Eight Critical Points for Mathematics

Chapter in Perspectives in Critical Thinking: Essays by Teachers in

Theory and Practice, edited by Dan Weil, 1999 Peter Lang.

Peter M. Appelbaum

Department of Curriculum and Instruction

William Paterson University

Wayne, NJ 07470

(973) 720-3123

(973) 720-3137 --fax

IN PRESS: DO NOT CITE WITHOUT PERMISSION

January 1999

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How am I to think of “critical thinking” in my classroom? I find this an overwhelming task at the current juncture in mathematics education. Once finding solace in the National Council of Teachers of Mathematics Standards’ (NCTM 1989, 1991, 1995) support for problem solving, reasoning, communication, and assessment that features these general goals, I now find myself caught in the cross-fire of a turf war among students’ expectations for a skill-based lecture format, parents’ desires that range from a delight in the “reform” movement to horror at the open-ended assignments coming home, standardized tests that have not yet caught up with reform, and a media-amplified backlash reminiscent of the seventies’ Back-to-Basics. I cherish the central place of mathematics in the school curriculum, and note the legacy of mathematical associations with critical thinking, having heard numerous clichés throughout my life that refer to mathematics as contributing clarity of thought, an appreciation for logic, and a propensity to analyze and generalize arguments and presumptions. Yet I also note with concern the historical inconsistency of any sort of “transfer of learning” of these skills to non-school experiences. And I further recognize the role of mathematics in perpetuating an ideology of “reason” that can contribute to regimes of truth and power rather than to a project of social justice (Appelbaum 1995, 1998; Mellin-Olsen 1987; Swetz 1987; Walkerdine 1987).

There has probably never been a time in the history of American education when the development of critical and reflective thought was not recognized as a desirable outcome of ... school. Within recent years, however, this outcome has assumed increasing importance and has had a far-reaching effect on the nature of the curriculum. (Fawcett, p.1)

Thank goodness such attention was paid to critical thinking back in the thirties when Harold Fawcett’s dissertation on his geometry class was printed as the NCTM yearbook. Sixty years later, we surely should be able to recognize innumerable examples of Fawcett’s attributes of a student “using critical thinking well;” such a student:

1. Selects the significant words and phrases in any statement that is important, and asks that they be carefully defined.

2. Requires evidence supporting conclusions he or she is pressed to accept.

3. Analyzes that evidence and distinguished fact from assumption.

4. Recognizes stated and unstated assumptions essential to the conclusion.

5. Evaluates these assumptions, accepting some and rejecting others.

6. Evaluates the argument, accepting or rejecting the conclusion.

7. Constantly reexamines the assumptions which are behind his or her beliefs and actions. (Fawcett, pp. 11-12, paraphrased)

The fact that this list is referenced as vital to our understanding as recently as 1993 (O’Daffer & Thomquist) would presumably attest to over a half-century of accumulated wisdom regarding how students studying mathematics involve themselves in a process described by Costa (1985) and Ennis (1985) as effectively using thinking skills to help one make, evaluate, and apply decisions about what to believe or do...

One problem is the continued undertheorizing of critical thinking in an individualized or egocentric and antisocial politics of education that echoes another early twentieth-century formulation often quoted: Robert Hutchins.

It must be remembered that the purpose of education is not to fill the minds of students with facts ... it is to teach them to think, if that is possible, and always to think for themselves (quoted in O’Daffer & Thomquist, p. 39).

An observation I want to stress in this essay is that I no longer construct “critical thinking” for myself as thinking “skills,” and find the notion that I “teach” critical thinking a barrier to successful experiences in my classroom. I prefer to take my students as critical thinkers who enrich their abilities and deepen their conceptions of themselves as thinkers through our efforts in class. This is an extrapolation from recent transformations of “problem solving” in mathematics. In 1980, the National Council launched Problem Solving as the number one “basic skill” for the eighties (a clever response to the Back-to-Basics movement). Horror of horrors: we found problem solving curricula sequencing the various problem solving skills and strategies added on to an already crowded smattering of mathematics, a whole new realm of opportunities for some students to feel good about themselves and others to “learn” that they are no good at solving problems. By the nineties we have the Standards’ presentation of problem solving not as a skill to be explicitly taught, but rather as the context through which all mathematics should be learned. Having tried this out I am moving on to “critical thinking” as the context rather than the objective of my classroom curriculum.

