Betsey Grobecker
Auburn University
Address for correspondence:
Department of Curriculum and Teaching
5040 Haley Center
Auburn University
Auburn, AL 36849-5212
E-Mail: grobebe@mail.auburn.edu
Abstract:
To adequately address the real complexity of learning problems, the field of special education needs to reassess what constitutes cognition and learning and the relationship between these two mental processes. With specific regard to the field of mathematics, our current tools and methodologies will be supported as valid 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 “transmitted” from a more knowing other. If however, we are willing to consider logical-mathematical knowledge as consisting of 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.
Piaget has contributed significantly to our understanding of the evolution of cognitive forms and structures of mental activity that are consistent with the principles of modern physics. For example, analogies between Prigogine’s work on “dissipative structures” and Piaget’s work on cognitive equilibria are evident: (a) Piaget conceived of cognitive structures as embodying dynamic equilibria that include interchanges with the outside; (b) these interchanges stabilize the structures through regulation; (c) equilibrium, in both cases, is characterized by a form of self-regulation; (d) the states, which have passed through a series of unstable states, are only understood on the basis of their past history; and (e) system stability is a function of its complexity (Piaget & Garcia, 1989).
Thus, Piaget conceived of cognitive structures as open systems of energy exchanges within and between systems. At the same time, structures must be closed (i.e., reversible) so that the elements under consideration can be related to each other and to the whole simultaneously. In fact, the evolution of operational structures involves a tension between the opening (synthesis) and closure (differentiation) of energetic patterns of activity that act to both conserve and transform operational structures (Piaget, 1980, 1985). Equilibration is the regulator of activity within and between individual and environmental energies. Specifically, as schemes extend and reorganize their form through a biological drive to assimilate the world, disturbances in the way reality is defined emerge. Such disturbance creates a psychic tension to resolve the conflict. If the newly emerging structures have gained enough stability, the disturbance will be confronted and reality redefined. However, if the structures do not have the degree of stability necessary to open to this disturbance without complete dissipation of their form and/or one cannot maintain the psychic tension necessary to sustain the process of change, neither reality nor schemes will be transformed. Thus, transformations in individual and environmental systems (i.e., symbol systems and institutions in society) do not have their source in separate isolated systems, but in the interactive relationship between these two forces (Piaget, 1995).
Unfortunately, research in learning disabilities (LD) has been dominated by reductionist theory. It follows that the assumptions that undermine what constitutes learning difficulties are flawed by the lack of attention to the complex, evolving, biological forms and structures of mental activity as well as the interdependent psychological forces that generate the means used to solve problems and the way those means are evaluated (Grobecker, 1996, 1998; Poplin, 1988; Heshusius, 1989; Reid, 1988; Thelen, 1989, 1990). As a result, children’s vital activity that gives meaning to symbols, as well as the interaction and transformation of this activity with environmental forces, are virtually ignored. The spontaneous nature of mental activity whose form takes shape through the processes of order and disorder that continually give way to each other within and between systems is, thus, given a static nature. Specifically, mental activity becomes governed by actions that break down dynamic activity into rigid, predetermined steps to achieve mechanical goals. As a result, we have created static, hierarchical steps that are removed from the complexity of human activity to explain what constitutes “normal” and atypical development.
Thelen (1989, 1990) argued that there is a need to rethink linear, unidimensional models of development that are so dominant if the study of learning disabilities. According to Thelen, there would be great benefit from studying human differences from the perspective of “open systems” of dynamic energy (i.e., systems composed of heterogeneous subelements that change over time). To do so, consideration has to be given to interactions within and between individual energy systems and their interaction with all other energy systems (e.g., the physical environment, society, etc.). Guess and Sailor (1993) claimed that integrating such a perspective with the study of learning differences presents both opportunities and challenges to: (a) define new directions in the conduct of research; (b) restructure the questions asked; and (c) provide a transition from empirical findings to practical applications.
This article contrasts the systems view of cognitive development to that of the information-processing perspective specific to the area of mathematics. The developmental evolution of logical-mathematical structures is then detailed, which includes protocols of students with LD and without LD (NLD). Implications for future research are summarized. [1]
Overview of Contrasting Theoretical Positions
Disabilities in mathematics are currently identified by a discrepancy between an intelligence quotient and achievement on a standardized mathematics test such as the Woodcock-Johnson Tests of Achievement (Revised) (WJTA-R). In other words, one has potential to acquire knowledge in mathematics but progress is hindered by a specific disability that may include: (a) fact retrieval and problems memorizing mathematics; (b) procedural deficits (e.g., counting strategies to solve problems); and/or (c) problems with spatial representation such as not being able to put numbers in the correct column (Geary, 1993).
