Constructivism (1998)
 Communicating Mathematics Using Heterogeneous Language and Reasoning
 John Pais
 1. Modeling the Learner's Learning

 Mathematics is the science of patterns. Richard Feynman described his own compelling scientific curiosity as simply his desire to "find out about the world," to discover and understand the patterns that exist in nature. In a profound insight describing how humans accomplish this task, Jean Piaget observed that "intelligence organizes the world by organizing itself." Precisely how humans accomplish this self-organization is still essentially unknown, but nevertheless has important consequences for structuring teaching-learning environments in which successful learning occurs. 

    Recently, some leading mathematics education researchers (see e.g. [3] and [6]) have focused their efforts on formulating, using, and testing models of cognitive processes that humans might use to successfully construct, in Piaget's sense of self-organization, their own mathematical concepts. These researchers emphasize a framework in which the learner's internal accommodation process has the primary role in the construction of knowledge.  In [3] p. 7, they describe "the essential ingredients" of their perspective as follows: 

An individual's mathematical knowledge is her or his tendency to respond to perceived mathematical problem situations by reflecting on problems and their solutions in a social context and by constructing or reconstructing mathematical actions, processes and objects and organizing these in schemas to use in dealing with the situations. 

    In contrast, other leading researchers (see e.g. [1] and [2]) employ an information-processing perspective in which both assimilation and accommodation processes play key roles in the learner's construction of knowledge. These researchers model the learner as an information-processing system that creates a knowlege representation by using both external instruction to assimilate information, and internal accomodation to structure and organize information. In [2] p. 14, they summarize essential aspects of their theoretical framework regarding how knowledge may be represented symbolically by the learner, and assert that the evidence (provided by current research in cognitive psychology) indicates the following: 

Symbols are much more than formal expressions. 

Any kind of pattern that can be stored and can refer to some other pattern, say, one in the external world, is a symbol, capable of being processed by an information-processing system. 

Cognitive competence (in this case mathematical competence) depends on the availability of symbolic structures (e.g., mental patterns or mental images) that are created in response to experience. 

Cognitive theories postulate (and provide evidence for) complex processes for transforming (assimilating and accommodating) these external representations to produce internal structures that are quite different from the external representations. 

 Today, instruction is based in large part on 'folk psychology.' To go beyond these traditional techniques, we must continue to build a theory of the ways in which knowledge is represented internally, and the ways in which such internal representations are acquired. 

 2. Communicating Mathematics

 The constructivist framework for mathematics education makes prominent the notion that each learner must actively construct her/his own mathematical concepts and that, ultimately, mathematical knowledge consists in the learner's individual ability to do mathematics in a given context, by purposefully re-constructing useful mathematical concepts and tools appropriate to the given context. 

    This view of learning entails that successful instructional strategies will de-emphasize the teacher-as-lecturer component of instruction and, instead, create an approach in which the primary focus is on each learner's construction of her/his own mathematical concepts. In fact, (radical) constructivism, as espoused by Ernst von Glasersfeld, seriously questions whether there can be any shared concepts and meanings at all between teachers and learners. Consider the following from [7], pp. 141-143: 

 ...A piece of language directs the receiver to build up a conceptual structure, but there is no direct transmission of the meaning the speaker or writer intended. The only building blocks available to the interpreter are his or her own subjective conceptualizations and re-presentations. 

...In communication, the result of interpretation survives and is taken as the meaning, if it makes sense in the conceptual environment which the interpreter derives from the given words and the situational context in which they are now encountered. 

...To put it simply, to 'understand' what someone has said or written implies no less but also no more than to have built up a conceptual structure from an exchange of language, and, in the given context, this structure is deemed to be compatible with what the speaker appears to have had in mind. 

 ...If, however, the participants adopt a constructivist view and begin by assuming that a speaker's meanings cannot be anything but subjective constructs, a productive accommodation and adaptation can mostly be reached. 

    Then, according to von Glasersfeld, communicating mathematics effectively is a difficult task due not only to its conceptual complexity and abstract nature, but also due to the intrinsic limitations of the mechanism of human communication. This state of affairs probably is no great shock to the experienced mathematics instructor, however it does provide explanation for any generated re-presentations (or memories) of situations in which there may have been a perplexing discrepancy between the constructions intended by the instructor and those actually created by the learner. 

    In a thoughful and thought-provoking article [11], William Thurston identifies this probem of communicating mathematics as crucial, and challenges mathematicians to address it by first asking (p. 162): 

How do mathematicians advance human understanding of mathematics? 

 and then answering partially that (p. 163): 

We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enablespeople to understand and think more clearly and effectively about mathematics. 

