## Thursday, September 30, 2010

### Why do students love this?

Last year, with my PreDP class (think Algebra I and II) I gave a class the Riddle of Diophantus and asked them to solve it. The students, with just a basic understanding of fractions, loved this problem and spent a good 40 minutes or so working on it. It allowed me to introduce algebraic expressions, substitution, and simple equations - as well as give the students so well-needed practice with fractions. This year, my colleague used it in her class with equal success. But what makes it work and engage students in a way that smaller problems rarely do?

Here are two versions:
"God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life." - from daviddarling
'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.' - from wolframalpha
Do you notice how the two versions differ? I don't know if this is because of translation issues or if there were several versions around already when these puzzles were compiled by Metrodorus.

Anyway, last year I had read "The Teaching Gap" and was all into giving students complex problems to work on individually and then in groups. I'm still into this approach, but am getting lazier with finding these problems (the Nrich website really makes all excuses sound silly). The idea was that complex problems challenge students and are thereby both more intrinsically fun, and better for learning.
This problem also has some other good qualities: it involves a translation from weird prose to math, which engages students who pride themselves on their language- and humanities-skills rather than math ability. And it's worder like a riddle - which triggers the "I'll beat it!" response in many people. It was also at the just right level of difficulty for my students. I find the hard part about finding/making problems is to gage the level of difficulty and the amount of scaffolding to provide.

One thing I try to keep in mind is what my colleague J calls the Wax-on-Wax-off (think "Karate Kid) pedagogy. Get them with the rich and complex problem, then follow it with direct instruction or some other ways of organizing and summing up the important concepts and methods introduced through the problem.
By the way, J really impressed me with this waxy-on-offy thing. He's just finishing a ph.d in the education of natural sciences and maths, and until this year hasn't spent much time teaching in "real" schools. Then off the bat he says something this awesome. Either he has some natural instinct for teaching, or it's really possible to learn something of practical significance in teacher education schools.