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In Math Cram Sessions, Solving for Why

nytimes.com

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Re: In Math Cram Sessions, Solving for Why

#11

Post by criddell »

golergka wrote: post #7
rb808 wrote: post #3

A good example of why well educated parents is more important than getting into the "the best schools".

Uhm, no - asking students to prove theorems and lemmas and answer all the "whys", instead of just applying a set of problem solving rules was one of the main reasons why the high school I went to was so good at math education. Our physics teacher constantly told us not to memorize the formulas, but to understand the logic behind them so we could just figure them out if we needed them. It's a better way to learn math…

I think the order related subjects are taught isn't always the best. Once I learned calculus, a lot of things I memorized in physics stopped being something I had to remember. Velocity is just dt/dt. Acceleration is just dv/dt. Give me a graph of velocities and I can find the distance traveled by integrating.

This became especially apparent during my first year of university where I took a macro economics course. Because I had already had a bit of calculus, so many of the mathematical problems in that course were almost trivial. Friends that didn't have any calculus found the course a lot more difficult because they had to memorize so many equations.

I have kids now and I watch them memorizing things and I can't help but think that's a waste of time. Remembering what the terms in the quadratic equation isn't nearly as important as remembering what the uses of the quadratic equation are. When do you deploy it? What does it give you? That's how I wish it was taught.

Re: In Math Cram Sessions, Solving for Why

#13

Post by chrisweekly »

ivansavz wrote: post #8

Yeah, private tutoring is the best. The custom, just-in-time curriculum that a tutor can prepare is the most efficient way to learn. I think equally important is the transmission of a relaxed, exploratory attitude towards the STEM subjects. When a learner sees their teacher take on positive attitudes towards complexity, they too develop this attitude and can go on to solve problems on their own. Unfortunately, this t…

Great post. FWIW in discussing this topic I've heard and now adopted the term "(in)numerate" vs "STEM-(il)literate" -- though the latter might be intended to be broader in scope?

Re: In Math Cram Sessions, Solving for Why

#14

Post by OskarS »

Koshkin wrote: post #6
somethingroma wrote: post #2

"This was the key. If you knew hhhhwwhy you didn’t have to memorize equations, or solve equations in the exact same way they did in the book..." I feel like this is a valuable lesson a lot of people could benefit from.

Knowing "why" is exteremely helpful in many cases. But let's not forget that, on the other hand, memorization plays a huge part in learning, and not only in mathematics. Learning "why" multiplication works will not help you to retain the multiplication table in your head. (Also, in many cases the answer to a question "why" you might get will be plain wrong - sometimes because the correct answer is too complicated, so…

I don't think that it's "memorization" that's the key: I think it's repetition. Learning anything, but especially mathematics, requires you to just sit down and solve problems over and over again. That is what builds the right connections in your mind, gives you a feel for the shape of the solution, and gives you good intuitions about problems.

Like: it's a pretty useless skill to have memorized what the anti-derivative of the cosecant function is. But if you solve enough integrals, and are familiar with the techniques, you can see the path to the solution of the problem, even without having memorized every single function and anti-derivative.

Re: In Math Cram Sessions, Solving for Why

#15

Post by gnulinux »

ivansavz wrote: post #8

Yeah, private tutoring is the best. The custom, just-in-time curriculum that a tutor can prepare is the most efficient way to learn. I think equally important is the transmission of a relaxed, exploratory attitude towards the STEM subjects. When a learner sees their teacher take on positive attitudes towards complexity, they too develop this attitude and can go on to solve problems on their own. Unfortunately, this t…

This is a minor point, but I found understanding the "discourse" behind a field super helpful. In abstract algebra my professor spent a lot of time why we study certain structures (groups, commutative rings, modules, fields) more than others. One time he was making this super frustrated remark that our book (Dummit&Foote) calls ring a non-unitary ring whereas in class he'll call ring as a ring with unity (i.e. a ring that has 1). He was frustrated because for him this was not an interesting distinction and "to have interesting results you need to introduce a 1 anyway". I think this sort of emotional and human responses emphasizes what is important to understand in that field and what is not. Some people saw algebra as a laundry list of structures with axioms, whereas it's just another way of solving problems and structure is the approach, so finding useful structures for the problem in hand is more important than studying all those structures, as if memorizing the periodic table.

Re: In Math Cram Sessions, Solving for Why

#16

Post by Buldak »

nimrody wrote: post #12
rb808 wrote: post #3

A good example of why well educated parents is more important than getting into the "the best schools".

No. It's an example of why it is important to have parents devote time to helping their children study. This used to be much more common in the past.

>This used to be much more common in the past.

Source?

Re: In Math Cram Sessions, Solving for Why

#17

Post by ummonk »

Koshkin wrote: post #6
somethingroma wrote: post #2

"This was the key. If you knew hhhhwwhy you didn’t have to memorize equations, or solve equations in the exact same way they did in the book..." I feel like this is a valuable lesson a lot of people could benefit from.

Knowing "why" is exteremely helpful in many cases. But let's not forget that, on the other hand, memorization plays a huge part in learning, and not only in mathematics. Learning "why" multiplication works will not help you to retain the multiplication table in your head. (Also, in many cases the answer to a question "why" you might get will be plain wrong - sometimes because the correct answer is too complicated, so…

You can just do a lot of multiplications, and let your brain cache the results over time.

Re: In Math Cram Sessions, Solving for Why

#19

Post by ummonk »

criddell wrote: post #11
golergka wrote: post #7

Earlier quoted context omitted.

Uhm, no - asking students to prove theorems and lemmas and answer all the "whys", instead of just applying a set of problem solving rules was one of the main reasons why the high school I went to was so good at math education. Our physics teacher constantly told us not to memorize the formulas, but to understand the logic behind them so we could just figure them out if we needed them. It's a better way to learn math…

I think the order related subjects are taught isn't always the best. Once I learned calculus, a lot of things I memorized in physics stopped being something I had to remember. Velocity is just dt/dt. Acceleration is just dv/dt. Give me a graph of velocities and I can find the distance traveled by integrating. This became especially apparent during my first year of university where I took a macro economics course. Bec…

Everyone should be well-versed in linear algebra and calculus before learning quantitative physics, economics, etc. It just makes the subjects so much easier.

Re: In Math Cram Sessions, Solving for Why

#20

Post by thomasfedb »

ivansavz wrote: post #8

Yeah, private tutoring is the best. The custom, just-in-time curriculum that a tutor can prepare is the most efficient way to learn. I think equally important is the transmission of a relaxed, exploratory attitude towards the STEM subjects. When a learner sees their teacher take on positive attitudes towards complexity, they too develop this attitude and can go on to solve problems on their own. Unfortunately, this t…

Private maths tutor here — what I find most gratifying about tutoring is revealing to my students how much they already know.

Often they'll know that they know the answer, but they don't realise that they also know why it's the answer!

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