How Craig Barton wishes he’d taught maths • Hacker News Board

Re: How Craig Barton wishes he’d taught maths

#11

Post by throwawaymath » Sat, Feb 09, 2019, 6:12 PM UTC

dxbydt wrote: post #5

burlesona wrote: post #4

Earlier quoted context omitted.

As a bumbling idiot who never did university level math, I don’t understand the question or the answer. (I did understand the probability one and knew the right answer to that at least.) I’m trying to catch up. Would you be kind enough to explain it?

you need closure on scalar mult. if (2,3,4) is a valid int triple & 1/3 is your scalar then (2/3,1,4/3) throws you out of the group so there goes your closure. unlike op, you don’t really need to know about fields to solve this.

While that's the right idea, I'd push back against not needing to know about fields since the scalars are just field elements. If you try to define a vector space over the integers, it's more accurate to say you can't choose 1/n as a scalar, because 1/n doesn't exist in your underlying field. Your closure ends before you even get to choose the element.

For students it might not be immediately obvious why that's a problem for vector spaces, but yes it does mean scalar multiplication won't be closed in the vector space. And more practically speaking, if you tried to solve a system of equations without invertible linear combinations, you'd have no linearity whatsoever. Elementary row operations likewise cease to be invertible, so matrix reduction isn't possible...the whole thing breaks down really.

Re: How Craig Barton wishes he’d taught maths

#12

Post by zazen » Sat, Feb 09, 2019, 6:22 PM UTC

keithpeter wrote: post #2

Quotes from OA that struck me as on the button... "A prejudice that was strongly confirmed was the value of mathematical fluency. Barton says, and I agree with him (and suggested something like it in my book Mathematics, A Very Short Introduction) that it is often a good idea to teach fluency first and understanding later." Agree fully with Barton and OA here. Until recently I taught GCSE Maths re-take students aged…

> Agree fully with Barton and OA here. Until recently I taught GCSE Maths re-take students aged 16 and over in a further education college. They were constantly tripping over really quite basic little skill issues and that prevented them from seeing how to tackle the longer and more complex problem solving questions.

I also agree fully. A little while back I did some support tutoring for A-level maths students. The number of students who turned up who mysteriously "had problems with longer questions"... I wish I'd known the example of calculating the perimeter of the rectangle with fractions. That would have really helped explain why the problem wasn't really the length of the question, it was the fact that the student had never properly learned the component skills separately.

Unfortunately, the problem of building impressive-looking edifices on shaky foundations is absolutely endemic in British high-school maths teaching. Thousands of students who never quite understood fractions are "learning" calculus through being taught recipes, and the easier exam questions are formulaic enough that they get through with Cs at least, without any mathematical understanding.

The A-level statistics modules, in particular, have very impressive _sounding_ syllabuses. Students learn T-tests, Chi-squared tests, all this sophisticated statistical machinery. If all these students really understood this stuff, Britain would have a vast army of highly trained statisticians. But nothing of the sort is true, of course: students are just learning a recipe for processing numbers. I can't imagine the carnage if a statistics exam asked the students to write an essay explaining the principle by which a T-test works.

Pardon my rant, this has been on my mind for a while.

Re: How Craig Barton wishes he’d taught maths

#13

Post by gizmo686 » Sat, Feb 09, 2019, 6:22 PM UTC

mlevental wrote: post #7

throwawaymath wrote: post #3

Wow, that vector space question is a great example. It’s the kind of thing that should be straightforward for anyone who has taken a linear algebra course, but I can also totally see students getting it wrong. This is especially the case because it’s actually very easy fundamentally (the set of all integers does not comprise a field, and so a vector space cannot be defined over it). But to my recollection, most of th…

I got thrown and I know a lot of math. I got thrown because the field the vector space is over is not what I focus on when thinking about the structure of the vector space. indeed it took me a second to even come up with a reason why the ground set has to be a field at all - so that you can undo scalings - and while it's a good reason it still feels trivial. edit: in fact i learned linear algebra from hoffman kunze

You can generalize vector spaces to work over arbitrary rings (in which case it is a module). I did have to do a double take on that question, because in my mind when I see someone talking about a vector space over a ring, I just silently translate it to a module (because 90% of the time, the person just misspoke, 9% of the time, even if they meant vectorspace, the immediate follow up would be "no, but does your point still stand of we consider it a module?", and maybe 1% of the time they are actually being adversarial.

