NJ: Merrill.

Kuhn, T. S. (1962). The structure of scientific revolutions. Chicago: University of Chicago Press.

Lamon, S. J. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal for Research in Mathematics Education, 27, 170-193.

Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, 16-32.

Mack, N. K. (1993). Learning rational numbers with understanding: The case of informal knowledge. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 85-105). Hillsdale, NJ: Erlbaum.

Meltzer, L., & Reid, D. K. (1994). New directions in the assessment of students with special needs: The shift toward a constructivist approach. The Journal of Special Education, 28, 338-355.

Mercer, C. D., Harris, C. A., & Miller, S. P. (1993). First invited response: Reforming reforms in mathematics. Remedial and Special Education, 14, 14-19.

Mercer, C. D., Jordan, L., & Miller, S. P. (1994). Implications of constructivism for teaching math to students with moderate to mild disabilities. The Journal of Special Education, 28, 290-306.

Miller, S. P., & Mercer, C. D. (1997). Educational aspects of mathematics disabilities. Journal of Learning Disabilities, 30, 48-56.

Murphy, L. A. (1997). Math reform in practice for higher- and lower-achieving children: Perspectives of children, parents, and teachers. Paper presented at the American Education Research Association, Chicago.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). Assessment standards for teaching mathematics. Reston, VA: Author.

Parmar, R. S., Frazita, R., & Cawley, J. F. (1996). Mathematics assessment for students with mild disabilities: An exploration of content validity. Learning Disability Quarterly, 19, 127-136.

Pascual-Leone, J. (1976). On learning and development, Piagetian style: A reply to Lafebvre-Pinard. Canadian Psychological Review, 17, 270-288.

Piaget, J. (1971). Biology and knowledge. Chicago: University of Chicago Press.

Piaget, J. (1973). To understand is to invent: The future of education. New York: Grossman.

Piaget, J. (1965). The child's conception of number. New York: W. W. Norton & Co.

Piaget, J. (1980). Adaption and intelligence: Organic intelligence and phenocopy. Chicago: University of Chicago Press.

Piaget, J. (1985). The equilibration of cognitive structures. Chicago: University of Chicago Press.

Piaget, J. (1987a). Possibility and necessity (vol. 1): The role ofpos- sibility in cognitive development. Minneapolis: University of Minnesota Press.

Piaget, J. (1987b). Possibility and necessity (vol. 2): The role of necessity in cognitive development. Minneapolis: University of Minnesota Press.

Piaget, J., Coll, C., & Marti, E. (1987). Associativity of lengths. In J. Piaget (Ed.), Possibility and necessity: The role of necessity in cognitive development (pp. 62-77). Minneapolis: University of Minnesota Press.

Post, T. R., Cramer, K. A., Behr, M., Lesh, R., & Harel, G. (1993). Curriculum implications of research on the learning, teaching, and assessing of rational number concepts. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 327-362). Hillsdale, NJ: Erlbaum.

Pothier, Y., & Sawada, D. (1983). Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14, 307-317.

Rivera, D. M. (1993). Third invited response: Examining mathematics reform and the implications for students with mathematics disabilities. Remedial and Special Education, 14, 24-27.

Saxe, G. B. (1988). The mathematics of street vendors. Child Development, 59, 1415-1425.

Sawada, P., & Caley, M. T. (1985). Dissipative structures: New metaphor for becoming in education. Educational Researcher, 14, 13-19.

Schifter, D. (1997). Developing operation sense as a foundation for algebra. Paper presented at the American Educational Research Association, Chicago.

Schoenfeld, A. H. (1994). A discourse on methods. Journal for Research in Mathematics Education, 25, 697-710.

Sherin, M. G. (1997). When teaching becomes learning. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago.

Sherin, M. G., Mendez, E. P., & Louis, D. A. (1997). A discipline apart: Mathematics as a challenge for FCL teachers. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago.

Sinclair, H. (1990). Learning: The interactive reaction of knowledge. In L. P. Steffe & T. Wood (Eds.), Transforming children's mathematics education: International perspectives (pp. 19-29). Hillsdale, NJ: Erlbaum.

Sinclair, H., & Sinclair, A. (1986). Children's mastery of written numerals and the construction of basic number concepts. In J. A. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 59-74). Hillsdale, NJ: Erlbaum.

Steffe, L. (1988). Children's construction of number sequences and multiplying schemes. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (vol. 2) (pp. 119-140). Reston, VA: The National Council of Teachers of mathematics.

Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4, 259-309.

Steffe, L. (1994). Children's multiplying schemes. In G. Harel &J. Confrey (Eds.), The development of multiplicative schemes in the learning of mathematics (pp. 3-39). Albany: State University of New York.

Swanson, H. L. (1993). An information processing analysis of learning disabled children's problem solving. American Educational Research Journal, 30, 861-893.

Vergnaud, G. (1982). A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 39-59). Hillsdale, NJ: Erlbaum.




Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 128-175). New York: Academic Press.

Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (vol. 2) (pp. 141-161). Reston, VA: National Council of Teachers of Mathematics, Inc.

von Glasersfeld, E. (1989). Cognition, construction of knowledge, and teaching. Synthese, 80, 121-140.

von Glasersfeld, E. (1990). Environment and communication. In L. P. Steffe & T. Wood (Eds.), Transforming children's mathematics education: International perspectives (pp. 30-38). Hillsdale, NJ: Erlbaum.

von Glasersfeld, E. (1995). A constructivist approach to teaching. In L. P. Steffe & J Gale (Eds.), Constructivism in education (pp. 3-15). Hillsdale, NJ: Erlbaum.

Wansart, W. L. (1990). Learning to solve a problem: A micro- analysis of the solution strategies of children with learing disabilities. Joural of Learning Disabilities, 23, 164-170.

Yackel, E., Cobb, P., & Wood, T. (1991). Small-group interac- tions as a source of learning opportunities in second-grade mathematics. Journal for Research in Mathematics Education, 22, 390- 408.

Notes The author would like to thank Dr. Lilliand Land, Coordinator

of Testing, Federal, and Kindergarten Programs, and Diane Simmons, Special Education Coordinator, Tallapoosa County Schools, for their support in the data collection.

Requests for reprints should be addressed to: Betsey Grobecker, Curriculum and Teaching, Auburn University, AL 36849.


I



Documents

Mathematics Reform and Learning Differences