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Math education: It’s not about numbers, it’s about learning how to think

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Re: Math education: It’s not about numbers, it’s about learning how to think

#111

Post by [deleted] »

J_Sherz wrote: post #77
MengerSponge wrote: post #73

Earlier quoted context omitted.

You couldn't finish your tests in high school, but you're good at math? Have you ever been evaluated for test anxiety? https://www.adaa.org/living-with-anxiety/children/test-anxie... There's no shame in it (although high schoolers can be assholes), and it can help your school accommodate your needs eg extra time to take your exams.

I've never been evaluated but I don't think test anxiety fits in my case - the speed issue was only ever in math / physics. My accuracy on the questions I got to was very high, I just couldn't go fast enough to complete enough questions. Same deal on SAT type math papers too.

[deleted]

Re: Math education: It’s not about numbers, it’s about learning how to think

#112

Post by [deleted] »

J_Sherz wrote: post #77
MengerSponge wrote: post #73

Earlier quoted context omitted.

You couldn't finish your tests in high school, but you're good at math? Have you ever been evaluated for test anxiety? https://www.adaa.org/living-with-anxiety/children/test-anxie... There's no shame in it (although high schoolers can be assholes), and it can help your school accommodate your needs eg extra time to take your exams.

I've never been evaluated but I don't think test anxiety fits in my case - the speed issue was only ever in math / physics. My accuracy on the questions I got to was very high, I just couldn't go fast enough to complete enough questions. Same deal on SAT type math papers too.

[deleted]

Re: Math education: It’s not about numbers, it’s about learning how to think

#113

Post by taco_emoji »

inanutshellus wrote: post #85
throwawayjava wrote: post #69

Earlier quoted context omitted.

Tying students' progress through the curriculum to their age is silly. If you think about it for a single second, the whole thing is totally absurd. Vague pre-puberty/tween/teenager distinctions make sense, but within each grouping/building? totally unmotivated. Yearly age-to-competency distinctions continue by sheer force of tradition, and are harmful to all but the totally mythological "average" student.

But it works at-scale, which is where the focus is. Bang-for-the-taxpayer-buck. Further, socially-speaking, being with your actual peers is extremely valuable, obviously. But yes, the ideal would be a tailored-to-each-child education. Personally, having seen the power of a good Montessori education, I think it's frickin' genius and should be the template for all education (specifically the grouping of 3 yrs together…

Not being argumentative -- is the current system actually cheaper? I'm not enormously familiar with Montessori, but is the student/teacher ratio vastly different, or are the various learning materials any more expensive over time than the books, computers, and reams of paper that traditional schools use?

Re: Math education: It’s not about numbers, it’s about learning how to think

#114

Post by ouid »

When people talk about the failure of mathematics education, we often talk about it in terms of the students inability to "think mathematically".

It's impossible to tell if students are capable of thinking mathematically, however, because I have not met a single (non-mathlete) student who could give me the mathematical definition of... anything. How can we evaluate student's mathematical reasoning ability if they have zero mathematical objects about which to reason?

Re: Math education: It’s not about numbers, it’s about learning how to think

#115

Post by mathperson »

acjohnson55 wrote: post #104
bitwize wrote: post #98

Earlier quoted context omitted.

> I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables. This was tried. It was called "New Math". Spectacular failure. Do you want to know what worked? Memorizing times tables. There's just no way around the fact that drilling is key to early mathematical lea…

It's way too early to judge "New Math" to be a failure. I think the idea of focusing on concepts over computation 100% in the right spot. The biggest problem to its acceptance is cultural. Parents don't feel comfortable with math concepts, and in my opinion, most of the negativity is coming from that insecurity. So they demand for things to be taught the "old way", even though that's produced a generation or two of m…

Not even wrong. You think New Math refers to the current modifications of common core. New Math actually refers to a series of cold war changes made in response to perceived russian scientific dominance.

https://en.wikipedia.org/wiki/New_Math

Re: Math education: It’s not about numbers, it’s about learning how to think

#116

Post by vonmoltke »

acjohnson55 wrote: post #104
bitwize wrote: post #98

Earlier quoted context omitted.

> I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables. This was tried. It was called "New Math". Spectacular failure. Do you want to know what worked? Memorizing times tables. There's just no way around the fact that drilling is key to early mathematical lea…

It's way too early to judge "New Math" to be a failure. I think the idea of focusing on concepts over computation 100% in the right spot. The biggest problem to its acceptance is cultural. Parents don't feel comfortable with math concepts, and in my opinion, most of the negativity is coming from that insecurity. So they demand for things to be taught the "old way", even though that's produced a generation or two of m…

I knew a guy with a Physics PhD who would agree completely with bitwize, so I don't think it is just parental insecurity.

