
In Math Cram Sessions, Solving for Why - danso
https://www.nytimes.com/2018/09/07/well/family/in-math-cram-sessions-solving-for-why.html
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ivan_ah
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 transfer of attitude works in the negative direction too.
If the teacher is out of their comfort zone (e.g. parent touching math for the
first time in 20 years), then the learner might pick up on the anxiety and
start to consider the subject to be unpleasant or hard---if your parent was
stressing out about helping you with algebra, then algebra must be a thing of
formidable complexity!

As another example, you can think of a teacher bored with the subject (e.g.
prof who is teaching this course for the 17th time this semester) who will
then transmit "this is stupid shit you have to know, because you're forced to
take this course" attitude and students will pick up on this too...

Luckily these days there are really good resources (youtube, free books,
interactive demos) so hopefully we'll have more STEM people in the future. We
won't make 100% of the population into STEM-experts, but it's not unrealistic
to hope that everyone can become at least STEM-literate. I'd like to believe
that I contribute to this with my books. This one in particular would be good
for adult readers who want to rekindle their relationship with the subject:
[https://www.amazon.com/dp/099200103X/noBSmath](https://www.amazon.com/dp/099200103X/noBSmath)

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saagarjha
> 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 really disagree with your choice of words here, specifically, “just-in-
time”. This makes it seem like the tutor is teaching things to the student
right before they need it for an exam, which is an unfortunate effect I’ve
seen happen because in the long run it doesn’t really teach anything. But
where a private tutor really does excel is having a better understanding of
how to tailor the curriculum to the student (this includes both simplifying or
making it more challenging, depending on the student and the topic).

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ivan_ah
Ah no, my intended meaning of “just-in-time” was definitely not in the sense
of just-in-time-for-the-exam. In fact I think there is very little use for
last-minute cramming... at most you can survive the test, but definitely not
build fluency or understanding.

I was using “just-in-time” in the sense of just-in-time-for-the-lesson, e.g.,
if the tutor wants to teach concept X which requires knowledge of prerequisite
concepts α1, α2, and α3, then the tutor can provide a quick review of these
concepts before explaining X. Tutors can do this "custom filling in of gaps,"
whereas in a groups setting the teacher might be forced to say "you should
know this already."

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unpythonic
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 a way to connect the
physical mechanics of a solution to the reasoning behind it: "How would you
explain to someone else why the fraction 1/2 greater than 1/4?" or "Why
doesn't angle-angle-angle show congruence but angle-side-angle does?"

I've also noticed that the Common Core brings in advanced topics earlier
without announcing to the student that it is an advanced topic. Ideas from
algebra are brought in at natural points of the discussion rather than making
a big deal of it. By the time that they realize they're learning algebra, they
are already into many of the "rules" that would have otherwise been taught by
rote. If you understand why you have to multiply both sides 5 to find out x/5
= 30, it feels much less arbitrary when the rules are made more explicit
later.

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sdenton4
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's still lingering bad PR
from the 'New Math' of the seventies, which made it somehow acceptable to make
the argument that we should stick to teaching math the same way forever.
(Especially for people who believe that math hasn't changed since Newton.)

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zrobotics
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?

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gnode
I think understanding the why of mathematics makes all the difference between
forgetting it soon after an exam, and being able to use it later in life.

I sometimes find myself needing to return to something I once knew in high-
school / university to solve some problem, not being able to remember the
specific methods, but the underlying bigger picture of the concepts comes back
to me, and I can make progress from there.

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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.

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ssivark
More importantly, it gives the reader a keen insight into the father's
perspective -- which the author is trying to convey, and not many people would
be familiar with.

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somethingroma
"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.

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Koshkin
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, sometimes for other reasons.)

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somethingroma
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 enjoyable to derive and 'see' why it works but also just used so
much. That said, I wouldn't encourage memorizing it without first
understanding it.

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PascLeRasc
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.

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somethingroma
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.

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sdenton4
It's not an aesthetic curiosity at all! 10 is equal to 1 mod 9... So suppose X
is written in base 10 (a0 * 10^0 + a1 * 10^1 + a2 * 10^3 +...) and you want to
find X mod 9. Then all of the 10^k's are just 1 (mod 9), so you just get the
sum of the digits.

So if X is divisible by 9, then the sum of the digits (mod 9) is zero.

Same works for 3 (x is div by 3 iff the sum of the digits id divisible by 3).
And 11 gets an /alternating/ sum of the digits, since 10 is -1 mod 11...

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rb808
A good example of why well educated parents is more important than getting
into the "the best schools".

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tesin
Isn't that just a matter of chicken and the egg timing tautology? The parents
are well educated presumably because they went to good schools.

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gaius
Not necessarily - many parents highly value education and encourage their kids
despite and/or because they don’t have much themselves. Recent immigrants are
a good example of this phenomenon.

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tesin
I don't disagree with that statement, but rb808 specifically cited "well
educated parents".

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bitlax
Feynman on why questions:

[https://www.youtube.com/watch?v=36GT2zI8lVA](https://www.youtube.com/watch?v=36GT2zI8lVA)

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jordigh
> 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?

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

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noobermin
That's a good point. I'm not one for details and I just assumed it above the
horizontal because it makes it a touch more interesting.

