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As a student in high school, I never bothered to memorize equations or methods of solving, rather I would try to identify the logic behind the operations and apply them. However, now that I've begun to teach Algebra in high school, I find it rather frustrating when students either a) memorize methods of solving the textbook problems or b) look for a general formula/method to "just plug in to"

I've tried to throw them curveballs as my old Algebra teacher did, but usually they just dismiss it as "a weird problem" and continue using whatever method they have been.

My objection to A is that it often impedes actual learning. Upon seeing a chunk of 6 similar problems in the textbook, many students just apply the same steps to every problem in the section (and usually get quite a couple wrong).

My objection to B is that from my experience, students who flat out memorize equations (like $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ for quadratic equations) often fail to extend the same logic (completing the square/simplification) when faced with different but similar problems. They also frequently misapply the "magic formulas" they were taught before (i.e. solving simple quartics $ax^4+bx^2+c$ with the quadratic formula) and needing plenty of prompting after the suggestion of substituting $x^2$.

This is the problem identified in this question and in particular the issue raised in this comment.

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up vote 5 down vote accepted

From my perspective, it is not just a simple "pro et contra memorization". Actually it is of great value to memorize formulas in general so that they are readily available. What may instead be the issues is rather

  1. How students memorize (reflected/un-reflected)
  2. What they do with the memorized stuff (reflect or not)

To go to the extreme, forbidding memorization would make you have to write up multiplication tables from scratch over and over again.

What I do as a teacher

Often times, I tell my students to first simply copy a proof from the textbook without thinking too much. This is a very first step of activating (even un-reflectedly) what the book says. This gets even the weakest of the students started. But copying was never the goal itself.

Next step is to copy the proof again while trying to figure out the steps. Also in particular identifying the first step that the student is unable to comprehend or uncertain about. If the students have good memories they might start to reflect upon which previous methods could be into play.

The ultimate goal is to break things down by reflecting until the memorizing can be condensed to a simple core of references to previous methods and an idea of the overall scheme of the proof.

My thoughts on the learning process

I think that memorizing and reflecting sometimes belongs to different situations when learning mathematics, as it does in other subjects.

Imagine you should learn to play the piano and went on forever analyzing and reflecting upon the way your fingers were acting. That would make the whole process slow and tedious. On the other hand, returning to the reflection of what your fingers do when playing at certain recurring occasions will be quite beneficial.

The same way I think about learning mathematics. Some of the time you should simply just do calculations with methods you have memorized, also to reinforce that you are capable of using those methods. At other times you should engage more deeply into reflection.

About curveballs

I used to throw those all the time too. I found out that it muddled the students distinction of when I was teaching them a method and when I was just throwing extra challenges for the especially gifted.

Also be careful how curved the ball is. If someone throw a ball at you, you will automatically at least consider catching it (or move). But being forced to move all the time instead of catching will enforce the idea that "this is something that other more talented than me would be doing". In other words "I am not good with maths".

Recently I have begun to throw the curveballs in a not so curved way and not aiming at anyone specific, but actually telling them "this is something you might try, but it is quite difficult". Then if someone catches my not-even-so-much-of-a-curveball they will feel "king of the world of mathematics".

I hope others will answer your question as well. I do not consider myself a very good or trained teacher. But the thoughts above corresponds to the experience that I have gained so far.

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I hope my English is not too bad... I am Danish so not a native speaker. – String Jan 8 '14 at 10:13
+1 for stating that memorizing has great value in general. – Georges Elencwajg Jan 8 '14 at 10:28

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