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I am teaching, among other things, a college algebra course this semester. In this course we do a lot of conceptual things but we also do some techniques to prepare students for calculus. One of the techniques is dealing with equations that involve squaring or rooting both sides.

Solve $(x - 3)^2 = 16$

When showing them how to solve this, we have the method: when you square root both sides, you get a plus or minus sign. This is the thing we need them to know. I can't have them going into calculus finding only one solution to an equation like this.

Conceptually, the reason this method is needed is that $\sqrt{x^2}$ is not x. Instead $\sqrt{x^2} = |x|$. However, this concept is significantly more difficult than the method, and you don't necessarily need the concept to complete the problems.


The advice I have received so far from other educators on this topic has been to sometimes skip the concept.


My question: Is there any good framework for deciding when it is a good idea to "skip the concept" and "just teach the method," knowing that students will appreciate the concept only much later in life, or possibly never?

I can't even tag this "undergraduate-education" because the question works for elementary schools too -- do you really show children why long division works? And if you do, do you intend them to retain that concept? Why is this skipping-of-the-concept sometimes good and sometimes bad?

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    $\begingroup$ I'm not sure this is a good example, because the concept seems pretty simple - if $a^2=b$, then $(-a)^2=b$ as well. $\endgroup$ – Alex Becker Mar 31 '14 at 15:28
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    $\begingroup$ I agree with @AlexBecker that the reason in this case is simpler than you're making it. The fact that $a^2 = 16$ if and only if $a=4$ or $a=-4$ is a basic feature of arithmetic, and no further explanation of why this works is required. More generally, it should be clear that $\sqrt{a^2}$ is either $a$ or $-a$, depending on whether $a$ is positive or negative. Bringing the absolute value into the explanation sort of muddles the issue. $\endgroup$ – Jim Belk Mar 31 '14 at 16:06
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    $\begingroup$ That being said, I think this is a fine question. For example, I certainly run into this sort of problem when teaching determinants in linear algebra. Sometimes it's just more important to teach how something works than to teach why it works that way. Even in calculus, the substance of the differentiation rules is much more important than the reasons that they are correct. $\endgroup$ – Jim Belk Mar 31 '14 at 16:08
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    $\begingroup$ I'm a little surprised at the initial belittling of the example I gave, on a math educators site. If you haven't taught this level of algebra to students, I'd suggest that you should be very careful about which concepts from this level of algebra you claim are "pretty simple!" In the same section of the text, the students learn to discard extraneous solutions caused by squaring both sides of an equation. This is caused by the same "pretty simple" issue, but we all should be able to recognize that, by the intermediate value theorem, each person has a moment where they only half-understand this. $\endgroup$ – Chris Cunningham Mar 31 '14 at 18:31
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    $\begingroup$ Barring any Eureka moments, of course. ;) $\endgroup$ – Chris Cunningham Mar 31 '14 at 18:32
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Framework for Concept Introduction: A general framework can be adapted from the Zone of Proximal Development or ZPD, which is an instructional concept (due initially to Lev Vygotsky) that continues to evolve, especially in its treatment within the world of education and relation to scaffolding.

The Wikipedia definition provided (quoting Vygotsky) is:

the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers

So the two constraints of (1) whether or not the concept falls within a student's ZPD, and (2) how much time can be spent on working with the student, are probably enough to give a very basic framework for introducing a concept. In particular, if either of these constraints is not satisfied, then the concept should be avoided for the time being.

This approach is admittedly simplistic, since there are lots of things that could be covered in a course, i.e., that are potentially within students' ken given enough time, but at the expense of other material. To develop such a framework further, then, one must at least have some idea of the course goals.

Frogs, Birds, and Chunking: The "frogs and birds" metaphor refers to a notion introduced by Freeman Dyson in a talk where he begins:

Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time. I happen to be a frog, but many of my best friends are birds. The main theme of my talk tonight is this. Mathematics needs both birds and frogs. Mathematics is rich and beautiful because birds give it broad visions and frogs give it intricate details. Mathematics is both great art and important science, because it combines generality of concepts with depth of structures. It is stupid to claim that birds are better than frogs because they see farther, or that frogs are better than birds because they see deeper. The world of mathematics is both broad and deep, and we need birds and frogs working together to explore it (p. 212).

