Posted by: Alexandre Borovik | August 25, 2008

This is my expanded and corrected comment to Susan Beckhardt’s post Musings on Math Education, part 1: Logarithms on a Pizza Box.

Perhaps my most formative mathematical experience was reading during summer vacations, aged 11, the books “Three Days in Karlikania” and “Black Mask of Al-Jabr” by Vladimir Levshin. The summaries of the books from Mathematical Fiction:

“Three Days in Karlikania”: Three children travel to Karlikania, an enchanted land populated by numerals. Here they befriend a slightly naughty zero (zeros are the babies of this magical country) and spend 3 wonderful days exploring mathematical ideas. Along the way they learn about infinity and the danger of dividing by zero, the laws of arithmetic, prime numbers and the sieve of Eratosthenes, the history of the number system, divisibility tests, and many many other ideas.

“Black Mask of Al-Jabr”: The 3 friends return to Karlikania. Their friend, the baby zero, is accosted by a mysterious x-shaped stranger, who challenges our heroes to recover his identity. Many adventures unfold, and the visitors to Karlikania master Pascal’s triangle, algebraic identities, the arithemetic and geometric means, and many other challenges. Finally the mysterious stranger is released from his enchantment; an equation is solved and the mask is removed. This is mathematical storytelling at its best!

A few months later, at school, I discovered that the images from the book were still alive in my brain and were telling me that the most natural way to deal with word problems in arithmetic was to compose a system of linear equations and solve it by formal manipulations. But I knew that I was not supposed to do that, so I did not try that in the class. Finally, I was told to take part in my school’s mathematics competition; one word problem on the list appeared to be slightly more difficult than the kind of problems we were normally given in our classes. I decided that this was giving me a right to try the “Black Mask of Al-Jabr” method and started composing an equation; I was pleased to see that it obviously reflected all the information which was given in the problem, and then helpfully recalled that for solution I had to move all unknowns to one side, taking care to change the signs when moving things over the equality sign. To my dissapointment, my solution was not accepted (and I did not get any prize) for two reasons:

(a) it was too early for me to attempt algebraic equations, they had not been on the syllabus yet;

(b) It did not matter that I got a correct answer since I solved the problem incorrectly anyway because I denoted the unknown by letter “a” instead of “x”.

Therefore I had to conceal for another year that I could actually use basic algebra. Interestingly, I had a clear understanding that the reason (b) was utter rubbish. I did not get, however, any moral trauma because my mother was a school teacher of great experience and warned me, on many occasions, that I had to treat all teachers as human beings and forgive them their weaknesses. A list of potential human weaknesses in my teachers was very explicitly formulated by my mother and included, but was not restricted to, drink problems, lack of education and basic stupidity.

Levshin’s was a wonderful book; I read it as a fairy tale, and it planted into my brain seeds of algebra. Of course, it worked on an emotional level because I had seen how my father solved, for my big brother, maths problems by composing systems of linear equations, and did that with obvious enjoyment. When reading Levshin’s fantasy, I knew that the book was explaining to me what my father did.