Students problems with reasoning, not exactly math

Question

Consider the following problem:

• Maria always buys ice-cream when she goes to the beach. She bought ice-cream today. So, she must have gone to the beach.

Obviously this statement is wrong. Maria could have gone to other place and bought an ice-cream. You don't need any math tool to arrive at this conclusion, all you need is reasoning.

However, several adults (18~50 years old) with difficulty in math, also have a really hard time to solve/understand such kind of problems. For them, learning math is the same as memorizing rules and formulas. Anything different than that (ex: reasoning) is extremely painful.

So, is it possible to make such students correctly answer GMAT style questions?

Answer 1

You can say that this is "just reasoning", but the truth is that this is a specific application of basic logic, in particular the implication (if/then) relation. I have a colleague with a PhD in logic who says, "Implication is tricky!" when I bring this up. And I do think that it's a major problem that schools don't teach basic logic as a high-school (or earlier) requirement; it really puts all their later coursework on a foundation of shifting sand without that.

If I had complete dominion at the community college where I teach, then personally I would mandate a 1-credit seminar in basic logic (at least drills with and-or-not-if/then statements) for incoming students. At times I've tried to find an hour in my basic math classes to work on this, but unfortunately at the moment other priorities take precedence for that time.

Some blog articles I've written on this subject:

• http://www.madmath.com/2012/07/teach-logic.html
• http://www.madmath.com/2013/11/branching-decisions-in-algebra.html
• http://www.madmath.com/2014/05/basic-logic-errors.html

Answer 2

There is a paradox here. Implication IS tricky. Part of the reason is it's REALLY HARD to avoid implicit assumptions in human scenarios. The only absolute way to avoid assumptions is to translate the English into meaningless symbols (p, q) and rigorously apply logic rules. The rules themselves are tricky. You pretty much need to prove them once to know they are correct, then apply them by rote from then on. Your example is on the cusp what most people can handle in their head. For anything much more complex, most of us will need to manipulate symbols using logical rules to "reason correctly."

In your example, some of your students will intuitively see the correct premise and conclusion. Others will need some help. All of us will need some help if the scenario gets complicated enough. (Or if it is sneaky enough to trick us into making unwarranted assumptions. It happens.) I'll probably lose points for politicizing this, but consider the recent debate on gay marriage. There were some not irrational arguments along the line of "If you want to have children, then you should be married." But there was an implicit assumption of bi-implication. Not once did I hear anyone--on either side of the debate--point out that this is not the same thing as saying "If you want to get married, then you should have children." So this sort of logical error is very easy to make, every by well-trained people.

I agree with Daniel and his blogs. The only solution is formal training in logic. If only to make people aware of how EASY it is to make mistakes.

If you're looking for the most bang for your teaching buck, what I have personally found the most useful is this:

1. What implication is: An if-then statement. Show how to identify the premise and the conclusion.

2. Show the differences between the INVERSE, CONVERSE, and CONTRAPOSITIVE. My first exposure to these three conditionals was a real eye-opener. It did more than anything else in my life to avoid the kind of error shown in your example.

Answer 3

This is a subtle issue. It goes to the heart of the difference between math and physics.

That A implies B does not entail that B implies A. One encounters frequently the errant reasoning that it does even among engineering students in the university (yesterday a student told me that because a matrix was diagonalizable it must be symmetric).

However, that A implies B and one has observed B provides evidence for believing A. This statement is somehow the basis for the scientific method.

In the example given in the original post, that Maria bought ice-cream today does provide some evidence that she might have gone to the beach today.

The difference is that between deductive reasoning and inference.

Inferential reasoning is more common, more natural, and more powerful (why does anyone believe Euclid's axioms?).

I quote from V. I. Arnold (Translation of the V. I. Arnold paper “From Superpositions to KAM Theory” (Vladimir Igorevich Arnold. Selected — 60, Moscow: PHASIS, 1997, pp. 727–740)), who in his inimitably polemical manner explains the issue better than I can anyway:

Now it became possible to apply the techniques developed in the problem of adiabatic invariants. As soon as I accomplished that, Kolmogorov suggested that I should submit the paper on perpetual adiabatic invariance to ZhETF, the main physical journal in the USSR. A few weeks later, M. A. Leontovich (who was, as far as I remember, a deputy to the editor-in-chief of ZhETF) invited me to his home (near the Atomic Energy Institute of the USSR Academy of Sciences) to discuss the manuscript. Having fed me, as usual, by boiled buckwheat and calling me, as usual, “Dimka” (M. A. called me in such a way until his death), Mikhail Aleksandrovich explained to me that the paper could not be published in ZhETF due to the following reasons.

1. The manuscript contained the words “theorem” and “proof” forbidden in ZhETF.
2. The manuscript claimed that “A implies B” while every physicist knew examples showing that B does not imply A.
3. The manuscript used the unintelligible terms “Lebesgue measure”, “invariant tori”, “Diophantine conditions”. Mikhail Aleksandrovich therefore proposed that I should rewrite the paper.

Now I realize how right he was in defending a physical journal from the Bourbaki-like mathematical jargon. For instance, indeed, while claiming that “A implies B” the author must point out explicitly whether the converse holds, otherwise any reader not spoiled by the mathematical slang would understand the claim as “A is equivalent to B”.

The issue is that mathematical deductive reasoning is rarely if ever applicable outside of mathematics where it should be replaced by the Bayesian paradigm of what Polya called plausible inference, of which it is an extreme case.

The moral for teaching is that the difference between necessary and sufficient conditions is not something to be passed over lightly and that confusion in regards to it is not necessarily evidence of stupidity. When one teaches that all symmetric matrices are diagonalizable one must remind students that not all diagonalizable matrices are symmetric. Moreover, all orthogonally diagonalizable matrices are symmetric ...