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I can never figure out what should be a memorization concept and what should be in a reference table. For example, in calculus, you are expected to memorize all the derivatives and integrals but in real life, you can always just look them up.

In some math classes, like stats and differential equations, you use statistical tables (and they also give you the formulas) and Laplace tables.

Is there a list of things?

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    $\begingroup$ For what it's worth, the z-score (and similar) tables that are used in statistics are slowly on their way out in the same way that sine tables are more-or-less gone from trig. Both are simply lookup tables for function values; there is very little value added to having a human in the loop. In contrast, choice of integration technique can yield very different seeming forms suited to different applications. $\endgroup$
    – Adam
    Apr 3 at 20:53
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    $\begingroup$ There is no such thing. The ability to memorize, and recall, material is closely related to intelligence. You might as well ask which openings and midgame positions in Chess should be memorized. $\endgroup$ Apr 4 at 11:37
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    $\begingroup$ @CarlWitthoft: If your goal was simply to play chess as effectively as possible, with no regard to artificial restrictions like "fairness" or "sportsmanship", you could skip a lot of the memorization and just have a good computer chess engine handy to evaluate positions for you and suggest promising lines of play. Obviously that would be cheating in a normal chess game, just like using a CAS would be cheating in most math exams. But actual working mathematicians nowadays use computers all the time, because for a lot of tasks (not all!) they're just better than the old Mk1 human brain. $\endgroup$ Apr 4 at 12:25
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    $\begingroup$ One of the advantages of memorization is learning intuition. Sure, you could just add things up with a calculator, but if you don't know your addition tables and the carry rule, it would be harder to find input mistakes or glitches, or even know one happened in the first place. Similarly, you could just grind your integrals through Mathematica, but if you don't know your antiderivatives and a few solving techniques, it would be harder to find input mistakes or glitches, or even know one happened. $\endgroup$
    – No Name
    Apr 4 at 13:46
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    $\begingroup$ One thing which could be very useful is to distinguish memorization from internalization. Memorization simply means you have it in your memory, and can regurgitate it on command. Internalization means you can do something with it... and I find those concepts less correlated than one might think. I internalized how to work with quadratic equations long before I memorized the quadratic equation (so much so that, when needed, I'd just derive it). By contrast, I memorized my Laplace transforms because I needed to apply them in that form -- I still don't feel I have internalized them! $\endgroup$
    – Cort Ammon
    Apr 4 at 21:00

6 Answers 6

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The goal of memorization is to reduce cognitive load. If a student plans on using derivatives as part of a larger task, and doesn't have them memorized, they need to interrupt their thought process either with looking up the derivative or working it out by hand. Whether this is useful for a student depends on what sort of applications of the derivative they plan on working with in the future, and so will vary from student to student.

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    $\begingroup$ I don't have an answer for this question, but this seems very narrow/specific for the amount of upvotes it got... $\endgroup$
    – Jasper
    Apr 4 at 8:41
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    $\begingroup$ I disagree. The point of memorization is to be able to associate concepts and values internally, thus leading to better understanding and (potentially) discoveries. $\endgroup$ Apr 4 at 11:39
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    $\begingroup$ @Jasper, I don't think this is as narrow as it seems at first glance. You could apply this to elementary students who are asked to memorize their times tables. Yes they could look it up, but if they are simplifying fractions or finding common denominators it will useful to have the multiplication/division facts at their fingertips so they don't have to interrupt their thought process to look up facts. $\endgroup$
    – Amy B
    Apr 4 at 13:38
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    $\begingroup$ Carl: this is an important point! Reducing the cognitive load involved is vital to being able to work with ideas internally, to form connections and make discoveries. $\endgroup$
    – TomKern
    Apr 4 at 16:05
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    $\begingroup$ I have come to agree with @Jasper that this post does not really answer the question. In particular, the 2nd sentence does not hold for the contexts posed in the question. The given examples suggest that reference tables are used once the immediate mathematical task grows sufficiently cumbersome or sufficiently secondary to the main purpose. However, it's not clear where the line should be drawn. $\endgroup$
    – Steve
    Apr 5 at 12:17
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The reasons are manifold.

