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Anecdotally, I've heard it said that in (Australian) grades 11 and 12 calculus needs to be taught in a procedural way involving merely recipes for doing calculus, rather than teaching for understanding common in lower grades.

In primary school there is a lot of emphasis on relational understanding, perhaps at the expense of rote learning (eg times tables) or procedures (eg long division). But on reaching senior school this is not expected.

Is there too much content in the curriculum? Is there an emphasis on exam preparation, including revision at the expense of understanding? Is the content not amenable to relational understanding (cf complex numbers just are)? Perhaps relational understanding only comes after a mastery of procedural techniques (of differentiation and integration). Does it need just to be taken on faith? What role does proof play? What role do algorithms play (Differentiation if easy but integration is hard)? Is relational understanding even necessary? - historically calculus was used successfully without proper underpinnings. Is instrumental understanding necessary without introducing further abstraction or esoteric machinery (logic and set theory)?

Is anyone aware of arguments or research supporting any of these claims?

A related question is regarding teacher (and student) attitudes to these things.

(Of course, I believe in the importance of imparting a relational understanding, but (for an essay I was writing) I was looking for some evidence of teachers attitudes regarding the necessity of teaching calculus in an instrumental way.)


I'm sorry if any of you misinterpreted my inquiry as dissent. I was merely looking for evidence of what teachers actually thought. I was looking for a representative sample of what teachers thought not just the opinion of a few.

Also I was looking for evidence to support the proposal that calculus needs to be taught in a particular way. I wasn't making that proposal!

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    $\begingroup$ It certainly is easier, but I'm sure it isn't the best stuff to teach. Given computer algeba systems and such, I'd even say it is almost useless. $\endgroup$ – vonbrand Apr 18 '14 at 1:29
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    $\begingroup$ I would be very interested to see research supporting this if it exists (or for gen-ed college courses in math). $\endgroup$ – Alexander Gruber Apr 18 '14 at 1:31
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    $\begingroup$ I'd even be interested to see research regarding the attitudes of teachers regarding instrumental vs relational teaching of high school calculus. $\endgroup$ – pdmclean Apr 18 '14 at 1:34
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    $\begingroup$ A minor quibble: You might wish to rephrase this question to ask for research related to this claim, rather than just research that supports the claim. $\endgroup$ – Benjamin Dickman Apr 18 '14 at 10:27
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    $\begingroup$ If you have found this "plenty of evidence" in a well-written study, then there should be some discussion of opposing views. $\endgroup$ – Benjamin Dickman Apr 19 '14 at 16:42
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Why does it matter what the research shows?

At best, a study might have found that students who were blindly fed recipes did better in the short term on examinations for calculus.

But then, what is the ultimate goal of education? For students to pass tests? I really hope not. The goal of any teacher, especially a teacher in mathematics, should first and foremost be to make the students understand why things are true, how they work, how to prove them... I personally feel that mathematics teachers have a moral duty to at least attempt to let their students really understand the material - otherwise the students are just being programmed like computers (they do it correctly, but they have no idea what they're doing).

calculus needs to be taught in a procedural way

This is simply not true. I know for a fact that it does not need to. I know this because I was once a grade 12 students who loved maths, and I know such students exist today. I know that for them, not only should it not be taught that way, it must not be taught that way!

Students want to understand. Deep down, all of us have this urge. Giving up and just asking how to get the answer is what a student does when they are frustrated and do not think they can overcome a conceptual difficulty. Teaching "procedurally" only encourages this kind of failure. If students can be taught that they have the inner potential to understand anything - and I believe all who like maths do - then they will take infinitely more away from a class.

Remember that mathematics did not come out of a text book. It was created by real people, who were trying to solve real problems, who often had to try hundreds of solutions before finally coming up with something that worked. They knew that it worked because they proved it. Mathematics is not a collection of magical formulas that "just work," and it should never be portrayed as such.

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    $\begingroup$ "Why does it matter what the research shows?" - It matters since it would be nice to have answers based on facts and research instead of personal opinion. Anecdotical evidence is something we should resort to if there are no other alternatives. $\endgroup$ – Roland Apr 22 '14 at 7:02
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    $\begingroup$ @Roland There is a distinction between personal opinion and logical argument. I am appealing to general principles of education that should be adhered to, and justifying my claims. One does not need to conduct a scientific study to ascertain that mathematicians like to understand what they are doing - they do. $\endgroup$ – user1161 Apr 22 '14 at 16:39
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    $\begingroup$ I agree with @Roland - I might add that the question itself, which asks for research for only one particular side, is also at odds with (my interpretation of) the site's goals. -1 and -1. $\endgroup$ – Benjamin Dickman Apr 23 '14 at 2:29
  • $\begingroup$ You should not discount the satisfaction derived from being able to "do" problems (like a recent question on how integration wizards do it over at MSE). Proving stuff is in the same region, just much harder. $\endgroup$ – vonbrand Apr 28 '14 at 3:33
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    $\begingroup$ @Roland: It is not axiomatic that educational research can answer every question about education, especially questions involving value judgments and philosophical issues about what education should be. $\endgroup$ – Ben Crowell May 28 '14 at 18:19

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