I agree with this manifesto wholeheartedly, just it’s too damn difficult to think of all the whys on the spot to get my son to be interested in that.
He’s doing “remote learning” at home, it’s basically Coursera-like lessons, since his school was hit by artillery during the first month of the war with Russia, and it still doesn’t work.
> It follows that we must not introduce any topic for which we cannot first convince the students that they should want to pursue it.
There are several good math related youtube channels and they have the advantage of allowing viewers to find questions | topics that they find of interest and then pursue .. which can lead to following for the latest releases and discussing with others that comment.
Matt is very engaging and there's a fair bit of back and forth from others on his channel comments which might take your son down a path of greater engagement and exploration.
If I understand this correctly, you cannot bend the approach to the teaching of any particular topic (e.g. what's in your son's homework this week). Rather you would have to bite the bullet and teach your own parallel course, abandoning the school curriculum to the teachers.
It doesn’t bring much understanding, though. It’s hard to create parallel curriculum for my kids, while I have my startup running. And the war isn’t helping either, with air alerts and rockets/drones trying to strike Kyiv seemingly every night.
Would be helpful to have concrete examples, directions, etc.
It was partly theatrical, but it seemed genuine. It was very engaging and it really prodded me to consider whether what he was telling us was true. It piqued my curiosity and motivated me to do my homework to watch someone so filled with energy trying to get a bunch of undergrads excited about multivariate calculus, especially at his age.
I think I speak for many here when I say that I'm more motivated to learn stuff when I can clearly articulate what benefit I will derive from having mastered the material. "Because it's going to be on the exam" never cut it for me. Unfortunately, I found that I was in a tiny minority of intellectually engaged students for the majority of my undergraduate years; the rest of my peers were much more interested in all-night cramming for the exam followed by a weekend of binge-drinking.
I don't know how well the approach in this essay would work for the general population, but it certainly sounds much more interesting to people who actually find studying mathematics enjoyable.
Judging from the rest of the website, the author appears to have some rather idiosyncratic opinions. For example, he seems to be unconvinced that rigour is an essential component of mathematics (even going as far as claiming not to understand what it means): https://intellectualmathematics.com/blog/what-is-rigour-anyw...
He also has his own take on the "two cultures" distinction often postulated for mathematicians, but also apparently assigning distinctly less value to so-called "lesser technocrats", seemingly going as far as calling Euler of all people a "technocrat" and calling into question the value of Euler's famous equation e^pi*i = 1 - why? https://intellectualmathematics.com/blog/four-types-of-mathe...
I see no indication that the author doesn't know the mathematics he's talking about, but I also feel like he's incredibly biased towards the particular way he likes doing mathematics without considering that there are legitimately valid different approaches to doing mathematics - probably even among students.
Modern formalism is because of calculus!… and the fact that contrary results were obtained regarding continuity and derivatives due to subtly different conceptions of the terms.
Enter the Weierstrass function:
> Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness.
But instead of saying "I'm not proving things rigorously because [reasons]", he's claiming that rigour doesn't really exist or has no importance which is kind of crazy for the reasons you mentioned.
But in skimming the book, it's just seems not very good.
Why not? Did you never experienced the differences in learning something because you had to vs. learning something because you are interested in it? I found the latter to be far more effective and the first mostly a sad waste of time.
School math was mostly wasted on me. So many hours for nothing (even though I had somewhat good grades). But when I have a specific coding problem now, that I can solve with math - then I see a reason and then I enjoy doing math, as it now has a purpose. If people think, I will never need that crap, than their brain will resist learning it. The result is wasted time and energy for everyone involved.
I think math instruction should take a cue from Pythagoras who, according to Iamblichus, taught by first paying his students a small amount for each successful learning accomplishment — until the students were older and wanted to pay him to learn more. It’s legendary, but the truth is that we shouldn’t expect kids to want to learn math.
Learning core skills should not rely on intrinsic motivation. It is so much easier to be intrinsically motivated once you have a base. Otherwise everything is just difficult and frustrating.
So I think this axioma is based on a previous one: everyone wants to be challenged, which I believe is not true. You have to give people a reason for that motivation first, to be able to challenge them. Otherwise you will make people feel insecure, because they will think that they’re not smart enough, and we are taught that maths = smart. (Maybe we have to tackle this social construct first.)
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But due to the force of all kinds of artifical force (passing the exam, learn by heart all the formula, paying the college debts,..), the quality of books is bad. It's more like a boring dictionary in most of cases.
The Math way of thinking is very different from software engineering though: On math, it's more about How things is reasoned about, rather than on result. It's how Math created Math itself.
When it comes to math, it's actually used by two distinct categories: real engineers and mathematicians. There's also physicists but they share the same trait with engineers: math is a tool and not a purpose. A mathematician will become very anal when you skip some tiny detail in a demonstration, like go from Taylor series expansion to Ito lemma by approximating dt^2 =~ 0 (I had this happen to me). An engineer (that is, me ;) couldn't give a funk since Ito lemma is a well known and already proven fact and they only use the quick derivation from Taylor expansion as a way to mentally remember the former when they need it for some actual, real-life use case.
Thanks for making this point. Many people tend to have a faulty view of education as some sort of comfortable path, where one's present proclivities are retained or even enhanced. However, if education is done right, then it is total revolution within the individual, destroying all sorts of faulty mental structures to build a better structure altogether.
I believe teachers who do not espouse challenge or do not put such challenge into daily practice, are of questionable value.
A better understanding of what education is about, from both teachers & learners is of great importance.
I have a weak interest in the best way to teach things. There are two interesting things in the U.K. around it. One is that methods for teaching children to read seem to have improved recently (that is, the results in tests used to measure reading ability have improved) and there has been a claim that this was mainly due to doing and then applying the results of some research into how to best teach it. The research was cheap and one wonders if it could be applied to more situations. Meanwhile in mathematics there is a regular desire from governments to improve the mathematics that is learned but it feels like the efforts don’t go so well. Usually any time mathematics education is in the news, Simon Jenkins will trot out the same ridiculous tired old article against mathematics education. The disconnect between the way that mathematicians and non-mathematicians think about mathematics education seems pretty bad to me.
Even for the most diligent students (which I was not), there will always be some dry material. Often, when I was struggling with writing up a difficult proof, I imagined him saying "BEHOLD," and it would cheer me up. Little things can make learning much more fun.
I'm grateful now, and was not at the time, to my educator mother for drilling me and drilling me again on addition, subtraction, multiplication, and division. She drilled me until we were both so frustrated we couldn't see straight.
Her relentlessness gave me a basic numeracy that set me free to explore conceptual math and actually have fun doing it. She did it for my sibs too. It made us kiddos capable of playing car-trip games like "spot the prime number on the license plate".
She knew there was no magic pedagogy to learning that basic arithmetic, just drilling. Now she was no mathematician herself. Mention the central limit theorem to her and you'd get "huh"? She studied classics in school. But she sure knew how to to teach.
