I’m just reading David Bessis’ book Mathematica (which I discovered through Substack) which spills the beans on how mathematicians really think, which has profound implications for how it is taught. I feel like I have wasted half a life by mistaking proficiency with mathematical formalism for mathematical aptitude. Perhaps I’m better at it than I had realised. I really recommend the book.
For short period I did a lot of differential equations and matrix maths. Then I escaped the end of the Apollo program into the loving arms of NASA where all that math disappeared in a puff of smoke. A perfect example of if you don't use it you. lose it.
One of my bees is that people who are pretty good at splitting a restaurant bill (or such) usually do it by what Feynman called “knowing numbers” - suppose you have a bill of $153.14 to split among 4 people, the first thing you think is that it’s close to $160 so you blurt out “it’s a bit less than $40 each.”
That’s the really useful skill, and we don’t teach it. We teach as though getting the exact sum for your account book is the thing that matters.
Also a great instance of mental versatility: you have different ways to solve different categories of problems. Sometimes an approximation is good enough. Or maybe it is the first step to a final solution.
I never got far with math (turned off forever by the "New Math" we were force-fed in the early 1960s in the US-). Am still mad about this!
But even I can figure out there is more than one way to manipulate numbers.
Using an overkill method = stupidity. Select the tool for the task.
Being able to do a thing does NOT mean you can teach the thing, or vice versa. We've all had many, many bad teachers who proved this! Three things are needed: understanding the thing, being able to do the thing, and being able to teach the thing.
What is it about math that makes it so hard for people to teach it? Or learners to learn it? Same with reading. What basic things don't we know about it? You would think these would be solved problems by now...
I go ice skating in a rink. At first the newbies try to use their skates like shoes, which does not work, so they slip and lose their balance. Eventually they notice you have to tilt the blade edges at an angle for them to grip the ice; then you can proceed from there. If only someone had told them that ONE thing! Same thing with skis: the edges of the skis grip the snow when you tilt the skis. With grip you can turn or stop. People give up skiing because they didn't learn this. What a waste!
You get good at "edging," you can dance!
We need math teachers who can teach us to dance with numbers. We need fascination, and joy.
At least kids learning to ski or skate find falling funny, but falling at math just makes you feel stupid. You think you are "no good" at math. No, the teacher was No Good at math teaching!
Mathematicians had the chance to get school math education right, or at least greatly improved, at the time of the Sputnik panic. Instead, under the influence of Bourbaki-style emphasis on starting from the fundamentals, we got the New Math (set theory without any knowledge of the C19 crises that motivated its development)
At least it has bequeathed us lots of Venn diagram memes.
I agree 1000%, and the past couple years, I have actually been teaching a seminar discussion course, centered on great texts, on key concepts in logic and math. I actually just got done grading PAPERS over GEOMETRY, and they were basically of the same quality level I've experienced in other settings. You may recall that my teaching situation is pretty unique, as I'm in a "Great Books" program where faculty are expected to be massively omni-disciplinary, but I daresay that they are getting more lasting value out of my class than they would out of a typical lecture/textbook course.
US secondary math educator of 40+ years here. I agree! In the courses at my school, we emphasized problem-solving and semi-guided exploration (gently nudging students in the right direction). So students sometimes struggled, as we hoped they would (that didn't mean bad grades). As a result, I'd get emails from parents complaining, "but you haven't taught them anything." I would reply, "I'm trying to teach them how to think."
It's amazing how quickly students can understand the basic ideas of calculus if you give them a graph on some graph paper and a ruler and ask them, "what's the slope of this graph when x =2, and what's the area between the curve and the x-axis between x = 1 and x = 6?"
I agree with the general thrust of this very much (kids who can pass exams but don’t know <blank> is exactly what the UK systems have ended up producing), but don’t idealise the English Language curriculum. Spelling and such is all well and good (though there’s nothing to make you a crack speller like being a reader) but it doesn’t even try to teach them to reason about language, which both doesn’t help with writing and, more importantly, means that they freeze when asked to do some reasoning, much like the polynomial-factoring kids can’t prove their way out of a paper bag. It’s not just maths, it’s a general malaise.
If you think maths teaching is bad, take a look at the curriculum for the Philosophy A-Level course (only one of the privatised(!) exam boards offers it). Half - half! - of the course, and of the exam marks, is given over to theology, which, to add insult as well as unwarranted existential implicature to injury, is described as 'Philosophy of God'. It also consists of rote learning arcane scholastic arguments.
The only reason I have seen suggested for this is that most philosophy teachers at 2ary level are also, or mainly, RE teachers. Whether that is true, and if so is the actual reason, I don't know. But it is insane.
It's actually only a quarter but that is still a huge proportion of the corse to dedicate to what is at best a philosophical backwater, especially approached in terms of learning mediaeval arguments by heart
I agree with you, but that is by introspection. You are confirming my personal prejudices because I personally find the abstract treatment of mathematical ideas easier than calculation. Ask me to calculate the determinant of a 5x5 by hand and I will very likely make a mistake, unless there are some friendly zeroes; but that is no bar to understanding what a determinant is.
