Rigorous mathematical thinking: mediated learning and psychological tools.

Abstract

The proposed rigorous mathematical thinking paradigm is based on two theories of learning: Vygotsky's sociocultural theory, with particular emphasis on his concept of psychological tools as mediators of cognitive processes, and Feuerstein's theory of Mediated Learning Experience. We examine the role of higher-order mental processes and psychological tools in rigorous mathematical thinking and analyze empirical data on cognitive and academic performance outcomes of the cognitively based program aimed at producing conceptual change in the underachieving students' comprehension of the mathematics concept of function.

Introduction

The proposed rigorous mathematical thinking (RMT) paradigm consists of two major components. The first in the RMT approach (Kinard, 2001; Kinard & Falik, 1999) drives concept development and applications in mathematics, science, and technology education. The second component is an instructional practice that implements RMT principles in classroom instruction. The RMT theory and its instructional pedagogy are based on two theories of learning: Vygotsky's socio-cultural (Vygotsky and his early followers used the term cultural historical) theory, with particular emphasis on his concept of psychological tools as mediators of cognitive processes, and Feuerstein's (1990) theory of Mediated Learning Experience.

In this article we will examine the role of higher-order mental processes and psychological tools in rigorous mathematical thinking that promotes students' conceptual change in classroom learning situations. In addition, we will describe and discuss empirical data on cognitive and academic performance outcomes of the cognitively based program aimed at producing conceptual change in the students' comprehension of the mathematical concept of function.

Need for Rigorous Thought in Mathematics

There is deep concern in the U.S. that American mathematics and science education is falling well behind that of other industrialized societies. This is manifested in poor performance and low academic achievement in mathematics and science for the vast majority of America's students, generally compared with students in other Western and industrialized Asian countries (Schmidt, 1998; Hoff, 2000; Eisenkraft, 2001; Peak, 1996; TIMSS Policy Forum, 1997; Office of Educational Research and Improvement, U.S. Department of Education, 1997; Schmidt & Valverde, 1997; Schmidt, McNight, & Raizen, 1996). Mathematics and science education are seen as cornerstones of adequate functioning in a technological society. The lack of rigorous thinking and problem solving skills in students, particularly with reference to the content of instruction, is a frequently identified concern (Kinard & Falik, 1999). Simply learning calculations and mechanical processes, without understanding and manipulating the deeper structures of thinking, is clearly not sufficient for competence.

Pointing out what appears to be a lack of focus on rigorous thinking during mathematics instruction for young children, Bybee and Sund (1982, p. 255) state that teachers "have children adding and substracting before they even know the meaning of number." In a paper given at the 89th Annual Meeting of the National Council of Teachers of Mathematics (NCTM), Ball (2002) emphasizes the need for student engagement in rigorous mathematics concept development beginning at an early age. Ball also expresses the urgency for quality teacher professional development that will facilitate the nurturing of students through a rigorous mathematics-learning environment.

The need for rigorous thinking is clearly revealed in a study by Stigler and Hiebert (1997a, 1997b) of eighth-grade mathematics lessons in Germany, Japan, and the United States as part of the Third International Mathematics and Science Study (TIMSS). TIMSS data show that U.S. eighth grade students scored below their peers from 27 nations in mathematics and below their peers from 16 nations in science (Peak, 1996; TIMSS Policy Forum, 1997; Office of Educational Research and Improvement, U.S. Department of Education, 1997; Schmidt & Valverde, 1997; Schmidt et al., 1996). Japanese students scored well above German and U.S. students, while German students moderately out-performed U.S. students.

The average international level, however, is also far from adequate. It is telling that a simple TIMSS task of finding the speed of a car when a graph of the functional relationship between the distance and time is given was correctly solved by only 54.3% of the eighth grade students internationally (Smith, Martin, Mullis, & Kelly, 2000).

The findings by Stigler and Hiebert indicate that there are several important differences in the way problem solving is taught in American, German and Japanese classrooms. For example, U.S. and German teachers lead students through the solution of example problems, while Japanese teachers, in contrast, have each student invent his or her own solutions and reflect on the solutions in an effort to promote understanding. Japanese teachers are significantly more likely to provide explicit links or connections between different parts of the same lesson. Less than one-fifth of U.S. teachers develop concepts when they include them in the lesson compared to more than three-fourths of German and Japanese teachers. None of the U.S. lessons include proofs, while 9% of the German lessons and 51% of the Japanese lessons include proofs. Japanese students devote the majority of their working time to inventing new solutions and engage in conceptual thinking about mathematics. U.S. and German students spend more than 90% of their working time practicing routine procedures compared to 40% for Japanese students. Thus, it appears that instructional practice which emphasizes the rigor of conceptual formation, logical reasoning, and creativity is more effective than teaching approaches that focus on routine practice and memory.

It is sometimes claimed that a higher level of math achievment in such authoritarian cultures as Russian or Chinese is due to rote learning that stymies students' independence and creativity. The following anecdote directly challenges this explanation. When Andrey Toom, a Russian mathematician came to America he expected U.S. students to be strongly orientated toward independent, critical thinking. Instead he was astounded to find only a very small number of students who had experienced real thinking and problem solving. He observed, "never before had I seen so many young people in one place who were so reluctant to meet challenges and solve original problems." (Toom, 1993, p. 12). When teaching business calculus, he reported "many of them would be most satisfied if the teacher simply repeated and explained what was written in the textbook and that he (the teacher) should teach from the text and give exams based on the text or similar problems" (p. 12). Toom concluded from his experience that his students' primary concern was their grades rather than the quality and depth of thinking and learning they needed to experience.