  1. One is cognitive load (@TomKern’s answer) and the distraction of looking things up sometimes causes the solution you’ve almost constructed to fall to pieces. This is usually the first thing I tell the students because, to their way of thinking, it has an immediate practical import, and to motivate them to do things that help them learn is my goal.

  2. Another is that if you don’t memorize things, you won’t know what to look up. Maybe some of us have never had a student who saw $\sin^2 x + \cos^2 x$ in the formula they had derived and didn’t know what to do. It’s just not obvious that there is anything to be looked up. In grad school, you learn some diligence and, if your advisor doesn’t have the answer, might spend some time each day doing (literature) research in the library, in the hopes that someone has published a paper on what to do. Undergraduates, at least the few who bring me such questions, do not have such perseverance, I’d say “luckily” because it’s a painfully slow process to find out something you should have already known.

  3. If you haven’t memorized things you use often, you will waste a lot of time looking them up. What you do changes as you go through life, so what you do often, that is, what you need to have memorized will also change. You need to have certain facts about first-semester calculus memorized in order to efficiently learn second-semester calculus. Similarly for higher-level courses. When a student gets a job, they will have to memorize new things, things they have to memorize just for that job and nothing else. It’s kind of like in calculus, except worse because of the narrowness of its scope and applicability. In a college friend’s first job, they (a team) had to go in and analyze a company in a week and make a proposal to a group that wanted to invest or buy the company. They had to memorize a ton a facts about this company that they would forget in a week or two, when they had to do it again for another company. You can’t keep such a job if you’re bad at memorizing. Students should take the opportunity to learn this useful skill, and teachers should help them with it.
    3’. Similarly, when one is speaking live with someone, like a boss or in a job interview, you don’t want to have to say, “Let me look that up.”

  4. Finally, a major reason is @CarlWitthoft’s point, that memorizing helps to relate concepts and discover relationships. The foregoing examples underscore this, but it is worth emphasizing because it relates to problem-solving, if I may call it that, at all levels, whether one is working at the frontiers of mathematics to prove new theorems, or for a business, or for a client. That some things are worth memorizing in certain situations means that there must be things that are not worth memorizing. What they are is particular to each situation and furthermore, particular to one’s future, which is uncertain. So teach people to have a solid foundation in basic principles, and they can learn what they need as time unfolds.

[Sorry about the switching around of “you,” “one,” “students,” etc. I seem unable to stick to a point of view this morning. The “you” here probably always means the learner, not the teacher, whether the learner is in school or working a job, even a teaching job. Funny thought just occurred to me: Imagine a teacher thought think-pair-share was a good thing to do and said to the class, “Let me just look that up.”]

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    $\begingroup$ I can vouch for your second reason from bitter experience. I initially avoided memorising all the various trigonometric identities (double-angle formulas &c), because I knew I could derive them when needed. I didn't realise that exam questions might need you to recognise them and apply them in reverse — something that can't be done as needed. (Luckily, I discovered this in mock exams, and was able to memorise what I needed before the real ones.) $\endgroup$
    – gidds
    Apr 4 at 16:46
  • $\begingroup$ @gidds: I contest your example: If you had only rote-memorised those identities for prompted regurgitation, you would not have acquired the ability to recognise them. The thing you actually learnt was the technique of applying a reverse identity and probably some patterns to spot right-hand sides of trigonometric identities. Once you had spotted such a case, you might as well have derived all of them from scratch to see which fits (which would be inefficient in an exam). If the exam had provided a table of identities, it would not have become easier with respect to the skills it should assess. $\endgroup$
    – Wrzlprmft
    Apr 5 at 6:32
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    $\begingroup$ Another example for point 2: if you haven't memorised (a+b)² = a² + 2ab + b², then you can always look it up or redevelop it manually when you have to develop (a+b)²; but you won't know what to look up if you have to factor a² + 2ab + b². There are many examples like this where one direction is easy, but the reverse direction is relatively hard even with a look-up table. $\endgroup$
    – Stef
    Apr 5 at 9:59
  • $\begingroup$ Excellent answer and summary. If this were my answer, I'd additionally tie in NoName's and CortAmmon's points about intuition and internalisation into #4, then bump it as the first point. -) $\endgroup$
    – ryang
    Apr 5 at 15:05
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I somewhat wish to contest Tom Kern’s answer:

The goal of most education is an internalised understanding, which in turn reduces cognitive load when building upon what you learnt (which in turn may be a deeper understanding of the same subject). Memorisation of some things almost inevitably happens on the way there, in particular when solving exercises. It may also be an important tool on the way (example below). Finally, seeking for efficient memorisation may structure one’s knowledge and expose patterns and similar.

