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Tim Gowers uses the differentiation of e^x as an example of something bright UK maths A-level students often don't understand fully: https://gowers.wordpress.c

Tim Gowers uses the differentiation of e^x as an example of something bright UK maths A-level students often don't understand fully: https://gowers.wordpress.com/2012/11/20/what-maths-a-level-doesnt-necessarily-give-you/ https://gowers.wordpress.com/2012/11/20/what-maths-a-level-d...

tomrod 5y ago
This article reminded me of my maths journey. I was mechanically dutiful as a student, and would make lateral connections but had a lot of patchwork understanding. It wasn't until I understood the derivations in Real Analysis that things started to click.
- alexilliamson 5y ago
  Yes exactly! Deriving calculus from set axioms truly opened my mind to math, and more generally critical thinking.
londons_explore 5y ago
It's because most exams and curriculums in the UK are so strictly defined that all questions are almost guaranteed to follow one of a small set of structures. And schools have figured out that rather than teaching the subject from first principles, it's easier to get students to get high grades by teaching them each of the structures. Eg. "Whenever there is a question about differentiating x^7, just put 7x^6 as the answer." They then get the students to try a few examples (x^3 becomes 3x^2, x^77 becomes 77x^76, etc), and thats the way every science-y subject is taught. I often think it leads to students who do well in exams, but can't solve many real world problems. It could be solved by having a part of every exam paper be never-seen-before applied problems. For example, for differentiation, one might ask "A road's height in meters as a function of the horizontal distance along the road in kilometers is defined as sin(x)cos(x)tan(x). At what points are the steepest uphills? Would you describe the slope of the road as 'very hilly', and why?"
- eigenket 5y ago
  I've seen pretty bright seeming UK university applicants able to do whatever you ask them but then completely shit the bed when you ask them to differentiate e^y with respect to y rather than e^x with respect to x.
  - ithinkso 5y ago
    Even worse, I've seen a lot of people that where convinced the derivative of f(x)=e^7 is e^7
    - londons_explore 5y ago
      That would trick me too... Exam questions never ask 'trick' questions like that where the answer is zero/infinity/undefined.
  - Rompect 5y ago
    I genuinely don't know whether the trick of that question is swapping the `e^x` with `e^y`, so just renaming a variable – or is `y` a function?
    - eigenket 5y ago
      You're differentiating the function f(y) = e^y with respect to y
- dan-robertson 5y ago
  I think the problem is they want calculus in the curriculum and it is too late to be able to put it in context. There are some great uses for calculus that are accessible to many high school students. In particular, with physics you usually learn about capacitors and nuclear decay. Both of these cases are basically solving the differential equation y' = ky but: - the physics course can’t depend on the concurrent maths course because you are allowed to take physics without taking maths, so you just learn weird equations full of exponential a instead of the ODE - I think the maths course doesn’t even teach differential equations. They are in FP1 (from a separate ‘further maths’ course) but definitely not in AS (penultimate year of school) maths. Possibly a few turn up in A2 (final year) but then they can’t have any good examples from physics because not everyone doing maths will be able to depend on knowledge about what a capacitor is or how nuclear decay works. But I guess population models might work. - there can be some better stuff in the further maths course (e.g. I think they might even have the ‘exponentiate a matrix’ solution to systems of first order linear ODEs)
