My next visit was to the BSME Mathematical Games, taught by Anna Kis, an experienced math classroom teacher who is currently earning an advanced teaching degree.
Anna’s class was the most delightful of my visit, full of playful approaches to teaching advanced mathematical concepts to 11 or 12 year olds. After this class, I was very curious what Anna taught for the rest of the semester, since this class session was just packed with interesting ideas and approaches for the classroom.
Anna’s lesson broke roughly into four parts: common multiples, congruence classes (mod 4), modular arithmetic, and then extensions. She made clever use of sound, color, patterns and drawings. She also created an interactive and engaging mood, and made expert use of questions. I also enjoyed having her jump outside of the lesson at times to highlight specific ideas from the practioner’s perspective.
Assignment: make color overlays of the 100 chart using different colors for multiples of 3, 4, 5, and 6. Observe patterns when two overlays are superimposed. In which numbers do two given overlays coincide?
Activity: three people repeat, “North, South, East, West” in turn. Then each person only speaks out loud when they say “North”.
Activity: volunteers play musical instruments on every 2nd beat, 3rd beat, and 5th beat. Variation: play on 2nd, 3rd, and 6th beats.
Discussion: what is the connection between the color overlays game and the music game?
Anna took care not to answer questions, thus allowing students to express and explain the mathematics in their own words. They were frequently asked to explain the math as an 11 or 12 year old would, so some had to express their more sophisticated mathematical knowledge in more common terms, and others had to work to find appropriate language for their observations.
Congruence Classes (mod 4)
Anna drew a number line and marked off the integers from 0 to 20.
Activity: color the multiples of 4 red.
A natural question arose: “Should we color zero?” Anna handled this question with a question: “What is a multiple of 4?”, and elicited some student definitions. The consensus definition was then applied to zero, and it was colored red.
Activity: color the other even numbers blue. How can we (as 11 or 12 year olds) define these numbers mathematically?
The class definition of even numbers that are not multiples of 4 was “4 box + 2”, where “box” could be any integer.
Question: if we color every 4th number the same color, how many colors would we need? Describe each color number.
Anna took out some mini post-its in four colors: red, blue, yellow and green, to represent the color numbers
Explain: yellow + green = red
Yellow numbers are 1 more than a multiple of 4, greens are 3 more. A generic diagram of yellow + green went up on the board, and group of four dots were circled, leaving spare groups of 1 and 3, which were combined, completing a red number.
Task: make a question for the others.
Question: what color is 201? How do you know?
Activity: Color Pascal’s Triangle with a secret rule. What is it? (It turned out to be coloring modulo 3.)
Magic trick: three envelopes labeled 1, 2, and 3 contain numbers. A volunteer chooses two envelopes and takes a number from each and adds them. The magician, upon hearing the sum, knows which envelopes were chosen. How is this done?
Nim Variant: a number line is drawn, and integers from 0 to 40 are marked off. Starting at 0, two players alternate advancing a pawn at least 1 and at most 4 n each turn. Whoever gets to 40 first wins.
These further variants each opened up to congruence arithmetic ideas from earlier in the lesson, and illustrated their importance and application.
I could imagine anything from Anna’s class fitting in very well with many US math circles I’ve seen. I would love to see approaches like this making their way into US math classrooms, and maybe BSME alumni will be the ones to do so!