Before You Learn Algebra From TabletClass, Read This

[Alfred Cromwell]

I have recently had the misfortune of encountering a rather popular mathematical YouTube channel called TabletClass Math, conducted by one John Zimmerman. My understanding is that he is an experienced and "credentialed" (not that this means much today) mathematics teacher, and has assembled an enormous audience by presenting elementary and secondary mathematics in what is repeatedly advertised as a “clear,” “simple,” and “step-by-step” manner.

This whole "step-by-step" business needs to stop.

The governing philosophy of TabletClass appears to be that mathematics ought to be rendered immediately accessible: difficult ideas are to be broken into manageable procedures, technical qualifications postponed whenever they interfere with comprehension, and students supplied with algorithmic methods by which the required answer may be obtained.

In other words, precisely the sort of pedagogical philosophy against which one ought to issue a warning.

A recent lesson on rational expressions provides an almost perfect specimen of this problem (I will link it down below).

We begin with the announcement that whenever one encounters the word rational in mathematics, one should “think of fractions.” This will not do here at City Tutoring.

One might have hoped that a lesson entitled Rational Expressions would begin by telling the student what a rational expression is. Instead we receive an association.

Rational means: think fraction. And what, then, is a rational expression?

We are informed that it is “basically a fraction” ("basically"? In a math lecture sir?) in which the numerator and denominator are “some sort of polynomials”, followed, astonishingly, by “or functions, whatnot.” You read that right gentlemen: "functions, whatnot."

There is something almost magnificent about the mathematical radius covered by those two words. If arbitrary functions will do, then apparently sin(x)/(eˣ + 1)
may wander into the theory of rational expressions merely because it has had the good fortune to be typeset with a horizontal fraction bar.

No!! No!! A rational expression is not defined by visual resemblance to a fraction.

One begins with polynomials over an understood coefficient domain and considers expressions of the form P(x)/Q(x), with Q(x) ≠ 0.

If one wishes to proceed from expressions to rational functions, one must then confront the equivalence relation by which distinct quotients represent the same element of the appropriate field of fractions.

These matters are not ornaments. They are a proper understanding, and teaching, of mathematics. Zimmerman only seems vaguely aware of this. He pauses to acknowledge that there are “some technical things about rational expressions.” Oh?
We await the technical things. They never arrive. Mr. Zimmerman made me wait, and that has now angered me.

Instead: “Basically…” Ah! That most industrious word in elementary mathematics.

Whenever a definition threatens to impose intellectual discipline, basically arrives with a broom.

We are subsequently transported to 24/60 because the chosen route into rational expressions is not algebraic structure but arithmetic familiarity.

Factor 24. Factor 60. Find common factors. “Cross-cancel.” Repeat.

The student is therefore encouraged to understand polynomial quotients principally by analogy with arithmetic fractions! That may be an analogy, but an analogy is not a definition, and a procedure is not an explanation!

Even the language of “cross-canceling” deserves scrutiny. Factors do not disappear because matching symbols have been spotted on opposite sides of a horizontal line. Cancellation is justified by multiplication by an inverse, and inverses exist under hypotheses.

The symbols are recording algebraic operations. They are not decorative marks to be struck from the page when identical shapes occur above and below.

This distinction becomes rather more than philosophical when Zimmerman reaches
(5c-1)(3c-4)/(5c-1)(2c+1). He observes the common factor (5c-1), “cross-cancels” it, and obtains 3c-4/2c+1. There follows the customary satisfaction. There you go.
Simplified. Thus these two written expressions are not simply interchangeable as functions without qualification. They agree on the domain of the original expression. The cancellation has changed the apparent natural domain of the representative.

This is precisely why mathematicians insist upon distinctions between expressions, rational functions as algebraic objects, and functions obtained by evaluation.

Many dumb people will dismiss this as “technical.” One can also dismiss foundations from a building as “technical.” The building will remain pleasantly simple until one attempts to stand inside it.

But the real philosophical character of the lesson becomes unmistakable during the factoring discussion.

