6 ms·

The Essence of Mathematics Through Elementary Problems

167 points by ArtWomb 5y ago

ArtWomb 5y ago
Maybe the best math text book I've ever read. Geared for secondary students (age 14-17). But I found it immediately engaging. And quite difficult to keep myself from solving just one more exercise ;)
- ravoori 5y ago
  Which one was the best you read?
  - kqr2 5y ago
    Comment is from the original poster so it's in the posted link
    - throwamon 5y ago
      OP said this may be the best math book they've read, so the best ever isn't necessarily a math book.
math-dev 5y ago
Is openbook == GPL?
- ljoshua 5y ago
  From the copyright notice inside the opening pages: > 2019 Alexandre Borovik and Tony Gardiner This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the work; to adapt the work and to make commercial use of the work providing attribution is made to the author (but not in any way that suggests that they endorse you or your use of the work).
  - math-dev 5y ago
    Thanks!
macintux 5y ago
From the description it reminds me of koans. Looking forward to this.
AdriftHedonist 5y ago
So the first questions are about committing to memory the multiplication tables up to 10, the squares up to 31 and cubes up to 11. How many people can actually do that?
- bigbillheck 5y ago
  I would expect that quite a few determined children could do so. The immediate next question is "first ten powers of 2" and not to gatekeep but surely vast swathes of the commentariat here knew most of those pretty young.
  - a_ghoul 5y ago
    I would argue that those powers of 2 are easier to commit to memory than squares up to 31 (from my experience only knowing the former).
    - gugagore 5y ago
      Powers of 2 certainly come up more often, e.g. memory sizes before everything had GBs of RAM.
    - throwaway316943 5y ago
      I don’t see why they should be any harder than any other type of memorization. Note that they are partially covered by knowing the times table up to 10.
    - bigbillheck 5y ago
      Why is '128' easier to remember than '121', '512' easier than '519', and '256' easier than, uh, '256'?
- elijaht 5y ago
  Quickly running through it I know from memory all but a couple of cubes, all but maybe ten squares, all of the multiplication tables. It would take maybe 30 minutes to get everything down I'm pretty handy with numbers but nothing special. It doesn't seem like that major a task
mrslave 5y ago
ePub all the PDFs! Book looks good though. Thanks for posting.
madhadron 5y ago
Oh, Borovik! Right, this just jumped to the top of my queue. I love his other works.
lcuff 5y ago
I confess I found it dismal that the first problem was to learn the multiplication tables. If I'm looking for a book to excite anyone, that's not where I start. I get that those operations quickly become important in algebra and number theory, but that's hardly the essence of mathematics. All that said, I'm not sure where I'd start for secondary students, but the interest generated by watching _ideas_ emerge in 3blue1brown videos is a stark contrast to this kind of start.
- svat 5y ago
  Note that the first problem is not actually to learn the multiplication tables! Problem 1 is to calculate a few things like 0.004×0.02 and 1.08÷1.2 in one's head; it's just that these real questions are placed in 1(b), with a precursor 1(a), purely to indicate prerequisites, that states "Compute for yourself, and learn by heart, the times tables up to 9×9." (Incidentally, even here the "compute for yourself" part is the real heart of the matter, though of course knowing the times table up to 9×9 is something very useful. You'll see a lot more connections once there's some "fuel", as mentioned in the preface and the introduction to the chapter.) In any case, don't let a cursory look at Problem 1 decide the nature of the book for you. There's more than meets the eye. I don't know much about Tony Gardiner but the other author, Alexandre Borovik, has a wonderful blog called "Mathematics under the Microscope" (https://micromath.wordpress.com/ https://micromath.wordpress.com/) (and also a book of that name); to everything he brings a profound perspective with a light touch, a unique combination. A couple of memorable posts I still remember from over a decade ago: • “Psychophysiology of blackboard teaching” https://micromath.wordpress.com/2010/09/24/psychophysiology-of-blackboard-teaching-2/ https://micromath.wordpress.com/2010/09/24/psychophysiology-... • “Achilles, Tortoise and Yessenin-Volpin” https://micromath.wordpress.com/2009/02/16/achilles-tortoise-and-yessenin-volpin/ https://micromath.wordpress.com/2009/02/16/achilles-tortoise... • All the posts about "a childhood story". Anyway, I haven't gone through this book so far, but skimming through some chapters and how they are put together, there is a lot of gold here. See for instance Problem 78, which is simply twenty opportunities to think about "3 - 1 = 2".
  - jvvw 5y ago
    Tony Gardiner was in charge of the maths olympiad training in the UK for many years - he produced some very good material for that including a book on maths olympiad preparation and the fact that he was one of the authors made me definitely interested in having a look at this.
nxpnsv 5y ago
Pretty cool, I would have love this in high school. My mathematics classes did nothing to nourish love for maths... luckily demoscene made teach myself a bunch..
Y_Y 5y ago
> Problem 92 Dad took our new baby to the clinic to be weighed. But the baby would not stay still and caused the needle on the scales to wobble. So Dad held the baby still and stood on the scales, while nurse read off their combined weight: 78kg. Then nurse held the baby, while Dad read off their combined weight: 69kg. Finally Dad held the nurse, while the baby read off their combined weight: 137kg. How heavy was the baby? It is never unwelcome to have a little humour in the maths book.
- diplodocusaur 5y ago
  If Lana was 31 years younger, she'd be 12 times as old as her baby. Lana is half as young as Robert, who is 44. Find the baby.
  - coolgeek 5y ago
    There are two Lanas. The first Lana is (probably) 43yo, and her child is 1yo. But they could (in theory) be 55 and 2, 67 and 3, 79 and 4, 91 and 5, 103 and 6 or 115 and 7. The second Lana is 22. Oh, wait... the task was to find the baby. Hey, what's a good regex for matching open tags except XHTML self-contained tags?
    - diplodocusaur 5y ago
      ?
Cadwhisker 5y ago
This looked very promising, starting from good foundations and moving on to more complex ideas/methods, even taking time to explain why the steps are useful. However one thing that spoils it for me is that it doesn't provide solutions to the problems. Some are explained in the text that follows, others are just "cross your fingers you got it right".
- svat 5y ago
  Each chapter ends with a "Comments and Solutions" in smaller print; as far as I can tell every single problem has solutions, sometimes multiple pages' worth.
  - Cadwhisker 5y ago
    Excellent; I missed this on my first pass. Many thanks.

