It's time to banish the "reverse chain rule"

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Sir Cumference
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Today in the Edexcel C4 exam students were presented with this integral

\displaystyle \int (e^x+1)^3 \ dx

From what I've seen so far it seems that loads of students used the "revere chain rule" incorrectly to get

\displaystyle \frac{1}{4}e^{-x}(e^x + 1)^4 + c

(or similar incorrect answers)

The Edexcel exam writers know full well when they wrote this paper that thousands of students were going to do this. So you could say the question is a good discriminator but the point is that Edexcel have created this confusion in the first place. Their C4 textbooks have a section on "reverse chain rule" before substitution is taught. They created this nonsense and it should be up to them to get rid of it but unbelievably the Edexcel textbook for the new spec also has a section on it!

So it's time for teachers to ignore the textbooks and stop teaching the "reverse chain rule". Teach substitution first and then show some standard results after this if you like or let students work them out themselves.

EDIT: To be a bit clearer, I’m mainly talking about f(ax+b) integration which Edexcel calls “reverse chain rule”.

EDIT2: Having looked again, the new Edexcel textbook has improved this (but it still has major problems) so I’m being a bit too hard on them
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Gregorius
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(Original post by Notnek)
...the point is that Edexcel have created this confusion in the first place. Their C4 textbooks have a section on "reverse chain rule" before substitution is taught.
This is new to me! What do they actually state in their textbooks?
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ghostwalker
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(Original post by Gregorius)
This is new to me! What do they actually state in their textbooks?
Don't know whether this is old spec or new spec textbooks, but here's the start and end of section:

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Gregorius
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(Original post by ghostwalker)
Don't know whether this is old spec or new spec textbooks, but here's the start and end of section:
Oh dear, that is a bit dim isn't it?
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Ollie1999
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I tried for soo long with the reverse chain rule it just was not working when checking so I just expanded the brackets
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old_engineer
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(Original post by ghostwalker)
Don't know whether this is old spec or new spec textbooks, but here's the start and end of section:
The material from the new Edexcel "Year 2" textbook is here (two subsections).

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DFranklin
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(Original post by Notnek)
[Edexcel] created this nonsense and it should be up to them to get rid of it but unbelievably the Edexcel textbook for the new spec also has a section on it!

So it's time for teachers to ignore the textbooks and stop teaching the "reverse chain rule". Teach substitution first and then show some standard results after this if you like or let students work them out themselves.
I would say: "I only regret that I have but one upvote I can give to your post".

But apparently PRSOM, so I can't even do that... :cry:
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ppdtg
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(Original post by Notnek)
.
...
Last edited by ppdtg; 1 year ago
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Prasiortle
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(Original post by Notnek)
So it's time for teachers to ignore the textbooks and stop teaching the "reverse chain rule". Teach substitution first and then show some standard results after this if you like or let students work them out themselves.
In my view the issue is a bit broader than this. You and I
both know that in the real world, integration is an enormously hard problem, in the sense that if you were to write down some random combination of functions, the probability that an antiderivative does not exist is close to 100%, hence why pretty much every application of mathematics makes use of numerical methods for integration. I think it's this fact that the treatment of mathematics at school level doesn't really make clear to students, because everything is presented as a series of formulae/algorithms/rules (e.g. Power Rule, Product Rule, Chain Rule, etc.), and this leads students to believe that there's a rule/algorithm for everything. Indeed, in this state of mind, it becomes natural to believe that any arbitrary composition of functions can be antidifferentiated, just as any arbitrary composition of (differentiable) functions can be differentiated. This is then further compounded by the fact that students rarely ever get a chance to work on problems which are either extremely hard or impossible, such that they never get to see anything which challenges their mistaken view.
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Pizza32
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Same
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Muttley79
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(Original post by Prasiortle)
I think it's this fact that the treatment of mathematics at school level doesn't really make clear to students, because everything is presented as a series of formulae/algorithms/rules (e.g. Power Rule, Product Rule, Chain Rule, etc.), and this leads students to believe that there's a rule/algorithm for everything. Indeed, in this state of mind, it becomes natural to believe that any arbitrary composition of functions can be antidifferentiated, just as any arbitrary composition of (differentiable) functions can be differentiated. .
Not in the classrooms in many schools - you are wrong. No good teacher would teach A level like this or slavishly follow a textbook.
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Muttley79
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(Original post by Notnek)
Today in the Edexcel C4 exam students were presented with this integral

\displaystyle \int (e^x+1)^3 \ dx

From what I've seen so far it seems that loads of students used the "revere chain rule" incorrectly to get

\displaystyle \frac{1}{4}e^{-x}(e^x + 1)^4 + c

(or similar incorrect answers)

The Edexcel exam writers know full well when they wrote this paper that thousands of students were going to do this. So you could say the question is a good discriminator but the point is that Edexcel have created this confusion in the first place. Their C4 textbooks have a section on "reverse chain rule" before substitution is taught. They created this nonsense and it should be up to them to get rid of it but unbelievably the Edexcel textbook for the new spec also has a section on it!

