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When I was learning the Pythagorean theorem(at time A), I was just told to memorize it. I used it often before trying to derive the equation(at time B), and I think actually I have forgotten the derivation now(denoted by time C). A is prior to B, and B is prior to C.

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I learned from The ABCs of How We Learn that chunking(in section D(deliberate practice)) takes two steps, namely proceduralization and automatization. In the first step, we need clear guidance(verbal control and explicit memory retrieval), but after that:

Interestingly, once procedures have been automatized, it is difficult for people to recover the original substeps.

While experts exhibit superior performance by definition, they may not be very good at decomposing and explaining their expertise.

Then my questions arise:

  1. I don't know if the derivation I did at time B is worthwhile?
  2. Have(had) I reached the step automatization in mastering the Pythagorean theorem at time C(between time A and time B, or between time B and time C)? What are the differences?
  3. If I will be unable to decompose and explain(in Daniel L. Schwartz's words in the book) the equation after automatization, can I just skip the derivation step and memorize the theorem(as a common sense, in my long-term memory) and apply it?
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    $\begingroup$ What do you mean by “mastering a theorem”? Proving or deriving a theorem is different from recognizing it, recalling it, or applying it. You may achieve different levels of proficiency at each skill, or proficiency at some but not all. $\endgroup$
    – Steve
    Apr 16, 2022 at 21:21
  • $\begingroup$ @Steve +1 So I think the theorem had not been automatized in my mind after time B. I think what you meant by different levels of proficiency is some theories like Bloom’s Taxonomy, but I don't know how to align the two steps of chunking with the six categories in that taxonomy yet. $\endgroup$ Apr 17, 2022 at 6:29
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    $\begingroup$ A general answer: Yes. You need to learn algebra (which includes equation derivation). This is the language of not only math, but also science and engineering. $\endgroup$ Apr 17, 2022 at 12:31
  • $\begingroup$ Let me guess, you learned it at age A = 15, you tried to derive it at age B = 20, and you are now C = 25? Diagram checks out. $\endgroup$
    – nanoman
    Apr 18, 2022 at 6:47
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    $\begingroup$ I don't think the derivative tag belongs on this question. $\endgroup$
    – shoover
    Apr 19, 2022 at 0:52

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I endorse Steve's comment about different skills and different levels of proficiency in different skills.

Also, if one at one time justified the Pythagorean theorem, but is no longer able to do so, I wouldn't say that the knowledge has been automatized; I'd say it's been forgotten as a result of disuse. Expertise in mathematics does become automatized, as it does in other areas of endeavor, but the automatizing is not forgetting why things are true or forgetting how to justify them. Proving statements and justifying steps are part of the everyday practice of the mathematics expert, and if one doesn't do those things, one isn't an expert.

On the other hand, automatization is manifested when an expert mathematician gravitates toward an effective proof strategy or a useful simplification without having to think about it, and without necessarily being able to consciously articulate how they knew the approach would be a productive one.

The question in the title is different from the one in the body of the post, but I should answer that one as well: yes, one does need to practice equation derivation.

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  • $\begingroup$ I have commented on Steve's comment, and I think I should delve deeper into the automatization step. $\endgroup$ Apr 17, 2022 at 6:32
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This question goes to the core of what mathematics is about. Benjamin Pierce wrote, "Mathematics is the science which draws necessary conclusions". So, deriving equations (as it's phrased here), is confirming that our conclusions are, indeed, necessary.

In theory, one could imagine an alternative path of faith-based/cargo-cult math, where people used principles or algorithms received from authority figures without justification. But that would miss the whole point of why mathematics is powerful, would lack any self-correction method, and likely wind up in a cul-de-sac where someone comes up with an incorrect principle and winds up disseminating a broken and malfunctioning process.

By way of analogy, one could likewise point at any physical or life science and say, "Why do I need experiments or evidence? Once the theory is written down, then I'll just memorize it."

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    $\begingroup$ I believe every mathematician also does some cargo cult. For instance, most would be unable to recite the basic principles of logic or set theory and how other conclusions follow from that. They just learned to use certain things without justification. $\endgroup$ Apr 17, 2022 at 5:05
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    $\begingroup$ @MichaelBächtold There is some truth in that, but I think the point is you don't really get a cargo cult if everyone keeps working their piece. So sure you get some mathematicians that couldn't manage to define basic set theory and build the reals from the basic axioms, (though I'm guessing a vast majority of decent research mathematicians would manage it given a couple months), but more importantly they don't apply the cargo-cult mentality to their own corner of the woods. And since you have overlapping areas of expertise and people are generally open and talk, you get self correction. $\endgroup$
    – DRF
    Apr 17, 2022 at 13:24
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    $\begingroup$ As a side note I've not done any actual math for over 10 years and I'm sure I could still manage to get to derivatives and integrals from union, power set, replacement, infinity, foundation, pairing, comprehension, extensionality and choice. Could do it without choice probably, but that's painful. $\endgroup$
    – DRF
    Apr 17, 2022 at 13:31
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I don't have access to the book mentioned in the OPs question so this answer will be based on the limited information present in the question.

There are many ways in which people "learn mathematics" and many different aspects of what is important depending on the learners focus and scope.

Some don't need to learn the fundamental intricacies of a theorem and instead want to focus on its use towards other things. Others choose to delve deep into those intricacies and find out how the theorem relates to the components it was derived from.

