MHB Interesting Blog Post for Math Teachers

[Ackbach]

Thought this Tim Gowers post was quite interesting.

 

[MarkFL]

The responses of the 17 year old in the blog seemed strikingly familiar! I, as most likely many of you, have had similar interactions with students on the forums.

 

[Fantini]

It is very interesting. It leads naturally to a self investigation as well: have we been taught this way? It's certainly a not so old problem. In my university at least, we are taught all deductions and proofs of almost all results, therefore I cannot attest to doing only the mechanical calculations without having seen the "why".

This applied to Calculus I only though. Multivariable Calculus, or Calculus II, is as Tim described. A few results were proved in the course but most were "accept and do it". If it serves as any excuse, multivariable calculus is quite hard and the conceptual basis requires so much of many different areas, such as multivariable analysis, topology and geometry.

Differential Equations, or Calculus III, is the embodiment of the text. Recipes for all basic differential equations, huge lists where all you have to do is apply the method again and again to get it imprinted on your brain (until immediately after the test, obviously).

I don't have much time now to continue this post, I'll see to it when I'm back. However it'd be very interesting if others shared their experience in this respect.

Cheers! (Wave)

 

[Ackbach]

Diff Eq is a tough one, in particular, because the "why's" are harder to come by. Why do you solve this type of DE this way? Because it works. Why does it work? Because some brilliant person (I mean that! Some of these methods are incredibly complicated and would have required an immense imagination to invent them.) dreamed up the method a long time ago. I'll give DE's this: it's not hard to check if a given solution solves the original DE.

I suppose you can go into existence and uniqueness theorems, and that's good and you should do that. But the fact is that there is no method of solving an arbitrary ODE, much less an arbitrary PDE. If there was, someone would have gotten the $1M prize from the Clay Institute for solving the Navier-Stokes equations. And we'd have heard about it.

The consolation of DE's is the application. No area of mathematics is more applied than DE's, because so many physical laws, once you've applied them in a particular circumstance, come out as a DE.

 

[CaptainBlack]

Thought this Tim Gowers post was quite interesting.

You point to the comments, which are quite interesting in themselves, but the blog post itself is also rather interesting, demonstrating something of the nature of the small group tutorial system practiced at some of our better institutions.

Tim's acount is very much like how many of us would like (and try) to conduct the help we give here. But it is also obvious that many students do not like it and prefer to have a list of "stuff" to memorise. This is not new, in the mid 80's I had an engineer working for me who complained (I don't recall to whom) that they were being asked to do stuff that was not in their degree course, notes or textbooks (I think we transferred them to the trials group where original though was not required).

(To balance the books somewhat I also had a good maths grad working for me who was so interested in understanding everything we were doing they never actually finished any work).

CB

 

[CaptainBlack]

You point to the comments, which are quite interesting in themselves, but the blog post itself is also rather interesting, demonstrating something of the nature of the small group tutorial system practiced at some of our better institutions.

Tim's acount is very much like how many of us would like (and try) to conduct the help we give here. But it is also obvious that many students do not like it and prefer to have a list of "stuff" to memorise. This is not new, in the mid 80's I had an engineer working for me who complained (I don't recall to whom) that they were being asked to do stuff that was not in their degree course, notes or textbooks (I think we transferred them to the trials group where original though was not required).

(To balance the books somewhat I also had a good maths grad working for me who was so interested in understanding everything we were doing they never actually finished any work).

CB

I know it is a breech of etiquet to reply to ones own post, but sometimes one must because of the way memory works.