However, memorisation alone rarely causes understanding or a sufficient reduction of cognitive load for application. In particular, students who only rote-memorise can (and often do) completely miss the goal of a course and can be completely lost when they need to build upon their understanding on the next level. And even if they can solve a task, they may need much more time to apply their memorised knowledge.

For example, when teaching quadratic equations, the goal is not that students can regurgitate a quadratic formula, but that they know when to apply it, transform a problem so they can apply it, exclude implausible solutions, get a better understanding of equations and logic in general, etc. Now, when learning all of these things through exercises and similar, having a quadratic formula readily available is extremely useful – so useful that you indeed might want to instruct students to memorise it as soon as it comes up (and motivate why). However, even if you don’t, students who seriously work on exercises and similar will inevitably memorise it over the course.

Thus, at the end of the teaching unit, not having memorised a quadratic formula is a strong indicator that a student also missed the goals of the course (or is severely underchallenged). However, the reverse does not hold: Students can have memorised that formula without reaching the goals of the course. An exam needs to assess whether students are able to properly apply the quadratic formula, not whether they can regurgitate it. It should not make much of a difference whether you write a quadratic formula on the blackboard or not – with the main exception probably being students who suffer from anxiety and for whom not having to memorise the equation will provide security and a ward against memorisation flukes.

I would therefore only instruct students to memorise something (or even assess their memorisation) if all of the following apply:

  • it helps achieving the actual learning goal (which is never the memorisation itself),
  • you expect that for some students memorisation as a side effect (of doing exercises and similar) will be not optimal,
  • you do not want to leave figuring out the above to the students, either because the usefulness of memorisation is not apparent a priori or they are not sufficiently independent learners yet. (Here, you need to compromise between your students learning math or learning how to learn.)

Otherwise, I do not see a need to make a distinction.

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  • $\begingroup$ thank you so much for your input, I very much agree with this. I realize that maybe it isn't a clear answer because while memorization is important - students fail to actually do it 'properly' so to speak $\endgroup$
    – Lenny
    Apr 5 at 11:50
  • $\begingroup$ @Lenny: I realize that maybe it isn't a clear answer – I am not sure what you are referring to. If there is anything you would like me to clarify, please let me know. $\endgroup$
    – Wrzlprmft
    Apr 5 at 12:22
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    $\begingroup$ @Lenny I worry that "memorisation alone" is sometimes taken as the focus of the question, rather than memorisation as a tool or activity among many tools and learning activities employed by a teacher and their students in a course. I took the question to mean, when, if ever, is memorization an efficient and effective aid in learning, and, when it is, how is it best employed? That's not how you expressed it, but I don't think a particular activity such as memorization can be adequately discussed outside the environment in which it is used. $\endgroup$
    – Raciquel
    Apr 5 at 14:36
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I completely agree with Tom's answer (the ratinonale). It is much easier to consult a table for something you learned before than for something you never learned.

But to answer the specific question. For a typical calculus student they should learn and have facility with all the standard integrals, methods in the general calculus book, during the course year. I.e. able to work problems and take tests, closed book and closed Mathematica. Maybe years later, they have to look up the arc length formula (or its derivation), but it will be much easier than if they never learned it, at all. I.e. No, "we could just look it up" is not the right approach when first learning the material.

(For the above, I'm assuming something like a general "101" class for STEM or undifferntiated students, using Thomas or the like. American examples, but philosophy applies generally.) There are a tiny amount of students taking crazy hard stuff at Harvard or CalTech...but that is not germane to the general answer. A more substantial portion will take calculus for business (or the like). To which the general philosophy still applies, but the amount of tricks and such will be smaller (no trig substitutions, e.g.)