There is one thing in common that both groups need to be sure of however, namely Logical Validity and Soundness. For example for a theorem $A \implies B$ they need to be able to recognize and ascertain the two following conditions to make good use of the theorem.

  • The scope in which the theorem is stated. They need to be able to recognize when they can use the theorem and when they cannot. In this case they need to be able to ascertain whether $A$ is true or not for their use case.
  • The trust that the theorem has been proved, i.e. trust that there is a valid chain of logical reasoning that establishes that from $A$ you can imply $B$. (That is, whenever A is true then B must also be true.)

With that in mind we can try to answer your questions. Question 1 and 2 uses subjective measures ("worthwhile" and "mastered") so they will also have subjective answers.


1. Was it worthwhile to derive it at time B?

Lets consider the equation here. You've invested time and mental resources to derive the theorem and made sure there are no errors in the reasoning. From this, what have you gained?

In terms of the two bullet points above, you've ascertained that, indeed, there is a valid chain of reasoning from $A$ to $B$. You've also hopefully been able to reinforce your understanding of when $A$ is true and what pitfalls may arise that makes it not true. So you've been able to strengthen your trust in both the validity of the theorem as well as your ability to recognize its scope. This trust is something you will remember.


2. When does automatization happen?

I (the answer writer) can't decisively determine this since I don't have access to the book. So this must be taken with a grain of salt.

However, from OPs own statement:

In the first step, we need clear guidance(verbal control and explicit memory retrieval)

It seems that proceduralization is the activity of explicitly recalling and following the recipe needed to get from the premises to the conclusion.

Automatization seems to be the process of these steps getting internalized and processed implicitly without need of conscious effort to follow them.

Thus I think the answer is that Automatization can happen in any/all of the steps A, B and C.


3. Is it okay to skip the derivation and remember only the theorem?

A theorem that we learn in a mathematics book is itself a "landmark" result in a big forest of logically valid chains of reasoning. They are mentioned because they are noteworthy and useful.

This means that learning only the theorem $A \implies B$ itself is great as long as you are able to trust your understanding of the two bullet points. You don't need to commit the derivation to memory as long as you know the theorem itself holds.

However, if you are tasked with convincing someone else that the theorem $A \implies B$ is true, you need to be able to convince them of the two bullet points. Commiting every single derivation to long-term memory is something that is seldom done but in this case you might need that information. Most people probably remember some landmark steps of the derivation but not every single step.

The theorem itself seems to be an automatization you're talking about if one is used to do every step of the derivation. It is useful to you and for getting the job done. If you cannot give the detailed derivation yourself then, you should maintain a list of references towards the derivations or know how to look them up again. A book or handwritten notes are "memory" too, even if they aren't in your head.

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So my answer is "both", not either/or. Belt and suspenders. So, yes absolutely, you should have some mathematical formulas in ready access memory (Pythagorean formula, quadratic formula, some (not all) trig identities, etc.). They are very frequently used and you will struggle if having to rederive them, each time.

Note this does not mean only that you memorize them by short term recall (e.g. the way a plebe memorizes the menu for noon meal). Perhaps starting a little with that (or for initial exams). But a lot of the memorization is by repeated use within drill problems. But it is still memorization, not dependent on derivation.

That does NOT mean there is no value in deriving key formulas once or even a few times. For one thing, you're using a different part of your mind and are practicing algebra manipulation (a general skill needed and one that many students are weak at). Also, it will give you more confidence in the formula itself. And even give you better confidence and faster recall if you have to look a derivation up, itself (e.g. using completing the square for some aspect of calculus or diffyQs). Or of course the ability to re-derive occasionally, under pressure; this is the case with some of the trig identities. I think most people make a basic subconscious decision of how much trig to memorize and how much to rederive as needed. And it does not need to be the bare minumum (e.g. definitions of the functions) as the "derive everything" adherents advocate, but can be a select few like sinsq + cossq...without every random cscsq thingie.

P.s. I see this tendency to a false either/or fallacy amongst many questioners and answerers here. I think this is an aspect of being Mathematics (algorithmically) minded, versus Educators (psychologically) minded and assuming that humans (basically smarter dogs!) learn in the manner of a computer program. Almost always we hear here, "memorization bad", "derivation good". But, like I said this is a fallacy, doesn't pertain to pedagogy of real, organic brained, humans. Both is good. It's like if you ask me if you should practice shooting or scrimmage for basketball...both, man!

P.s.s. The "do we need to" (first three words of question title) is an obvious question, but also shows the wrong emphasis. Instead of trying to figure out the bare minimum, embrace doing extra (in all facets of learning). Of course, you can overdo it, it is an optimization problem. But in general, for most students, for most "non masters", they should be looking to do more (of whatever, drill, short term memorization, derivations), rather than less. "And", not "instead". Don't be so concerned about wasted effort. It will all pay off.

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The goal of learning mathematics is not to learn various theorems or to be able to use them.

Rather, the goal of learning mathematics is to learn how to derive new facts from old ones. The point is the skill of deriving, not the facts you derive.

Mathematics is a humanities subject, not a science one. When you study literature, you quickly realize memorizing an analysis of some poem done by someone else is kind of pointless; the idea is for you to come up with your own analysis, possibly of different poems. Mathematics is the same. Chunking and automizing facts makes sense for subjects where the point is the facts; mathematics is not one of those subjects.

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