scythe 5y ago
I was expecting this to happen because the proof that the limit at zero of (e^h - 1)/h = 1 is tricky — nope, the student doesn't recognize the derivative formula in the first place.
Aardwolf 5y ago
> The particular topics he wanted me to cover were integrating log x, or ln x as he called it What's wrong with calling it ln x? The way this is written in the article implies there's something weird about calling it that. The name 'log' can mean log2, log10 or natural logarithm depending on the field. Removing ambiguities from math notation should be considered a good thing. The author expressed a worry about math education. Consider that a clear non ambiguous notation would help.
- dan-robertson 5y ago
  Most mathematicians use log to mean either natural log, or sometimes log in the relevant base (e.g. 2 if you are talking about information theory). In school (and engineering or physics I guess) you often are made to use ln for natural log and you are taught a way to pronounce that name (somewhere between lun and l’n) It feels like the point is “this person had not been exposed to university style mathematics”.
  - SilasX 5y ago
    "Yes, how dare someone have a different context than me [in which ln x is correct and log x is not]." Similar to those who mock people for saying a word incorrectly that they only learned from reading.
  - Aardwolf 5y ago
    > It feels like the point is “this person had not been exposed to university style mathematics”. Imho math is about logic and reasoning, not about what group you're part of
    - jfengel 5y ago
      The group is a bigger deal than you might expect. There are an infinite number of true theorems, almost all of which are boring. Mathematicians decide what is interesting, and that's not a matter of logic. A computer can bang out new theorems at light speed but nobody cares. Mathematics, like science and programming, is as much about humans as about the raw logic and data. You're welcome to be a group of one and please only yourself. But then you wouldn't care if it were published, and it wouldn't be unless you showed it to someone and they took an interest.
Aerroon 5y ago
I think the reason for this is that derivation from 'first principles' isn't really done. You'll do it once or twice in the intro to derivatives and that's it. The other 40 hours you spend on derivatives won't even touch it. The issue with being able to derive the formulas for derivation yourself is that it's not very useful. You simply don't have time to make those derivations during a test. It's like trying to use grammar rules in a conversation - conversations happen at a pace where you cannot apply grammar rules. You'll just have to know the patterns. You learn things in school to do a test. The usefulness of the vast majority of the knowledge they attain is purely to help them do the test. Later in life you might wish you knew more about this or that, but that's not at all apparent to the student.
mabbo 5y ago
I had a wonderful grade 12 calc teacher in high school who taught everything from first principles. I would leave his class feeling like I had gone to the gym from my brain. Despite his incredible teaching, I only pulled off a low 70s grade in the class. So I retook the course the next year. Taught by a new teacher fresh from teachers college, theoretically with a specialty in math since they were teaching an upper level math course. I don't think the new teacher even knew how to do derivatives from first principles. Just rote memorization of the different types of differentiation. I got an A in that class the second time, having learned nothing.
vmilner 5y ago
[I should add that this was posted in 2012, and differentiation from first principles was apparently emphasised far more in the syllabus in 2017 onwards.]