Note also that calculus class and the manipulation of hand solving equations drives great facility and familiarity with algebra and trig. As a tangential benefit.

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    $\begingroup$ Having a list of all the standard integrals is fine as long as you have done the work to work them out as well. "Here are the integrals, prove them." Particularly the case for the inverse trig and inverse hyperbolics -- and be rigorous with the ranges. $\endgroup$ Apr 4 at 22:28
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I'm not an educator, but I think there's a simpler answer than those given before: Memorize the things you'll need to use frequently, look up the ones that are less common.

This is somewhat related to TomKern's answer, as frequent interruptions to look something up will cause the student to lose their train of thought in the larger problem.

In practice, this is likely to happen naturally for many (most?) students. At the beginning they have nothing memorized, and they'll have to look up or work out everything. Some of these things will come up frequently, and they'll remember the result from previous lookups.

I was never good at deliberately trying to memorize things. But when you use concepts in context frequently, they eventually become second nature.

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    $\begingroup$ It's not so much "memorise the things you need to use frequently", as "If you use a thing frequently enough, you will remember it." $\endgroup$ Apr 4 at 21:15
  • $\begingroup$ @PrimeMover That's my point in the third paragraph. The question appears to assume the teacher tells the students to "Memorize these". Like when I was in high school, I had an English teacher who required us to memorize all the prepositions, and quizzed us on it each week. $\endgroup$
    – Barmar
    Apr 4 at 21:20
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    $\begingroup$ @PrimeMover The word "enough" forestalls any criticism of the sufficiently of a use-only approach, but I doubt it's the most efficient way to learn in all cases. I've certainly had students fail to recall things even though they should have had enough practice, even ones who have done more than the assigned homework. In some cases, I doubt the student would have been able to do enough, even if an ideal "enough" existed for them. All needed help learning how to commit things to memory, and the latter group needed something more besides. $\endgroup$
    – Raciquel
    Apr 5 at 15:03
  • $\begingroup$ @Raciquel But would forcing them to memorize allow them to learn it any better? If they don't grasp it, it's hard to overcome that. $\endgroup$
    – Barmar
    Apr 5 at 15:16
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    $\begingroup$ @Raciquel: I'm not sure how to force someone to memorize – “Open your books on Page 1 and memorize the copyright information. You will be quizzed on this.” $\endgroup$
    – Wrzlprmft
    Apr 5 at 15:26
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For example, in calculus, you are expected to memorize all the derivatives and integrals

Since when? A student who memorizes these things is memorizing instead of learning. I mastered calculus by understanding the subject and I gave no effort to memorizing. I know the derivatives and integrals because I learned them; if I had memorized them instead then I would not know them.

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    $\begingroup$ I know the derivatives and integrals because I learned them; if I had memorized them instead […] – This seems to use a different definition of memorise than dictionaries and everybody else here. You may have memorised as a side effect of actively working with them rather than deliberate rote memorisation, but you still committed them to your memory somehow. Otherwise you wouldn’t know them. $\endgroup$
    – Wrzlprmft
    Apr 7 at 6:43
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    $\begingroup$ @MichaelHardy, do you need to consult a table to determine $\frac{\mathrm{d}}{\mathrm{d}x}\left(x^2\right)$? $\endgroup$ Apr 7 at 9:34
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    $\begingroup$ @JoelReyesNoche : No, I don't. What is the point of your question? $\endgroup$ Apr 7 at 16:17
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    $\begingroup$ @MichaelHardy, right now, how do you (Michael Hardy) know how to find the derivative of $x^2$ if you (Michael Hardy) didn't memorize it? If you remember it, then you must have memorized it, right? $\endgroup$ Apr 8 at 0:44
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    $\begingroup$ @BCLC, the terms that you mention are relevant. Michael Hardy, the point I'm trying to make is that while one can memorize something and not understand it, it is also possible to memorize something and still understand it. $\endgroup$ Apr 8 at 1:52

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