I agree with you that procedural knowledge is frequently based upon memorization. However you then use this other term: understanding. Are you sure that understanding is distinct from memorization of lots of related concepts and then drawing inferences of the relations between those concepts? Possibly understanding is the memorization of certain concepts which can be applied to a variety of other concepts.
Edit—To put this another way it seems like you’re saying we focus too much on crystallized intelligence and not enough on fluid intelligence however it seems to be harder to increase fluid intelligence, and it seems to me that increases in crystallized intelligence can at least partly compensate for deficiencies in fluid intelligence.
I guess these things are not sharply divided, but for me the difference between “memorization” and “understanding” is whether the fact is isolated or connected to a larger network, and whether if you forget it you have a chance to rediscover it.
As a consequence, when people use “memorization”, the more they know, the harder it becomes, because they have a longer list of facts to remember. While if they have “understanding”, the more they know, the larger is the existing network where they can plug the new things. Human memory is built in a way that makes it easier to recall things that are connected to other things.
A good way to teach math is to have students discover various things, starting at elementary school. You achieve it mostly by providing problems where the solution is already within their reach. The art is to take each “inferential step” and try to split it into multiple smaller substeps whenever possible; and then you just create a list of problems that make the student discover the individual substeps one by one. (The other important aspect is continuous debugging: you ask the students to explain their solutions by “thinking loudly” i.e. being explicit about everything they do, and you observe whether their generated model is correct. If they mess up something, e.g. create a wrong generalization, it is best to provide a problem where their approach will be obviously wrong, so they notice and correct themselves. Another method, if at least some students in the class got the model right, is to let the class debate.)
If you do it right, not only will the students get deep understanding, but they will also like the math, because it will feel like something they discovered for themselves (the IKEA effect?), instead of something that an authority told them to believe (suggesting lower status).
Unfortunately, for many teachers it is difficult to use this method correctly, if they are used to teach (and learn) by memorization. This is a recursive problem, because to use this method properly, you must understand why it works, instead of just trying to immitate the steps, immitate them incorrectly, and then see that your students got stuck and didn’t discover the thing they were supposed to discover. At that point there are two options: (a) say “screw it” and give the information to the students explicitly to memorize, and keep telling everyone that the new system doesn’t work; or (b) keep waiting until the magic happens, without doing the necessary steps that would make it happen, and later have the parents complain that their child is in the third grade and still cannot do the basic addition.
Are you sure that understanding is distinct from memorization of lots of related concepts and then drawing inferences of the relations between those concepts?
Some people will memorize a lot and still fail to draw the right inferences. Sometimes because they memorized some parts incorrectly, or missed/forgot some important parts. Sometimes it is compartmentalization; it doesn’t occur to them that some things can also be used in different contexts. Drawing some inferences from the very beginning is a more reliable approach.
it seems like you’re saying we focus too much on crystallized intelligence and not enough on fluid intelligence
I am not sure I use these terms correctly, but seems to me that the process of discovery of some concept requires fluid intelligence, but from then on, the concept itself becomes a part of crystallized intelligence. Actually, it seems the other way round: having the students draw inferences in many little steps requires them to only use a little fluid intelligence at a time; but remembering many unconnected facts and then having to discover the important patterns on their own would require greater fluid intelligence, but only once in a while. Many conveniently small gulps, versus a few large ones that make most people choke.
Ironically, there are two groups of students that seem to achieve the greatest benefits from the teaching method I am trying to describe here—those who do the mathematical olympiad, and those who completely suck at math. For the former, having a method that relies more on understanding than on memorization, allows them to go much further, and to use the skills in unusual contexts. For the latter, having a method that leads them in little steps is the only way to learn anything, instead of remaining stuck at the very beginning.
It is frequently the average student who complains about the method as unnecessary. Because the average students are already good at memorization (that’s what they do all the time at school), and the curricullum is more or less designed to fit into their memory (if that’s the style that most teachers and students use, expecting to learn more would be unrealistic).
I agree with you that procedural knowledge is frequently based upon memorization. However you then use this other term: understanding. Are you sure that understanding is distinct from memorization of lots of related concepts and then drawing inferences of the relations between those concepts? Possibly understanding is the memorization of certain concepts which can be applied to a variety of other concepts.
Edit—To put this another way it seems like you’re saying we focus too much on crystallized intelligence and not enough on fluid intelligence however it seems to be harder to increase fluid intelligence, and it seems to me that increases in crystallized intelligence can at least partly compensate for deficiencies in fluid intelligence.
I guess these things are not sharply divided, but for me the difference between “memorization” and “understanding” is whether the fact is isolated or connected to a larger network, and whether if you forget it you have a chance to rediscover it.
As a consequence, when people use “memorization”, the more they know, the harder it becomes, because they have a longer list of facts to remember. While if they have “understanding”, the more they know, the larger is the existing network where they can plug the new things. Human memory is built in a way that makes it easier to recall things that are connected to other things.
A good way to teach math is to have students discover various things, starting at elementary school. You achieve it mostly by providing problems where the solution is already within their reach. The art is to take each “inferential step” and try to split it into multiple smaller substeps whenever possible; and then you just create a list of problems that make the student discover the individual substeps one by one. (The other important aspect is continuous debugging: you ask the students to explain their solutions by “thinking loudly” i.e. being explicit about everything they do, and you observe whether their generated model is correct. If they mess up something, e.g. create a wrong generalization, it is best to provide a problem where their approach will be obviously wrong, so they notice and correct themselves. Another method, if at least some students in the class got the model right, is to let the class debate.)
If you do it right, not only will the students get deep understanding, but they will also like the math, because it will feel like something they discovered for themselves (the IKEA effect?), instead of something that an authority told them to believe (suggesting lower status).
Unfortunately, for many teachers it is difficult to use this method correctly, if they are used to teach (and learn) by memorization. This is a recursive problem, because to use this method properly, you must understand why it works, instead of just trying to immitate the steps, immitate them incorrectly, and then see that your students got stuck and didn’t discover the thing they were supposed to discover. At that point there are two options: (a) say “screw it” and give the information to the students explicitly to memorize, and keep telling everyone that the new system doesn’t work; or (b) keep waiting until the magic happens, without doing the necessary steps that would make it happen, and later have the parents complain that their child is in the third grade and still cannot do the basic addition.
Some people will memorize a lot and still fail to draw the right inferences. Sometimes because they memorized some parts incorrectly, or missed/forgot some important parts. Sometimes it is compartmentalization; it doesn’t occur to them that some things can also be used in different contexts. Drawing some inferences from the very beginning is a more reliable approach.
I am not sure I use these terms correctly, but seems to me that the process of discovery of some concept requires fluid intelligence, but from then on, the concept itself becomes a part of crystallized intelligence. Actually, it seems the other way round: having the students draw inferences in many little steps requires them to only use a little fluid intelligence at a time; but remembering many unconnected facts and then having to discover the important patterns on their own would require greater fluid intelligence, but only once in a while. Many conveniently small gulps, versus a few large ones that make most people choke.
Ironically, there are two groups of students that seem to achieve the greatest benefits from the teaching method I am trying to describe here—those who do the mathematical olympiad, and those who completely suck at math. For the former, having a method that relies more on understanding than on memorization, allows them to go much further, and to use the skills in unusual contexts. For the latter, having a method that leads them in little steps is the only way to learn anything, instead of remaining stuck at the very beginning.
It is frequently the average student who complains about the method as unnecessary. Because the average students are already good at memorization (that’s what they do all the time at school), and the curricullum is more or less designed to fit into their memory (if that’s the style that most teachers and students use, expecting to learn more would be unrealistic).