Dyson is discussing mathematicians, but you may wish to consider whether or not the discussion can be applied to students of mathematics, as well. Something like: Can students in mathematics classes be meaningfully divided into tadpoles and hatchlings?

My sense is that the answer, generally, is no: There might exist certain preferences, but early on students are not to be classified in this way. Instead, you must elect to provide a mix of "concepts that unify our thinking" and "details of particular objects." To share my own perception (and perhaps out myself as somewhat ranine), I think one must alternate between minor details, as objects are introduced, and broader comments as students' arsenal of examples grows.

With regard to "chunking," I refer to studies on chessboard perceptions; an early one, comparing the ability to reconstruct positions among players at different levels, can be found here. In this study and others that follow, it seems that experts are able to "chunk" concepts in a different way than novices. This can be helpful in situations that rely on one's memory capacity, but is also helpful more generally when clumping together broad ideas that can then be deconstructed.

To clarify: I bring up these three items (frogs, birds, chunking) to suggest that students be taught about new objects and some of the related details (frogs), and that more general conceptual explanations (birds) be introduced to unify the objects in a way that allows for better chunking. This allows for mastery to be attained in a domain (here, mathematics).

Loose Ends: Your question is actually four different ones.

  1. Is there any good framework for deciding when it is a good idea to "skip the concept" and "just teach the method," knowing that students will appreciate the concept only much later in life, or possibly never?

  2. Do you really show children why long division works?

  3. And if you do, do you intend them to retain that concept?

  4. Why is this skipping-of-the-concept sometimes good and sometimes bad?

I hope that I have provided helpful thoughts about 1, though I make no remarks here about utility; this is a needed consideration, but difficult to pin down when you do not know students' future paths. The answers to 2 and 3 are yes, assuming that the students have sufficient procedural fluency for this concept to be within their ZPD.

As for 4, skipping a concept can be bad because (as indicated above) it robs students of the opportunity to gain the bird's-eye-view that leads to chunking and, eventually, mastery/expertise. Skipping a concept can be good because, in most cases, an individual item covered can be generalized far beyond what could reasonably be expected in any single course. (Consider an example like Bloch's principle.)

As a final note, I think a good example of a concept that should be initially skipped is why you can write any rational number in lowest terms. It is true that, for a given rational number, you will be able to do this. However, to assert this for all rational numbers, one will need approximately the strength of the Principle of Mathematical Induction (PMI) or the Well-Ordering Principle (WOP). When teaching students (whether in elementary school or a community college developmental mathematics course) about rational numbers and discussing the process of simplification, I would not start talking about PMI or WOP.

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  • $\begingroup$ Induction and Well-Orderings are hard, but Least Numbers are pretty easy: "If some natural number has a property, there is a least natural number with that property." E.g.: If there is an even natural number (like 6), there is a least even natural number (like 2). It even has the rhyming name of the "minimal criminal". Starting from that principle, the existence of lowest terms is about as easy as proofs get. $\endgroup$ – user173 Apr 15 '14 at 2:36
  • $\begingroup$ If you want to talk about any arbitrary natural number, you already will need induction because natural numbers are inductively defined. But besides that, you don't need induction for lowest terms. Just consider all possible fractions equal to the original and with denominator not larger than the original and take the one that has the smallest denominator. It is just a finite set for any given instance and it is easy for students to see that there must be a lowest possible denominator. No induction is actually necessary for any particular given fraction. $\endgroup$ – user21820 Jun 6 '14 at 13:13
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I think the precise wording of the question accidentally prejudices the potential answers. Namely, at all levels of fanciness-of-math, and at all levels of development of kids'/peoples' thinking, the thread of "how can we do these things" and the thread of "why did that work to accomplish our goals?" and the thread of "should I be curious about both discovery of, and substance of, discovery of amazing algorithms or principles ??!??" should not be put in apparent conflict.

At every stage of development, few people are able to "ask truly insightful questions", and, perhaps, to understand answers to those questions which they haven't managed to ask.

Nevertheless, I think mathematics teachers should encourage questioning, while not making it "mandatory". Sketch explanations and history without grilling kids on it. Make it clear that mathematics is a human endeavor, can be legitimately questioned, but/and has many good answers. Many kids will not care much, but the meta-point that asking questions is legit is important to make by itself.