Psychological Tools and Mediated Learning Experience

One of the starting points of the RMT approach is the claim that students' failure in mathematical problem solving often stems from the lack of more general cognitive skills rather than specific mathematical operations. It is not enough, however, just to recognize that the culprit is a general cognitive skill. Unlike traditional educational approaches as well as some of the constructivist theories that take the students' cognitive status as naturally given, we believe that such a status depends on education and can be modified by education. In this we follow a well-known Vygotskian thesis that education is a driving force of cognitive development (Vygotsky 1978; see also Kozulin, 1998).

In Vygotsky's sociocultural theory, cognitive development and learning are operationalized through the notion of psycholocial tools. Psychological tools first appear as external symbolic tools available in a given culture. Among the most ancient of these symbolic mediators, Vygotsky (1978, p. 127) mentioned "casting lots, tying knots, and counting fingers." Casting lots appears in a situation when the uncertainty of decision caused by the presence of two equally strong opposing stimuli is resolved by an application of the artificial and arbitrary tool--die. The individual links his or her decision to the answer given by a die, thus artifically resolving a situation that cannot be solved in a natural way. Tying knots exemplified the introduction of an elementary mnemonic device to ensure the retrieval of information from the memory. Finger counting demonstrates how an ever-present object (fingers) can serve as an external symbolic tool that organizes cognitive functions involved in elementary arithmetic operations. Cultural historical development created a wide range of higher order symbolic tools including different signs, symbols, writing, formulae, graphic organizers, etc. Cognitive development and learning, according to Vygotsky, depends on the child's mastery of symbolic mediators, their appropriation and internalization in the form of inner psychological tools.

Because symbolic tools are usually taught in the context of content material, teachers rarely distinguish between difficulties caused by the lack of content knowledge and difficulties that originate in the students' poor internalization of such general symbolic tools such as, for example, a table. The lack of symbolic tools becomes apparent only in special cases, such as a case of immigrant students who come to the middle school without prior schooling. For these students use of table, as a means of organizing and forming relationships among data, is in no way a natural tool of their thought, because nothing in their previous experience is associated with this artifact (Kozulin, 1998). It would be incorrect to assume, however, that students with standard educational background spontaneously appropriate and internalize psychological tools. For many underachieving students only a special cognitive intervention built around symbolic tool leads to their acquisition.

Feuerstein's (Feuerstein et al., 1980) Instrumental Enrichment (FIE) cognitive intervention program offers one of the richest sources for the acquisition of symbolic tools and operations associated with them. The program demonstrated its effectiveness in significantly improving problem solving skills in learning disabled, underachieving and culturally different students (see Kozulin, 2000). The FIE program includes 14 booklets of paper-and-pencil tasks that cover such areas as analytic perception, comparisons, categorization, orientation in space and time, syllogisms and some others. These booklets are called instruments because they help to "repair" a number of deficient cognitive functions.

Teaching the FIE program is based on Feuerstein's (1990) notion of Mediated Learning Experience (MLE). Feuerstein postulated that the quality of child-adult interaction depends on the presence of certain criteria of MLE. Among the most important of these criteria are Intentionality & Reciprocity of interaction, its Transcendent character (i.e. having significance beyond a here-and-now situation), and the Mediation of Meaning. Studies that follow this paradigm (Feuerstein, Klein, Tannenbaum, 1991; Kozulin & Rand, 2000) focus on the presence of the specified MLE parameters in child-adult interaction and the consequences of the absence or insufficient amount of MLE for the child's cognitive development and learning.

The RMT approach, thus, is based on the notion of psychological tools, both general and mathematically specific, and the notion of MLE interaction that dicates the didactics of teaching both general cognitive strategies and specific mathematical material. The pedagogical process of the RMT paradigm applies the criteria of MLE to guide and nurture students to appropriate and utilize psychological tools to concomitantly develop cognitive functions and mathematical concepts. The ongoing intentionality of the mediator is to mentally arouse and drive students to: (a) become clearly aware of the structural nature of each psychological tool and to utilize the tool according to its structure-function relationship; (b) define and label each cognitive function or specific thinking action they must activate as they use the tool to perform FIE tasks; and, (c) apply the psychological tool and the emerging cognitive functions to discover underlying principles connected with the FIE tasks and transcend the specifics of these tasks to systemically construct elements of mathematical concepts. We describe this pedagogical process as cognitive conceptual construction. In what follows we will demonstrate how the RMT approach helps in advancing the notion of mathematical function in underachieving students.

Zone of Proximal Development and Scientific Concepts

Another concept that we use as a theoretical foundation of our work is the Zone of Proximal Development (ZPD) (Vygotsky 1986; 1998). Though the notion of ZPD can be interpreted in a number of ways, two aspects are particularly important in the present context.