1. When I was doing A-levels back in the late 60's, the only course on which it was intimated that you were required to think for yourself was Eng-Lit. (though we were taught derivatives using difference quotients etc)

2. A-level maths was routine and uninteresting, only when I got to university and saw real maths did it become interesting (as in non-trivial). (note: I was intending to do theoretical physics and was enrolled for joint maths and physics. I dropped physics after the first year because it was just not interesting enough - or rather not as interesting as real mathematics)

CB

 

[Fantini]

Diff Eq is a tough one, in particular, because the "why's" are harder to come by. Why do you solve this type of DE this way? Because it works. Why does it work? Because some brilliant person (I mean that! Some of these methods are incredibly complicated and would have required an immense imagination to invent them.) dreamed up the method a long time ago. I'll give DE's this: it's not hard to check if a given solution solves the original DE.

I suppose you can go into existence and uniqueness theorems, and that's good and you should do that. But the fact is that there is no method of solving an arbitrary ODE, much less an arbitrary PDE. If there was, someone would have gotten the $1M prize from the Clay Institute for solving the Navier-Stokes equations. And we'd have heard about it.

The consolation of DE's is the application. No area of mathematics is more applied than DE's, because so many physical laws, once you've applied them in a particular circumstance, come out as a DE.

I've come to terms with DEs because I realized how much trouble they are. The problem is not understanding that it is incredibly hard to solve any given DE, but rather that there is no underlying motivation as to why someone should learn to solve them. This touches the point of applications: a serious introduction to DE must show how naturally they show up in such a variety of contexts: Physics, for starters, economy, mathematical modelling in general. There are plenty of examples! Why not use them to your advantage?

More than also motivating the study of solutions to DEs, I find the most intriguing part of this area is interpreting the situation and setting up the equation, rather than arriving at a specific solution (partially because probably there isn't one). There are many paths to analyzing DEs which are just as important, sometimes a lot more. Shouldn't this be mentioned and somewhat explored in those courses? I firmly believe so.

You point to the comments, which are quite interesting in themselves, but the blog post itself is also rather interesting, demonstrating something of the nature of the small group tutorial system practiced at some of our better institutions.

Tim's acount is very much like how many of us would like (and try) to conduct the help we give here. But it is also obvious that many students do not like it and prefer to have a list of "stuff" to memorise. This is not new, in the mid 80's I had an engineer working for me who complained (I don't recall to whom) that they were being asked to do stuff that was not in their degree course, notes or textbooks (I think we transferred them to the trials group where original though was not required).

(To balance the books somewhat I also had a good maths grad working for me who was so interested in understanding everything we were doing they never actually finished any work).

CB

The balance of which you speak is hard to achieve, but immensely satisfying in my humble opinion. If you are completely careless about how things in your surroundings work, you lack the minimum curiosity necessary to advance past natural limitations that will be placed around. If you try to fully understand everything you will never take any steps, because a certain degree of acceptance is required to move forward. (This may have felt tautological, but nevertheless I decided to comment.)

As you've said it yourself, it is not new. I strongly believe the reason that such mindset exists is precisely because you can keep the "is-this-going-to-be-on-the-test-or-not", getting "good" grades with poor comprehension of the material. People will not question your knowledge, but rather the numbers/letters assigned to it. They would rather trust your A than test if you actually earned it.

I know it is a breech of etiquet to reply to ones own post, but sometimes one must because of the way memory works.

1. When I was doing A-levels back in the late 60's, the only course on which it was intimated that you were required to think for yourself was Eng-Lit. (though we were taught derivatives using difference quotients etc)

2. A-level maths was routine and uninteresting, only when I got to university and saw real maths did it become interesting (as in non-trivial). (note: I was intending to do theoretical physics and was enrolled for joint maths and physics. I dropped physics after the first year because it was just not interesting enough - or rather not as interesting as real mathematics)

CB

Obviously I have no authority, but your contributions are most fruitful. By all means, do breech! (Clapping) I am not familiar with this A-level math courses you have all spoken of, we don't have that here in Brazil (as far as I know). Your situation is not at all uncommon: I had considered theoretical physics for sometime before I chose mathematics. Upon entering university, I wasn't disappointed about my decision: the basic physics courses were horrible, to say the least. Consequently, I despised physics for a while until my taste came back, together with the understanding that the subject is intrinsically beautiful, disregarding the awful institute that carries the burden of teaching said courses.

