The Essence of Mathematics Through Elementary Problems(openbookpublishers.com)
openbookpublishers.com
The Essence of Mathematics Through Elementary Problems
https://www.openbookpublishers.com/product/979
31 comments
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.
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?
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?
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 ;)
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?
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.
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).
Powers of 2 certainly come up more often, e.g. memory sizes before everything had GBs of RAM.
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.
Why is '128' easier to remember than '121', '512' easier than '519', and '256' easier than, uh, '256'?
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
I'm pretty handy with numbers but nothing special. It doesn't seem like that major a task
We had to memorize the squares up to 21 in grade school.
The book is absolutely right though -- memorizing squares/cubes/2^n is a math superpower. It's extremely useful when solving textbook problems or when just exploring math on your own. I've seen the powers of 3 so often they've immediately become recognizable, 27,81,243, just as I'm sure most people immediately recognize 81 as a square when they see it -- it just "pops out" from the page.
This holds for simple identities too. Having things like x^2 - y^2 = (x+y)(x-y) immediately come to mind when you see just one half of the identity is ridiculously useful.
The book is absolutely right though -- memorizing squares/cubes/2^n is a math superpower. It's extremely useful when solving textbook problems or when just exploring math on your own. I've seen the powers of 3 so often they've immediately become recognizable, 27,81,243, just as I'm sure most people immediately recognize 81 as a square when they see it -- it just "pops out" from the page.
This holds for simple identities too. Having things like x^2 - y^2 = (x+y)(x-y) immediately come to mind when you see just one half of the identity is ridiculously useful.
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.
I like to point out to people that to learn the times tables up to 10, there are only really 36 facts to learn, since anything times 1 or 10 is trivial, and multiplication is commutative. (That's already a nice little problem to look at, to show that it is 36.)
And then there are many interesting tricks to approach those 36. For example, the 'fingers' trick for the 9 times table. (Which leads to consideration of why do the digits of a multiple of 9 always add up to a multiple of 9?) Muliplying by 5 is half of multiplying by 10. Multiplying by 4 is doubling twice, etc.
Before you know it, you just have to drill the ones most common to choke on (like 7x7, 7x8).
And then there are many interesting tricks to approach those 36. For example, the 'fingers' trick for the 9 times table. (Which leads to consideration of why do the digits of a multiple of 9 always add up to a multiple of 9?) Muliplying by 5 is half of multiplying by 10. Multiplying by 4 is doubling twice, etc.
Before you know it, you just have to drill the ones most common to choke on (like 7x7, 7x8).
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/) (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-...
• “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".
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/) (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-...
• “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".
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.
Mathematician's Delight by W. W. Sawyer seems a good book.
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".
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.
Excellent; I missed this on my first pass. Many thanks.
Oh, Borovik! Right, this just jumped to the top of my queue. I love his other works.
From the description it reminds me of koans. Looking forward to this.
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..
Is openbook == GPL?
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).
> 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).
Thanks!
ePub all the PDFs!
Book looks good though. Thanks for posting.
Book looks good though. Thanks for posting.
It is never unwelcome to have a little humour in the maths book.