Re: How Craig Barton wishes he’d taught maths

#14

Post by ganzuul » Sat, Feb 09, 2019, 6:24 PM UTC

keithpeter wrote: post #2

Quotes from OA that struck me as on the button... "A prejudice that was strongly confirmed was the value of mathematical fluency. Barton says, and I agree with him (and suggested something like it in my book Mathematics, A Very Short Introduction) that it is often a good idea to teach fluency first and understanding later." Agree fully with Barton and OA here. Until recently I taught GCSE Maths re-take students aged…

Personally I find understanding, or trying to understand, a very good mnemonic for remembering things.

Re: How Craig Barton wishes he’d taught maths

#15

Post by gizmo686 » Sat, Feb 09, 2019, 6:26 PM UTC

dxbydt wrote: post #5

burlesona wrote: post #4

Earlier quoted context omitted.

As a bumbling idiot who never did university level math, I don’t understand the question or the answer. (I did understand the probability one and knew the right answer to that at least.) I’m trying to catch up. Would you be kind enough to explain it?

you need closure on scalar mult. if (2,3,4) is a valid int triple & 1/3 is your scalar then (2/3,1,4/3) throws you out of the group so there goes your closure. unlike op, you don’t really need to know about fields to solve this.

If the triple is defined over the integers, why would you allow 1/3 as a scalar in the first place.

By this logic, should R not be a vector space, as it is not closed under scalar multiplication by i, or Q not be a vector space, as it is not closed under scalar multiplication by sqrt(2)?

Re: How Craig Barton wishes he’d taught maths

#16

Post by keithpeter » Sat, Feb 09, 2019, 6:40 PM UTC

zazen wrote: post #12

keithpeter wrote: post #2

Quotes from OA that struck me as on the button... "A prejudice that was strongly confirmed was the value of mathematical fluency. Barton says, and I agree with him (and suggested something like it in my book Mathematics, A Very Short Introduction) that it is often a good idea to teach fluency first and understanding later." Agree fully with Barton and OA here. Until recently I taught GCSE Maths re-take students aged…

> Agree fully with Barton and OA here. Until recently I taught GCSE Maths re-take students aged 16 and over in a further education college. They were constantly tripping over really quite basic little skill issues and that prevented them from seeing how to tackle the longer and more complex problem solving questions. I also agree fully. A little while back I did some support tutoring for A-level maths students. The n…

Going back around the millennium or before when I last taught A level maths at college, we had them in over the summer before term started for a two week intensive algebra and basics course.

Seemed to help.

The original author (Tim Gowers, a Fields medallist and professor of mathematics at Cambridge) has a totally hilarious blog post about being asked to coach a teenager doing A level maths...

https://gowers.wordpress.com/2012/11/20/what-maths-a-level-d...

Re: How Craig Barton wishes he’d taught maths

#17

Post by zazen » Sat, Feb 09, 2019, 6:59 PM UTC

keithpeter wrote: post #16

zazen wrote: post #12

Earlier quoted context omitted.

> Agree fully with Barton and OA here. Until recently I taught GCSE Maths re-take students aged 16 and over in a further education college. They were constantly tripping over really quite basic little skill issues and that prevented them from seeing how to tackle the longer and more complex problem solving questions. I also agree fully. A little while back I did some support tutoring for A-level maths students. The n…

Going back around the millennium or before when I last taught A level maths at college, we had them in over the summer before term started for a two week intensive algebra and basics course. Seemed to help. The original author (Tim Gowers, a Fields medallist and professor of mathematics at Cambridge) has a totally hilarious blog post about being asked to coach a teenager doing A level maths... https://gowers.wordpres…

Thanks for linking that, it's a great read. I really should read more of Gowers' posts.

