One of my most memorable courses in undergraduate was taught by a spritely lecturer who must've been in his 80s. He'd make a statement or proposition, write it on the board, then quickly turn around, point at a student somewhere in the classroom with an open palm, and shout, "do you believe this?"
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.
> My second axiom concerns the process of learning. It says: We learn when we are challenged, when we push ourselves. If you’re not stuck you’re not learning. If it’s not a struggle you’re not doing it right.
It follows that we must always look for new points of view and pursue open-ended questions. The role of the teacher is not to make life easy for the student by giving crystal clear lectures and predictable tests. Instead the role of the teacher is to guide and encourage the student’s own process of learning by setting suitable challenges and by stimulating thought and reflection.
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.
It's small, but one of my favorite math professors in college, when he could tell the class was losing focus on the lecture, would begin proofs with a shouted "BEHOLD x!" instead of the more traditional "let x..." It always woke people up.
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 convinced by this... as long as we make a distinction between mathematics and arithmetic.
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.
I find this axiomatic approach shallow and unworkable. You will only reach kids that already don’t need anything. Ie, those who are already self motivated. Those kids aren’t the issue.
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.
Yeah I can’t fully agree with axioma #2. While people learn when you challenge them, every person reacts differently when they struggle. And there is this thing, anxiety, that makes people insecure. Mathematics anxiety is even a thing [1].
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.)
The first bit is interesting. I really didn't get excited about math until I had a TI89, and then I was obsessed with reverse engineering it. Eventually, I would build my own computer algebra system while doing math grad school.
I do definitely agree that something is lacking in current mathematical education, but calling your own counter-program "intellectual mathematics" is a bit ... on the nose.
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.
Talking about not understanding rigor in the context of calculus is silly.
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.
I read the manifesto and somewhat disagree with the claim "It follows that we must not introduce any topic for which we cannot first convince the students that they should want to pursue it."
But in skimming the book, it's just seems not very good.
"I read the manifesto and somewhat disagree with the claim "It follows that we must not introduce any topic for which we cannot first convince the students that they should want to pursue it.""
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 also took a brief look at the calculus "book" (more a set of lecture notes) and far from being the revolutionary novel approach that it claims to be I found it fairly... standard?
Math was born out of reality needs, not from thin air.
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.
Well for starters programmers are not engineers, calling them that is just putting lipstick on a pig, a feel-good title like "chief corporate luminosity enhancer" instead "window cleaner", calling chiropractors "doctors", putting police-like uniforms on mall shop security guards, etc. Pathetic and lame but the term has stuck.
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.
Is there something similar to this, but for school-grade mathematics?
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.
Depends upon your son but perhaps youtube + forum engagement with others centred about math channels that have lower+upper school and university material.
> 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.
"In Intellectual Mathematics a topic is introduced only when the student can be convinced of the value of doing so."
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.
oofnik|2 years ago
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.
throw_pm23|2 years ago
atomicnature|2 years ago
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.
dan-robertson|2 years ago
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.
thewakalix|2 years ago
jstrieb|2 years ago
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.
OliverJones|2 years ago
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.
dr_dshiv|2 years ago
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.
titaniczero|2 years ago
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.)
——-
[1] https://en.m.wikipedia.org/wiki/Mathematical_anxiety
mathgladiator|2 years ago
jmount|2 years ago
Tainnor|2 years ago
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.
zmgsabst|2 years ago
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.
https://en.wikipedia.org/wiki/Weierstrass_function
WoahNoun|2 years ago
But in skimming the book, it's just seems not very good.
hutzlibu|2 years ago
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.
Tainnor|2 years ago
revskill|2 years ago
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.
MichaelRo|2 years ago
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.
murkt|2 years ago
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.
defrost|2 years ago
> 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.
https://www.youtube.com/user/standupmaths
alimw|2 years ago
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.
danielmarkbruce|2 years ago
edtechdev|2 years ago
neonate|2 years ago
mif|2 years ago