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.  and ) 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  p. 7, they
describe "the essential ingredients" of their perspective as follows:
In contrast, other leading researchers (see e.g.
and ) 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 
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:
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 , pp. 141-143:
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 ,
William Thurston identifies this probem of communicating mathematics as
crucial, and challenges mathematicians to address it by first asking (p.
and then answering partially that (p. 163):
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):
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):
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 ,
in which the primary medium of communication that is chosen is the computer
language ISETL (pp. 15, 21, respectively):
To be continued ...
 John R. Anderson, Lynne M. Reder, and Herbert A. Simon. Applications and Misapplications of Cognitive Psychology to Mathematics Education, 1996 (to appear). http://sands.psy.cmu.edu/personal/ja/misapplied.html
 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.
 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.
 Jon Barwise and John Etchemendy. Computers,Visualization, and the Nature of Reasoning, 1996 (to appear). http://csli-www.stanford.edu/hp/index.html
 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.