By Arnold Cusmariu
Travis Meier, opinion editor for The Washington Post, got on LinkedIn to brag about his first WaPo column titled “The trouble with schools is too much math,” calling it a “major moment.”
As the author of the recently published Logic for Kids, I thought I’d take a close look.
Alas, the article is nothing to brag about. While Meier makes some good points, he also harbors serious misconceptions about logic, overlooks serious problems with mathematics instruction, and would do away with a key component of mathematics with us since Euclid. I sent him an email suggesting he study my book. No reply as of this writing.
Meier may well be right that most people “have no use for imaginary numbers or the Pythagorean theorem” and that “more than three-fourths of the population spends painful years in school futzing with numbers.” Many computations can be done by simple calculators and, in technical jobs, by sophisticated software such as Matlab. The convoluted mess known as Common Core, which has caused massive headaches among students, parents, and teachers alike, goes unmentioned, however. My critique of Common Core is here and here.
Meier’s proposal that schools “end useless math requirements” will elicit howls of protest (laughter?) from the education establishment in this country and, indeed, probably worldwide as well. He wants to dump those requirements in favor of “something more useful.” Like what? Meier answers, “applied logic.” What’s that? Let’s take a close look at Meier’s exposition of this idea, quote:
This branch of philosophy [logic] grows from the same mental tree as algebra and geometry but lacks the distracting foliage of numbers and formulas. Call it the art of thinking clearly.
- Logic is not a branch of philosophy or any other subject. It’s a subject in its own right. See my book or any standard logic textbook, e.g., Hurley.
- The association with philosophy is because credit as “the first logician” usually goes to the philosopher Aristotle (384-322 BC), who devised syllogistic logic, which boils down to Venn diagrams.
- Logic is about methods for assessing the correctness (validity) of arguments regardless of subject matter, including all of mathematics and, by implication, science.
- Logic didn’t “spring from the same mental tree as algebra and geometry.” Rather, logic sits in judgment of them.
- Stated in simple terms, logical reasoning means using the word “therefore” correctly. My book explains in ways kids can understand what is involved in using this word correctly.
- To refute Meier’s other metaphor, that logic “lacks the distracting foliage of formulas,” the reader need only consult a symbolic logic textbook.
- Logic isn’t an art, of any kind. It’s as rigorous as anything in mathematics; in fact, even more so because logic seeks to represent mathematical argumentation in a way that renders validity (or invalidity) apparent, for which purpose powerful symbolic resources are necessary.
But wait, there’s more, quote:
“Logic teaches us how to trace a claim back to its underlying premises and to test each link in a chain of thought for unsupported assumptions or fallacies. People trained in logic are better able to spot the deceptions and misdirection that politicians so often employ. They also have a better appreciation for different points of view because they understand the thought processes that produce multiple legitimate conclusions concerning the same set of facts.”
- “Tracing back to premises” is another metaphorical muddle. Logic shows, rather, how an argument can move forward from premises to conclusion, making sure that each step in the inference chain exemplifies a rule of inference.
- Steps in an inference chain are checked against rules of inference to determine whether (a) the correct rule was applied and (b) whether the rule was applied correctly.
- The term “fallacy” is ambiguous between formal and informal fallacies.
- A formal fallacy is committed when (a) the wrong rule of inference is applied, or (b) a rule of inference is misapplied. Logic can test for both as explained in my book as well as standard logic textbooks.
- Testing for informal fallacies means determining which fallacy out of a known list has been committed, e.g., false cause, poison pen, hasty generalization, appeal to authority, and so on. Standard logic textbooks have lists of informal fallacies. These errors are less cut-and-dried and as such are more difficult to prove than formal fallacies.
- An argument can be logically correct even though (a) it’s premises are false and its conclusion is true; or (b) all components of the argument are false.
- However, an argument cannot be logically correct if its premises are true but its conclusion is false.
- These are key distinctions that Meier’s gloss obscures.
Too much “futzing with numbers” is not the problem with mathematics instruction. The problem is exemplified by this astonishing admission in a recent geometry textbook, quote:
We have said that theorems are going to be proved by logical reasoning. We have not explained what logical reasoning is, and in fact, we don’t know how to explain this in advance. As the course proceeds, we will get a better and better idea of what logical reasoning is, by seeing it used, and best of all by using it yourself. This is the way that all mathematicians have learned what a proof is and what it isn’t. (My italics).
As a mathematics major in college, I can attest from personal experience that the logical structure of mathematical proofs is never made explicit. What we get is a proof sketch, which leaves it to students to fill in the logic blanks (good luck) and repeat the reasoning process for other theorems (good luck). Here is a sample, from a well-known analysis textbook by Edmund Landau (page 9):
Theorem 12: If x < y, then y > x.
Proof: Each of these means that y = x + v for some suitable v.
Got that? A logically explicit version of Landau’s proof is on pp. 271-275 of my article on logic in mathematics education, available here.
Finally, it seems not to have occurred to Meier that his proposal to end mathematics requirements means doing away with teaching mathematical proofs, a key component of mathematics since Euclid; and, paradoxically, also teaching the logic behind proofs, however cursorily. Maybe Meier should try proving the Pythagorean Theorem using “practical logic.”