The first aspect refers to those cognitive functions of the student that have not been yet completely developed, but are emerging at the time of assessment or/and teaching. These functions are "invisible" in a sense that they cannot be revealed through static testing procedure. On the other hand, when mediation is provided students display some elements of these functions, which, being included in overall mediational activity, elevate the overall performance level. Vygotsky believed that functions that at a given developmental point appear only in a cooperative context, at the next ("proximal") point will become approrpiated by the student him or herself. Current research does not have an agreed upon list of functions whose development can be profitably described through the notion of ZPD. Some authors (e.g. Chaiklin, 2003) insist that the notion of ZPD is relevant only to the major changes in the child's development reflected in the transition from one age period to another. Other authors (e.g. Wells, 1999) suggest using the notion of ZPD in any situation when the dynamic process of acquisition of new skills is taking place. We base our own application of the notion of ZPD on the well-known statement of Vygotsky (1986, p. 185) that not every educational intervention has an immediate developmental impact. Only some of the learning activities contribute to cognitive development: rather than just forming particular operations they promote more general cognitive abililities. It is important to recognize that ZPD is not a static "developmental fund" that can be simply activated by the proper interaction. ZPD is a dynamic formation that changes depending on both the natural development of the child and the deliberate educational intervention. One of the important outcomes of ZPD-oriented teaching is the creation in students of the propensity toward collaborative learning, which in turn leads to greater awareness of "what they already know" and "what they still don't know". Students learn how to identify what they need for successful solution of the problem and how to request the missing information from the teacher. In the context of our study the notion of ZPD is applied to a broad range of cognitive functions that are promoted by the mediated learning interactions and by the acquisition of psychological tools. The development of these more general functions in its turn supports the formation of a more specific concept of mathematical function.

The second aspect of ZPD relevant to a present discussion is the interaction between students' spontaneous concepts and scientific concepts mediated by the teacher (Vygotsky, 1986; Karpov, 2003). According to Vygotsky, concepts that emerge from the children's everyday experience are experientially rich and sometimes quite practical in the context of daily life, e.g. "sun rises in the morning". These concepts, however, are unsystematic and often appear at variance with scientific understanding of the natural phenomena. Scientific concepts mediated by the teacher are logical and systematic, but their verbal or formulaic form is often too abstractive and detached from the students' own experience. The ZPD is identified as a "space" within which a dialogue between these two types of concepts is taking place. The task of a mediator is to present an abstractive scientific concept as a schema that can organize and transform the students' spontaneous concepts. In our case the task is to create such a schematic model of function that on the one hand conforms to the mathematical meaning of function, but on the other helps students to relate to their existent everyday experiences of functional relations. The work with FIE pages provides an important intermediary between the language of mathematics and the hands-on experience with functional relations.

The Concept of Mathematical Function

A particular instance of the lack of rigorous thinking in mathematics classroom is the static view and superficial understanding of the concept of function which appears to be typical among students completing mathematics courses from elementary school through graduate and professional programs. One student, when reflecting on his high school mathematics experiences, stated, "The concept of what a function was and how it could be used was never completely understood. I feel that teachers whom I had never went deeply enough into the concept.... Most of the experiences that were given were of the plug and chug variety. For example, f (-9) = x + 2 = -7. The notion of independent and dependent variables was never spoken about. It was vaguely touched upon when I went to college, which included junior college and a major university."

Wilson (1994) reported a case study with a student who was a pre-service teacher in mathematics education and had taken several university courses for which the concept of function was primary. The case study revealed that this pre-service teacher viewed functions as static mechanisms to carryout mere numerical operations or computational activities through equations or algebraic expressions. Wilson presented specific data that provided evidence of the teacher's deficiencies in her understanding of function. Moreover, Wilson also cited a study by Even (1989, 1993) which indicates that the lack of deep understanding of the concept of functions is widespread among preservice teachers. Thus, we can see that even some math teachers who acquire advanced education in the mathematics content discipline are only prepared to teach the important concept of function from a superficial algorithmic, procedural perspective without engaging students to do deep thinking and building the structural aspects of concept formation.

Study Description

Our thesis is that students' difficulty in mathematics does not originate with their abilities to perform mathematical operations, which are basically procedural in nature. Instead, we propose that the major source of student inadequacy stems from their lack of the cognitive prerequisites that drive both their understanding and the development of these procedures and the underlying mathematical concepts. The MLE interaction and FIE, as a rich source of cognitive processes and a mechanism for cognitive practice with concept formation, can play critical roles in producing these prerequisites. Therefore, we have formulated the hypotheses given below.

Hypotheses

Hypothesis A: Eighth grade students who receive regular academic instruction with some exposure to MLE will improve in their general cognitive performance, as measured by pre- and post-testing using the Otis-Lennon School Achievement Test (OLSAT) Form F (Harcourt Brace, 1996).

Hypothesis B: Eighth grade students who receive MLE-based academic instruction in math that includes the acquisition of psychological tools through FIE lessons with transition to the Algebra curriculum will improve both in their general cognitive performance, as measured through pre- and post-testing using the OLSAT (Harcourt Brace, 1996) Form F, and their understanding of the concept of function, as measured by specially designed standards-based pre- and post-content performance tests.