So it's time for teachers to ignore the textbooks and stop teaching the "reverse chain rule". Teach substitution first and then show some standard results after this if you like or let students work them out themselves.
Many teachers DO ignore the textbook - why is it Edexcel's fault? It is mentioned elsewhere too - it is just poor teaching and poor understanding.
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Prasiortle
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(Original post by Muttley79)
Not in the classrooms in many schools - you are wrong. No good teacher would teach A level like this or slavishly follow a textbook.
The majority of mathematics teachers are not "good teachers". If they were, we wouldn't be so far down the PISA rankings. As Notnek informed me of before, the majority of mathematics teachers in the UK don't even have a degree in mathematics.
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Sinnoh
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Hang on, I don't get it. Aren't you supposed to be able to use all the different methods of integration anyway? Aren't you meant to have learnt all this by the time you do the exam? Putting a chapter on Reverse Chain Rule doesn't prevent people from learning integration by substitution, especially when it's specified in the textbook that it only works for linear functions. The textbook is not to blame, poor teaching/understanding is.
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username3694440
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Every integral in c4 can be checked with a silver calculator, so a lot of these mistakes could easily be avoided if there was a bit more emphasis on checking techniques. I know its a bit late now, but for anyone sitting the exam next year- get good at checking your work, you can confirm you're right in basically every integration question.
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Radioactivedecay
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I agree but it is also partly the teachers fault. Our teachers stated to us over and over that this method will only work if and only if the integral of the inside function is a constant. This should've been made clear by all teachers
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Muttley79
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(Original post by Prasiortle)
The majority of mathematics teachers are not "good teachers". If they were, we wouldn't be so far down the PISA rankings. As Notnek informed me of before, the majority of mathematics teachers in the UK don't even have a degree in mathematics.
That old chestnut! Have you actually ever looked at the PISA questions or how schools are selected? In some countries schools compete as to who is tested but in this country schools are chosen randomly. In one of the tests the worst performing school in our LA was selected ...

We actually do well in problem solving questions

There is no-one in my department without a maths degree nor in many of the schools I know.
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Prasiortle
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(Original post by Muttley79)
That old chestnut! Have you actually ever looked at the PISA questions or how schools are selected? In some countries schools compete as to who is tested but in this country schools are chosen randomly. In one of the tests the worst performing school in our LA was selected ...

We actually do well in problem solving questions

There is no-one in my department without a maths degree nor in many of the schools I know.
See https://www.thestudentroom.co.uk/sho...8#post78156532, where Notnek explicitly says "the majority of maths teachers do not have maths degrees".

You can quibble with the PISA methodology all you want but the fact is that even if you look at other metrics like our performance in the International Mathematical Olympiad, we're simply not that good at maths as a nation.
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Muttley79
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(Original post by Prasiortle)
See https://www.thestudentroom.co.uk/sho...8#post78156532, where Notnek explicitly says "the majority of maths teachers do not have maths degrees".

You can quibble with the PISA methodology all you want but the fact is that even if you look at other metrics like our performance in the International Mathematical Olympiad, we're simply not that good at maths as a nation.
The IMO is a tiny proportion of people who have to be at school who do the UKMT challenges! What a silly argument. I don;t know where Notnek got his data from but no-one can do a PGCE to teach Maths without a degree that is at least 50% maths or they have to do a SKE.Many school don;t even interview someone without a maths degree to teach.
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Prasiortle
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(Original post by Muttley79)
The IMO is a tiny proportion of people who have to be at school who do the UKMT challenges! What a silly argument. I don;t know where Notnek got his data from but no-one can do a PGCE to teach Maths without a degree that is at least 50% maths or they have to do a SKE.Many school don;t even interview someone without a maths degree to teach.
Well, what Notnek wrote, and he seemed pretty sure about it, was "The stats aren't always clear but probably around 50% of maths teachers have a maths related degree so that would include e.g. physics/engineering etc.", which would imply that the other 50% of maths teachers have a degree in something completely unrelated to maths, because "most people with maths degrees ... can get higher paying jobs elsewhere", presumably meaning in finance, data science, programming, etc.
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