11
	Mathematics as a Way of Thinking about Things
	Albert A. Cuoco
	Education Development Center

I didn't always feel this way about mathematics. When I started teaching high school, I thought that mathematics was an ever-growing body of knowledge. Algebra was about equations, geometry was about space, arithmetic was about numbers; every branch of mathematics was about some particular mathematical objects. Gradually, I began to realize that what my students (some of them, anyway) were really taking away from my classes was a style of work that manifested itself between the lines in our discussions about triangles and polynomials and sample spaces. I began to see my discipline not only as a collection of results and conjectures but also as a collection of habits of mind.

     This realization first became a conscious one for me when my family and I were building a house at the same time I was researching a problem in number theory. Now, pounding nails seems nothing like proving theorems, but I began to notice a remarkable similarity between the two projects. The similarity did not come from the fact that house-building requires applications of results from elementary mathematics (it does, by the way); rather, house-building and theorem-proving are alike, I realized, because of the kinds of thinking they require. Both require you to perform thought experiments, to visualize things that don't (yet) exist, to predict results of experiments that would be impossible to actually carry out, to tease out efficient algorithms from seemingly ad hoc actions, to deal with complexity, and to find similarities among seemingly different phenomena.

     This focus on mathematical ways of thinking has been the emphasis in my classes and curriculum writing ever since, and I'm now convinced that, more than any specific result or skill, more than the Pythagorean Theorem or the fundamental theorem of algebra, these mathematical habits of mind are the most important things students can take away from their mathematics education (see Cuoco, Goldenberg & Mark, 1996; Cuoco, 1995; and Goldenberg, 1996 for more on this theme). For all students, whether they eventually build houses, run businesses, use spreadsheets, or prove theorems, the real utility of mathematics is not that you can use it to figure the slope of a wheelchair ramp, but that it provides you with the intellectual schemata necessary to make sense of a world in which the products of mathematical thinking are increasingly pervasive in almost every walk of life. This is not to say that other facets of mathematics should be neglected; questions of content, applications, cultural significance, and connections are all essential in the design of a mathematics program. But without explicit attention to mathematical ways of thinking, the goals of "intellectual sophistication" and "higher order thinking skills" will remain elusive.

     The habits of mind approach seems to be gaining acceptance among other mathematics educators. Everybody Counts (NRC, 1989) describes it this way: "Mathematics offers distinctive modes of thought which are both versatile and powerful. . . . Experience with mathematical modes of thought builds mathematical power--a capacity of mind of increasing value in this technological age. . . ."

     A curriculum that uses workplace and everyday tasks to support the goal of developing mathematical thinking is less likely to use the tasks as the curriculum; it is less likely to let the message "high school graduates should be able to solve problems like these" evolve into "high school graduates should be able to solve these problems." Conversely, a curriculum firmly rooted in concrete problems is less likely to turn the goal of developing mathematical habits of mind into a "mathematics appreciation" curriculum, that studies little more than lists of mathematical ways of thinking. The dialectic between problem-solving and theory-building is the fuel for progress in mathematics, and mathematics education should exploit its power. Problems can be both sources for and applications of methods, theories, and approaches that are characteristically mathematical. For example, through the work of Descartes, Euler, Lagrange, Galois, and many others, techniques for solving algebraic equations developed alongside theory about their solutions. (See, e.g., Kleiner, 1986.)

     What does it mean to organize a curriculum around mathematical ways of thinking? One way to think about it is to imagine a common core curriculum for all students lasting through, say, grade 10. Students would work on problems, long-term investigations, and exercises very much as they do now, except the activities would be aimed at developing specific mathematical approaches. In contrast to other kinds of organizers currently in use (applications, everyday situations, whimsy, even computational skill), the benchmark for deciding whether or not to include an activity in a curriculum would be the extent to which it provides an arena in which students can develop specific mathematical ways of thinking, such as:

• Algorithmic thinking: Constructing and using mechanical processes to model situations.

• Reasoning by continuity: Thinking about continuously varying systems.

• Combinatorial reasoning: Developing ways to "count without counting."

• Thought experiment: Learning to imagine complex interactions.

• Proportional reasoning: Thinking about scaling, area, measure, and probability.

• Reasoning about calculations: Developing algebraic thinking about properties of operations in various symbol systems.

• Topological thinking: Generalizing notions of closeness and approximation to non-metric situations.

     These themes would run throughout the K-10 experience. They would be discussed explicitly in class, in diverse contexts, while students were working on problems. For example, an investigation involving topological reasoning might ask students to improve on the way users are allowed to organize their desktops in Macintosh and Windows environments.

