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

Re: How Craig Barton wishes he’d taught maths

#41

Post by zazen » Sat, Feb 09, 2019, 9:01 PM UTC

dxbydt wrote: post #37

gizmo686 wrote: post #34

Earlier quoted context omitted.

>A working definition of a space might be - you have a member in that space, you can get to every other member by just scalar mult. Isn't this a 1 dimensional space. Eg. Consider the vector space R2 over R. If you have the vector (1,0), there is no way to arrive at the vector (1,1) through just scalar multiplication.

if you don’t give me basis how’ll i span the space ?

You talked about having "a member" and getting to every other member with scalar multiplication only. But even given a 2-member basis for R2, how are you planning on using 2 members, with scalar multiplication only?

I'm afraid you have badly misremembered this stuff.

Re: How Craig Barton wishes he’d taught maths

#42

Post by dxbydt » Sat, Feb 09, 2019, 9:07 PM UTC

zazen wrote: post #41

dxbydt wrote: post #37

Earlier quoted context omitted.

if you don’t give me basis how’ll i span the space ?

You talked about having "a member" and getting to every other member with scalar multiplication only. But even given a 2-member basis for R2, how are you planning on using 2 members, with scalar multiplication only? I'm afraid you have badly misremembered this stuff.

you are right.

Re: How Craig Barton wishes he’d taught maths

#43

Post by ajross » Sat, Feb 09, 2019, 9:09 PM UTC

throwawaymath wrote: post #31

ajross wrote: post #21

Earlier quoted context omitted.

> why would you allow 1/3 as a scalar in the first place. Because it's a definitional thing. A "scalar" is routinely defined as a real number, not an integer. And you're absolutely right that it makes no sense, which is the whole point of the multiple-choice question. Four of those answers are plausible, the other requires you to make assumptions (like a redefinition of scalar) not in the question as posed.

Huh? A scalar is very specifically a field element by definition. This is why it's important to specify the field you're working with when you talk about a vector space - a scalar is not going to be a real or a complex if your field isn't R or C. If you've seen someone define a scalar as a real number, that's really only because they're informally stating their underlying field is R.

I keep seeing you people in this thread and wondering, with all respect, what planet you're coming from.

The whole purpose of this exercise is to see if there was a way to come up with a straightforward, reasonably informal, multiple choice question that would expose a fundamental understanding in basic university math concepts like "vector space" in the same way we see in primary math.

And instead all you people want to do is natter over the ways in which someone could cleverly make the "wrong" answer right. It's... beyond missing the point, it's actively working against the whole goal of the exercise.

Re: How Craig Barton wishes he’d taught maths

#44

Post by dxbydt » Sat, Feb 09, 2019, 9:10 PM UTC

zazen wrote: post #40

dxbydt wrote: post #28

Earlier quoted context omitted.

This isn’t quite right. When I personally learnt these things in an undergrad program in math in the US, we learnt monoids. Then we learnt semigroups. Then groups. Then abelian groups. Then vector spaces. Then on the midterm we got questions exactly like the one we are debating here - is this guy a vector space, is that guy a semigroup, is that guy abelian etc. At that point, none of us knew what a ring was, what a f…

>If you have (2,3,4) and want to navigate to (5,6,7) who is also in your space and you have scalar mult as your tool of choice then mult with 2 gets you to (4,6,8) but then you are stuck. Soon you realize no matter what you do you can’t navigate that space without fractions. One of us is very confused. It seems to me that I also can't get from (2,3,4) to (5,6,7) by pure scalar multiplication even if fractions are all…

you can change the direction, -1 is a scalar. but yeah, you are right about the rest. cheers!

Re: How Craig Barton wishes he’d taught maths

#45

Post by gizmo686 » Sat, Feb 09, 2019, 9:24 PM UTC

ajross wrote: post #43

throwawaymath wrote: post #31

Earlier quoted context omitted.

Huh? A scalar is very specifically a field element by definition. This is why it's important to specify the field you're working with when you talk about a vector space - a scalar is not going to be a real or a complex if your field isn't R or C. If you've seen someone define a scalar as a real number, that's really only because they're informally stating their underlying field is R.

I keep seeing you people in this thread and wondering, with all respect, what planet you're coming from. The whole purpose of this exercise is to see if there was a way to come up with a straightforward, reasonably informal, multiple choice question that would expose a fundamental understanding in basic university math concepts like "vector space" in the same way we see in primary math. And instead all you people wan…

Because you are going to have students who mark the answer as correct, and you need to be prepared to explain to them why it is wrong. In addition, you explanation of why it is wrong should be accurate, and should not suggest that other correct answers are also wrong. Returning to the original question, why is it that Z3 is not a vector space, but Q3 is. If you say that neither of these are vector spaces, then you have a misunderstanding about what a vector space is which the question would miss because the author forgot to include Q3 as an option.

By itself, this is a minor complaint (you cannot include every example in you choices, although I do think that an example which could not be viewed as an R-vector space would be good to include). However, when you explain why Z3 is not a vector space, your explanation must be correct. An explanation which also excludes Q3 is incorrect.

Re: How Craig Barton wishes he’d taught maths

#46

Post by blattimwind » Sat, Feb 09, 2019, 9:58 PM UTC

theontheone wrote: post #27

I consider myself pretty strong at math (in university right now) and I was stumped by the vector space question. I never considered, actually, what domain scalars should be drawn from. Wikipedia says "the scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields."

Without context it isn't a well-posed question; the important context of that question is the teaching material itself, which would have introduced scalars as reals in this case (I assume), thus establishing what is meant by "obvious" or "usual" [object].

