- Mathematics under the Microscope, to be published by the American Mathematical Society. By a kind permission of AMS, a pre-publication version 0.99 is available for non-commercial personal use without redistribution or creation of derivative work.
And a more recent one and still unfinished,
- Shadows of the Truth: Metamathematics of Elementary Mathematics, working draft 0.69 of 25 Dec 2008.
Both books can be downloaded for free – just click on the links above. If you already have the books then check the recent posts on the right, they may provide extra material and discussion.
Only my readers can judge whether my books are good or bad. But they have no analogues — otherwise what was the point of writing them? I value your comments; please use this post as an entry point.
And please send me your personal stories.
***
Unless otherwise expressly stated, all original content of the book ”Shadows of the Truth: The Metamathematics of Elementary Mathematics” is created and copyrighted ©2008 by Alexandre V. Borovik and is licensed for non-commercial use under a
As a very young lad walking home from school in the euclidean grid of streets that are the suburbs of Chicago, I thought about avoiding sidewalk cracks: “Step on a crack, break you back.” Somehow I knew, or had been told by an older brother that lines were infinite. I reasoned that I didn’t know if sidewalk cracks were perfectly regular, and the cracks running north to south from a block to the east might extend to my path. It was Chicago and from the point of view of a child, virtually infinite, so there clearly was no way to avoid sidewalk cracks. I missed an opportunity to become obsessive compulsive.
During the new-math era, we were taught algebraic concepts before having mastered arithmetic. In our French class we were to play zoot for multiples of 7 and the class (a small rural private school in Georgia [southern USA]) of 4 students it was difficult to keep count of where the multiple of seven was. It turned out (5th grade) that none of us had learned our multiplication tables. We had to be taught late.
I don’t think multiplying 2 digit numbers ever was adequately explained, but we did learn that. In my own case, poorly. So I like stick methods and lattice methods for place holding.
In 6th grade, it was time to move to a new town and a new school. In the private school we were learning to add rational expressions — a task that is easier than adding fractions — and in the new school, we were learning to add, subtract, multiply and divide rational numbers.
In both cases, I had good teachers. The new teacher taught me the algorithms to multiply, divide and add fractions, and get caught up with the standard curriculum in the city pubic school.
I can’t say that the training in either case was bad. Once in public school, I learned to look for common factors before dividing fractions, and with the standard tricks for identifying multiples of three, seven, and eleven, I learned to factor pretty quickly.
I have fairly good number recognition (which is being shot by having cell phones remember them for me), and as a teen, I used to pride myself on memorizing license numbers at a single glance — once collected from a hit and run accident because of this, but the number was easy ***360.
I think that number sense and recognition for 4 digit numbers can help tremendously in teaching multiplication of 2 digit numbers, and that basic algebraic facts such as factoring quadratics can be internalized before proof via attempts to give alternative algorithms — or to choose the method dependent on the numbers involved.
Scott: my warmest thanks! But what is zoot?
Members of the class count one-by-one 1,2,3,4,5,6,zoot,
8,9,10,11,12,13,zoot,
etc. in French, pronouncing the swear word at multiples of the base and at any number containing the base … ,69,zoot,zoot,…,zoot,80
Very interesting. In my old blog I wrote about Vladimir Radzivilovsky who trained very young children to do counting in twos, threes, etc. forward and backward, with performance timed. In base 7 case, it would be counting
7, 14, 21, 28, 35, 42, 49, 56, 63, 70 and back.
The ultimate aim of the the exercise was to put a child in position when he/she composed times tables and naturally memorised/interiorised them.
The zoot game strikes by its unproductivity:
13, zoot, 15, 16, zoot, 18, etc.
unnecessary complicates the pattern.
Right, but it was to teach us to count in French. The French teacher’s husband was the math teacher. I am a firm believer in count-bys. I taught my own children their multiplication table via count bys, and I am even in favor of count-by 1/12:
1/12,1/6,1/4,1/3,5/12,1/2,
7/12,2/3,3/4,5/6,11/12,1
Of course, I cannot do this with out visualizing an egg cartoon.
When the boys were growing up we had scattered count-by sheets taped to the kitchen walls. They were in no apparent pattern. The 7s line was adjacent to the 3s line. So the boys observed that 3s backwards is 7s forwards.
