I also like Jo Boaler's approach to school-level mathematics: https://www.youcubed.org/ (the Ideas section is probably the one to look at, although looking at examples of activities may give more of a flavour - the Mindset Mathematics set of books has some great things in it).
I think much comes down to why we teach mathematics at school. Some basic numeracy, geometry and statistics are good for everyday life, but for most people there probably isn't much they need beyond that, so the question is whether we are teaching it as an art or as a tool for the sciences. And there is a large intersection there to muddy the waters. I always suspect too that many people would really enjoy university-level mathematics who never discover that they would because they are only exposed to school-style mathematics. I was lucky that I encountered the more beautiful aspects of mathematics early on - and that I enjoyed them!
That was an interesting article. I liked the author’s description of trigonometry: “Two weeks of content are stretched to semester length by masturbatory definitional runarounds.”
(I think Juliette Culver and I knew each other in Oxford in the 90s; if so, hello again.)
Yes I think we did! I had a different surname then!
Lovely to hear from you - my mathematics these days is mostly doing Olympiad mentoring for the UK Maths Trust and showing my two boys (10 &12) bits of mathematics I think they will find interesting!
Thanks for these recommendations, I’ll definitely have a look!
And yes I completely agree - the “why” of learning maths shapes “how” it’s taught. No doubt that Wolfram is biased because of his computational background (symbolic computation isn’t standard in school yet, as he essentially assumes, but I suppose his argument is the world is getting more computational than less, hence applications of maths should account for that).
I completely understand about people missing out on university level mathematics because of the way it’s taught in school. With the benefit of what I know now, sadly I think I’m one of those people.
By the way the introduction to the Mindset Mathematics series is a decent introduction to Jo Boaler's stuff and I think can be read on the ebook samples.
Computing the problem is the last step, often skipped in school. This is because students are given a list of quadratics or probability questions, which can be boring and a common mistake in school. Applied maths is essential for several reasons. First, it eliminates the problem of trigonometry, catering to the majority of students who want to study pure or abstract maths at university. Second, applied maths is future-proof, as the tools of computation have changed over the last 50 years, from simple calculators to computers and artificial intelligence.
Thanks for reading! I'd say computing is the 3rd step, because making sense of the answer should be the last step, which is indeed often missed out. I agree making maths more applicable would eliminate the problem of trigonometry.
Thanks for the tag. Are you familiar with the 1968 paper “On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities” by J. M. Hammersley? (It is easy to find by Googling.)
> This eliminates the problem which most kids have with things like trigonometry. You'll often hear students say: "What's the point of learning trigonometry when I'll never have to use it ever again?" This might be true. But what if we gave a senior-year baseball player the question: "what angle do you need to hit the baseball for it to land in the stands?" Maybe they'll care about the answer. It might even spark their interest in mechanical engineering.
This is a colossal mistake. What you are describing is a physics problem for college undergrads requiring a solid understanding of Newtonian principles, not a problem for kids. There is a lot of very useful math out there, but kids' objection that they will never need trigonometry in real life is generally correct.
Fair enough. It’s on me to have come up with a better example. The point I wanted to illustrate is that the utility of a tool is explained by context, and not by the tool itself - kids in school don’t get to see this context in a relatable way when they learn maths.
But yeah fair enough trig being useless for most and it is probably over represented in current curriculums as a result.
Thanks for taking the time to read my article, share your thoughts and help me think about how to improve my argument!
Your article wasn't bad at all! I do have pretty strong opinions about the subject, though. I'm a physicist who married another physicist and had six kids, we know way more math than anyone we know, and we're frustrated that most of what our children learn in school is either useless, propaganda, or both.
If you're going into STEM, OK, learn math, but don't learn it with any applications at all, they're just distractions to those who know it will pay off in a career. Nobody else needs any math at all beyond about 9th grade. Cooking, driving, managing a budget, animal husbandry, and a heck of a lot of tech awareness, like how to write a simple computer program, how to avoid getting sucked into Youtube rabbit holes, and how to maintain a healthy, non-addicted relatoinship with social media - all of that is way more important than geometry, trigoometry, or algebra II.
Thanks for mentioning in your recommendations, Zan - I really appreciate it.
Also, I really enjoyed your essay. I think you make some good points. And I agree with the idea that wherever possible kids should be taught things in a way that is applicable to real world problems.
Thanks for mentioning me, Zan! I don't think I ask ChatGPT for answers, though 😎 I have a conversation with it about optionality and Stoicism - and I challenge it when it becomes a bit to dogmatic in it's loyality to Taleb's book.
To add on to your concept of relevance for math, there's so much in finance that can fit the framework. The magic of compounding, the fact that being up 50% and then down 50% means that you're down 25%. the concept of present value, and so on.
Absolutely! Ideas of NPV, risk-adjusted rates, annuities and perpetuities go so far in explaining compound interest and exponents in a meaningful way.
Even if students don’t want to go into finance - setting up questions about mortgage payments, or savings accounts in personal finance (for example) would still use all these tools.
I’ve only been on Substack now for just a short time. I have discovered a very friendly, thoughtful body of writers who enjoy supporting one another’s work. I am honored to have somebody like yourself enjoy my family writing. I was terrified no one would want to subscribe to my work. Again, I can’t thank you enough! 💝
If you haven't read it, then I Lockhart's Lament is worth reading: http://worrydream.com/refs/Lockhart-MathematiciansLament.pdf
I also like Jo Boaler's approach to school-level mathematics: https://www.youcubed.org/ (the Ideas section is probably the one to look at, although looking at examples of activities may give more of a flavour - the Mindset Mathematics set of books has some great things in it).