Hypothesis C: Eighth grade students who receive MLE-based academic instruction in math that includes FIE lessons with transition to the Algebra curriculum will demonstrate greater conceptual change in their understanding of function than eighth grade students who receive regular academic instruction in the Algebra curriculum, as measured by specially designed standards-based pre- and post-content performance tests.

Participants

The above hypotheses were tested with two classes of eighth grade students at a school located in the inner city of a large metroplitan area in the Midwest of the U.S. One class was designated as the comparison group while the other class was considered as the treatment group. The authors were informed by the principal of the school that students were randomly assigned to the two classes before the academic year started. The treatment group consisted of 26 African American students (14 males and 12 females). The comparison group consisted of 27 African American students (15 males and 12 females). Prior to the initiation of the study, teachers of the two classes made the collective general assessment that students in the comparison group were generally more mature and, overall, had higher grades than students in the treatment group.

Pre-tests and Post-tests

Cognitive Ability. Both the treatment group and the comparison group were administered a cognitive ability pre-test just prior to the intervention and a cognitive ability post-test immediately following the intervention. The Otis-Lennon School Achievement Test (OLSAT) Form F (Harcourt Brace, 1996) served as both the pre- and post-cognitive test. The OLSAT covers the following cognitive areas: Verbal Comprehension; Verbal Reasoning (arithmetic reasoning, logical selection, word/letter matrix, verbal analogy, verbal classification, and inference); Figural Reasoning (figural analogy, pattern matrix, and figural series); and Quantitative Reasoning (number series, number inference, and number matrix). The OLSAT assesses most of the mental processes that we have identified as essential for rigorous mathematical thinking.

Math Concept. Both the treatment group and the comparison group were administered specially designed standards-based pre- and post-content performance tests on the mathematical concept of function. Mathematics performance related to understanding the concept of function was evaluated on the pre-test using a task that presented a growing pattern in a figural modality that consisted of the basic unit of two squares, with one on top of the other, and with a shaded isosceles triangle on top (see Fig. 1).

The growth in the pattern was conveyed in moving from left to right, with a basic unit being added for each stage of growth. Each stage was labeled "House (no.)" with the number of the house increasing sequentially by one. This figural presentation was followed by the following test items:

[FIGURE 1 OMITTED]

1. Describe in detail each pattern you see in this picture of houses.

2. Organize your data from these patterns using a table. Show your table below.

3. Predict what House 7 will look like. Describe it.

4. Describe in words a rule that can be used to predict what House n will look like.

5. Write this rule using symbols.

6. Is this a function? Why or why not.

7. Describe the two variables. How they are related to each other? How do they differ from each other?

The post-test presented a different pattern, which started as a square and grew by adding the number of squares equivalent to each shape or stage number (see Fig. 2). The test items were the same as those given for the pre-test. The potential threats to the validity of the study stemming from the use of a different pattern between the pre-test and the post-test would appear to be minimal for the following reasons.

Each pattern is a simple progression which grows from stage to stage by the addition of small numbers of squares. Thus, the same shape and process are involved in the growth of each progression. In addition, the growth process in each pattern takes place without a change in the orientation of the squares being added.

[FIGURE 2 OMITTED]

The following rubric was constructed to calibrate each response in terms of students' demonstrated understanding of the concept of function:

5.0 -- Fully Demonstrated

4.0 -- Mostly Demonstrated

3.0 -- Partially Demonstrated (with deficiencies)

2.0 -- Showed Little Understanding

1.0 -- Showed No Understanding

An attempt to establish the reliability of the above coding for the rubric consisted of requesting three mathematics educators to independently grade a battery of four students' papers, which included all test items from both the pre-test and the post-test, using the rubric coding and determining the variability of their scorings for each test item. All three mathematics educators gave equivalent scores to 39% of the items while at least two of three mathematics educators gave equivalent scores to 79% of the items. For almost all cases where the mathematics educators differed in their scoring, the coefficient of variation ranged from 10-24%.

Description of Interventions

Both groups received instruction in the regular mathematics curriculum with an Algebra component. The treatment group received nine and a half hours of the RMT intervention over a six-week period from a trained RMT teacher along with the regular mathematics curriculum from their regular math teacher who was not trained in FIE/MLE. This group was also taught science by a teacher who was trained in the theory and practice of MLE. This science teacher did not teach the treatment group FIE.

Each RMT intervention session was for a duration of about 30 minutes with an average of about three sessions per week. Completed FIE pages and worksheets were collected perodically and chronicled for each student. Students were required to make a daily written entry of reflection in their journals using precise language that was emerging during the engagements. Journals were collected at the completion of the intervention.

The comparison group received regular instruction in the mathematics curriculum with an Algebra component during the same six-week period that the treatment group received the RMT intervention. The mathematics teacher who taught the comparison group the regular mathematics curriculum was the same person who taught the treatment group the regular mathematics curriculum and who was not trained in FIE/MLE. During this period the comparison group was also taught science by the same science teacher who taught science to the treatment group and who was trained in the theory and practice of MLE. This science teacher did not teach the comparison group FIE.