The phrase "memory works far better when you learn networks of facts" was a happy find - I've never been able to express that idea so concisely.

I remember discovering they'd moved "differentiation from first principles" away to a further-maths module, as if it's a peripheral, difficult little oddity for the keen kids to hear about. It was the surest, saddest sign that the powers that be had given up on genuinely educating the average A-level maths student.

Re: How Craig Barton wishes he’d taught maths

#18

Post by skh » Sat, Feb 09, 2019, 7:00 PM UTC

gizmo686 wrote: post #15

dxbydt wrote: post #5

Earlier quoted context omitted.

you need closure on scalar mult. if (2,3,4) is a valid int triple & 1/3 is your scalar then (2/3,1,4/3) throws you out of the group so there goes your closure. unlike op, you don’t really need to know about fields to solve this.

If the triple is defined over the integers, why would you allow 1/3 as a scalar in the first place. By this logic, should R not be a vector space, as it is not closed under scalar multiplication by i, or Q not be a vector space, as it is not closed under scalar multiplication by sqrt(2)?

Something being sidestepped in the post you responded to is that what is being talked about is a valid algebraic object with lots of structure to it. It’s called a module which you can think of as a sort of vector space. It’s just that the scalars may not have the property that they have multiplicative inverses. (I’m deliberately focusing on rings that are integral domains for the nitpickers.). When talking about these objects you have to include the underlying scalar set.

For instance the real numbers are a vector space over the rationals. They are a different vector space over the reals. They are not a vector space over the complex numbers and are not a vector space over the integers. But they are a module over the integers. But not a module over the complex numbers.

Re: How Craig Barton wishes he’d taught maths

#19

Post by joe_the_user » Sat, Feb 09, 2019, 7:03 PM UTC

mlevental wrote: post #7

throwawaymath wrote: post #3

Wow, that vector space question is a great example. It’s the kind of thing that should be straightforward for anyone who has taken a linear algebra course, but I can also totally see students getting it wrong. This is especially the case because it’s actually very easy fundamentally (the set of all integers does not comprise a field, and so a vector space cannot be defined over it). But to my recollection, most of th…

I got thrown and I know a lot of math. I got thrown because the field the vector space is over is not what I focus on when thinking about the structure of the vector space. indeed it took me a second to even come up with a reason why the ground set has to be a field at all - so that you can undo scalings - and while it's a good reason it still feels trivial. edit: in fact i learned linear algebra from hoffman kunze

As I remember, most first courses in linear algebra don't deal with the subject as anything like abstract algebra and aren't going to be targeting students who can distinguish a ring from a field. Essentially, until you can past the calculus sequence, most math in the US is centered on calculation with only a few forays into proofs and definitions.

Re: How Craig Barton wishes he’d taught maths

#20

Post by ajross » Sat, Feb 09, 2019, 7:05 PM UTC

throwawaymath wrote: post #11

dxbydt wrote: post #5

Earlier quoted context omitted.

you need closure on scalar mult. if (2,3,4) is a valid int triple & 1/3 is your scalar then (2/3,1,4/3) throws you out of the group so there goes your closure. unlike op, you don’t really need to know about fields to solve this.

While that's the right idea, I'd push back against not needing to know about fields since the scalars are just field elements. If you try to define a vector space over the integers, it's more accurate to say you can't choose 1/ n as a scalar, because 1/ n doesn't exist in your underlying field. Your closure ends before you even get to choose the element. For students it might not be immediately obvious why that's a p…

> I'd push back against not needing to know about fields since the scalars are just field elements.

The point was that you don't need to know the jargon of "field" and the full set of implications. It's enough to know that multiplying integers by non-integer scalars can give non-integers, which means that "scalar multiplication" can produce a thing that is not a "triple of integers". So it's not a well defined vector space operation.

No need for "field" or "closure" or any other jargon not in the question as posed.