On Dec 26, 2013, at 3:41 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> From the outline and description it looks like a basic introductory course to logic and proof, much as used to be taught in 10th grade. Devlin recognizes the market opportunity created by the death of proof-based geometry in high school, and how to repackage that into his constantly repeated catch-phrase "mathematical thinking", so that it sounds "new! improved!" and sells better. > > Cheers, > Joe N
On the surface that would be a good description, an introduction to logic. The textbook (you can try googling it) has 4 chapters.
1. What is mathematics 2. Getting precise about language 3. Proofs 4. Proving results about numbers
You are right, it is high school-ish, actually a bit less. While we never had a single course in the subject (logic), take the set theory, truth tables, axioms of algebra and proof based geometry we did have, and any one of us would zip through this material in a week. You are also right about the grade level. I might have said 9th to 10th grade, but we are certainly in the same ball park. But beneath the surface it isn?t ?introductory? in the sense that after you take this course you then move on to an intermediate course. After this course, if you have a genuine interest, you would have to take a genuine introduction to logic course, all over again, with all the necessary development that this one lacks.
But Devlin describes this as a ?transitional? course rather than just an introduction to 10th grade logic which it is. And in that sense, what it really lacks, which I will explain later, is analysis. A transitional course between school math and math major math shouldn?t be about 10th grade logic. It should be about analysis.
I generally use three phases to describe mathematical awareness.
I. Determinism II. Consistency III. Analysis
Put simply -
In arithmetic we accept the operations and operands because they work and by ?work? I mean that they are deterministic. Like a function, they produce a single result and always the same result. If addition said that 3 + 5 is sometimes 8 and sometimes 6, it wouldn?t be of much use to us. Determinism isn?t something we have to think about, it?s built in. As long as there is a ?right? answer we are satisfied. And rightly so, we do not yet have enough substance or experience to think otherwise.
However, as we extend our journey further into the set of real numbers we start to face choices. Determinism still rules and there will always be a ?right? answer. But unlike the explicit mappings in arithmetic, we are now faced with actual choices in the path to the right answer and we make those choices such that they are consistent and do not break the determinism of mathematics. It is consistency that tells us that a minus times a minus is a positive, because if it were not, mathematics would be inconsistent and not deterministic. It is consistency that tells us that a number raised to a negative power is equal to the reciprocal of that number raised to a positive power.
The final stage, analysis, is the formal theory of mathematics.
Mathematics is strange in that the theory comes last. In the physical sciences, like chemistry and physics, we can get right to the formal theory relatively quickly because these are theories of the physical world with which we have plenty of experience. We cannot do this with mathematics because mathematics is about real numbers and despite their name, they don?t exist. We cannot begin with the theory of something that is neither tangible nor even intangible, yet. Thus, many years are spent familiarizing ourselves with the behavior of these unreal real numbers through reasoned exercises in elementary and applied subjects like arithmetic, algebra, geometry and calculus. And there is (was) plenty of logic involved, implicitly and explicitly. But not formal theory.
The transition that the aspiring mathematician faces isn?t a transition from rote procedure to 10th grade logic. And shame on Devlin for perpetuating this absurd myth in order to support the cockeyed fraud that college and education have become. What kind of aspiring mathematician would not already have a personal familiarity with logicality by college? That would be like an aspiring musician not already having a personal familiarity with musicality by college. Yeah, I know there are such students today, in college. But they aren?t aspiring, not in the truthful use of the word.
The transition that the aspiring mathematician faces is a transition from years of just exposure to the behavior of real numbers to developing an actual formal theory of real numbers. It isn?t going to be about advanced elementary algebra or advanced elementary calculus. It is now about formal algebra and formal calculus, which are very unlike their elementary namesakes. It is now about analysis. And analysis is a very different experience than what took place before analysis.
You don?t see a lot of analysis in high school. There have been several suggestions for this.
1. The students are not mature enough. 2. There isn?t enough interest. 3. There aren?t enough qualified teachers.
I used to buy (1) but not any more. I think a lot of people base (1) on what I call the epsilon-delta phenomena. The near-total shutdown (in even good students) that occurs in calculus class when you try to introduce limits using the formal epsilon-delta definition. But pulling a very formal definition like epsilon-delta right out of the blue, with no prior development, is hardly a good indicator of whether students are mature enough for analysis. I think in a proper setting, like an introductory analysis class where formality is developed in successive bites, students could succeed. We certainly had the logic skills for it, by the 10th grade. Also to note, many advanced students run out of math before the 12th grade, so they have the time to take analysis, if it were offered.
I think the real reason that analysis is rare in high school is the lack of interest by the schools, lack of qualified teachers and the fact that the vast majority of math-paths are applied math based, not pure math based.
But if high schools were specialized, then those factors could be easily overcome.
 Example of inconsistency
Given that 2 * 2 is 4 and assuming -2 * -2 is -4
a. Then 4 / 2 = 2 (if c = ab then a = c/b and b = c/a, arithmetic) b. And -4 / -2 = -2 (same reason as above) c. So far so good, and now we must decide what 2 * -2 is. 4 or -4? d. If -2 * 2 is 4 then 4 / 2 = -2 (this is inconsistent with a) e. If -2 * 2 is -4 then -4 / 2 = 2 (this is inconsistent with b)
Choosing (accepting) the right path during this portion of the journey isn?t about deriving or basing the choices on fundamental axioms like it is in analysis. It is about rejecting inconsistent choices. That does require reasoning but it isn?t the same exercise that analysis is.
 Formal Thinking Skills
One of the things that struck me most when I started to review college introductory physics courses was the lack of the formal skills that my peers and I took for granted in the 9th and 10th grade. And I see this in many new grads. They can be talking about a subject that they took, that they think they know, that I have no familiarity with, and within the course of a single conversation, I will be correcting them. I can take what they are only regurgitating and make sense out of it, which can be unnerving to them for sure. This ability comes from formal thinking skills, which are sorely lacking in education today.