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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

#41

Post by Koshkin »

somethingroma wrote: post #27
PascLeRasc wrote: post #24

Earlier quoted context omitted.

The best is the 9's table, how all the digits add up to 9. My mom's an elementary teacher and there's always a few third graders who figure that out on their own and love it.

Additionally 9 * n = [n-1, 10-n] for n = 2-11; where n-1 is the digit in the 10's place and 10-n in the single place. This just an aesthetic curiosity. I know the pattern continues for larger n I've just never bothered to generalize it. Also never compared it to other bases.

> [n-1, 10-n] for n = 2-11

I think the usual 10 * n - n makes more sense and is much easier to remember.

Re: In Math Cram Sessions, Solving for Why

#42

Post by kaitai »

threatofrain wrote: post #28
unpythonic wrote: post #22

As much as the Common Core standards are criticized here in California, I have to say that the emphasis on the "why" behind every operation is really fantastic. It's usually in the last section of each homework, but it provides a good discussion point when going over the homework each night. Asking students to answer questions like the following are easy to zip through, but they provide a good place to pause and find…

There's so much misinformation on the Common Core that many people don't realize how much of a modest adjustment it was from before. Another way of summarizing the changes made by the Common Core, aside from your note that it focuses more on why, is that it advances the math curriculum by about half a year.

There is indeed so much misinformation! People should read the standards themselves, and also differentiate between the Common Core standards and the curricula being sold to districts.

Re: In Math Cram Sessions, Solving for Why

#43

Post by tzs »

unpythonic wrote: post #22

As much as the Common Core standards are criticized here in California, I have to say that the emphasis on the "why" behind every operation is really fantastic. It's usually in the last section of each homework, but it provides a good discussion point when going over the homework each night. Asking students to answer questions like the following are easy to zip through, but they provide a good place to pause and find…

One can make a good argument that the "why" of math is really the only reason to teach it as a standard subject required for everyone, at least beyond just elementary arithmetic.

We like to pretend that all kinds of jobs require knowing algebra and geometry, but no, but that is greatly exaggerated.

What you learn when you study those subjects that you have a decent chance of actually using on the job is how to reason.

There were a couple of interesting essays on this point by Underwood Dudley.

One is called "What is Mathematics For?" (which he admits is click-bait, and it should really be called "What is Mathematics Education For") from the May 2010 "Notices of the AMS" [1].

The other is called "Is Mathematics Necessary?" [2], which I believe is an earlier, shorter version of the what later became the AMS article.

[1] https://www.ams.org/notices/201005/rtx100500608p.pdf

[2] http://www.public.iastate.edu/~aleand/dudley.html

Re: In Math Cram Sessions, Solving for Why

#45

Post by Koshkin »

jordigh wrote: post #40

> A ball is thrown off a building at a speed of 15m/s and at 30 degrees to the horizontal. If the building is 100m tall, how far from the base of the building will the ball land? g=9.8m/s² 280.17 metres, right?

Depends on whether the angle is above or below the horizontal. (The ball won't make it this far regardless.)

Re: In Math Cram Sessions, Solving for Why

#46

Post by somethingroma »

aj7 wrote: post #39
somethingroma wrote: post #9

Earlier quoted context omitted.

Memorizing multiplication tables is over rated. Seeing connections between those early multiples up to 12 is more interesting. Can be helpful I suppose in factorization. Maybe I just don't trust my memory (as in how am I sure that 7*8 is 56) and why I despise rote memorization and rather retain memory from use and practice. Regardless a good example of something to remember in math is the quadratic formula Still enjo…

No. If you can’t multiply in your head, you are crippled in any quantitative reasoning. You have interjected too many steps in estimating, calculating, judging, etc. This is not to say there aren’t useful, non-quantitative pursuits.

I don't understand what you're saying or objecting to.

Incase this is what you mean: remembering the multiplication table is the antithesis of multiplying in ones head.

Re: In Math Cram Sessions, Solving for Why

#47

Post by Koshkin »

ummonk wrote: post #17
Koshkin wrote: post #6

Earlier quoted context omitted.

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.

Spending time memorizing the multiplication table, then, is more efficient on both accounts: you will not need to do "a lot of multiplications" to begin with, and it takes less time for your brain to "cache the results".

Re: In Math Cram Sessions, Solving for Why

#48

Post by zrobotics »

sdenton4 wrote: post #25
unpythonic wrote: post #22

As much as the Common Core standards are criticized here in California, I have to say that the emphasis on the "why" behind every operation is really fantastic. It's usually in the last section of each homework, but it provides a good discussion point when going over the homework each night. Asking students to answer questions like the following are easy to zip through, but they provide a good place to pause and find…

I haven't worked in an american classroom, but as a math educator, thought the CommonCore looked great when it was first rolling out. Math education is (kind of strangely) a bit of a battlefield in the US (maybe not so strangely, when everything else seems to be, too)... I think a big part of it is prevalent math anxiety - perhaps embarrassment at seeing unfamiliar things on the kid's homework. Along with this, there…

Agreed, I can't stand the way we teach math in the US. I love math enough to have accidentally gotten a minor in it, but especially high school does nothing but train kids to hate math. My calculus class in high school was a net negative, since teaching to the test just caused many of my friends to switch away from STEM majors in college (if Calc 1 was that horrible, why become an engineer ahd have to suffer even more?).

One of these friends went on to become a forensic accountant, so she obviously couldn't hate math that much, but that calculus class was traumatic enough that she summarily dismissed a biology major in college. I think one of the main pain points, which my dad was able to tutor me through, was understanding why. Learning calculus (or any math, for that matter) as just a list of mechanical steps is awful, whereas learning why we perform the mechanical steps allows one to glimpse the beauty behind math. I wonder how many children we've ruined this beautiful subject for?

Re: In Math Cram Sessions, Solving for Why

#49

Post by j2kun »

> Or why his first-generation ABC (“American-born Chinese”) children were more interested in sports or the humanities than studying fractions and common denominators.

This could have just as easily been written: "Or why his first-generation ABC children were more interested in putting a ball in a hoop and learning languages that nobody speaks anymore than in noticing and thinking about the patterns and structure in the world around them."

Framing the same underlying idea can disparage one thing while elevating another.

Re: In Math Cram Sessions, Solving for Why

#50

Post by Koshkin »

somethingroma wrote: post #46
aj7 wrote: post #39

Earlier quoted context omitted.

No. If you can’t multiply in your head, you are crippled in any quantitative reasoning. You have interjected too many steps in estimating, calculating, judging, etc. This is not to say there aren’t useful, non-quantitative pursuits.

I don't understand what you're saying or objecting to. Incase this is what you mean: remembering the multiplication table is the antithesis of multiplying in ones head.

> remembering the multiplication table is the antithesis of multiplying in ones head

But it's not! Because in order to be able to multiply numbers in your head efficiently you must remember the multiplication table.

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