This discrepancy formula is predicated on the assumption that the operations of addition, subtraction, multiplication, and division exist as hierarchically ordered linear skills that are independent of each other and acquired by “transmission” from a more knowledgeable other (Hofmiester, 1993). Because the learner has no preknowledge of concepts, a necessity is created to clearly explain and reinforce concepts while focusing student attention to what is relevant in tasks so that changes in them occur (Engelmann, Carnine, & Steely, 1991). In effect, the learner is attempting to extract qualities from what the teacher says, which is facilitated by the replication of sameness in strategies students use as well as in the content of the curriculum that students interact with (Carnine, 1993, 1997). Explicitly taught self-regulation skills that are practiced enable the learner to better monitor, evaluate, and revise strategies while learning (Case, Harris, & Graham, 1992; Dixon, 1994; Englert, Tarrant, & Mariage, 1992; Hutchinson, 1993; Jitendra & Hoff, 1996; Mercer, Jordan, & Miller, 1994; Miller & Mercer, 1997). This teaching is facilitated by posing questions to inform the teacher how well the child understands the logical orders of the presenter so that procedural errors can be corrected (see Grobecker, in press, for an extended discussion of this position).
In contrast to the information-processing perspective of mathematics development, constructivist theory holds that 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 coordination of action-schemes and from the ensuing, coherent, deductive modes of reasoning . . .” (Sinclair, 1990, p. 27). The coordinations of action schemes, as a biological process, are 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 enables the truth of one’s actions to be validated in a transforming reality (Sinclair, 1990).
Due to the fact that logical-mathematical knowledge evolves by its coordination (assimilation) and differentiation (accommodation) of forms and structures of organizing activity as puzzlements are resolved, 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 activity generated to achieve such aims, and the results of their activity (Steffe, 1994). The pedagogical challenge becomes one of facilitating children’s ”reflective abstractions from, and progressive mathematization of, their initially situated activity” (Cobb, Yackel, & Wood, 1992, p. 23). A critical element in the facilitation of these reflections 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). 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 mental activity can support (von Glasersfeld, 1990). Ultimately, children’s available mental operations impose constraints both on what can be placed in the “zones of potential construction” and on situations that can be presented to them (Steefe, 1992).
When children require external focus to 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 predetermined goal results in “context- and problem-specific routines and skills rather than insight, self-confidence, 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. “The child will become discouraged and lose concentration and all spontaneous curiosity. The equilibrium of the child’s schemes—be they practical, representative, or conceptual schemes—is likely to be upset at all levels” (Bovet, 1981, p. 6).
Thus, it is possible that mathematics “disabilities” are created by stretching children’s minds to perform operational processes that they are unable to reflect upon intelligently. When learning has, as its starting point, the child’s logical orders and self-regulation of his or her learning activity, a specific skill disability cannot exist independent of cognitive structures. The discrepancy formula has survived because tests of intelligence fail to reflect the inherent biological and psychological activity that organizes the world through anticipated perception (Grobecker, 1998). As such, intelligence quotients support the stimulus-response frame of reference which continues as the prevailing view in behavioral science in America (Bond, 1995). The process of standardization in general (including standardized tests of mathematics) has created superficial tools because extremely complex and dynamic learning activity acquires a rote, mechanistic nature by its dependence on a “normal” curve (Sawada & Caley, 1985). In other words, the adult observes a fragment of behavior removed from children’s logical orders such that their deep, transformative organizing activity is left undiscovered both by the adult and the children.