Results

Emergence of Cognitive Functions and Concept Formation for Treatment Group

The RMT activities were mediating students to: (a) define the problem (figure out what had to be done) on each FIE page; (b) carefully analyze each psychological tool on the page; (c) determine the relationship between the use of the tool and solving the problem in order to initiate the process of appropriating the tool according to its structure-function relationship; (d) utilize the tool to perform the FIE tasks on the page; (e) identify and define the cognitive functions being used and how they are being used specifically to perform the tasks; (f) share and reflect on different strategies that were used, challenges encountered, and ways in which these challenges were addressed; (g) apply psychological tools and the emerging cognitive functions to discover underlying principles connected with the FIE tasks; and, (h) transcend the specifics of these tasks to systemically construct elements of mathematical concepts.

Mediating transcendence involved guiding students through worksheets that were specially designed by the authors to engage students in conscientiously practicing formation of conceptual elements of a mathematical function by the joint use of psychological tools and cognitive functions. These conceptual elements were: (1) change within the context of conserving constancy; (2) changeability; (3) interdependence; (4) cause-effect relationship; (5) variable; (6) functional relationship; (7) independent and dependent variables; (8) one-to-one corespondence; and, (9) ordered pairs. A major outcome from the RMT intervention was the interaction between the enhancement of students' cognitive strategies and problem solving skills and the emerging development of the mathematics concept of function. The following vignettes illustrate this process.

The concept of constancy was first introduced to students as a part of the work with FIE tasks "Organization of Dots" (see Fig. 3--In Fig. 3 only the unconnected dots and the square and right isosceles triangle models are given and students must find a way to replicate the models by connecting the dots). Through reflection on this work, students were mediated to decontextualize from the specifics of the FIE tasks to start generalizing to further construct the concepts of constancy within the contexts of dynamic change, variable, and functional relationship (Fig. 4-6). An excerpt from the class' engagement while reflecting on Page 1 of "Organization of Dots" presents supporting evidence. The mediator, who is the RMT teacher, is very intentional to have the students focus on the concepts of change and constancy in order to mediate transcendence beyond the FIE tasks to the construction of the idea of a variable and the relationship between independent and dependent variables.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Mediator: Now that you have completed this page [Fig. 3], what changes do you observe?

Student 2: The figures kept changing their positions ... I mean the angles, how they are turned.

Mediator: Could we say that the orientation of the figures changed?

Student 11: Yes, that's a good word.

Student 2: Yeah. That's what I'm saying.

Student 10: Well, I noticed that the figures got more entangled.

(Note: A subgoal of the FIE program is to help students acquire concepts, labels, and vocabulary in order to master the FIE tasks. In building RMT, this subgoal becomes a major and an ongoing focus of the mediation. The mediator intentionally demands that students draw from their prior knowledge to construct a broad repertoire of new concepts, labels, and vocabulary in order to precisely and completely describe objects, situations, and phenomena. This focus mediates meaning for students, which fuels their motivation to develop more precise and accurate language. During earlier sessions, students experienced tasks with the "overlapping of objects" and other concepts, and thus were mediated to advance their vocabulary to more precisely and accurately describe and label these concepts. At this point, they are drawing from their newly created experiences and using the language that continues to evolve).

[FIGURE 5 OMITTED]

Student 12: Yes. There is greater intrusion of figures into each other's space.

Student 7: I agree. I would say that the figures overlap more as we move down the page.

Student 14: The dots keep getting closer together as we move down the page, too.

Mediator: I hear you saying that the orientation of the figures, the closeness of the dots, and the overlapping of figures are changing as we go down the page. Since these things are changing what are they?

Pause.

Mediator: Well, are they staying the same? Are they constant?

[FIGURE 6 OMITTED]

Student 1: No, We just said that they are changing.

Mediator: So when something changes it does what?

(Note: The mediator intends to mediate deeper and more precise meaning).

Student 4: It expands.

Student 11: It should shrink.

Mediator: So what is a word that covers both expanding and shrinking?

Student 5: It transfers?

Student 11: It modifies?

Mediator: Good! Now what is another word that means it changes or it modifies?

Student 2: I know. It varies.

Mediator: Great! So something that varies is a what?

Several Students almost in unison: It's a variable!

Mediator: What are the variables here?

Student 7: The variables are the closeness of the dots and the overlapping of the figures.

Mediator: That's correct.

Student 15: You know, I think there is a functional relationship between the closeness of the dots and the overlapping of the figures.

Mediator: That's powerful. Let us call the closeness of the dots the proximity of the dots. Is the proximity increasing or decreasing as we move down the page?

Student 3: The proximity is increasing.

Mediator: What about the distance between the dots.

Student 9: The distance from dot to dot is decreasing as we move down the page.

Student 15: So there is a functional relationship between the proximity of the dots and the overlapping of figures.

Student 3: The proximity of the dots is the controlling variable. It is the independent variable. The overlapping of figures is the dependent variable.

Mediator: That is terrific!

Student 6: I see layers of functional relationships.

Mediator: Tell us more.

Student 6: We organized the loose dots onto functional relationships using the psychological tools. These tools function the ways they do because of their structure and the functional relationships that build their structures.

Student 4: Now we have created a new functional relationship between the proximity of the dots and the overlapping of figures.

Student 12: Yeah. I think there is another functioning going on. We have been networking our cognitive functions to help us comprehend all that is happening.

Mediator: This is fantastic. How do these layers of functional relationships differ? Is one more abstract than the others?