When swimming a mile, I usually do 72, 25 yard laps and find myself computing how many are left. I mentioned this to a very athletic student who looked at me increduously… I guess in weight lifting you only have to count to 10.
There’s a beer drinking game that follows the same logic, that I played growing up. All participants stand in a circle, preferably around the keg. One person starts by saying “One” and the person to his immediate left was say “two” and to that persons left “three” and so on. When we reach 7, multiples of 7 or any number with a 7 in it, that number is replaced with the word “buzz”. If you mess it up and don;t say buzz, you chug.
There are several advanced versions of the game also like “Buzz, Bozz, Bizz”, which is played the same as above but Bozz is replaced for 8’s and Bizz for 9’s.
For an interesting twist we used to change the direction (clockwise to counterclockwise and vice-versa) on every “Buzz”.
I didn’t subscribe to “count-by” sheets when I was younger. As a matter of fact, I don’t even know what they are. I learned “Gazintas”. As in two “Goes in to” ten, five times. Hey I’m from a coal mining town in North Eastern Pennsylvania, where there was much more emphasis put on drinking than multiplication.
Dear Alexandre,
Just finished the first round of reading of your wondeful book.
As a former “docent” of Mathematics at MIIT (Moscow Institute of Rairoad Engineers, currently – “University”) I really admire your comments on the way of teaching Mathematics. I used to teach there during years 1971 – 1987, when due to certain policy at the dapartment of Mathematics and Mechanics we were lucky to have great students in the department of Applied Mathematics.
It worse mentioning that many of our graduates successfully work in different fields of mathematics and its applications all over the world. Some of them after graduating worked with V.I.Arnold and Ya.G.Sinai. You probably know it yourself.
I also had an opportunity to work with A.D. Myshkis and Ye.S.Ventzel, as well as A.A.Yushkevich, F.I.Karpelevich and L.Ye.Sadovsky, who was a chairman of our department at that time. All of them were great educators and taught me a lot.
I read your book without stopping, which is definitely necessary in this kind of reading, sometimes just skipping points reqiring more thinking. So I will definitely re-read it, now with lower speed and therefore more likely with more joy.
Thanks again for the pleasure I had reading the book.
Michael Shtilman.
Sasha,
I want to relate another story that was very influential to me. I have written about it elsewhere and it contains many nice personal reflections.
My mother’s diamond is a large stone. My wife wears it now. It has a flaw in it that can be seen with a jeweler’s magnifying glass. It is called a nine cut, and diamonds are rarely cut in this fashion now a-days. In the kitchen of our house, I asked her about the nine cut. I was either in second or third grade. I have been having a hard time reconstructing the time frame. I should have been older, but I know when she died and when she was healthy enough to be out and about in the kitchen.
She gave me a ruler and a protractor. We counted out 40 degrees, and she showed me how to draw a nine cut pattern. I played with the protractor and discovered the differences between the complete graphs on an even number of vertices and on an odd number of vertices. I learned that to form the graph, I could start from an initial vertex and draw n-1 lines, then move to the next vertex and draw n-2 lines, etc. Most of these discoveries were trial and error. There were much erasing and an in-exactitude caused by my young hands learning how to hold a ruler and stabilizing it.
The “bubble wrap” aspects of these exercises were very calming. Observations were being made and connections were formed that influenced my own mathematical life. I still enjoy meditating on the higher dimensional simplices, the hypercubes, the hyper-octahedra, etc.
As a child, I did not know that I could have counted the triangles, the tetrahedra, etc that were contained therein. I did not know that I could have counted the tilings of the regular n-gon and found the Catalan numbers. No one in my family or school knew that such secrets were there to be discovered and could have been at the grasp of a child. My mother, perhaps, knew that there were deeper ideas within. She did read to me (at an earlier age) from “A Wrinkle in Time,” before her eyesight was too bad.
My three older brothers are computationally competent. In fact the eldest is a quite successful engineer. She obviously had more influence over them than me.
A compass, a ruler, a protractor, a pencil, and plenty of paper: these tools in a child’s hands can unlock many secrets.
Somehow your home page is cross-linked to this “Welcome” post, making it impossible for someone to browse your more recent posts.
Denise: point taken. AB
[...] Morsa kuramı, Mathematical Foundations of Quantum Mechanics, Topics in number theory və s. Aleksand Borovikin maraqlı dersi gözlənilir: Elementary Mathematics from a higher point of view. Sonuncunun [...]