tomrod 5y ago

This article reminded me of my maths journey. I was mechanically dutiful as a student, and would make lateral connections but had a lot of patchwork understanding. It wasn't until I understood the derivations in Real Analysis that things started to click.

alexilliamson 5y ago
Yes exactly! Deriving calculus from set axioms truly opened my mind to math, and more generally critical thinking.

londons_explore 5y ago

It's because most exams and curriculums in the UK are so strictly defined that all questions are almost guaranteed to follow one of a small set of structures. And schools have figured out that rather than teaching the subject from first principles, it's easier to get students to get high grades by teaching them each of the structures. Eg. "Whenever there is a question about differentiating x^7, just put 7x^6 as the answer." They then get the students to try a few examples (x^3 becomes 3x^2, x^77 becomes 77x^76, etc), and thats the way every science-y subject is taught. I often think it leads to students who do well in exams, but can't solve many real world problems. It could be solved by having a part of every exam paper be never-seen-before applied problems. For example, for differentiation, one might ask "A road's height in meters as a function of the horizontal distance along the road in kilometers is defined as sin(x)cos(x)tan(x). At what points are the steepest uphills? Would you describe the slope of the road as 'very hilly', and why?"

eigenket 5y ago
I've seen pretty bright seeming UK university applicants able to do whatever you ask them but then completely shit the bed when you ask them to differentiate e^y with respect to y rather than e^x with respect to x.
- ithinkso 5y ago
  Even worse, I've seen a lot of people that where convinced the derivative of f(x)=e^7 is e^7
  - londons_explore 5y ago
    That would trick me too... Exam questions never ask 'trick' questions like that where the answer is zero/infinity/undefined.
- Rompect 5y ago
  I genuinely don't know whether the trick of that question is swapping the `e^x` with `e^y`, so just renaming a variable – or is `y` a function?
  - eigenket 5y ago
    You're differentiating the function f(y) = e^y with respect to y
dan-robertson 5y ago
I think the problem is they want calculus in the curriculum and it is too late to be able to put it in context. There are some great uses for calculus that are accessible to many high school students. In particular, with physics you usually learn about capacitors and nuclear decay. Both of these cases are basically solving the differential equation y' = ky but: - the physics course can’t depend on the concurrent maths course because you are allowed to take physics without taking maths, so you just learn weird equations full of exponential a instead of the ODE - I think the maths course doesn’t even teach differential equations. They are in FP1 (from a separate ‘further maths’ course) but definitely not in AS (penultimate year of school) maths. Possibly a few turn up in A2 (final year) but then they can’t have any good examples from physics because not everyone doing maths will be able to depend on knowledge about what a capacitor is or how nuclear decay works. But I guess population models might work. - there can be some better stuff in the further maths course (e.g. I think they might even have the ‘exponentiate a matrix’ solution to systems of first order linear ODEs)

scythe 5y ago

I was expecting this to happen because the proof that the limit at zero of (e^h - 1)/h = 1 is tricky — nope, the student doesn't recognize the derivative formula in the first place.

Aardwolf 5y ago

> The particular topics he wanted me to cover were integrating log x, or ln x as he called it What's wrong with calling it ln x? The way this is written in the article implies there's something weird about calling it that. The name 'log' can mean log2, log10 or natural logarithm depending on the field. Removing ambiguities from math notation should be considered a good thing. The author expressed a worry about math education. Consider that a clear non ambiguous notation would help.

dan-robertson 5y ago
Most mathematicians use log to mean either natural log, or sometimes log in the relevant base (e.g. 2 if you are talking about information theory). In school (and engineering or physics I guess) you often are made to use ln for natural log and you are taught a way to pronounce that name (somewhere between lun and l’n) It feels like the point is “this person had not been exposed to university style mathematics”.
- SilasX 5y ago
  "Yes, how dare someone have a different context than me [in which ln x is correct and log x is not]." Similar to those who mock people for saying a word incorrectly that they only learned from reading.
- Aardwolf 5y ago
  > It feels like the point is “this person had not been exposed to university style mathematics”. Imho math is about logic and reasoning, not about what group you're part of
  - jfengel 5y ago
    The group is a bigger deal than you might expect. There are an infinite number of true theorems, almost all of which are boring. Mathematicians decide what is interesting, and that's not a matter of logic. A computer can bang out new theorems at light speed but nobody cares. Mathematics, like science and programming, is as much about humans as about the raw logic and data. You're welcome to be a group of one and please only yourself. But then you wouldn't care if it were published, and it wouldn't be unless you showed it to someone and they took an interest.

Aerroon 5y ago

I think the reason for this is that derivation from 'first principles' isn't really done. You'll do it once or twice in the intro to derivatives and that's it. The other 40 hours you spend on derivatives won't even touch it. The issue with being able to derive the formulas for derivation yourself is that it's not very useful. You simply don't have time to make those derivations during a test. It's like trying to use grammar rules in a conversation - conversations happen at a pace where you cannot apply grammar rules. You'll just have to know the patterns. You learn things in school to do a test. The usefulness of the vast majority of the knowledge they attain is purely to help them do the test. Later in life you might wish you knew more about this or that, but that's not at all apparent to the student.

mabbo 5y ago

I had a wonderful grade 12 calc teacher in high school who taught everything from first principles. I would leave his class feeling like I had gone to the gym from my brain. Despite his incredible teaching, I only pulled off a low 70s grade in the class. So I retook the course the next year. Taught by a new teacher fresh from teachers college, theoretically with a specialty in math since they were teaching an upper level math course. I don't think the new teacher even knew how to do derivatives from first principles. Just rote memorization of the different types of differentiation. I got an A in that class the second time, having learned nothing.

vmilner 5y ago

[I should add that this was posted in 2012, and differentiation from first principles was apparently emphasised far more in the syllabus in 2017 onwards.]