167 points by ArtWomb 5y ago

Maybe the best math text book I've ever read. Geared for secondary students (age 14-17). But I found it immediately engaging. And quite difficult to keep myself from solving just one more exercise ;)

ravoori 5y ago
Which one was the best you read?
- kqr2 5y ago
  Comment is from the original poster so it's in the posted link
  - throwamon 5y ago
    OP said this may be the best math book they've read, so the best ever isn't necessarily a math book.

math-dev 5y ago

Is openbook == GPL?

ljoshua 5y ago
From the copyright notice inside the opening pages: > 2019 Alexandre Borovik and Tony Gardiner This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the work; to adapt the work and to make commercial use of the work providing attribution is made to the author (but not in any way that suggests that they endorse you or your use of the work).
- math-dev 5y ago
  Thanks!

macintux 5y ago

From the description it reminds me of koans. Looking forward to this.

AdriftHedonist 5y ago

So the first questions are about committing to memory the multiplication tables up to 10, the squares up to 31 and cubes up to 11. How many people can actually do that?

bigbillheck 5y ago
I would expect that quite a few determined children could do so. The immediate next question is "first ten powers of 2" and not to gatekeep but surely vast swathes of the commentariat here knew most of those pretty young.
- a_ghoul 5y ago
  I would argue that those powers of 2 are easier to commit to memory than squares up to 31 (from my experience only knowing the former).
  - gugagore 5y ago
    Powers of 2 certainly come up more often, e.g. memory sizes before everything had GBs of RAM.
  - throwaway316943 5y ago
    I don’t see why they should be any harder than any other type of memorization. Note that they are partially covered by knowing the times table up to 10.
  - bigbillheck 5y ago
    Why is '128' easier to remember than '121', '512' easier than '519', and '256' easier than, uh, '256'?
elijaht 5y ago
Quickly running through it I know from memory all but a couple of cubes, all but maybe ten squares, all of the multiplication tables. It would take maybe 30 minutes to get everything down I'm pretty handy with numbers but nothing special. It doesn't seem like that major a task

mrslave 5y ago

ePub all the PDFs! Book looks good though. Thanks for posting.