Re: Math education: It’s not about numbers, it’s about learning how to think

#117

Post by e12e »

adamnemecek wrote: post #57
redial wrote: post #40

Earlier quoted context omitted.

Symbols are symbols, are symbols. Instead of trying to compress everything into a very limited set of "daily language" symbols we should be trying to expand our ideas, including our language(s).

Let's add Chinese, Amharic and Arabic characters. Why not, expanding is expanding.

Arabic numerals are already widely used in mathematics...

Re: Math education: It’s not about numbers, it’s about learning how to think

#118

Post by Jtsummers »

dahart wrote: post #91
Jtsummers wrote: post #76

Earlier quoted context omitted.

> The symbols we use in math aren't to be confused with the ideas, they are shorthand. Ideas and symbols are independent, and we still have to explain what Greek letters do using English anyway. We use Greek letters for convenience, not for expressive power. Writing "∫ x ∂x" is only faster and more compact than writing "The integral of the equation x over x", it is not an improvement of the concept, right? It is not,…

Very true, good point, it's much easier and more convenient to do algebra on the symbolic expression. There is a reason math has evolved the way it has. Not using any symbols would suck, no question. But, nothing about writing math in text prevents manipulating the text, right? You can do it, it would just be laborious and exhausting for people who are familiar with the use of symbols.

It wouldn't just be laborious for those of us familiar with the symbolic notation, it'd be laborious for the new students as well.

Try a "solve for x" problem with the equation described in prose (still using short variable names and numbers though):

  x added to 4 all divided by 3 is equal to 16.
What's the next step in solving this? First, because we're abandoning the conventions of symbolic notation the groupings of equation parts (terms and expressions) becomes a bit hairier. I've tried to be clear, and I think in this instance it is, that "divided by 3" applies to "x added to 4". So we know that we should multiply both "sides" by 3. I put sides in quotes because, well, now there are no sides. An equally valid prose description is:

  one third of x plus four is the same as sixteen
If we lack conventions for expressing these statements we could probably come up with an absurd number of equally valid prose presentations of this one expression. This makes reasoning about the next steps much, much harder, as we first have to parse a near arbitrary expression and then discern the author's intent (and seek clarification for more complex ones).

  the result of dividing four and x by three is sixteen.
By the time you settle on a subset of english for prose math specification you'll be down to such a small or specialized portion of the language that you might as well add back in the symbols and at least lose the English dependency for your work so you can collaborate with people around the world.

There's a reason a good portion of elementary mathematics was spent on "word problems", where we reduced problems like this down to their essential equations (a process of abstraction, though not described to us as such). Usually you'd see the above as something like:

  Andrew has some number of apples. Billy gives him 4
  more. He splits the apples evenly between himself,
  Billy, and Charlie. Everyone has 16 apples, how many
  did Andrew have to start with?
All that other crap is noise. Just like in the prose version of the equation "the result of ..." has no value in trying to solve the problem.

Re: Math education: It’s not about numbers, it’s about learning how to think

#119

Post by archeantus »

If we want to teach people how to think, I propose that math isn't the best way to do it. I can't tell you how many times I complained about how senseless math was. The real-world application is very limited, for the most part.

Contrast that to if I had learned programming instead. Programming definitely teaches you how to think, but it also has immense value and definite real-world application.

Re: Math education: It’s not about numbers, it’s about learning how to think

#120

Post by ebola1717 »

acjohnson55 wrote: post #104
bitwize wrote: post #98

Earlier quoted context omitted.

> I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables. This was tried. It was called "New Math". Spectacular failure. Do you want to know what worked? Memorizing times tables. There's just no way around the fact that drilling is key to early mathematical lea…

It's way too early to judge "New Math" to be a failure. I think the idea of focusing on concepts over computation 100% in the right spot. The biggest problem to its acceptance is cultural. Parents don't feel comfortable with math concepts, and in my opinion, most of the negativity is coming from that insecurity. So they demand for things to be taught the "old way", even though that's produced a generation or two of m…

idk, I studied higher level math very seriously, and even there, practicing computations is pretty key to understanding the underlying concepts. Concepts and abstractions are important, but they don't stick without practice.
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