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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

#161
post #43
post #19

Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday stru…

To this point, I have been doing a plenty of 1on1 lessons with kids trying to catch up with Math in high school/college/matura exams and one this I have noticed is that things which are problematic to these people are things which were covered quite well already in elementary school. Later the curriculum only expands on these problems so it's even harder and harder to catch up. I have been working with college freshm…

Not all are created equal. Our children are unique.

One of my children showed natural talent in language at 9 months with no prompting. This was brought to our attention by childcare staff. Another of our children showed a natural talent with mathematical concepts at about a year old.

Even as they grew, our linguist struggled with math (for years) and our procedurally oriented child struggled with language (for years).

To this day, these two children maintain these core differences. It took at least 6 years for our linguist to crack basic arithmetic (even basic addition) which was at least several years behind our proceduralist.

I found out later that some leading child psychologists recognise different brain types in very young children (exactly as I found). An Internet search on brain types of children will show some high profile child psychologists who talk about this in depth, despite some strong "opinions" (ie. devoid of evidence) that oppose these studies.

Our linguist, with minimal pressure, has developed into a strong mathematician (at least grades wise) but to this day has never demonstrated anywhere near the natural ability of our proceduralist.

I have drawn the same conclusions with my own siblings and my wife's siblings. At a very young age, our own strengths become apparent without intervention. I am very glad I never pushed my kids to be equal (or even close to equal) in all skills. I consider most uses of the word "equal" worrisome (except for equal opportunity, a concept frequently downplayed in the last decade or so. Even Zuckerberg's famous open letter was unclear on such a fundamental concept).

OBS: when I asked our linguist to step through basic math, they understood the concept but could not do the work independently. Someone in this thread described a similar story for their child and attributed this to a lack of "confidence". For my child, I wholly reject that it was confidence related. When things clicked for our linguist, they clicked. If anything, our linguist's ability to crack the basics took patience on my part. I wanted my child to succeed quickly but I restrained myself (thankfully).

People need to realise that not all kids are the same. We have innate strengths. We have different learning styles, different learning rates, and different interests and motivations. I strongly reject the modern populist theory that we are all equal in ability and I believe we do significant harm because of this factoid. The motives behind this factoid concern me deeply.

If I could offer one piece of advice, your child(ren) are unique. Don't ever let anybody tell you that your child's strength or weakness comes from social conditioning. The only social conditioning cones from extreme behaviour (eg. Heavy handed forcing of "equality" under the banner of political correctness is extremely harmful, rather than focussing on potential and opportunity. This heavy handedness is also driving some extremely destructive social engineering under the banner of "equality". If you are watching academic trends you should be horrified as a patent).

I always encourage(d) play at a young age (physical activity, math games, language games). However, if you make this more than games (if you call this teaching and you start to measure), you set kids up for failure, especially when many children need patience and time.

According to PISA rankings, most western countries (especially English speaking) are not the top performers. I have hinted my beliefs of the root cause of this in this post. I predict most western countries will slip in ranking even further (especially English speaking countries). If things play as I expect, the slip will be significant in the next 10 years.

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

#162
post #11

Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically . The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day…

Many other commenters have pointed out how great examples and ties to the real world are; the love of applications is well-represented in this discussion. But I'll throw out beauty as an underrated motivator!

I love math because it's beautiful, primarily, and this isn't represented often in elementary or high school education. The idea of "function" or the idea of "constructive proof of bijection" are beautiful concepts. The patterns of math are beautiful (so beautiful I created an adult coloring book called Math with Crayons that focuses on Aztec diamonds, fully packed loops, lozenge tilings of hexagons known as totally symmetric self-complementary plane partitions -- you know, the stuff people are actually doing research on today!). Numbers to me are kind of boring. But how many kids get to draw fractals in class and then talk about the concept of dimension, and then argue about how one could rigorously define dimension, with the example of trying to define dimension for fractals as an edge case? How many kids get to draw non-crossing partitions and Dyck (lattice) paths and then try to prove that they are both enumerated by the Catalan numbers? How many kids get to think about the mathematical structures you allude to, and play with the fundamentals of category theory? Well, very few. And why?

