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Lesh, R., Cramer, K., Doerr, H., Post, T., Zawojewski, J., (2002) Model Development Sequences. In Lesh, R., Doerr, H. (Eds.) Beyond Constructivism. Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching. Lawrence Erlbaum Associates, Mahwah, New Jersey.


Chapter 2

Model Development Sequences


Richard Lesh
Purdue University
Kathleen Cramer
University of Wisconsin–River Falls
Helen M. Doerr
University of Syracuse
Thomas Post
University of Minnesota
Judith S. Zawojewski
Purdue University


Please note - At this time, only the introduction and summary of this chapter are online. We are seeking copyright permission to include the entire chapter.

There are additional resources available for this chapter at

This chapter describes instructional modules that are based on a models and modeling perspective, and that were designed to meet goals that are unusual compared with those driving the development of most commercially produced materials for instruction or assessment. First, the modules were designed to provide rich research sites for investigating the interacting development of students and teachers. Therefore, they are modularized so that components can be easily deleted, extended, modified, or resequenced to suit the needs of researchers (or teachers) representing a variety of theoretical perspectives, purposes, and student populations. Second, to make it possible to observe processes that influence the development of students’ and teachers’ ways of thinking, the modules were designed to be thought revealing (Lesh, Hoover, Hole, Kelly, & Post, 2000) and to be efficient for producing maximum results using minimum investments of time and other resources. Consequently, from the perspective of teachers, they have the unusual characteristic of seeming to be small-but-easy-to-extend rather than being large-and-difficult-to-reduce. Third, they were designed to emphasize important understandings and abilities that are needed for success beyond schools in a technology-based age of information.

Even though many of the big ideas that are especially powerful in everyday situations have long traditions of being treated as foundation-level ideas in elementary mathematics, it will be clear, in transcripts that are given throughout this book, that many others are not. Also, the activities that we’ll be describing often give special attention to levels and types of understanding (and ability) that seldom have been emphasized in traditional textbooks, tests, or teaching. Consequently, by emphasizing an unusually broad range of understandings and abilities, a broader range of students typically emerge as being highly capable; and, many of these students are surprisingly young or were formerly labeled "below average" in ability or achievement. In short, a goal of these modules has been to help researchers investigate the nature of situations in which surprising students produce surprisingly sophisticated results at surprisingly young ages, in surprisingly brief periods of time, and with surprisingly little direct instruction.

Our experience suggests that one of the most effective ways to achieve the preceding goals is to put students in situations where their everyday knowledge and experience enables them to clearly recognize the need for the constructs that they are being challenged to produce — and where a lack of facility with esoteric facts and skills does not prevent them from using their knowledge and experience to develop the required conceptual tools.


The kind of model-development sequences described in this chapter are designed to be used in research, as well as in assessment or instruction. Furthermore, they are designed to focus on deeper and higher-order understandings of the conceptual schemes that underlie a small number of especially powerful constructs in elementary mathematics–rather than on trying to cover a large number of small facts and skills. Yet, skill-level abilities are not neglected because it is relatively easy to link model development sequences to instructional materials that deal with these latter types of ideas and abilities. In particular, model development sequences are modularized to make it easy for researchers or teachers to add, delete, modify, or re-sequence their components.

  • For example:

It is possible to use model-eliciting activities (i.e., model-construction activities) as stand-alone problem solving experiences–perhaps being preceded by a warm-up activity and followed by student presentations or discussions focusing on response assessment.

  • It is possible to use model-eliciting activities (or model-adaptation activities) as performances assessments–and to use them somewhat like pre-tests (or post-tests) preceding (or following) a traditional instructional unit (or a chapter in a book) in which the relevant construct is emphasized. In this case, warm-up and follow-up activities might not be used.


  • It is possible to have students engage in a complete model-development sequence–and to use traditional paper-based or computer-based materials as supplementary resources, where needed, for specific students to focus on specific facts and skills.


In general, in the preceding sequences, model-eliciting activities and model-exploration activities are designed for students to work together in teams consisting of three to four students each; other activities, such as the presentations and discussions, also are intended to emphasize student-to-student or student-to-teacher interactions as much as interactions with concrete materials. In fact, even in model-adaptation activities, where students often work individually, rather than in teams, the goal is to develop conceptual tools that are sharable, transportable, and reuseable for a variety of purposes in a variety of situations. Therefore, important social dimensions of conceptual development are not neglected.

Unlike constructivist teaching materials in which carefully guided sequences of questions provide the only means of leading students to assemble and adopt conceptual systems of the type the author has in mind, model development sequences put students in situations where they must express, test, and modify, revise, and refine their own ways of thinking during the process of designing powerful conceptual tools that embody constructs that students are intended to develop. In short, students adapt their own ways of thinking rather than adopting the author’s (or teacher’s) ways of thinking, and the adaptation (modification, extension, and revision) of existing conceptual systems is given as much attention as the construction (or assembly) of conceptual systems that are assumed to be completely new to the student(s). Sometimes, it is useful for students to invent their own language, diagrams, metaphors, or notation systems that express their ways of thinking–in the presentations and discussions that follow model-eliciting activities, as well as in other activities that occur in model development sequences. But, in other cases, and especially in model-exploration activities (where a primary goal is to introduce students to powerful language, diagrams, metaphors, or notation systems), it often is not necessary to expect students to invent conventions that took many years to develop in the history of mathematics and science. Concerning the artificial introduction of socially accepted language, symbols, and other representational media, dangers that arise result from the facts that it is easy to introduce language and symbols whose meanings presuppose the existence of conceptual schemes that students have not yet developed. The result is that students often sound like they understand constructs that they in fact do not. But, this reason for being conservative about the introduction of standard language, notations, and conventions is a very different than the notion that the value of these media depends mainly on votes from some group of people–rather than depending on their power and utility that they provide for the underlying conceptual systems they are intended to embody. In addition to focusing on powerful constructs and conceptual systems, the kind of activities that are emphasized in Model development sequences are intended to go beyond isolated problem-solving experiences that are intended mainly as vehicles for emphasizing problem-solving processes.