Student 10: The functional relationship coming from the networking of the proximity of the dots with the overlapping of figures is more abstract than the psychological tools.

Student 8: The networking of the cognitive functions is the most abstract.

Student 3: I agree. Using these cognitive functions is a powerful tool. Can we call this a psychological tool?

Student 5: It sure has me thinking at a high level.

At this point the mediator asked the students were there functional relationships in real life. In response one student stated,

  I woke up this morning and it was snowing, real big, fluffy flakes.
  It's been snowing since midnight. The lady who gave the weather
  forecast on TV said there was a good chance for snow because of what
  the temperature and relative humidity were going to be. This was
  around 6:30 in the evening and I had no idea it was going to snow. But
  sure enough it snowed, and it's still snowing. I think there is a
  functional relationship between temperature, humidity, and the chance
  of snowing. The pressure might have something to do with it too. I'm
  not sure.

This sample of classroom interaction demonstrates the effect of the MLE dynamic in stimulating student engagement in thinking to bring about conceptual change.

The conceptual understanding of a functional relationship between variables was enhanced when students appropriated and began using a table as a psychological tool. The following is an excerpt from one of the sessions in the intervention that focused on part of Page 3 (see Fig. 7) of "Orientation in Space I" FIE tasks.

[FIGURE 7 OMITTED]

Mediator: What is the structure of every table or chart that you have seen? What is every table composed of?

Student 1: Columns.

Mediator: Yes. In which direction do the columns run?

Student 8: They go up and down.

Student 3: That's vertically.

Student 5: They also contain rows that run horizontally.

Mediator: Very good! Now tell me, what comes at the top of every column in every table?

Student 3: There is a heading.

Mediator: What is this heading doing with every item listed under the heading?

Student 12: It's grouping them into a category.

Student 6: Yeah. It's forming a set.

Mediator: Are you telling me that every item under the heading belongs to the heading?

Student 2: Yes. The heading is a superordinate category.

(Note: The mediator has been very intentional to mediate meaning by asking a series of very pointed questions to get students to draw from their knowledge base in order to invest in producing precise and clear details about the structure of every table or chart. At the same time he is mediating transcendence by leading them to do inductive thinking--move from the specifics of individual tables to a generalization of the structure of every table. Students produced the term "superordinate category" during the first session of the intervention when they were mediated to start comparing according to superordinate concepts. Now he is going to mediate transcendence to the formation of the concept of "variable").

Mediator: Great! Now think a little further. Do the items under the superordinate category change in some way?

Student 3: So the superordinate category is a variable since its value can change.

Mediator points to the table on the overhead projector. What are the variables in this table?

Students: "Position, Object, Side of Person".

Mediator: What are the possible values to the variable "position"?

Students collectively: A, B, C and D.

Mediator: What are the possible values to the variable "object"?

Students: House, Tree, Flowers, and Bench.

Mediator: What are the possible values to the variable "side of person"?

Students: Front, Back, Right, and Left.

Mediator: As we mentally move across one row from left to right from column to column what are we doing?

Student 8: We are creating relationships.

Mediator: That's great!

Student 1: I see something. When I choose a position and an object I can use my hypothetical thinking to determine the side of the person.

Student 7: So the side of the person will be fixed.

Student 1: Exactly.

Student 10: The position, object, and side of person are in a functional relationship.

Mediator: That's powerful! Now can we find a functional relationship in every table?

Student 12: Yes. If the information in the table is logical and makes sense there has to be a functional relationship.

Mediator: Will this be true for a table with science data, history information, or economic statistics?

Students: Yes!!

This engagement helped students to start appropriating a table as a general tool to help them organize data and form relationships among data. When they start using this tool to organize and represent relationships among mathematical data it becomes a mathematically specific psychological tool.

Evidence of Student Transcendence for Treatment Group

One activity that required transcendence from the FIE tasks was a growing pattern of squares (see Fig. 8). Students were instructed to use their cognitive functions and psychological tools to define the patterns and derive a mathematical function. One student's work is displayed in Fig. 8-10.

Discussion of Results

The hypotheses of this study were formulated to scientifically evaluate the efficacy of the RMT paradigm and its interactional dynamic, MLE. The statistical difference between the pre-test and post-test mean values on the cognitive ability test for the comparison group, who were exposed to MLE instruction in their science class, gave support to our claim that the MLE dynamic in instruction enhances cognitive development.

Central to the theory and instructional practice of the RMT paradigm is the application of the MLE dynamic to lead students to appropriate psychological tools of FIE to perform the FIE tasks and to bring about transcendence to the formation of mathematics concepts. We claimed, therefore, that RMT intervention concomitantly increases student cognitive ability and conceptual understanding of mathematical function. This claim was supported by the strong statistical significance (p < 0.005 and p < 0.0005 on the one-tailed test) in the difference between the mean values of the pre-test and post-test for the cognitive ability test and the math concept test, respectively, for the treatment group.

Pre and Post Tests

Results on the cognitive ability test for the comparison group are given in Table 1. The mean value for the post-test is statistically higher than the mean value for the pre-test (p < 0.05 on the one-tailed test). This outcome appeared to support Hypothesis A, which stated that eighth grade students who receive regular academic instruction with some exposure to MLE will improve in their general cognitive performance.