For that matter, with a talented expositor, the "why" can very often be made far more interesting than "drill in execution of algorithm". In my own direct experience, going back to childhood, the execution was not difficult, and was completely boring apart from the "points scored" with authorities for perfect papers, for being quick in "blackboard races", and so on. And, yes, it is sometimes (not always) useful to have a quick mind and good memory... but those are not things that are directly taught, and by now it seems to me perverse to "test" or "teach" in ways that too heavily reward inalterable natural features.

Although I suspect many kids are ignoring what the teacher's saying anyway, statements of principle that apparently legitimize skepticism or disinterest might be good PR for "the Establishment", in any case, as opposed to continued emphasis on obedience to authority.

The point that various algorithms actually succeed, not merely because of some authority, but because of fact, should be the goal. And that we might like to have an acquaintance with these remarkably reliable truths.

So I'd not want to pre-emptively refuse to explain the "why"... at least acknowledge that there is an answer, and a good answer, to the "why", for any interested parties. And the answers are "right" not by authority of the teacher, but from reality. In fact, this is an amazing point in comparison and contrast to much else that people see these days (as well as days long gone).

So: opportunities for further explanations for interested parties, and admission in general that there are explanations for those interested. Emphasis that it's not "rules from authority", but "rules as description of reality".

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Here are my responses for the examples you and Jim Belk gave:

For $(x-3)^2=16$, I would explain by graphing $y=(x-3)^2$ and discussing it.

For long division, I would not explain, but use that time for calculator-based examples. (If Justice Scalia has been on the Supreme Court for 10,000 days, how many years is that? If a pest advances by 100 ft every day, how many miles will it advance in a year?)

For determinants, I would promise an explanation much later: "if you want to know why $\det(AB)=\det(A)\det(B)$, you can look up exterior algebras, or go to math grad school and get a good explanation there."

You can probably reason other cases by analogy with these examples.

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    $\begingroup$ Upvoting for graphing. A lot of students that struggle with formulas really seem to benefit from graphs. $\endgroup$ – Brian Rushton Apr 1 '14 at 2:03
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    $\begingroup$ For determinants, why is a quick sentence or two about first stretching volumes by a factor of det(A) and then by det(b) inappropriate? $\endgroup$ – Steven Gubkin Apr 11 '14 at 22:34
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    $\begingroup$ @StevenGubkin, I doubt that many students in linear algebra would get much from a quick sentence or two on that topic. I am thinking particularly of the econ majors I taught. $\endgroup$ – user173 Apr 12 '14 at 2:48
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The criteria for deciding when to teach the "why" in addition to the "how" has to be how complicated the "why" is relative to what the students can reasonably handle. What the students can reasonably handle is dependent both on their age and their educational background.

For elementary age kids, the "why" is almost never called for. They would not appreciate it, almost certainly wouldn't understand it, and have no need of it. I remember the John Milton quote, "...forcing the empty wits of children to compose Themes, verses, and Orations, which are the acts of ripest judgement and the final work of a head filled by long reading, and observing, with elegant maxims, and copious invention. These are not matters to be wrung from poor striplings, like blood out of the nose, or the plucking of untimely fruit." - from "Of Education".

The analogy in mathematics is that the "why" should not be forced on very young children.

For middle school on up, the "why" is much more important. The kids are going to ask about the why, and it is definitely time to begin asnwering those questions. Even then, in middle school, following the logic of the mathematics is more important than begin able to come up with the logic themselves. The imagination, that most important faculty of the mathematician, is somewhat dormant in the middle school years. Without it, you simply cannot write proofs.

Finally, in the high school years, you have kids who have hopefully learned how to follow logic, but now have recovered somewhat of their imaginations, and can see how to get from A to B.

These ideas, by the way, do not originate with me, except insofar as I have applied them to teaching mathematics. The ideas themselves are from the Dorothy Sayers speech "The Lost Tools of Learning", which you can find online.

