
Connections between Abstract Algebra and High School Algebra (2015) - jpelecanos
https://blogs.ams.org/matheducation/2015/12/10/connections-between-abstract-algebra-and-high-school-algebra-a-few-connections-worth-exploring/
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galobtter
"For example, we teach students in high school that if the product of two
polynomials is zero, then to solve we set each one separately equal to zero.
Yet this does not hold with nonzero numbers. For example, working in
polynomials with real coefficients, we know that f(x) * g(x)=0 implies either
f(x) = 0 or g(x) = 0. Yet it is not the case that if f(x) * g(x) = 4, then
either f(x) = 2 or g(x) = 2."

Does this really require knowing abstract algebra? Seems obvious to anyone
doing any sort of multiplication that if the output is 0 then one of
variables/functions has to be 0, if it is nonzero then the variable/function
can be anything but 0.

~~~
dboreham
Reading this I thought the same thing. Then I remembered one of my high school
teachers telling the class that Bertrand Russell wrote a multi-volume book
with the goal to prove that 1+1 = 2. At that point I realized that you don't
need to work in abstract algebra in the high school mathematics curriculum,
but rather you need teachers who have a deep understanding of mathematics.
Unfortunately given economic reality that's hard to achieve in the present
day.

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kindfellow92
You only need multi volumes to prove 1 + 1 = 2 if you start with first order
logic. If you start with the rules of basic arithmetic, it takes less than a
page.

There is no fundamental difference when changing your starting assumptions
other than one set of assumptions might prove more things than the other. From
the perspective of single proof, either is equally as good. We axiomatically
know basic arithmetic to be true, just as we axiomatically know first order
logic to be true.

~~~
dboreham
Boy, I didn't realize I needed to set the explicit "irony flag" on that post!
The idea is that my math teacher was making a math joke, stimulating his
students' curiosity to think about deep things like "what does it mean to add
and how do you prove things that seem intuitive" and informing the class that
people have written serious and long books on the foundations of mathematics.
None of my kids' math teachers (so far) have made any math jokes...

~~~
posterboy
It's not really a joke, just an understatement. Very dry humor indeed. Could
be modesty or exageration, but at its core, it's true.

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vinchuco
6 4 / 1 6 = 6̶ 4 / 1 6̶ = 4 / 1

2 6 / 6 5 = 2 6̶ / 6̶ 5 = 2 / 5

9 5 / 1 9 = 9̶ 5 / 1 9̶ = 5 / 1

...

[1]
[https://en.wikipedia.org/wiki/Anomalous_cancellation](https://en.wikipedia.org/wiki/Anomalous_cancellation)

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ivan_ah
Very cool: closure + associativity + identity + inverse properties we use for
addition and multiplication = group structure. See
[https://en.wikipedia.org/wiki/Group_(mathematics)#Definition](https://en.wikipedia.org/wiki/Group_\(mathematics\)#Definition)

Simplify and connect for the win!

~~~
cmpb
Yep! And I really wish this concept (of bringing together all of those
properties, and exploring some of the resulting structures) was taught
(specifically including the introduction of the term "group") in high school
algebra. I think that would reduce a lot of the friction between learning
secondary and post-secondary mathematics.

~~~
llamaz
It _is_ taught to 16 year olds in Australia, if they elect to take math C
(math a/B is compulsory where math A is remedial). Everyone hated it and
wondered what the point was (including me - it was only when I did a course on
abstract algebra in university and learned about quotient groups that I was
converted)

Also covered in math C is the computational parts of linear algebra (e.g.
Gaussian elimination, Cramer's rule).

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jonsen
Great. Teach kids some rote manipulations. Things they will have forgotten if
they later stumble upon some math. Or will have to relearn in college math.

Will the same mistake happen with the learning to code in school movement? Or
will there be a sensible connection between abstract computer science and high
school coding?

~~~
noxToken
I think the learning to code movement is a general waste of time. Sure, offer
them the class if the child is interested, but shoveling kids through a
mandatory Scratch or Python course? Are we really supposed to believe that the
ability to create Pong in Python is time better spent than something like auto
skills?

Computer literacy (not just how to use the Office suite) and very basic IT
(passwords, security culture, generalized troubleshooting, basic networking,
etc.) would take a child much further as a general purpose class. Think about
every time a family member has called and complained that they've lost
internet connectivity. It turns out they unplugged something or flipped a
hardware switch. Teaching students these basic parts of troubleshooting would
be a boon.