[FIGURE 8 OMITTED]

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[FIGURE 10 OMITTED]

The comparison group was taught eighth grade science by a teacher who was trained in the theory and practice of MLE and who used MLE at times during her instructional delivery. Probably because of the cognitive focus of the MLE process, these students' cognitive abilities were enhanced.

Results on the cognitive ability test for the treatment group are given in Table 2.

Although analyzing data from the cognitive ability test across groups was not relevant to testing Hypotheses A and B, a comparative analysis showed that the mean value on the pre-test for the comparison group was statistically higher (p < 0.025 on the one-tailed test) than the mean value on the pre-test for the treatment group. In addition, the mean value on the post-test for the comparison group was statistically higher (p < 0.05 on the one-tailed test) than the mean value on the post-test for the treatment group. Although the treatment group scored lower in cognitive ability than the comparison group both before and after the intervention and had a gain score that was not statistically higher than the gain score of the comparison group, its gain in cognitive ability was accompanied by a strong statistically significant (p < 0.0005 on the one-tailed test) gain in conceptual understanding regarding mathematical function. Further, the comparison group made a change in cognitive ability similar to that of the treatment group, but they made virtually no change in conceptual understanding regarding function.

These outcomes suggest that cognition alone does not bring about conceptual change, but that cognitive change is an essential factor in driving conceptual understanding. The concomitant emergence of cognitive improvement and conceptual change in mathematics that took place through cognitive conceptual construction in the intervention for the treatment group is a significant finding that should be important for mathematics education.

The results on the math concept test on function for the treatment group are given in Table 3. As stated above, the mean value of the post-test is statistically higher (p < 0.0005 on the one-tailed test) than the mean value of the pre-test.

As stated above regarding the cognitive ability test results for the treatment group, the mean value of the post-test is statistically higher (p < 0.005 on the one-tailed test) than the mean value of the pre-test (see Table 2). These two findings appear to support Hypothesis B, which stated that eighth grade students who receive MLE-based academic instruction in math that includes the acquisition of psychological tools through FIE lessons with transcendence to the Algebra curriculum will improve both in their general cognitive performance and their understanding of the concept of function.

The results on the math concept test on function for the comparison group are given in Table 4. There was virtually no difference between the pre-test mean value and the post-test mean value.

The comparison group scored higher than the treatment group on the mathematical concept pre-test because of the comparison group's success on items regarding rudimentary aspects of patterns. However, the comparison group showed significant deficiencies on most of the advanced test items. The treatment group showed significant deficiencies on all items on the mathematical concept pre-test, including those dealing with the rudimentary aspects of patterns. For both groups prior to the intervention the concept of function was, at most, limited to an equation by which to plug in one unknown number and chug out another. There was very little or no understanding of the dynamic nature of a variable.

The treatment group's mean gain score on the math concept test was statistically higher (p < 0.0005 on the one-tailed test) than the mean gain score for the comparison group. These data appear to support Hypothesis C, that eighth grade students who receive academic instruction in mathematics on the concept of function through the cognitive conceptual process of the RMT paradigm will demonstrate greater conceptual development about function than students who receive regular academic instruction in mathematics on the concept of function.

The MLE Didactic

The MLE dynamic was essential throughout the intervention with the treatment group in bringing about student engagement in the appropriation and utilization of psychological tools. The first of the MLE criteria observed in the teaching/learning interactions was Intentionality & Reciprocity (for detailed analysis of these criteria, see Feuerstein, 1990). The teacher's intention to help students in appropriating psychological tools provided by the FIE program expressed itself in constant focusing on the instrumental aspect of FIE tasks and the cognitive consequences of their appropriation. The reciprocity of interaction was reflected in the students' awareness of the ultimate goal of interaction and the teacher's sensitivity to students' responses and initiatives. The second MLE criterion, Transcendence, revealed itself in the teacher's and students' understanding that FIE tasks formed a basis for the development of cognitive skills that were much broader in their nature and application than a concrete "Organization of Dots" or "Orientation in Space" task. The transcendent character of interaction was achieved through constant emphasis on the generalizable aspects of the actions performed and tools acquired and their "bridging" to everyday experiences of students. Intentionality and Transcendence were intimately connected to the third criterion of MLE, Mediation of Meaning. Mediation of Meaning reflected the motivational aspect of learning interaction, captured by the ubiquitous students' question, "Why do we do this?". As seen in the transcripts of the lessons, the RMT teacher skillfully guided students into recognizing the meaning of the operations they performed and the goals they had to set for themselves in order to solve the problem.

Application of MLE criteria was also essential in helping students to become aware of their thinking and to create the need to be persistent and vigilant in producing deeper understanding. Several changes in the quality of student engagement were observed as the RMT sessions advanced.

The first was the progressive development of a synergy of student collaboration and classroom participation. There was an emergence of a social dynamic during classroom discussions, as revealed in the vignettes, and when students sought mediation from and provided mediation to each other when confronted with very challenging tasks. This social dynamic and sharing express student reciprocity to the mediator's ongoing intentionality to incite students to become co-constructors of their own knowledge production and learning. They speak also to the mediation of emotional meaning and its impact on mediating students to take on the intentionality of the mediator.