I think much comes down to why we teach mathematics at school. Some basic numeracy, geometry and statistics are good for everyday life, but for most people there probably isn't much they need beyond that, so the question is whether we are teaching it as an art or as a tool for the sciences. And there is a large intersection there to muddy the waters. I always suspect too that many people would really enjoy university-level mathematics who never discover that they would because they are only exposed to school-style mathematics. I was lucky that I encountered the more beautiful aspects of mathematics early on - and that I enjoyed them!
That was an interesting article. I liked the author’s description of trigonometry: “Two weeks of content are stretched to semester length by masturbatory definitional runarounds.”
(I think Juliette Culver and I knew each other in Oxford in the 90s; if so, hello again.)
Yes I think we did! I had a different surname then!
Lovely to hear from you - my mathematics these days is mostly doing Olympiad mentoring for the UK Maths Trust and showing my two boys (10 &12) bits of mathematics I think they will find interesting!
Such great serendipity! Glad I very passively facilitated this reconnection :)
Thanks for these recommendations, I’ll definitely have a look!
And yes I completely agree - the “why” of learning maths shapes “how” it’s taught. No doubt that Wolfram is biased because of his computational background (symbolic computation isn’t standard in school yet, as he essentially assumes, but I suppose his argument is the world is getting more computational than less, hence applications of maths should account for that).
I completely understand about people missing out on university level mathematics because of the way it’s taught in school. With the benefit of what I know now, sadly I think I’m one of those people.
I think you probably are!
By the way the introduction to the Mindset Mathematics series is a decent introduction to Jo Boaler's stuff and I think can be read on the ebook samples.
Thanks Juliette - I'll definitely take a look into this
You should check out @holymathnerd if you haven't found her on Substack yet.
Interesting! Never come across her but will subscribe now - thanks Michael
Computing the problem is the last step, often skipped in school. This is because students are given a list of quadratics or probability questions, which can be boring and a common mistake in school. Applied maths is essential for several reasons. First, it eliminates the problem of trigonometry, catering to the majority of students who want to study pure or abstract maths at university. Second, applied maths is future-proof, as the tools of computation have changed over the last 50 years, from simple calculators to computers and artificial intelligence.
Thanks for reading! I'd say computing is the 3rd step, because making sense of the answer should be the last step, which is indeed often missed out. I agree making maths more applicable would eliminate the problem of trigonometry.
Thanks for the tag. Are you familiar with the 1968 paper “On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities” by J. M. Hammersley? (It is easy to find by Googling.)
> This eliminates the problem which most kids have with things like trigonometry. You'll often hear students say: "What's the point of learning trigonometry when I'll never have to use it ever again?" This might be true. But what if we gave a senior-year baseball player the question: "what angle do you need to hit the baseball for it to land in the stands?" Maybe they'll care about the answer. It might even spark their interest in mechanical engineering.
This is a colossal mistake. What you are describing is a physics problem for college undergrads requiring a solid understanding of Newtonian principles, not a problem for kids. There is a lot of very useful math out there, but kids' objection that they will never need trigonometry in real life is generally correct.
Fair enough. It’s on me to have come up with a better example. The point I wanted to illustrate is that the utility of a tool is explained by context, and not by the tool itself - kids in school don’t get to see this context in a relatable way when they learn maths.
But yeah fair enough trig being useless for most and it is probably over represented in current curriculums as a result.
Thanks for taking the time to read my article, share your thoughts and help me think about how to improve my argument!
Your article wasn't bad at all! I do have pretty strong opinions about the subject, though. I'm a physicist who married another physicist and had six kids, we know way more math than anyone we know, and we're frustrated that most of what our children learn in school is either useless, propaganda, or both.
If you're going into STEM, OK, learn math, but don't learn it with any applications at all, they're just distractions to those who know it will pay off in a career. Nobody else needs any math at all beyond about 9th grade. Cooking, driving, managing a budget, animal husbandry, and a heck of a lot of tech awareness, like how to write a simple computer program, how to avoid getting sucked into Youtube rabbit holes, and how to maintain a healthy, non-addicted relatoinship with social media - all of that is way more important than geometry, trigoometry, or algebra II.
Thanks for mentioning in your recommendations, Zan - I really appreciate it.
Also, I really enjoyed your essay. I think you make some good points. And I agree with the idea that wherever possible kids should be taught things in a way that is applicable to real world problems.
Thanks again, Zan.
Thanks for mentioning me, Zan! I don't think I ask ChatGPT for answers, though 😎 I have a conversation with it about optionality and Stoicism - and I challenge it when it becomes a bit to dogmatic in it's loyality to Taleb's book.
To add on to your concept of relevance for math, there's so much in finance that can fit the framework. The magic of compounding, the fact that being up 50% and then down 50% means that you're down 25%. the concept of present value, and so on.
robertsdavidn.substack.com/about
Absolutely! Ideas of NPV, risk-adjusted rates, annuities and perpetuities go so far in explaining compound interest and exponents in a meaningful way.
Even if students don’t want to go into finance - setting up questions about mortgage payments, or savings accounts in personal finance (for example) would still use all these tools.
Why, thank you so very, very much, Zan! 🤗
I’ve only been on Substack now for just a short time. I have discovered a very friendly, thoughtful body of writers who enjoy supporting one another’s work. I am honored to have somebody like yourself enjoy my family writing. I was terrified no one would want to subscribe to my work. Again, I can’t thank you enough! 💝
I am glad you are finding your community here! It's great to have you
Thanks for the shout out Zan.
Least I could do!