Comments on: Calculus Teaching Mistake: The “Generalized Chain Rule”

By: sexy Costumes

sexy Costumes — Sat, 03 Dec 2011 13:57:43 +0000

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[...]Calculus Teaching Mistake: The “Generalized Chain Rule” « Paul Raff[...]…

By: David

David — Sat, 29 Oct 2011 03:01:59 +0000

I agree with Paul’s sentiments. For some reason, text books ( and more critically lecturers) often teach multiple subset rules rather than one superset rule. In my opinion it is best to state and prove a superset theorem, and then demonstrate how each of the subset rules is an application of the initial theorem. This allows students (who are so inclined) to really understand the maths, rather than remembering rules. This builds flexible and robust knowledge, while wrote learning is brittle outside of the confines of the recognised examples. There is another subtle problem with teaching multiple subset rules. It implies they are necessary, i.e. it implies the student has missed something that distinguishes these cases from the more general case. That mightn’t be concerning for someone who knows they aren’t necessary. But for a student learning this for the first time he must think that they are necessary. This increases the burden on memory, and for all but the most diligent students (who will seek out the unifying principle independently) it gives them a sense that they are missing something diagnostic. This means they will lack confidence in recognition of the problem type at exam time. Unfortunately the problem is rife. During my finance days at university we were told to remember at least 5 different formulae for annuity valuations, when the sum of a geometric progression was the only tool required. As another example I have often wondered why y = e^(f(x)) dy/dx = f’(x)e^(f(x)) is taught in preference to the more general: y = a^(f(x)) dy/dx = f’(x)a^(f(x))logbase e (a) {my apologies for notation but I am no html expert}. As Paul notes, both cases should be shown to have a connection with the chain rule. My hope is simple: that writers of text books prove the superset rule first, and then when writing the subset rules, draw attention to how they are no more than specific cases of that superset rule.

By: Charlie

Charlie — Sun, 21 Aug 2011 09:27:08 +0000

Brings back college days – Calculus was my favorite subject, but now I wonder how I actually made it through. I am now going to play around with the chain rule for a bit and see how it stacks up. Thank you for reigniting the old math spark!
Charlie

By: mustafa

mustafa — Thu, 14 Jul 2011 07:08:46 +0000

i am student at hte college, so i have a problem to understand the concept of chain rule in Leibniz notation , it doesnt make sanse ??

By: David

David — Thu, 02 Jun 2011 06:36:16 +0000

I really wonder why most teachers requires students to do the long method of computation when in fact, there are simpler calculations that will result to the same answers. I think this is where students start to be confused with the subject.

By: Sophia

Sophia — Wed, 01 Jun 2011 17:17:49 +0000

In standard calculus you will note that the derivative of a function is a multiplier for a linear term in the variable. The n-th derivative is the multiplier for an n-th degree term. Effectively then the first derivative times the variable is a linear function, and higher order n-th degree functions.

By: Mark Timothy

Mark Timothy — Wed, 02 Feb 2011 18:23:45 +0000

I always fear mathmatics and i think i have a phobia of it. I do not know how people understand these stuff.

By: Sam

Sam — Wed, 05 Jan 2011 21:20:40 +0000

Pavlovian? This is the sort of thing that kills mathematics for so many otherwise interested kids.

That said, I will admit that it’s fun for me when I get to the “Pavlovian” point, that is, I’ve mastered the use of a concept and now I don’t have to think about it.

But I have to get there myself, in this case, by seeing the chain rule and then doing a bunch of exercises. If you try to beat it in, you’re going to lose me.

By: Kellen Myers

Kellen Myers — Mon, 25 Oct 2010 06:50:13 +0000

I will point out that Strauss, Bradley, & Smith is the other Calc textbook at Rutgers, and this text also has several listings of rules that are no more than special cases of rules they’ve already learned.

In particular, p 140 features the “Extended Derivative Formulas” each of which is a combination of one of the power, trigonometric, exponential, or logarithmic rules plus the chain rule.

This semester alone, at least three or four students have asked me whether they should use these rules rather than the chain rule.

These are certainly useful examples, but I think I agree that presenting them as additional rules for derivatives is not helpful – it is, I would reckon, somewhat harmful.

By: Jon Rogawski

Jon Rogawski — Wed, 17 Mar 2010 03:39:40 +0000

Paul – I agree that simpler is better, so I thought about this quite a bit. In the end, I felt it was important for students to learn to apply the Chain Rule by pure reflex for those special cases that come up repeatedly – after all, developing the right pavlovian reflexes is also part of learning mathematics. But there are no absolutes in pedagogy, and I certainly acknowledge your point (BTW, for the 2nd Edition currently in progress, I did reduce this slightly).