     After a decade of this core curriculum, students could choose from a set of electives that would vary from school to school and from year to year. Courses in probability, geometry, physics, history, algebra, cryptography, linear algebra, art, data-analysis, accounting, calculus, computer graphics, trigonometry, and whatever else interests teachers and students are all candidates. If students have a solid foundation in mathematical thinking, they will be prepared for a wide array of high-powered courses designed to meet the interests and needs of the entire spectrum of students. This is a genuine alternative to the current system of tracking: it would give students a choice and a chance to pursue their interests (16-year-old students do have well-developed interests). But no matter what choices they made, students would be assured of a substantial mathematics program that built on a core curriculum centering around mathematical habits of mind.

     Such a curriculum would help students develop general strategies for doing mathematics, establish underlying mathematical (not just contextual) connections among the tasks, and help students develop the intellectual prowess necessary to deal with the kinds of problems they'll face after graduation. For example, a strand on algorithmic thinking would be a good context for investigating problems such as Lottery Winnings (p. 111) or Buying on Credit (p. 87). Whereas the contextual similarity of these tasks is evident even at a superficial level, they also share a deeper mathematical similarity based on a kind of algorithmic thinking that is somewhat removed from the mathematics backgrounds of most adults.

     Show a group of eighth graders a table like Table 11-1. Then ask these eighth graders to describe what is going on. Their responses will be quite different from those of most adults who have been schooled in algebra. Adults immediately search for a "rule"--a procedure that can be performed to the "Input" column to produce the "Output" numbers. (In this case, multiplying by 5 and subtracting 1 does it). Young students are much more likely to see other patterns (the last digits on the right, for example), and very often they'll notice that every number in the right-hand column is 5 more than the one preceding it. This is the germ of recursive thinking, a very important way of looking at things. Rather than extinguish it during high school, a strand on algorithmic thinking would develop it in tandem with the more traditional "closed form" (multiply by 5 and subtract 1) way of modeling the data. Recursive approaches are ideal ways to build spreadsheets and model processes using computer algebra systems like Mathematica. And investigating the connections between recursive and closed form models can become a theme that organizes a great many of the topics in traditional high school mathematics.

Table 11-1: An input/output table
	Input Output 1 4 2 9 3 14 4 19 5 24 6 29

     Recursive thinking also gives students genuine intellectual power. Listen to a group of adults discussing the question, "How does the bank figure out the monthly payment on my car loan?" You'll hear qualitative statements, but you'll seldom hear a satisfactory mathematical description of what goes on behind the button on the calculator. Students accustomed to thinking in algorithms would ask themselves how the bank constructs a spreadsheet for computing the balance owed at the end of each month. They'd articulate an algorithm something like, "The amount you owe at the end of a month is the amount you owed at the beginning, plus 1/12 of the yearly interest on that amount, minus whatever you make for a payment." This simple model is easily executed on a spreadsheet, and it quickly leads to an algorithm for calculating the monthly payment on a loan. This can be refined in calculus to the method that is used in practice, and it can be modified well before calculus is known to handle tasks like those in this volume.

     The usefulness of this kind of algorithmic thinking transcends the analysis of a particular context; algorithmic thinking is used by chefs, construction workers, librarians, and people surfing the Internet. A curriculum that focuses on developing similar mathematical habits will go a long way toward achieving the goal of preparing students for challenges that don't yet exist. And it offers a mathematical framework that meets the goal of providing tasks that prepare students both for the world of work and for postsecondary education, that "exemplify central mathematical ideas," and that "convey the rich explanatory power of mathematics."

References

Cuoco, A. (1995).

Some worries about mathematics education. Mathematics Teacher, 88(3), 186-187.

Cuoco, A., Goldenberg, E. P., & Mark, J. (1996).

Habits of mind: An organizing principle for mathematics curriculum. The Journal of Mathematical Behavior, 15(4), 375-403.

Goldenberg, E. P. (1996).

"Habits of mind" as an organizer for the curriculum. Boston University Journal of Education, 178(1), 13-34.

Kleiner, I. (1986).

The evolution of group theory: A brief survey. Mathematics Magazine, 59(4), 195-215.

National Research Council. (1989).

Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press.

Albert A. Cuoco is Senior Scientist and Director of the Mathematics Initiative at the Education Development Center (EDC). Before coming to EDC, he taught high school mathematics for 24 years to a wide range of students in the Woburn, Massachusetts, public schools, chairing the department for the last decade of his term. A student of Ralph Greenberg, Cuoco received his Ph.D. in mathematics from Brandeis in 1980. His mathematical interest and publications have been in algebraic number theory, although his recent work in high school geometry is gradually convincing him that geometric visualization has a place in mathematical thinking.