I have argued that, from a systems approach to development, the source of learning problems lies in the equilibration cycle in that the spiral of mental activity in LD children is not as expansive as that of their same-aged peers as it winds itself upward and outward by the exercise of its forms (Grobecker, 1996, 1998). This qualitative difference in the equilibration cycle in children with LD creates a vulnerability in their system that could easily magnify and disrupt all developmental levels to follow. For example, their systems may experience an excessive degree of agitation when structures are dissipating their forms for reorganization onto higher-order levels. As a result, cognitive systems may favor a return to their initial state rather than sustaining the tension necessary to reorganize their structures onto higher-order levels (Grobecker, 1998; Thelen, 1989). Such periods of system reorganizations have been described as particularly sensitive for children with LD (Reid & Stone, 1991). Thus, the cognitive systems of these children are less complex, preventing them from acting on and transforming material forms using age-related higher-order structures. Further, because the nature of the equilibration cycle is evolving and generative, their cognitive systems limit the number of interactions entering into the systems that can be coordinated with other schemes.
Psychic tensions necessary to create and sustain the disturbance for change in structures of mental activity can also be overly stressed due to emotional forces encountered when energy is extended into the world. Children’s vital activity is charged with emotion and the tension it calls out to give it form in reality creates the activity of reflection (Dewey, 1934/1980). Significant stressors (e.g., frustration and discouragement) encountered when constructing meaning create a lack of desire to extend one’s energy into the world resulting in qualitatively different structures (Piaget, 1972; Thelen, 1989, 1990). Specifically, the child’s system creates a need to set up rigid boundaries in its structures to keep negative influences out. In turn, openness to perturbations is decreased. Without perturbations, no tensions are created to stimulate a need to solve problems and evaluate one’s actions on objects. And without meaningful problems to be resolved, there is minimal exercise of cognitive activity, which enables the expansion and growth between subsystems and the system as a whole (Grobecker, 1998).
Due to the fact that the development of cognitive systems is stochastic in its nature and intertwined with highly complex psychic forces, their evolution in each individual are unpredictable. Further, the degree of disorganization in mental structuring activity that affects children’s ability to engage in reflective abstraction when asked to infer meaning to problems presented varies along a continuum. Also varying along a continuum is the degree of psychic strength and personal will necessary to sustain the tension involved with the process of change.
To investigate this premise specific to the area of mathematics, we need to better understand the evolution logical-mathematical structures in children with NLD as well as developmental trends in LD children. This understanding can only be gained by activities that provide the opportunity to observe children’s self-generated biological activity (i.e., mental reflections) when engaged in activities that allow for the investigation of means-ends consequences as thinking is formulated (Grobecker, 1998). The task that follows is an example of such an activity and provides insight into the accompanying description regarding the development of logical-mathematical structures in children with and without LD.
The Evolution of Logical-Mathematical Structures
The task children participated in was designed by Sinclair (Piaget, Grize, Szeminiska, & Bang, 1977), and modified by Clark & Kamii (1996). In this study, children in grades 2, and 4 - 7 with LD and NLD were provided with three fish in which the second and third fish were two and three times the size of the first fish respectively (Grobecker, in press). The children were told, “This fish (pointing to B) eats 2 times what this fish (pointing to A) eats, and this big fish (pointing to C) eats 3 times what the little one (pointing to A) eats.” After one fish receives chips, children determine the amount of chips for the remaining two fish. The five problems presented were: (a) 1 cheerio to A (1, 2, 3), (b) 4 cheerios to B (2, 4, 6), (c) 9 cheerios to C (3, 6, 9), (d) 4 cheerios to A, (4, 8, 12), and (e) 7 cheerios to A (7, 14, 21). If the child failed to independently group the cheerios they were asked, “Is there any particular way you could put the cheerios to show that this fish (B) eats two times what this fish (A) eats and this fish (C) eats 3 times what this fish (A) eats?” If the child answered problem three incorrectly (3, 6, 9) a counter suggestion with the correct solution was offered that included the rearrangement of the cheerios into the correct multiplicative groupings.
A description of all participating students is presented in Table 1. Four of the students in grade six were not classified as LD but were receiving basic skills instruction in mathematics. Approximately 84% of the classified students were receiving some type of special education intervention in mathematics. In Table 2, the number of students in each group achieving at the various developmental levels detailed below, are provided.