A claim to be made again and again is the haphazard association of mathematics with any sort of rationality or clarity of thought. The links we make are based on a cultural convention of mathematical activity in schools as opposed to any universal quality of mathematics that the subject exemplifies (Appelbaum 1995; Hersh 1997; Pinxten et al. 1983; Rotman 1993). This claim has recently been well made in different terms by Heinrich Bauersfeld (1995). As Bauersfeld tries to research children’s thinking, “certain problems arise with the structuring of the internal process:”

How does a child learn to correct an inadequate habit of constructing meaning? Teachers can easily correct the products, but there is no direct access to the individual (internal) processes of constructing. Thus, on the surface of the official classroom communication, everything can be said and presented acceptably, but the hidden strategies of constructing may lead the child astray in other strategies or in the face of even minor variations. (p. 285)

When it comes down to it, writes Bauersfeld, “ ... there is no help from mathematics itself, that is, through rational thinking or logical constraints, as teachers often assume. Mathematics does not have self-explaining power, nor does it have compelling inference; for the learner there are only conventions.” (p.287) For me, any claim to critical thinking is not unleashed by the mathematics; it is an attribute of the classroom activity, or a description of pedagogical dynamics.

Yet another driving feature of my current understanding of critical thinking is the dangerous pleasure it affords as an “objective” toward which I steer my students, and for which I reward them. In the words of Alfie Kohn (1993), “a brief smile and nod are just as controlling as a dollar bill -- more so, perhaps, since social rewards may have a more enduring effect than tangible rewards.” (p.31) I now work at reminding myself that critical thinking is not something my students need to be tricked into performing, but rather a process they will go through as human beings as long as my organization of classroom activity does not stifle it, reward it, or distort it.

It is necessary, according to Erna Yackel (1995), “for teachers to understand that students’ activity is reflexively related to their individual contexts, and that the teacher contributes, as do the children, to the interactive constitution of the immediate situation as a social event.” (p.158) Alan Schoenfeld (1989), meanwhile, collects research to support the notion that mathematics can be taught in a problem-based way so that students experience the subject as a discipline of reason developed because of the need to solve problems and for intellectual curiosity. Jack Lochhead (1987), on the other hand, stresses attitudes over methods, encouraging us to have students choose or construct their own problem solving techniques, rather than follow a specific method; students are forced through a structure to choose and evaluate their method. And the ever-quoted “bible” of contemporary mathematics education, the NCTM’s Curriculum and Evaluation Standards (1989) chimes in:

A climate should be established in the classroom that places critical thinking at the heart of instruction ... To give students access to mathematics as a powerful way of making sense of the world, it is essential that an emphasis on reasoning pervades all mathematical activity. (p. 25)

So how am I to think of critical thinking in mathematics? I imagine curves of motion through space, each of which is a trace of the above approaches, and each of which has particular moments of critical change. In geometry we sometimes speak of “critical points” of a curve, and as I look back over my shifting pedagogy I can identify such points in the flow of my classroom life, points at which the flow has a sudden shift in acceleration toward a critical thinking classroom. What follows is a list of eight critical points in the historical trace of my teaching/learning strategies, each of which invite considerations of how to enrich critical thinking in the teaching and learning of mathematics in schools.

1.Treat Mathematical Actors as Mathematical Critics

I used to have my students invent their own procedures and algorithms, and to always search for another way to do a problem. The result was presentation of multiple perspectives on a single situation. This helped to establish mathematics as a humanly constructed technology of meaning which I hoped would lead to two results: (a) a view of oneself as a maker of meaning; and (b) a view of mathematics as made by people, and thus subject to the same critique as other human endeavors, according to criteria of value.

Now I recognize that it is not enough to provide a forum of presentation or solipsism. Dewey admonished that a democracy provides not just access but the opportunity to be heard. Students in my class not only explain their strategy or procedures. They now have to use another person’s strategy or procedure in a similar problem/situation, and participate in a discussion that notes the strengths and weaknesses of each. Students are asked to identify a situation in which they would use each strategy offered (for example, to explain to a younger child, to impress a town council member during a presentation on a local issue, to calculate most quickly, to be most sure of their result...). Now my students perceive mathematical thinking through multiple perspectives, and can articulate a plausible reason for selecting each perspective over others in particular contexts of use.

For example, we once needed to determine how many bags of concrete mix to purchase if one bag would fill two square feet of area (three inches thick was the recommendation), and the area was a rectangle measuring three feet by five feet. Kudan suggested eight bags, because 3X5=15 and 15/2=7½, and he figured he would need to have to buy whole bags. Marlee came to the same conclusion by reasoning in terms of a ratio: 1:2 is equivalent to 7½:15. Xandie drew a picture of a three by five rectangle, drew lines at every foot to create a fifteen square grid inside the rectangle, and proceeded to color in two squares at a time, counting up to seven, which left an empty square; the empty square called for another bag, making the total eight bags. Pearline skip counted by twos on her fingers until she got to fourteen, and then figured another bag would make eight.

