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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?
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.
The reasons are manifold.
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.
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.
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.”
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.”]
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:
Otherwise, I do not see a need to make a distinction.
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.
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.
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.