On one hand, I think it prudent to allow for the possibility that some people don't think or learn the way I do, and calculation-oriented texts have a place. On the other, hard experience has taught me that I am no special snowflake and whatever peculiarity I have will be shared by some community; it is gratifying to be confirmed in this instance.
Coincidentally, I am currently working my way through Jacob Klein's "Greek Mathematical Thought and the Origins of Algebra", which in part examines this theme from a philosophical rather than pedagogical perspective.
On the one hand, I was a little confused by the bit about Direct Instruction and its relationship to getting "a classroom full of kids who can factor polynomials and solve trigonometry problems". But it's OK now, I read the Wikipedia article and figured it out: two separate things.
ON THE OTHER HAND, I've been groping blindly in the darkness in search of that bit where one's brain "interprets [INSERT KNOWLEDGE/COMPETENCY HERE] as an injury and attempts to heal from it" for as long as I can remember. Thank you!
IIRC, a few years ago a study of the statistics used in journal papers (medical) indicated that the wrong statistics were used at least 30% of the time (about the same as the non-repeatability of the results). As regards Math, at least for some math, like calculus, one can get a close result by brute force programming - eg differentials and integrals. Applying formulae, (and manipulating them) is not hard, but creating them from a model of the data is very difficult for me. For statistics, "p-hacking" is now trivial given the software to easily compute p-values using different statistical tests. Data mining for p-values is also possible, although the best medical journals now demand that the experiment aim is provided before the experiment is done to eliminate this nonsense.
I've had a similar thought about science, that "the use and importance of the scientific method" and "the body of knowledge about the world that we've discovered to date" are distinct things, not always well served by trying to teach them together.
I think there has been some improvement over time, as it's finally been recognised that calculators mean much less need for arithmetical skill.
One thing that I've always been struck by. My mother used to tell me how she had been made to do long compound interest calculations in pounds, shillings and pence, with zero marks if you didn't get it exactly right, and no discussion of the underlying principles.
But even before calculators this could have been made very easy with
1. A book of tables to convert L/s/d into pennies and back again
2. A book of log tables, so that the calculation is reduced to two multiplications and some conversions back and forth
As mysterious as the inner workings of a calculator for most, but quick and reliable.
I’m just reading David Bessis’ book Mathematica (which I discovered through Substack) which spills the beans on how mathematicians really think, which has profound implications for how it is taught. I feel like I have wasted half a life by mistaking proficiency with mathematical formalism for mathematical aptitude. Perhaps I’m better at it than I had realised. I really recommend the book.
For short period I did a lot of differential equations and matrix maths. Then I escaped the end of the Apollo program into the loving arms of NASA where all that math disappeared in a puff of smoke. A perfect example of if you don't use it you. lose it.
One of my bees is that people who are pretty good at splitting a restaurant bill (or such) usually do it by what Feynman called “knowing numbers” - suppose you have a bill of $153.14 to split among 4 people, the first thing you think is that it’s close to $160 so you blurt out “it’s a bit less than $40 each.”
That’s the really useful skill, and we don’t teach it. We teach as though getting the exact sum for your account book is the thing that matters.
Also a great instance of mental versatility: you have different ways to solve different categories of problems. Sometimes an approximation is good enough. Or maybe it is the first step to a final solution.
I never got far with math (turned off forever by the "New Math" we were force-fed in the early 1960s in the US-). Am still mad about this!
https://www.youtube.com/watch?v=W6OaYPVueW4
But even I can figure out there is more than one way to manipulate numbers.
Using an overkill method = stupidity. Select the tool for the task.
Being able to do a thing does NOT mean you can teach the thing, or vice versa. We've all had many, many bad teachers who proved this! Three things are needed: understanding the thing, being able to do the thing, and being able to teach the thing.
What is it about math that makes it so hard for people to teach it? Or learners to learn it? Same with reading. What basic things don't we know about it? You would think these would be solved problems by now...
I go ice skating in a rink. At first the newbies try to use their skates like shoes, which does not work, so they slip and lose their balance. Eventually they notice you have to tilt the blade edges at an angle for them to grip the ice; then you can proceed from there. If only someone had told them that ONE thing! Same thing with skis: the edges of the skis grip the snow when you tilt the skis. With grip you can turn or stop. People give up skiing because they didn't learn this. What a waste!
You get good at "edging," you can dance!
We need math teachers who can teach us to dance with numbers. We need fascination, and joy.
At least kids learning to ski or skate find falling funny, but falling at math just makes you feel stupid. You think you are "no good" at math. No, the teacher was No Good at math teaching!
Mathematicians had the chance to get school math education right, or at least greatly improved, at the time of the Sputnik panic. Instead, under the influence of Bourbaki-style emphasis on starting from the fundamentals, we got the New Math (set theory without any knowledge of the C19 crises that motivated its development)
At least it has bequeathed us lots of Venn diagram memes.