Their teachers don't know any of that stuff.

If teachers only know computation, that's all they can teach.

If we had teachers who really had breadth and depth in math teaching math, I think we could start teaching concepts, structures, and beauty very early. I've had conversations about the basic ideas of category theory (relations not objects!) with many kinds of people. This is not out of the grasp of a kid. Set theory -- not out of the grasp of kids. The very idea of what a number is: in high school I got to take a seminar with other high school students on this, looking at John Stuart Mills and Frege and Russell and Whitehead and all this thinking about what a number is. But this is extraordinarily unusual material for a high school or even college student. Of course it's difficult to deal with these notions later, then -- people have been trained out of their natural curiosity and into a view of math as computation at worst and manipulation to find x at best, rather than seeing it as a way of dealing with beautiful structures.

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

#163
post #149
post #141

Earlier quoted context omitted.

Then maybe instead of calling math fancy arithmetic, we should call arithmetic "hard math"? :P Seems like you agree more than disagree; math and arithmetic aren't easily separable, if you're being serious instead of cheeky?

Well, I was trying to make two points. 1) Arithmetic (number theory) is far harder than most people realise, to the point that its considered one of the most complicated mathematical disciplines. 2) Most subject areas in math bear little to no resemblance to arithmetic, so it doesn't make sense to call all math 'Fancy Arithmetic'.

Sure, understood and I agree. I was also trying to make one point: all math is built on arithmetic operations if you dig down to first principles, so it does make some sense to call all math 'fancy arithmetic'. There is no math at all without the concept of addition sitting somewhere under it.

But I'm not actually proposing that, just sharing a cheeky point of view to contrast @taneq's boss, so don't take me too seriously. ;)

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

#164
post #63

Earlier quoted context omitted.

Schools (at least around here) do not hold kids back anymore. No matter what. Some of them get behind in one or more subjects (math, say) as early as 1st or 2nd grade and all their later teachers are stuck trying to give them the remedial math help the desperately need, while teaching their other students on-grade-level material, and also trying to teach the remedial kids enough of the new stuff that their scores on…

Yeah. To my mind, the problem is that learning is very individualized (not even taking into account learning disabilities!), and yet mainstream U.S. pedagogical theories really only target the middle of the bell curve. What's needed -- at least for a huge chunk of students -- is something more like Montessori, where a learner who's struggling can receive individualized attention from a teacher or a peer , and where t…

Montessori is awesome when done right. It's also easy to get wrong. Moving an entire educational system over to it is a formidable task.

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

#165
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. Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi wit…

> 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…

Memorizing time tables and similar rote learnign means that precisely kids who could be good at math hate it. Meanwhile, those who have good memory and sux at problem solving think they are good at it.

People who were taught by memorising go into outrage a new type of exercise is introduced. Suddenly thinking is needed and that is bad in their eyes. Not exactly success. Meanwhile, you can memorize time tables in later age if you decide it is useful (people rarely do).

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

#166
post #144

Earlier quoted context omitted.

It is a US thing. And it's because until middle school (age 11-13) your teacher is likely not that familiar with math (they have taken math courses, but likely little formal math beyond geometry and trigonometry, perhaps some courses on math pedagogy). So for them, math is all about the numbers, they barely got to the more abstract concepts. In middle school in the US you start getting teachers specialized in their t…

What kind of education is required from teachers in US? In Poland it is Bachelor's degree in relevant topic (for elementary and middle school) or Master's degree (for high schools) plus a course in pedagogy. Most teachers however get Master's degree even if not required (they get salary increase for that). It is hard for me to imagine math teacher not good in match themselves.

