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:

"Why are textbook presentations of mathematics so difficult to follow? The layman might get the idea that a skillful mathematician can sight-read a page of mathematics in the way that Lizst sight-read a page of difficult piano music. This is rarely the case. The absorption of a page of mathematics on the part of the professional mathematician is often a slow, tedious, and painstaking process.

The presentation in textbooks is often 'backward.' The discovery process is eliminated from the description and is not documented. After the theorem and its proof have been worked out, by whatever path and by whatever means, the whole verbal and symbolic presentation is re-arranged, polished and re-organized according to the canons of the logico-deductive method."

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 re-construction of the theory-building. (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 hi-lighter). 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 pre-historic.) 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 definition-theorem-proof 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:

1. Abstractness (and hence wide applicability)

By abstract I mean "context-free." When we say "1 + 1 = 2," we don't mean "1 cat + 1 mouse = 2 animals." To us, "1," "2," "+" and "=" are abstract concepts which fit together in "1 + 1 = 2." This relation, however, manifests itself in a multitude of contexts -- dollars, days, dogs -- what have you.

Here is another example: When we study the equation " ax² + bx + c = 0," we don't need to specify what "x"  represents before we can write down the solution. Once we know how to solve this equation, we can use it in economics, engineering, biology -- what have you.

This is the reason we see familiar mathematics in unfamiliar places all the time.

2. Objectivity

By objective I mean there is "universal agreement" (or at least near universal agreement). Everyone agrees that "1 + 1 = 2." In fact, mathematics is possibly the only discourse where there is a real sense of universal agreement.

This is the reason mathematics is so successful in facilitating the study of so many diverse fields. One can argue that mathematics is the most successful language ever invented.

3. Permanence   

It is hard to imagine that any time in the future we will abandon "1 + 1 = 2." Again I can't think of any other field having this attribute.

These are important attributes indeed. So how should one formulate a methodology for mathematics that would best capture these unique qualities?

One such methodology -- and this was formulated by the ancient Greeks and is still overwhelmingly prevalent today -- is the axiomatic-deductive system. (However, I must caution that we do not enjoy universal agreement here -- this is not mathematics after all!) At the heart of the axiomatic-deductive system is the mathematical proof.  

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:

1. fundamental concepts (definitions),
2. fundamental hypotheses (axioms),
3. previously established results (theorems), and
4. logically correct inference,

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 by-the-large 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 axiomatic-deductive system. Here are Euclid 's original axioms (they have later been improved):

Postulate 1.

To draw a straight line from any point to any point.

Postulate 2.

To produce a finite straight line continuously in a straight line.

Postulate 3.

To describe a circle with any center and radius.

Postulate 4.

That all right angles equal one another.

Postulate 5.

That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Common notion 1.

Things which equal the same thing also equal one another.

Common notion 2.

If equals are added to equals, the wholes are equal.

Common notion 3.

If equals are subtracted from equals, the remainders are equal.

Common notion 4.

Things which coincide with one another equal one another.

Common notion 5.

The whole is greater than the part.

What is striking about these axioms is how utterly intuitive and non-controversial 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 axiomatic-deductive 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 ().