Table 1 | Mean ages, IQ scores, WRAT Reading and Woodcock-Johnson Mathematics Standard Scores, and Number of Males/Females for Each Student Group |
Children with NLD | |||||||
Grade | Mean Age | IQ | WRAT-R | Calculation | Word Prob. | M | F |
---|---|---|---|---|---|---|---|
Second | 7.8(0.4) | - | 107(9.2) | 100(0.0) | 113 (9.8) | 2 | 1 |
Fourth | 9.9(0.8) | 108(5.7) | 109(7.1) | 101(14.9) | 120(5.7) | 1 | 1 |
Fifth | 11.2(0.3) | 104(5.5) | 101(7.4) | 103(14.0) | 114(9.9) | 7 | 3 |
Sixth | 11.8(0.4) | 110(6.2) | 109(6.5) | 104(8.8) | 116(7.9) | 7 | 3 |
Seventh | 12.6(0.3) | 108(3.8) | 110(8.2) | 102(6.7) | 103(5.6) | 2 | 3 |
Children with LD | |||||||
Grade | Mean Age | IQ | WRAT-R | Calculation | Word Prob. | M | F |
Second |
7.8(0.7) |
96(9.9) |
93( 9.2) |
97(1.4) |
101(3.5) |
1 |
1 |
Fourth |
9.5(0.5) |
102(4.2) |
88( 2.8) |
104(0.0) |
99(10.6) |
1 |
1 |
Fifth |
11.1(0.4) |
94(3.7) |
83(11.6) |
94(15.5) |
96(8.4) |
6 |
2 |
Sixth |
11.8(0.5) |
104(8.7) |
92(13.9) |
90(7.0) |
105(7.5) |
9 |
2 |
Seventh |
12.5(0.3) |
103(9.4) |
99(5.5) |
94(3.7) |
108(14.1) |
5 |
1 |
Table 2 | Number (and Percent) of Students at Each Grade Level that Achieved at Each of the Seven Levels on the Fish Task by Grade Level for Each Student Group |
Children with NLD | |||||||
Fish Level | |||||||
Grade | I | II | III | IV | V | VI | VII |
---|---|---|---|---|---|---|---|
Second |
1(33) | 2( 67) | |||||
Fourth |
2(100) | ||||||
Fifth |
5(50) | 4(40) | 1(10) | ||||
Sixth |
1(10) | 7(70) | 2(20) | ||||
Seventh |
1(20) | 1(20) | 3(60) | ||||
Children with LD | |||||||
Grade | I | II | III | IV | V | VI | VII |
Second |
1(50) |
1(50) |
|||||
Fourth |
2(100) |
||||||
Fifth |
2(25) |
2(25) |
3(38) |
1(12) |
|||
Sixth |
1( 9) |
2(18) |
4(37) |
3(27) |
1( 9) |
||
Seventh |
2(32) |
1(17) |
1(17) |
1(17) |
1(17) |
Young children’s desire to understand quantity is first noted in their attempts to associate a tag to object quantities. As 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. As such, they fail to act on object quantities using reversible structures of operational thought. Such thinking is due to a lack of coordination between the parts to each other and the whole and the simultaneous ability to deduce composite unit structures (i.e., unitizing) from the elements contained within those parts.
Through continued engagement in activities that provoke a need to exercise reflection, this activity expands thereby enabling the abstraction of successive subgroups and, later, reversible structures of operational logic. When logical-mathematical activity has evolved to additive structures of operational thought, children reflect on amount as a numerical composite whose parts are also composite units; that is, the sets that combine to form the larger set (Behr, Harel, Post, & Lesh, 1994; Lamon, 1994; Steffe, 1994). Because the composite units or sets are compared successively, the inclusion relations between the sets can only be represented on one level of abstraction (Clark & Kamii, 1996; Piaget, 1987a; 1987b). The two protocols that follow are examples of students transitioning into operational thought (level 1 - no serial correspondence or serial correspondence with qualitative quantification) and in the early stages of additive thought structures (level 2 - additive thinking with a numerical sequence of +1 or +2) respectively. Note that the child at level 2 experiences no puzzlement when the counter suggestion is given. This lack of puzzlement typified children’s reactions to the counter suggestion at these early developmental levels.
(Grade 5, LD ) For 4 to B, he gave C 7, changed the amount to 8, then gave A 2. _2, 4, 7_8_ “What do you have there?” "Eight." "You put 8 instead of 7?" "Yeah." "Why did you feed this Little Fish 2?" "Because it’s like almost its size." "Why?" "‘Cause it’s like almost its size and its stomach isn’t really that big." "It’s almost the same size?" "Yeah." "Why did you feed this Big Fish here 8?" "That one got more. I thought it could eat more. Maybe it could eat more. Yeah, it has a lot of room." "How many more did you give it?" "It ate one more. I gave it like about 3 more." "Three more?" "I thought like this one _B_ got 4 so I thought maybe it could get a little higher." "So you just made it higher?" "Yeah." "And you picked, how many more did you pick to decide to make it 8?" "Three more." He placed the cheerios randomly on and under the fish. "Why did you put them that way?" "Because it shows the size. It’s something small."