Zimmerman presents the student with 15c^2-23c+4 and discusses different ways of factoring it. He is sympathetic to students who find guessing difficult. He therefore presents a systematic procedure: multiply the leading coefficient and constant term, enumerate factor pairs, locate two whose sum produces the middle coefficient, split the middle term, group, and factor. Thus: 15(4)=60, and since (-3)(-20)=60,
-3+ (-20) = -23, one rewrites 15c^2-23c+4 as 15c^2 - 20c -3c + 4, then groups:
5c(3c-4)-1(3c-4), and finally obtains (5c-1)(3c-4).

As a computational algorithm, perfectly serviceable.

But then comes the sentence which might reasonably be engraved above the entrance to the entire pedagogical enterprise: “As long as you can factor, that’s what counts.” No, Mr. Zimmerman. That is precisely what does not count. Or, more charitably, it is not all that counts.

A student who has memorized a reliable succession of transformations may certainly produce the correct factorization. But mathematics is not distinguished from clerical competence by the successful production of the final line.

Why does this procedure work? Why does finding two integers with the appropriate product and sum permit the middle term to be decomposed? Why does grouping subsequently produce a common binomial factor? What algebraic property is being exploited? In what ring is the factorization occurring? To what extent is such a factorization unique? What role is played by the units of the coefficient domain?

These questions are not extracurricular enrichment for students who have finished the “real” mathematics early. They are the beginning of understanding what the calculation means. Zimmerman repeatedly reassures his audience that they need not guess if they simply “run this little procedure.” And therein lies the difficulty.

The student is trained to associate mathematical competence with possession of an executable algorithm. One is reminded less of mathematical reasoning than of operating instructions for agricultural equipment. And yet the instructor repeatedly insists that this is understanding.

It is especially revealing that Zimmerman narrates the student's internal monologue throughout: “You're thinking…” “You're looking at…” “I'm thinking…” “You might be saying…” The entire lesson is organized around the psychological management of the learner confronting a calculation. This is doubtless comforting. It is not, however, a substitute for organizing mathematics around mathematical necessity.

The structure of the subject should determine the exposition, not the anticipated panic level of the sweaty nervous young man or lady holding the pencil.

There is even a “911 algebra emergency” if the student cannot factor the quadratic.

I would suggest that the emergency lies elsewhere. The emergency is the possibility of producing students who can successfully manipulate algebraic notation for years without ever being forced to ask what mathematical objects that notation denotes.

They may learn, above all, that whenever mathematical precision begins to feel uncomfortable, one can postpone it until some unspecified later course. Then these same students eventually encounter abstract algebra, real analysis, or topology and realize with astonishment that definitions suddenly matter. Of course they matter.
They always mattered. Mr. Zimmerman merely declined to mention them!

To be clear: there is nothing objectionable about calculation. There is nothing objectionable about examples. There is nothing objectionable about making mathematics intelligible to struggling students. Indeed, a good exposition can do all three. What is objectionable is the quiet substitution of procedural fluency for mathematical understanding, followed by the suggestion that the distinction is merely “technical.” A student should certainly know how to simplify for example (15c^2-23c+4)/(10c^2+3c-1).

But he should eventually understand why the simplification is legitimate, what remains invariant during it, what domain restrictions survive it, and what algebraic object is represented before and after cancellation. Otherwise we have taught him to move symbols. We have not yet taught him algebra.

One must therefore resist the seductive proposition that because a method is understandable, reproducible, and effective on an examination, it constitutes an adequate mathematical explanation. Mathematics is not a collection of reliable tricks for persuading symbols to yield answers. It is the study of structures, relations, operations, invariants, and consequences compelled by definitions. The calculation serves the structure. The structure does not exist merely to justify the calculation after the student has passed his examination.

And perhaps, before announcing that rational expressions are “basically fractions” involving “functions, whatnot,” spare thirty seconds for the apparently unfashionable practice of giving a definition.

We mathematicians are therefore owed something more than another reassuring appeal to “step-by-step” instruction. We demand an accounting. Mr. Zimmerman should explain why precision was knowingly set aside, why domain restrictions were permitted to disappear without ceremony, and why procedure was repeatedly allowed to masquerade as understanding.

Failing that, an apology to the mathematics he has so casually glossed over would not be excessive. If Mr. Zimmerman happens to read this post, I invite him to come over into the comment section and apologize or at least explain why his teaching is so sloppy.

Alfred Cromwell
Founder & President of City Tutoring

[Alfred Cromwell]

Here is the lesson: https://www.youtube.com/watch?v=tne46_wCxhs