madhadron 5y ago

Oh, Borovik! Right, this just jumped to the top of my queue. I love his other works.

lcuff 5y ago

I confess I found it dismal that the first problem was to learn the multiplication tables. If I'm looking for a book to excite anyone, that's not where I start. I get that those operations quickly become important in algebra and number theory, but that's hardly the essence of mathematics. All that said, I'm not sure where I'd start for secondary students, but the interest generated by watching _ideas_ emerge in 3blue1brown videos is a stark contrast to this kind of start.

svat 5y ago
Note that the first problem is not actually to learn the multiplication tables! Problem 1 is to calculate a few things like 0.004×0.02 and 1.08÷1.2 in one's head; it's just that these real questions are placed in 1(b), with a precursor 1(a), purely to indicate prerequisites, that states "Compute for yourself, and learn by heart, the times tables up to 9×9." (Incidentally, even here the "compute for yourself" part is the real heart of the matter, though of course knowing the times table up to 9×9 is something very useful. You'll see a lot more connections once there's some "fuel", as mentioned in the preface and the introduction to the chapter.) In any case, don't let a cursory look at Problem 1 decide the nature of the book for you. There's more than meets the eye. I don't know much about Tony Gardiner but the other author, Alexandre Borovik, has a wonderful blog called "Mathematics under the Microscope" (https://micromath.wordpress.com/ https://micromath.wordpress.com/) (and also a book of that name); to everything he brings a profound perspective with a light touch, a unique combination. A couple of memorable posts I still remember from over a decade ago: • “Psychophysiology of blackboard teaching” https://micromath.wordpress.com/2010/09/24/psychophysiology-of-blackboard-teaching-2/ https://micromath.wordpress.com/2010/09/24/psychophysiology-... • “Achilles, Tortoise and Yessenin-Volpin” https://micromath.wordpress.com/2009/02/16/achilles-tortoise-and-yessenin-volpin/ https://micromath.wordpress.com/2009/02/16/achilles-tortoise... • All the posts about "a childhood story". Anyway, I haven't gone through this book so far, but skimming through some chapters and how they are put together, there is a lot of gold here. See for instance Problem 78, which is simply twenty opportunities to think about "3 - 1 = 2".
- jvvw 5y ago
  Tony Gardiner was in charge of the maths olympiad training in the UK for many years - he produced some very good material for that including a book on maths olympiad preparation and the fact that he was one of the authors made me definitely interested in having a look at this.

nxpnsv 5y ago

Pretty cool, I would have love this in high school. My mathematics classes did nothing to nourish love for maths... luckily demoscene made teach myself a bunch..

Y_Y 5y ago

> Problem 92 Dad took our new baby to the clinic to be weighed. But the baby would not stay still and caused the needle on the scales to wobble. So Dad held the baby still and stood on the scales, while nurse read off their combined weight: 78kg. Then nurse held the baby, while Dad read off their combined weight: 69kg. Finally Dad held the nurse, while the baby read off their combined weight: 137kg. How heavy was the baby? It is never unwelcome to have a little humour in the maths book.

diplodocusaur 5y ago
If Lana was 31 years younger, she'd be 12 times as old as her baby. Lana is half as young as Robert, who is 44. Find the baby.
- coolgeek 5y ago
  There are two Lanas. The first Lana is (probably) 43yo, and her child is 1yo. But they could (in theory) be 55 and 2, 67 and 3, 79 and 4, 91 and 5, 103 and 6 or 115 and 7. The second Lana is 22. Oh, wait... the task was to find the baby. Hey, what's a good regex for matching open tags except XHTML self-contained tags?
  - diplodocusaur 5y ago
    ?

Cadwhisker 5y ago

This looked very promising, starting from good foundations and moving on to more complex ideas/methods, even taking time to explain why the steps are useful. However one thing that spoils it for me is that it doesn't provide solutions to the problems. Some are explained in the text that follows, others are just "cross your fingers you got it right".

svat 5y ago
Each chapter ends with a "Comments and Solutions" in smaller print; as far as I can tell every single problem has solutions, sometimes multiple pages' worth.
- Cadwhisker 5y ago
  Excellent; I missed this on my first pass. Many thanks.