Critical
Thinking in
Mathematics
This is a rough summary of what I talked about in the "Critical Thinking and Pedagogy" discussion group meeting. I hope to write this up more properly, and would greatly appreciate your comments, questions and debate to help me flesh ideas out more fully. I was supposed to say something about "critical thinking in mathematics." Since our main interest is undergraduate teaching and learning, my intention was to attempt to address the perennial complaint by professors of the lack of critical engagement in the great majority of undergraduate learners of mathematics. My talk focused on the reading of mathematics textbooks. To get to the issue I want to look at, I will start with a quote from the book The Mathematical Experience by Davis and Hersh:
Borrowing my humanities colleagues' parlance, the reading of mathematics is "reading of the closest kind." In textbooks, mathematics is usually presented in the following sequence: definition  theorem  proof . However, the definition is often crystallized only after the whole theory has been developed; thus, complete understanding of a definition often requires a reconstruction of the theorybuilding. (A host of critical thinking skills would come in at this point, but, as I said earlier, I only wanted to address why reading mathematics requires such tremendous effort in this presentation.) This is the students' greatest problem: they don't realize this (many students read mathematics books with a hilighter). As a result, they really don't understand the definitions enough to have a chance to engage with the ideas and concepts critically. Let's look at an example. "1 + 1 = 2" is a mathematical fact. How did this come about? (This is of course prehistoric.) Here is my conjecture: First, we have some idea of quantities ("manyness"), and then we have the idea that lumping two quantities together gives us an even larger quantity  the idea of addition. How do you make this precise? You invent numbers. Thus, the concept of "1" and "2" are invented after we have the concept of addition, and "1" and "2" are defined exactly such that "1 + 1 = 2." Think about how we learnt counting as kids. We learnt it exactly this way. There is first the concept of "1". Then "2" is "1 + 1," and "3" is "2 + 1," etc. Basically we have no problem with that. If, however, you insist that "1 + 1 = 2" be presented in the definitiontheoremproof canon, we can do it like Russell and Whitehead in their Principia Mathematica  by filling 362 pages! The fact that the concepts of "1" and "2" are so incredibly profound makes truly critical discussions of "1 + 1 = 2" almost impossible. So why is mathematics presented in this seemingly absurd way? Is it because mathematicians just wanted to give us a hard time? Before I answer this question, I want to point out something obvious: Mathematics is the only subject all pupils the world over learn for often more than 10 years. Why? Because mathematics is very useful. Probably the only school subject more useful is language (but different peoples learn different languages). Why is mathematics so useful? I would like to single out three attributes of mathematics:
Let's say we wish to know if the mathematical statement 'if p then q ' holds. The process of determining the truth or falsehood of this statement using only:
is called a mathematical proof. Within this system, mathematics knowledge is characterized thus: "If you accept the axioms (axioms are neither true nor false) and if we agree to use the same logic (there are actually different brands of logic, but we do practice bythelarge the same logic), then you must necessarily accept these results." (Peirce defined mathematics as the "science of making necessary conclusions.") This sets mathematics apart from other disciplines because almost no other discipline could articulate a set of useful axioms. Euclid 's Elements is the exemplar of the axiomaticdeductive system. Here are Euclid 's original axioms (they have later been improved): Postulate 1.
Postulate 2.
Postulate 3.
Postulate 4.
Postulate 5.
Common notion 1.
Common notion 2.
Common notion 3.
Common notion 4.
Common notion 5.
What is striking about these axioms is how utterly intuitive and noncontroversial they are  it is so easy to accept them. At the same time, anyone familiar with Euclidean geometry should be amazed at their incredible richness in terms of the results that could be generated from this set of axioms. These are necessary qualities of a good set of axioms. The axiomaticdeductive methodology highlights the mode of inquiry of mathematics is formal and deductive . The three characteristics of mathematics  abstractness, objectivity and permanence  thus follow. Mathematics is written not for pedagogy, but ("everlasting") posterity. At first I wanted to add the adverb "unfortunately" in the last sentence, but deleted it on second thought. We should just be aware of this peculiarity of written mathematics, especially when it comes to teaching. Contributed by Peter Pang, University Scholars Programme ().

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Scholars Programme, and the Centre for Development of Teaching and Learning(CDTL) © Centre for Development of Teaching & Learning, CDTL 2003, All rights reserved. 