For all grade levels a BS and certifications is all that's required in most of the US. Sometimes a specialized degree is required or helpful (this is regional) for certain areas. Like a degree in primary education may get you a job as a 1st grade teacher in some state where a degree in English wouldn't (because that degree, ostensibly, conveys the knowledge and ability to specifically help young students like identifying learning disabilities and modalities, or some roles bordering on social services like identifying psychological trauma). At the higher grades (starting at 6th grade or so in some areas, but more often just high school) a degree in the subject matter is preferred or required. But this can often be worked around by demonstrating capability (certification exams and classes) or having enough course work in the subject (like most STEM degree holders have enough math to qualify as a math educator).

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

#167
post #145

Earlier quoted context omitted.

∆ - delta - d as in difference ∑ - sigma - s as in sum ∫ - long s - s as in sum It's not totally arbitrary.

I could use alpha - a as in add. I could use sigma - s as in subtract. I could use iota - I as in integrate. Of course it's arbitrary.

You claimed the symbols were arbitrary, they are (often) not. What you demonstrate here is not that the symbols are arbitrary, but that the concept (sound as abbreviation) maps to more than one potential meaning and the choice of which one is standard or convention is arbitrary.

That is correct, but at this point it is standard to see sigma as summation (or in some other contexts as set or space, but it's also clear in those contexts which meaning is appropriate, certainly an algebra student seeing sigma notation will not also be seeing sigma used for "set" or "space" in that same course).

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

#168
post #11

Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically . The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day…

Learning math is like learning a language. It simply takes a lot of time and effort. If you read the rules of German grammar cover to cover, that will not prepare you to speak German; in order to be fluent, you MUST practice, over and over again, using and interacting with German.

Likewise for math. There is no "new book" to learn math that is going to blow the cover open on everything. It's a problem of motivating people to become fluent in math. I would like to see discussion in those terms -- fluency, because that's what it is.

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

#169
post #55

Earlier quoted context omitted.

Henri Poincaré once said: "Mathematics is the art of giving the same name to different things" I know the temptation is great to give lots of examples while teaching math, but you get the risk of only teaching little examples to students and not the abstract vision that math education demands. Take your quadratic equations, it's all nice to demonstrate how coefficients shape the plot. But when you start talking about…

As a counterpoint to that, I flat-out despised abstract Math until I got to college precisely because I found it useless. At best I knew I was probably going into engineering in college and I'd been told "engineers use calculus". That was my sole motivation to learn anything about abstract math, and it was pretty bad. It didn't help that the "examples" given in class were mundane toy problems (ex: "how long will this…

This is especially a problem in college. Few math profs seem to enjoy teaching and seem reluctant to ground their subject using concrete illustrations. Without these, there are no tangible points of reference, aside from derivations from known equations. But mere transformations rarely offer insight without adding some sort of context.

I think the crux of the problem in teaching math is the need to illustrate each concept from MULTIPLE perspectives: theoretical, tangible, incremental, graphical, dynamic, etc. It's only by connecting the concepts to manifestations that most of us appreciate the meaning of math. Few are satisfied by disembodied equations or proofs. Most of us need some form of grounding using what we already know to make sense of new concepts. Otherwise math is just castles in the air.

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

#170

Earlier quoted context omitted.

In theory it shouldn't matter but it does. We brainwash students to think in terms of y being a function of x. y = f(x) If you give a problem with x = f(y) it really confuses people. If I label the horizontal axis y and the vertical axis x in the cartesian plane then confusion ensues. t = b(a) where t is the velocity and a is time That's confusing. I suppose it's analogous to me saying "The male, fluffy, white, large…

If this is enough to "really confuse" someone than there's a serious problem with understanding VS rota learning.

You're totally right... but the previous poster is too. I encounter many students who really freak out at f(t) = y vs f(t) = x vs f(x) = y. Obviously they don't understand very well. I go back and forth between using notation as a crutch (always stay consistent so we can focus on the topic instead of the notation) and trying to really wean students from notation-dependence (let's do f(smily face) = y today! let's rotate all the axes and draw this with the z-axis sticking out!).

When people have very fragile understanding, they may be genuinely unsure if derivative with respect to x is computed the same way as derivative with respect to n. After all, x is always a real-valued variable and n is always a non-negative integer.... right?

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