(Grade 2, NLD) For 9 to C she fed B 8 and A 7 _7, 8, 9_. She placed the cheerios in two uneven lines under B and one line under A. "OK, how come 8 for the Middle Fish?" "Because if this _C_ was 9 that would if that one _B_ would be 8. This would have to be 9 x it." "Ok, that makes it 9 x it?" "This _B_ would be 8 and I guess you would take away 1 because this fish _C_ eats 3 and this _B_ eats 2, and this eats 1." "Ok, and how come you decided 7 for that fish _A_?" "Because if this _B_ was 8 and this _A_ would be 7 because this, this is 7 and this would be 7 x as much as 8." "Would you leave the cheerios like this under the big fish?" They are randomly distributed. "They’re alright." After the counter she responded "I wouldn’t do that." "Why not?" "Because 9, this would be 9, and then this _B_ would be 8 and this _A_ would be 7." "So you think she’s wrong then?" "Yes."
The new power of reversible structures enables children to coordinate more complex grouping relationships as they participate in tasks that involve sharing, dealing, magnifying, and creating multiple sets of equal groups. An elementary form of distributive reasoning gradually emerges and is referred to as a "splitting structure" (Confrey, 1994; Confrey & Smith, 1995; Steffe, 1992). In a splitting structure, children are aware of a contained unit by creating a unit of units that is iterated (i.e., distributed) 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 reapplication of that process) in which 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 now 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. The protocol of the child below is an example of level 3 thinking (additive thinking involving +2 for B and +3 for C), in which some initial attempts at the splitting structure are evident in the way she groups the cheerios.
(Grade 5, LD). For 4 to A, this student ended with the solution of 6 and 9 to B and C respectively _4, 6, 8_10_9_. "This one [B] would get 1, 2, 3, 4, 5, 6 and this one [C] would get 1, 2, 3, 4, 5, 6, 7, 8." "Why would the Middle Fish get 6?" "Because 4, the Middle Fish eats 2 x what this is [A] so I put 2 more here. And then this one [C] eats 3 times as much as the Little Fish does so 1, 2, 3, 4, 5, 6. Actually, it should be 3 more so this [B] would be 6, 6 here and 3 more. That would be 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So then it could be this one has 4, this eats 2 more so 4 add 2 and this one you put 6 and add 3 more it’s to. "Ok. So I see you have this is your 4 and 2 more here and what’s here?" "It’s my 6 and 3 more here." She took away 1 cheerio to make it 9. She has the cheerios in a group of 4 and a group of 2 under fish B. Under fish C is a group of 6 and a group of 3.
Continued efforts to resolve each newly created conflict enables structures to expand to such a degree that they are reorganized into second degree higher-order relationships and are referred to as multiplicative structures of thought. Advancement to this level involves the conceptual coordination of multiple composite units that are nested withing each other (Chapman, 1988; Clark & Kamii, 1996; Lamon, 1996; Piaget, 1987a, 1987b; Steffe, 1988, 1992, 1994; Vergnaud, 1983, 1994). In the fish task, if A=7 and B =14, the elements of 14 are coordinated as: (a) 14 (1 units), (b) 2 composite units each consisting of 7 1-units _2(7 units)_; and (c) 1 composite 2-unit consisting of 2 of the 3, 7 1-units _1(2(7 units))_ or 2/3 of 21 (e.g., Lamon, 1994). Thus, in multiplicative structures of thought, the distributive property relates multiples of the same composite units where all of the relationships are considered simultaneously. However, two transitional levels of behavior (4 and 5) were noted in the fish task prior to the abstraction of equivalent, nested composite unit structures across the fish.
At level 4 (transitional additive to multiplicative), students use the additive solution of level 3, but also use the logic of multiplicative thinking in at least one other problem. When children do consistently abstract second-order composite unit structures (level 5 - doubling), they direct much of their attention to the action of splitting by halving or doubling these units rather than distributing an equivalent unit across the fish n times as noted in the protocol to follow.