Students discussed their solution strategies. The group noted that Xandie’s and Pearline’s methods were similar in that they counted the bags needed, one visually and one numerically. The students liked Marlee’s ratio approach the best because it seemed the simplest to do, even though it was the most challenging to understand why it worked. Kudan’s strategy seemed fastest and the most reasonable one to use in a situation where the numbers involved might be cumbersome. In another problem, if the numbers were something like 53 feet by 294 feet and 3 inches, and a bag covers 7 and a half square feet, they felt that the counting strategies would be too confusing, and the ratio too difficult to solve. This group of students went on to work through another similar problem using all four strategies each, including in their journals thought on what audience would most appreciate each strategy as an explanation.

2.Make a choice, Pursue it, and Consider the Consequences

I used to structure my activities to include a collection of critical thinking skills, in order to facilitate my students’ development and refinement of these skills: comparing, contrasting, conjecturing, inducing, generalizing, specializing, classifying, categorizing, deducing, visualizing, sequencing, ordering, predicting, validating, proving, relating, analyzing, evaluating, and patterning (O’Daffer & Thomquist).

Now I no longer view my job as a trainer in skills. I instead recognize that my students come to me with varying inclinations to use skills of critical thinking in school contexts. I presume that critical thinking is a trait of human experience. I can take advantage of this trait and make critical thinking the context through which mathematics is learned. I provide open-ended situations in which students use the above skills of critical thinking to draw a mathematical conclusion or accomplish a mathematical task. I ask the students to design the questions and investigations themselves. Student then must choose one question or investigation, work together based on their selections, and report to the classroom community as they see it would make an impact. Here they must choose as well. We discuss whether their choices of question, investigation, timing and format of reporting were good ones, and how they made these decisions.

My class once was investigating calculator patterns. We selected a starting number from 0 to 9, and an adding constant from 0 to 9. Students would enter some starting number, and add to it the constant number; then repeatedly adding the constant number, patterns emerged in the one’s digits on the calculator screen. After initial explorations, ideas for investigation were collected in a class discussion: odd versus even constant or starting numbers; the relationship between the constant and the length of the resulting pattern; the effect of a starting number on a particular constant chosen; and so on. Students chose to work in groups based on which investigation seemed most interesting or promising to them.

Another day we explored which four-sided shapes could make a square shadow when held up to a light source. Groups investigated the relative importance of angles, parallel sides, lines of symmetry, and distance from the light source. Several groups split into two research teams that either worked abstractly or preferred an experimental approach, cutting out shapes and tilting them against a light source to see the shadows produced. This was a great activity that led to many insights in geometry. But most fascinating was the class’ conclusion that teams working abstractly were able to understand the significant issues more readily than the people who had worked with the actual shadows.

3.Obsess About Functional Relationships

I used to collect data in explorations and support students’ identification of patterns in the data, encouraging them to search for more than one pattern or to articulate more than one rule or description of the same pattern.

Now my students are pressed to go further by recognizing how changes in one or more categories of data are related to changes in other categories. In an investigation of the behavior of bouncing balls, students studied the fact that the ratio of the height a ball is dropped from to its return height is consistent and a special property of a ball (this ratio is called a “coefficient of restitution”). Research groups collected data on weight, circumference of the ball’s equator, density, and heights of bounce, comparing data across these different categories of measurement. Another group explored heights at which the ratio no longer held, depending on the different characteristics that had been measured.

On another day this same class was studying water drops. After measuring the rate of absorption of a drop of water for different materials, one group switched to maximum number of drops a material could absorb. By changing their variables, they were able to convince the class of the importance of their research for athletic clothing, sanitary napkins, and Band-Aids.

Analyzing a survey of interest in new bike racks versus new stall doors for the second floor bathrooms, my class noted confusion over how race was defined in their survey. These students suggested that affirmative action forms and surveys unwittingly perpetuate an image of “minority” by lumping together some groups into one big category while dividing other groups into specific categories. (For example, Dominican-American and Haitian-American were important distinctions in this school, whereas Italian-Americans, Polish-Americans, recent Russian immigrants, and some Hispanic students would all identify themselves as “white.”) Class members felt that the “minority” status of some groups should be questioned.

Strategy games also offer an opportunity to understand the relationships among variables. Mancala is a game I often use: it involves moving “stones” in and out of “pots.” Usually there are four stones in each pot to start, and a typical board has six pots on each player’s side. Playing the game with standard rules offers numerous opportunities for strategy and decision discussion, especially when we expand the conversation by shifting the goal of play: to win, to lose, to keep the game lasting as long as possible, to “tie.” But we can also study how changes in variables -- the number of stones in each pot to start, the number of pots, the direction of move on each turn -- effect the strategies for a game “well played.”