Models & Modeling as Foundations for the Future in Mathematics Education

Document Type

Presentation Abstract

Presentation Date

11-21-2005

Abstract

My talk will be about a book that is currently in press, which was edited by myself, Eric Hamilton, and Jim Kaput. The title of the book is the same as the title of my talk - above. The project that gave rise to this book was born due to a variety of related concerns of the editors. One was that so much of curriculum reform in mathematics education has attempted only to make incremental improvements in the traditional curriculum that we have inherited, and has largely ignored the voices of some very important stake holders - notably people in professions that are heavy users of mathematics, science, and technology. What might be possible, we asked, if we erased the board of past curriculum goals and began afresh to re-conceive "the 3Rs" for the twenty-first century? And, what new possibilities might arise if we quit ignoring the views of those who are not school people or professors of mathematics?

The general questions that we addressed also were stimulated by the following kinds of observations. In fields ranging from aeronautical engineering to agriculture, and from biotechnologies to business administration, outside advisors to future-oriented university programs increasingly emphasize the fact that, beyond school, the nature of problem solving activities has changed dramatically during the past twenty years. For example, powerful tools for computation, conceptualization, and communication have led to fundamental changes in the levels and types of mathematical understandings and abilities that are needed for success in such fields. … The following questions arise.

What is the nature of typical problem-solving situations where elementary-but-powerful mathematical constructs and conceptual systems are needed for success in a technology-based age of information? What kind of "mathematical thinking" is emphasized in these situations? What does it mean to "understand" the most important of these ideas and abilities? How do these competencies develop? What can be done to facilitate development? How can we document and assess the most important (deeper, higher-order, more powerful) achievements that are needed: (i) for informed citizenship, or (ii) for successful participation in the increasingly wide range of professions that are becoming heavy users of mathematics, science, and technology?

Authors in this book agreed that such questions should be investigated through research - not simply resolved through political processes (such as those that are used in the development of curriculum standards or tests). We also agreed that researchers with broad and deep expertise in mathematics and science should play significant roles in such research - and that input should be sought, not just from creators of mathematics (i.e., "pure" mathematicians), but also heavy users of mathematics (e.g., "applied" mathematicians and scientists). This is because the questions listed above are about the changing nature of mathematics and situations where mathematics is used; they are not simply questions about the nature of students, human minds, human information processing capabilities, or human development.

We also ask:

Why do students who score well on traditional standardized tests often perform poorly in more complex "real life" situations where mathematical thinking is needed? Why do students who have poor records of performance in school often perform exceptionally well in relevant "real life situations?

These latter questions emerged because many participants in our work shared the experience of encountering our former mathematics students when they appeared several years later in courses or jobs where the mathematics that we tried to teach them would have been useful. In some cases, we have been discouraged by how little was left from what we thought we had taught. On the other hand, we often were equally impressed that some students whose classroom performances were unimpressive went on to develop a great deal from seeds that we apparently helped to plant.

Upon further reflection and research about the preceding issues, most of us gradually developed the opinion that, for most topics that we have tried to teach, the kind of mathematical understandings and abilities that are emphasized in mathematics textbooks and tests tend to represent only a shallow, narrow, and often non-central subset of those that are needed for success when the relevant ideas should be useful in "real life" situations. For example, in projects such as Purdue University's Gender Equity in Engineering Project, when students' abilities and achievements were assessed using tasks that were designed to be simulations of "real life" problem solving situations, the majority of understandings and abilities that emerged as being critical for success included were not among those emphasized in traditional textbooks or tests. Consequently, when we recognized the importance of a broader range of deeper understandings and abilities, a broader range of students naturally emerged as having extraordinary potential. Furthermore, many of these students came from populations that are highly under represented in fields that emphasize mathematics, science, and technology; and this was true precisely because their abilities were previously unrecognized. … Such observations return us to the following fundamental question:

What kind of understandings and abilities should be emphasized to decrease mismatches between: (i) the narrow band of mathematical understandings and abilities that are emphasized in mathematics classrooms and tests, and (ii) those that are needed for success beyond school in the 21st century?

Many people assume that students simply need more practice with ideas and abilities that have been considered to be "basics" in the past. Others assume that old conceptions of "basics" should be replaced by completely new topics and ideas (such as those associated with complexity theory, discrete mathematics, systems theory, or computational modeling). Still others assume that new levels and type of understanding are needed for both old and new ideas. Examples include understandings that emphasize graphics-based or computation-based representational media. … My own perspectives lean toward this third option - without denying the legitimacy of the other two. But, this is what I intend to talk about during my visit to the University of Montana. So. no attempt will be made to resolve such issues here. Let me simply point out that, when issues of this type were discussed by authors in our book, three levels of students were given special attention.

  • Undergraduate students preparing for leadership positions in fields, such as engineering, where mathematical and scientific thinking tend to be emphasized.
  • Middle-school students who, with proper educational opportunities, could have the potential to succeed in universities such as Purdue or Indiana University, which specialize in a variety of fields that are increasingly heavy users of mathematics, science, and technology.
  • Teachers (as well as professors and teaching assistants) of the preceding students.

For K-12 students and teachers, questions about the changing nature of mathematics (and mathematically thinking beyond school) might be rephrased to ask: If attention focuses on preparation for success in fields that are increasingly heavy users of mathematics, science, and technology, how should traditional conceptions of the 3R's (Reading, wRiting, and aRithmetic) be extended or reconceived to prepare students for success beyond school in the 21st century?

Additional Details

Monday, 21 November 2005
4:10 p.m. in NULH

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