    Thurston specifically identifies the ineffectiveness of standard instruction in mathematics classrooms as a communication problem, and suggests, since humans are acustomed to using all their mental facilities to understand and communicate, that a greater emphasis be placed on fostering wider, one-on-one channels of communication that operate efficiently in both directions (pp. 166-167, with a slightly different order below): classrooms, ...we go through the motions of saying for the record what we think the students 'ought' to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models.  ...We assume the problem is with the students rather than with communication... 

Mathematics in some sense has a common language: a language of symbols, technical definitions, computations, and logic. This language efficiently conveys some, but not all, modes of mathematical thinking. 

...One-on-one, people use wide channels of communication that go far beyond formal mathematical language. They use gestures, they draw pictures and diagrams, they make sound effect and use body language. Communication is more likely to be two-way, so that people can concentrate on what needs the most attention. With these channels of communication, they are in a much better position to convey what's going on, not just in their logical and linguistic facilities, but in their other mental facilities as well. 

    The difficulty of achieving teacher-learner communication, which, as we have seen, follows from constructivist epistemology, is certainly supported and reinforced by Thurston's observations above. In addition, Thurston further underscores the importance of mathematicians seriously addressing the problem of effectively communicating their language and mental infrastructure, both to students and other mathematicians (p. 168): 

We mathematicians need to put far greater effort into communicating mathematical ideas. To accomplish this, we need to pay more attention to communicating not just our definitions, theorems, and proofs, but also our ways of thinking. We need to appreciate the value of different ways of thinking about the same mathematical structure. 

We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics... This entails developing mathematical language that is effective for the radical purpose of conveying ideas to people who don't already know them. 

 3. Heterogeneous Language and Reasoning

 Thurston's notion of "developing mathematical language for the radical purpose of conveying ideas to people who don't already know them" can be seen in the constructivist approach that is implemented in [3], in which the primary medium of communication that is chosen is the computer language ISETL (pp. 15, 21, respectively): 

Our main strategy for getting students to make mental constructions proposed by our theoretical analysis is to assign tasks that will require them to write and/or revise computer code using a mathematical programming language. 

It is important to note that the reason we are using ISETL here is because there are essential understandings we are trying to get students to construct as specified by our genetic decomposition [of a concept], and we cannot do this with many other systems. For example, writing a program that constructs a function for performing a specific action in various contexts tends to get students to interiorize that action [conception of function] to a process [conception of function]. 

  To be continued ... 


[1]  John R. Anderson, Albert T. Corbett, Kenneth R. Koedinger, and Ray Pelletier. Cognitive Tutors: Lessons Learned. The Journal of Learning Sciences, 4, 167-207, 1995. 

[2]  John R. Anderson, Lynne M. Reder, and Herbert A. Simon. Applications and Misapplications of Cognitive Psychology to Mathematics Education, 1996 (to appear).

[3]  Mark Asiala, Anne Brown, David J. Devries, Ed Dubinsky, David Mathews, and Karen Thomas. A Framework for Research and Curriculum Development in Undergraduate Mathematics Education. In Jim Kaput et al. (eds.), Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education 6, 1-32, The American Mathematical Society, Providence, Rhode Island, 1996. 

[4]  Jon Barwise and John Etchemendy. Visual Information and Valid Reasoning. In Walter Zimmermann and Steve Cunningham (eds.), Visualization in Teaching and Learning Mathematics, 9-24, The Mathematical Association of America, Washington, DC, 1991. 

[5]  Jon Barwise and John Etchemendy. Computers,Visualization, and the Nature of Reasoning, 1996 (to appear).

[6]  Ed Dubinsky. Reflective Abstraction in Advanced Mathematical Thinking. In D. Tall (ed.), Advanced Mathematical Thinking, 95-123, Kluwer, Dordrecht, The Netherlands, 1991. 

[7]  Ernst von Glasersfeld. Radical Constructivism: A Way of Knowing and Learning. Falmer Press, New York, 1995. 

[8]  Saunders Mac Lane. Mathematics Form and Function. Springer-Verlag, New York, 1986. 

[9]  John Pais. Calculus for Kinetic Modeling. Interactive MathVision, St. Louis, MO, 1996-1998. 

[10]  Anna Sfard. Operational Origins of Mathematical Objects and the Quandary of Reification-- The Case of Function. In Guershon Harel and Ed Dubinsky (eds.), The Concept of Function: Aspects of  Epistemology and Pedagogy, MAA Notes 25, 59-71, The Mathematical Association of America, Washington, DC, 1992. 

[11]  William P. Thurston. On Proof and Progress in Mathematics. Bulletin of the American Mathematical Society 30 (2), 161-177, 1994.