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    $\begingroup$ I disagree wrt elementary students. I think we should be teaching elementary kids how numbers work, and giving them a good sense of numeracy, not teaching recipes for how to carry out a meaningless algorithm. $\endgroup$ – PurpleVermont Apr 11 '14 at 21:02
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    $\begingroup$ @PurpleVermont: The wording "teaching recipes for how to carry out a meaningless algorithm" simply assumes the algorithms are meaningless. The standard algorithm for multiplying numbers or for long division is far from meaningless: they allow you to carry out two of the most important operations in arithmetic! I'm sorry, but I can't teach calculus to kids who can't multiply. They must be so familiar with the basic operations, that they can handle the higher abstractions that calculus requires. I don't object to teaching numeracy, but you have a false dilemma. $\endgroup$ – Adrian Keister Apr 17 '14 at 11:18
  • $\begingroup$ I disagree that you need kids who can carry out long multiplication and long division to be able to learn calculus. Anything more than simple mental math they can do on a calculator. You want them to be familiar with the meaning of multiplication and division, not the standard algorithms. If you could only have one (numeracy or facility with the algorithm), I'd suggest you pick numeracy :) $\endgroup$ – PurpleVermont Apr 17 '14 at 23:25
  • $\begingroup$ @PurpleVermont: I still insist it's a false dilemma. Moreover, I don't think the data agrees with your position. You might find this post interesting: joelonsoftware.com/articles/GuerrillaInterviewing3.html. Scroll down about 2/3 of the way when the author starts talking about Serge Lang. $\endgroup$ – Adrian Keister Apr 18 '14 at 3:12
  • $\begingroup$ I don't think teaching the reasons precludes a student becoming good at doing the basics quickly. Yes, by the time they get to calculus, I'd expect the students to be able to do basic algebra quickly and effortlessly. But I'd also expect them to be able to explain why what they are doing is "legal". If they don't know that, it will likely come back to bite them later. I agree, btw, that it's a false dilemma. I was simply rejecting the claim that elementary students don't need to learn the "why" of what they are doing. I think the ones who do learn it will be best off, ultimately. $\endgroup$ – PurpleVermont Apr 19 '14 at 2:50
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I see the solution of $(x-3)^2=1$ as $x-3 = \pm 1$ hence $x = 3 \pm 1$ as the proper use of a theorem. In particular:

Theorem: If $a,b \in \mathbb{R}$ and $a^2=b$ then $a = \pm \sqrt{b}$.

Proof: Suppose $a,b \in \mathbb{R}$ and $a^2=b$. Observe $a^2 \geq 0$ hence $b \geq 0$ thus $b=(\sqrt{b})^2$. Note then $a^2-\sqrt{b}^2=0$ hence $(a-\sqrt{b})(a+\sqrt{b})=0$. But, the product of two real numbers is zero iff one of the factors is zero. Thus, $a-\sqrt{b}=0$ or $a+\sqrt{b}=0$. Consequently, $a = \pm \sqrt{b}$. $\Box$

On occasion you may encounter an instructor who says $x^2=4$ implies $x = \pm 2$ is not good work. Instead, you must show $x^2-4=(x-2)(x+2)=0$ thus $x = \pm 2$. Personally, I reject this philosophy because I think as we mature we ought to learn from our previous work and as we converse with individuals of similar training there is no need to rehash things which are already settled.

When you first learn factoring you ought to be forced to prove this theorem a few times. However, sometime, not too long after that, we should also learn that those details are so obvious that there is no common expectation those details are shown. In the same way, we teach difference quotients to begin our discussion of derivatives. But, only a silly person insists to use difference quotients and limits on problems which allow algebraic techniques from the family of theorems we develop early in calculus.

I want students who leave my courses to know for certain the best way to calculate standard problems. If a substandard method is needed to motivate the definition then I want to make certain the demarcation between the motivating substructure and the optimal (theorem-based) calculation is crystal clear. I see the same for algebra: there are tricks (theorems) which help us to see past multiple steps which are closely tied to axioms of real numbers. If you can jump reliably from one point to another then it hardly makes sense to tip-toe just because you can.

In fact, it is just as important to know theorems as to be able to prove them. This continues to be true at higher levels of mathematics. Unfortunately, this reality is sometimes clouded by our tendency to over-emphasize the definition-proof aspect of the subject over the intuition-theorem-application of higher courses.

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