"Wales's Strictest Headteacher"
I agree 1000%, and the past couple years, I have actually been teaching a seminar discussion course, centered on great texts, on key concepts in logic and math. I actually just got done grading PAPERS over GEOMETRY, and they were basically of the same quality level I've experienced in other settings. You may recall that my teaching situation is pretty unique, as I'm in a "Great Books" program where faculty are expected to be massively omni-disciplinary, but I daresay that they are getting more lasting value out of my class than they would out of a typical lecture/textbook course.
US secondary math educator of 40+ years here. I agree! In the courses at my school, we emphasized problem-solving and semi-guided exploration (gently nudging students in the right direction). So students sometimes struggled, as we hoped they would (that didn't mean bad grades). As a result, I'd get emails from parents complaining, "but you haven't taught them anything." I would reply, "I'm trying to teach them how to think."
It's amazing how quickly students can understand the basic ideas of calculus if you give them a graph on some graph paper and a ruler and ask them, "what's the slope of this graph when x =2, and what's the area between the curve and the x-axis between x = 1 and x = 6?"
I agree with the general thrust of this very much (kids who can pass exams but don’t know <blank> is exactly what the UK systems have ended up producing), but don’t idealise the English Language curriculum. Spelling and such is all well and good (though there’s nothing to make you a crack speller like being a reader) but it doesn’t even try to teach them to reason about language, which both doesn’t help with writing and, more importantly, means that they freeze when asked to do some reasoning, much like the polynomial-factoring kids can’t prove their way out of a paper bag. It’s not just maths, it’s a general malaise.
“sustained only by the personality problems which made it slightly less painful to force myself through the bloody stuff than to admit I was wrong”
That’s how most of us mathematicians do maths as well
If you think maths teaching is bad, take a look at the curriculum for the Philosophy A-Level course (only one of the privatised(!) exam boards offers it). Half - half! - of the course, and of the exam marks, is given over to theology, which, to add insult as well as unwarranted existential implicature to injury, is described as 'Philosophy of God'. It also consists of rote learning arcane scholastic arguments.
The only reason I have seen suggested for this is that most philosophy teachers at 2ary level are also, or mainly, RE teachers. Whether that is true, and if so is the actual reason, I don't know. But it is insane.
It's actually only a quarter but that is still a huge proportion of the corse to dedicate to what is at best a philosophical backwater, especially approached in terms of learning mediaeval arguments by heart
I agree with you, but that is by introspection. You are confirming my personal prejudices because I personally find the abstract treatment of mathematical ideas easier than calculation. Ask me to calculate the determinant of a 5x5 by hand and I will very likely make a mistake, unless there are some friendly zeroes; but that is no bar to understanding what a determinant is.
On one hand, I think it prudent to allow for the possibility that some people don't think or learn the way I do, and calculation-oriented texts have a place. On the other, hard experience has taught me that I am no special snowflake and whatever peculiarity I have will be shared by some community; it is gratifying to be confirmed in this instance.
Coincidentally, I am currently working my way through Jacob Klein's "Greek Mathematical Thought and the Origins of Algebra", which in part examines this theme from a philosophical rather than pedagogical perspective.
On the one hand, I was a little confused by the bit about Direct Instruction and its relationship to getting "a classroom full of kids who can factor polynomials and solve trigonometry problems". But it's OK now, I read the Wikipedia article and figured it out: two separate things.
ON THE OTHER HAND, I've been groping blindly in the darkness in search of that bit where one's brain "interprets [INSERT KNOWLEDGE/COMPETENCY HERE] as an injury and attempts to heal from it" for as long as I can remember. Thank you!
IIRC, a few years ago a study of the statistics used in journal papers (medical) indicated that the wrong statistics were used at least 30% of the time (about the same as the non-repeatability of the results). As regards Math, at least for some math, like calculus, one can get a close result by brute force programming - eg differentials and integrals. Applying formulae, (and manipulating them) is not hard, but creating them from a model of the data is very difficult for me. For statistics, "p-hacking" is now trivial given the software to easily compute p-values using different statistical tests. Data mining for p-values is also possible, although the best medical journals now demand that the experiment aim is provided before the experiment is done to eliminate this nonsense.
I've had a similar thought about science, that "the use and importance of the scientific method" and "the body of knowledge about the world that we've discovered to date" are distinct things, not always well served by trying to teach them together.
I think there has been some improvement over time, as it's finally been recognised that calculators mean much less need for arithmetical skill.
One thing that I've always been struck by. My mother used to tell me how she had been made to do long compound interest calculations in pounds, shillings and pence, with zero marks if you didn't get it exactly right, and no discussion of the underlying principles.
But even before calculators this could have been made very easy with
1. A book of tables to convert L/s/d into pennies and back again
2. A book of log tables, so that the calculation is reduced to two multiplications and some conversions back and forth
As mysterious as the inner workings of a calculator for most, but quick and reliable.
I think you’re probably onto something here.