Re: How Craig Barton wishes he’d taught maths

#47

Post by dorchadas » Sat, Feb 09, 2019, 10:03 PM UTC

a_c wrote: post #29

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…

The math education I experienced focused heavily on "how". "How" such and such operation arrive to its conclusion and "how" such and such operation fulfil some "rules". Seldom does it touch on "why". Why certain notion, like linear algebra, heck, maybe even negative number, exists in first place. Procedures like negative times negative gives positive number. Yeah sure, but why? What does that mean really. I think the…

I'm a high school math and science teacher, and I completely agree this is a problem. Several of my students struggled greatly with simple arithmetic until I set down and gave them reasons for why a negative times a negative was a positive (which meant I also had to explain to them why multiplication works like it does).

I personally think part of the problem is how elementary education is structured. At least in my state, elementary school teachers are expected to be extreme generalists, and only have to take a math class or two -- and then nothing above simple college algebra (which is fine). But they always complain about how difficult it is, and they don't understand math, often taking the state's required exam multiple times because they can't pass math. Yet these are the people we allow to teach kids math; it's a huge issue when they're being taught math by people who don't understand why it works, only the algorithms they've memorized.

Coincidentally, this is also why the 'new math' was so lambasted -- these teachers (and often, parents) don't understand how numbers work, thus they think it's useless to teach kids to subtract 20 and add 2 instead of subtracting 18. Despite the fact that one is much easier to do mentally, and allows you to get a good sense of how subtraction and addition interact.

I'm not a fan of charter schools, but if I had money, I'd start an elementary charter school where the subjects were taught by people who understood that and not generalists. And students would get like multiple hours of recess a day, especially those first few years. Just pure, unstructured play time. But that's a rant for another day.

Re: How Craig Barton wishes he’d taught maths

#48

Post by agumonkey » Sat, Feb 09, 2019, 10:43 PM UTC

Math is done very very wrong, I don't think most teachers know enough math (sorry for that dubious and bold claim).

As a computer guy who hates state machines and was always obsessed with math, I feel that just about everything about maths is taught wrong from the get go.

Just the other day I learned about something inductive function got me curious about: linear ordering of structures as proof of termination. Turns out it's been studied in math for long: it's called a well-order. Fine.. thing is we're taught about linear recursion in HS .. but we have no pragmatic notion of induction except ~~ P n-1 => P n ~~ It's so cryptically compressed that I suspect no student beside aspies and other prodigies can have the slightest clue about that. Yet it's so important (and so obvious when shown).

Re: How Craig Barton wishes he’d taught maths

#49

Post by RBerenguel » Sat, Feb 09, 2019, 10:51 PM UTC

agumonkey wrote: post #48

Math is done very very wrong, I don't think most teachers know enough math (sorry for that dubious and bold claim). As a computer guy who hates state machines and was always obsessed with math, I feel that just about everything about maths is taught wrong from the get go. Just the other day I learned about something inductive function got me curious about: linear ordering of structures as proof of termination. Turns…

Induction is baked in the most “common” way of defining the naturals (Peano axioms). IIRC, it’s the definition I got for the “proper naturals” when I was in HS (but, my Maths teacher was a mathematician, and was who got me interested in them).

Re: How Craig Barton wishes he’d taught maths

#50

Post by mncharity » Sat, Feb 09, 2019, 11:06 PM UTC

> One question I had in the back of my mind when reading the book was whether any of it applied to teaching at university level. I’m still not sure what I think about that. There is a reason to think not, because the focus of the book is very much on school-level teaching, and many of the challenges that arise do not have obvious analogues at university level. [...] I think at Cambridge almost everyone would get this question right (though I’d love to do the experiment). But Cambridge mathematics undergraduates have been selected specifically to study mathematics. Perhaps at a US university, before people have chosen their majors, [...] More generally, I feel that there are certain kinds of mistakes that are commonly made at school level that are much less common at university level simply because those who survive long enough to reach that stage have been trained not to make them.

Note the "I think [...] almost everyone would get this question right (though I’d love to do the experiment)". This is a familiar state. Widespread. Call it, teachers who have not yet had their "oh shit!" moment.

One of the blog comments points at Eric Mazur's (Harvard, physics) oft-repeated talk "Confessions of a Converted Lecturer". Who describes the first time he gave students a Force Concept Inventory. Worried about wasting their time with such easy questions. :) Unaware physics education research was about to become a focus of his career.

Many have been surprised by "Minds of Our Own" (1997) https://www.learner.org/resources/series26.html The short (3 min) introductory video shows MIT and Harvard students struggling to light a bulb with a battery and a wire. Full episodes are below (by clicking on "VoD" buttons).

Harvard Center for Astrophysics has both first-tier astronomy and astronomy education programs. When meeting a new CfA graduate student, I've a little drill, prompting for the color of the Sun, and then of sunlight. They almost always get the first wrong, and then get a conflict, often with a nice "oh, wait, that doesn't make sense does it" moment. The collision of two bits of non-integrated and flawed understanding. Of the few who get it right, halfish (but small N) learned it from CfA instruction on common misconceptions in astronomy education, rather than from their own astronomy education.

But perhaps mathematics is doing better at robust integrated understanding than are astronomy, physics, chemistry, biology and medical school. It seems possible at least.

It's not just people who have had, or not had, their "oh shit!" moment. Professions too. Medicine realizing that medical errors were a major cause of mortality. Realizing even cheap easy universally-approved interventions (aspirin for ER chest pain) weren't consistently being executed. Realizing other industries had decades of experience on how to pursue quality, to which medicine had been oblivious. When the New York Times babbles about "Truth" and "The Journalism You Deserve", I shake my head and think, there's a field that has no clue how badly it's doing, how much work on process quality it's unaware of; a field that has not yet had its "oh shit!" moment.