The second change was students' increased interest in performing the FIE tasks and connecting to mathematics concepts. This is an aspect of mediating meaning and transcendence.

A third change was a significant increase in students' confidence. Students who refused to respond during earlier sessions were volunteering to express themselves during the latter sessions. In addition, this increased confidence was expressed through increases in the volume and quality of students' chronicled writing samples. This change expresses the role of the mediation of cognitive meaning in bringing about emotional meaning, which can stimulate deeper cognitive engagement.

These changes collectively give expression to our notion of intellectual rigor that has its origin in rigorous engagement which is initiated and perpetuated by the MLE dynamic.

Enhancement of Student's ZPD and Transition to a New Zone of Actual Development

One of the goals of the present study was to evaluate the impact of the RMT approach on students' ability to receive mediation and in this way to expand their ZPD. Transcripts of the lessons and samples of students' work indicate that the majority of students in the treatment group successfully acquired a broad range of cognitive tools that helped them to progress from a vague and unarticulated notion of mathematical function to a notion that is much closer to the relevant scientific concept. Students demonstrated conceptual and terminological sophistication that clearly was out of their reach at the beginnning of intervention. One should realize, however, that not all cognitive and conceptual structures that emerged during the mediation became crystallized by the end of the intervention period. Many of them still remain within the students' expanded ZPD and have not been transferred yet to the students' zone of actual performance revealed in the post-test results. At the same time if one compares the progress made by the students' in the treatment group, who studied at a lower level than their peers in the comparison group, one may clearly see that they benefited from learning in ZPD and enhanced their actual performance. Fig. 11 presents these findings in a schematic way.

The comparison group improved its cognitive performance as evaluated by the OLSAT battery, but showed practically no advance in mathematical concepts. In the treatment group, on the contrary, the advancement of mathematical concepts was significantly greater than the improvement of cognitive performance, which in turn was greater than in the comparison group. These results can be interpreted in the following way. The comparison group worked mostly within its zone of actual performance, gaining some cognitive skills through science instruction given by the MLE trained teacher. In the absence of systematic exposure to psychological tools and the lack of bridging to match curriculum, students were limited in the ability to use psychological functions maturing in the ZPD. The treatment group students received intensive mediation of psychological tools that both boosted their general cognitive skills and their mathematical concepts. We attribute this to the enhancement of the students', ZPD and active involvement of those cognitive processes that in the absence of mediation would have remained dormant. The observed performance of comparison and treatment groups clearly reflects Vygotsky's understanding of the difference between already formed and emergent cognitive functions. Without the appropriation and utilization of psychological tools aimed at cognitive conceptual construction, conceptual change would have remained absent for the treatment group, thus seriously impeding these students' propensity to master the very important mathematical concept of function.

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James T. Kinard, Institute for Cognitive Literacy, Chicago

Alex Kozulin, The International Center for the Enhancement of Learning Potential, Jerusalem

Table 1 Summary of Results on Cognitive Ability Test (OLSAT) for
Comparison Group (n=23)

Item                Pre-test  Post-test  Gain

Arithmetic Mean     43        49         +6
Standard Deviation  11.7      10.2
Median              48        49
Range               61-23=38  66-29=37   19-(-7)=+26

Note. The difference between the mean value for the pre-test and the
mean value for the post-test on the cognitive ability test for the
comparison group is statistically significant (p < 0.05) on the one-
tailed test.

Table 2 Summary of Results on Cognitive Ability Test (OLSAT) for
Treatment Group (n=24)

Item                Pre-test  Post-test  Gain

Arithmetic Mean     35        43         +8
Standard Deviation  9.8       9.1
Median              34        43
Range               56-14=42  59-28=31   28-0=+28

Note. The difference between the mean value for the pre-test and the
mean value of the post-test on the cognitive ability test for the
treatment group is statistically significant (p < 0.005) on the one-
tailed test.

Table 3 Summary of Results on Math Test on Concept of Function for
Treatment Group (n=20)

Item                Pre-test     Post-test    Gain

Arithmetic Mean     1.9          2.8          +0.9
Standard Deviation  0.60         0.79
Median              1.7          2.6
Range               3.4-1.0=2.4  4.4-1.5=2.9  2.5-0.1=+2.4

Note. The difference between the mean value of the pre-test and the mean
value of the post-test on the Math Concept Test for the treatment group
is statistically significant (p < 0.0005) on the one-tailed test.

Table 4 Summary of Results on Math Test on Concept of Function for
Comparison Group (n=24)

Item                Pre-test     Post-test    Gain

Arithmetic Mean     2.7          2.8          +0.1
Standard Deviation  0.7          0.5
Median              2.5          2.6
Range               4.3-1.6=2.7  3.8-2.0=1.8  0.6-(-0.2)=+1.8

Note. There is virtually no difference between the mean value of the
pre-test and the mean value of the post-test on the Math Concept Test
for the Comparison Group.

Comparison Group
% Increase in Cognitive Ability               13.9%
% Increase in Understanding Function Concept   3.7%

Treatment Group--RMT
% Increase in Cognitive Ability               22.8%
% Increase in Understanding Function Concept  47.3%

Figure 11 Percent increases between pre and post tests on cognitive
ability and mathematics concept on function.

Note: Table made from bar graph.