(Grade 6, LD). For 4 to A, he fed B 8 and C 16 _4, 8, 16_. "This one _B_ 8." "Why?" "Because you doubled the 4. And you give this one _C_ 16 ‘cause you double the 8." He put all the cheerios in one group under each fish. "Any way the cheerios should go under the fish?" He rearranged them into a row of 4 under A, two rows of 4 under B and two rows of 8 under C. "Why in rows like that?" "No particular reason." "So this fish [B_] eats 2 x what the Little Fish eats and this fish [A] eats 3 x what the Little Fish eats?" "Yes."
The great majority of students at this level liked the counter suggestion better. However, their justifications suggest that they attended mainly to the ease of counting and/or to the fact that the groups could be subtracted or divided evenly. Thus, there was no evidence that they assimilated the higher-order composite unit structures from the problem.
The student in the protocol to follow is transitioning into level 6 (correct problem solution but lack of logical consistency in grouping the cheerios). This student’s thinking provides significant insights into how the distributive-algebraic and the splitting-analytic processes are interdependent and support each other (Lamon, 1996). Specifically, for problem 4 (4, 8,12) she clearly thought about four higher-order composite units as noted by her reference to an "extra one" as a unit of four. However, she reflected on and modified her reasoning while acting on the splitting structure of 4.
(Grade 6, NLD) For 4 to A, she initially put out 8 to B and 16 to C and then changed C to 12 as she reflected on the problem _4, 8, 16_12_. At first, the cheerios were randomly placed. "So that’s _C_ 16 there?" "Yes." "How are you going to fix those?" She began to put the cheerios out in groups of 4. "Oh, wait, I have an extra one." She took away the extra 4 from C. "Why do you think you had 16 first?" "I quickly thought that I thought 4 x 4 =16." "You thought 4 x 4?. . . Why . . . ?" "It’s one of them that I know very well" . . . "Why do we have them the way you put them out?" "When you say that this one _A_ would get 4 then that one _B_ double this one _A_ and this one _C_ gets triple this one _A_."
The following student likes the grouping of the cheerios in the counter better. Although he tends to correctly group the cheerios after the counter, this grouping is preferred because it makes the cheerios easier to count and not due to a more coordinated abstraction of the problem components, which is similar to level 5 children. Such reasoning requires the evolution of proportional structures.
(Grade 6, LD). (3, 6, 9)"What do you think about his idea?" "It’s good." "Why?" "‘Cause he explained it nicely and he’s right." "Do you think your fish should have the cheerios like that or do you want to keep them like that?" The cheerios were randomly distributed and he rearranged them into groups of 3. "Do you think that’s better?" "Yes." "Why?" Because now they’re like all in rows and you can count easier." "Ok, how would you count?" "You count in rows of 3."
What differs between simpler multiplicative structures and more complex structures of proportional reasoning is that proportional reasoning involves an abstraction between two second-order relationships simultaneously rather than a relationship between two concrete objects or two directly perceivable quantities (Piaget & Inhelder, 1975; Lesh, Post, & Behr, 1988). Thus, the student cognitively coordinates multiples of different composite units (Lamon, 1944) and reflects upon the equation A/B = C/D as a dynamic transformation where the structural similarity on both sides of the equation is attended to. This coordinated abstraction between the two complex systems of relationships enables the child to change any element of an equation to compensate for a change in another element. Lamon (1994) referred to this process as using the same scalar operator (i.e., ratio) in both measure spaces of the equation such that ratio is invariant across situations (Confrey & Smith, 1995).
At the highest level scored (7) all problem solutions are correct as well as the groupings of the cheerios. The logic of students is concise in that they express only pertinent points. However, in the protocol of the student to follow, she believes that other groupings do not distort the problem. Thus, while she appears to have an intuitive sense of invariance of groupings across the fish, her structures of proportional reasoning were not yet fully evolved and stabilized.
(Grade 6, NLD) For 7 to A, she fed B and C 14 and 21 _7, 14, 21_ . . . "I see you have them in groups of 7. Would it make sense to group them any other way and still have this _B&C_ being 2 x and 3 x as big as this one _A_? "I guess you could group them in twos if you want to count in twos. But it would take a longer time than if they were in sevens." "If you did that, would that still be 2 x what this Little Fish eats?" "Yeah, they’re just grouped differently." Why don’t you try that?" These two _A&C_ wouldn’t work with this ‘cause they’re odd numbers and this one wouldn’t work because 14 is an even number." "What would your times problem be though, to show that this _B_ is 2x what this _A_ eats if you group them like that?" "I guess it wouldn’t cause it’s hard to find a way to make it so it looks like its 2 x bigger this one _B_ gets 2 _groups of 7_. ‘Cause these two _A & C_ are odd. So they can’t go in groups. Like they’re hard to chop.