 

[Jameson]

Great comments so far.

Only thing I really have to add is that I think the main importance of a slower approach is that teachers can push for students to view the topics through an analytical approach at an earlier point in their math development. Some form on analysis can be introduced in basic algebra even but in my experience the idea of proof or analysis isn't focused on in the US system until finishing the calculus series. For me that was in linear algebra but it might be different for others.

Single variable calculus doesn't require knowledge of geometry or algebra that hasn't been seen before so it's definitely both possible and feasible to derive formulas, focus on concepts over memorization, etc. As Fantini said though the geometry of multivariable calculus jumps a bunch of levels and I don't know if spending a lot of time in the theory of the Jacobian for example (forgive me if there is a better example) becomes worth it at the time in the timeline of one semester. So perhaps there becomes a time when one can say they appreciate that someone else proved this or that but knowing the proof backwards and forwards isn't necessary. That might be true for some and never true for others but that point should certainly be beyond single variable calculus.

Somewhere between being able to prove everything and memorizing everything is where different people need to be, but we are definitely on the latter side right now in the US and it's a serious problem.

 

[kanderson]

It's like this with everyone in math classes. "I REALLY NEED THE A!" sort of thing. Just outright memorisation. I try to do stuff on my own time. Don't have much time sadly.

 

[Jameson]

It's like this with everyone in math classes. "I REALLY NEED THE A!" sort of thing. Just outright memorisation. I try to do stuff on my own time. Don't have much time sadly.

If you are solidly ahead of the material you study in math class then I would talk to your teacher about it. Demonstrate that you're really interested in math and already know these topics so would like to continue moving forward at an accelerated pace. At my high school the math nerds studied in a supply closet by themselves but not all teachers would be ok with that. Maybe you can ask for different homework and tests. Math teachers usually love a student who loves math at your age. :)

 

[kanderson]

My math teacher doesn't like me.

 

[Fantini]

My math high school teacher didn't like me as well. Not saying it is the case, but some of them apparently dislike the fact that some students can become equal in skill in their area. That's not a bad thing, if you have the whole rest of body of knowledge to rest upon. You're not a teacher because you know how to calculate things faster (most of the time), but because you are able to see the interconnections between everything and how it all fits nicely. That's more important than, say, being able to $7 \times 7$ matrices multiplication in your head in a couple of minutes.

 

[Ackbach]

In my opinion, the younger grades, say, ages 6 through 12, are when kids don't mind memorizing things, and they're incredibly good at it (try playing the Memory game with a 10-year-old. You'll lose.) On the other hand, they don't much relish close reasoning. So, have them memorize the standard math facts: multiplication tables (up to 15, because then hex arithmetic becomes much easier), addition, subtraction, division. Then, when the kids reach about 12 years old up through, say, 15 or 16, they tend to be extremely argumentative. They want to beat everyone in an argument. They like logic and reasoning, but their imaginations are not so alive - so teach them the logic of how those math facts they've memorized fit together. Then, when they get to be 15 or 16, their imaginations come alive again, and they are asking the big questions, and they want to express themselves. So teach applications, problem-solving (which requires enormous imagination), and presentation skills (presenting your solution in front of a live audience, but also things like $\LaTeX$ and good ways to write mathematics).

One important point: the teacher has to love math, and pass that love on to the kids.

Then you're teaching "with the grain", and I think the results would be tremendous. This approach, I think, answers the vast majority of the criticisms I hear, including:

1. Kids these days can't add! (Solved in the first stage.)
2. Kids these days can't solve problems! (Solved in the third stage.)
3. Kids these days can't follow an argument! (Solved in the second stage.) Also heard as, "Kids these days can't prove anything!"
4. Kids these days hate math! (Solved by all three stages being "with the grain", and taught by a teacher who loves math and loves the students.)