No specific developmental anomalies in the protocols are evident in the children with LD (e.g., inability to generate strategies and regulate problem-solving). However, an ANCOVA conducted on the dependent variable of level achieved on the fish task using IQ as a covariate and group as a factor for children in grades 5 - 7 showed significant differences for group (F(1, 47) = 5.36, p < .02) with more NLD students achieving at the level of multiplicative structures. [2] On the WJTA-R, all of the children with LD achieved within one standard deviation of the mean even though classroom performance suggested otherwise. Further, 69% of the children in both groups who used additive thought structures in the fish task provided correct answers to one digit multiplication problems and 21% solved two-digit multiplication problems on the Calculation test of the WJTA-R. These findings are consistent with past research investigating additive and multiplicative structures (Grobecker, 1997) and spatial-thought structures (Grobecker & De Lisi) in children with LD.
The performance differences between the fish task and the standardized test, which measures the acquisition of "transmitted" facts, demonstrates that we can show children higher-level strategies that they can "learn" through repetition and practice of similar formulas and strategies as suggested by Carnine (1991, 1993, 1997) and Bley and Thornton (1995). However, there are negative consequences to this practice. For example, students in both groups achieving at the lower levels often used the word "times" but acted upon the fish problem using additive thought structures. Even the reorganization of the cheerios that was accompanied by a verbal explanation of the correct solutuion in the counter, failed to disturb the thinking of these children due to the limitations of their mental operations. Thus, while explicitly taught words and procedures resulted in the correct answers on tests that measure such teaching methodology, the degree of intelligent learning that occurs is questionable. Specifically, the transmission of explicitly taught skills removes thinking from children’s logical orders and the self-regulated, problem-solving behavior that enables children to anticipate, evaluate, and extend their biologically-based structures of organizing activity. As a result, knowing will not be generalized to problems whose format differs significantly from the context in which it was learned nor is there a purpose to mathematics beyond doing well in explicitly taught formal schooled tasks.
Psychologically, the limited intellectual purpose for task engagement results in decreased levels of motivation and levels of arousal. Specifically, the psychic tension that is intertwined with children’s vital logic is stultified (Dewey, 1933). The boundaries of a system have become too inflexible to open to new possibilities, thus preventing an extension of energy into the world to assimilate and reconstruct this observation. To open to this new possibility is to create a psychic tension to change; a notion that is too threatening to a system that has shut itself down to perpetuate the known. Thus, transformations in reality that result in deeper reflections and more thoughtful, flexible, adaptive actions are replaced with rigidity in routines out of a need for sameness and the accompanying fear of the new.
Most likely these dynamics have resulted in the "misguided" effort to teach these children self-regulated skills and to emphasize redundancy in practice that removes learning from children’s vital activity. Devoid of an assimilative base (Gallagher & Wansart, 1991) to guide reflections, errors serve no purpose to the child or the teacher. Further, these errors have the potential to show distortions from subject content as originally "transmitted" to children because the skill consisted of static bits of information in its conception. Thus, distortions that are labeled as disabilities, are not inherent solely in the learner, but are created in the relationship between the learner and the demands of the instructional process. Similarly, lack of motivation and task arousal are inherent in the quality of cyclical energy exchanges between the self and the social and cultural forces that the self interacts with and creates (or conversely, perpetuates).
Conclusion
To adequately address the real complexity of learning problems, the field of special education needs to reassess what constitutes cognition and learning and the relationship between these two mental processes. With specific regard to the field of mathematics, our current tools and methodologies will be supported as valid 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 "transmitted" from a more knowing other. If however, we are willing to consider logical-mathematical knowledge as consisting of 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 children’s difficulties in mathematics 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.
Notes
Acknowledgement
I would like to thank Dr. Jeff Gorrell, Associate Dean and Dr. Richard Kunkel, Dean, Graduate School of Education,for helping to fund the trip to Geneva.
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