Month: January 2018

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.

Multiples

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.

Congruence Arithmetic

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?

Extensions

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!

 

My second BSME visit was to a partner school. Every week the BSME students have a visit to one of several partner schools around Budapest. The list is very interesting, and includes many English language schools, elite specialized schools, a school for the blind, and many more.

I was part of a visiting team, invited to learn about BSME, and the program arranged a special visit for us to the Petrik Lajos Bilingual Vocational School of Chemistry, Environmental Protection and Information Technology. This school was founded in 1879, so like many Hungarian schools, has an aura of tradition surrounding it. Many of Hungary’s chemists are educated and trained here. We were hosted by Márti Barbarics, who besides teaching math, is working towards a doctorate in math education.

Márta is piloting a problem-based math curriculum that seeks to actively engage students. This is a curriculum that is based on the Pósa Method, a project that is led by Péter Juhász, another BSME instructor (more on that in a later post). Márta described the project as quite an innovation, as compared to the traditional curriculum.

This was a curious fact, in light of Hungary’s fame and tradition in using problem solving to nurture the mathematical interest and expertise of some of the world’s most famous mathematicians and scientists! In fact, many in the US math circle movement trace the origins of math circle back to Hungary (http://mathcircle.berkeley.edu/program) while others point to Bulgaria, Romania and Russia.

Much of the problem solving method that have had a huge impact on US math education in past decades (and problem solving was quite fashionable, for a time) is encapsulated in George Pólya’s 1945 book, How To Solve It (https://en.m.wikipedia.org/wiki/How_to_Solve_It).

Pólya (http://www-history.mcs.st-andrews.ac.uk/Biographies/Polya.html) ended up at Stanford University. You might be interested in the Stanford Mathematics Problem Book (https://books.google.com.ar/books/about/The_Stanford_Mathematics_Problem_Book.html), or in that the Stanford Math Circle (http://mathcircle.stanford.edu) was founded in 2005.

As another sidetrack, one of Pólya’s most famous talks is, “Let us teach guessing”, and you might be able to find the video online. The workshops at HCSSiM start off with this exploration, called “The Watermelon Problem”, and I’ve find that it is a wonderful way to get the conversation started. [I also found out this fall from a delightful math seminar talk at Bard College, delivered by Moshe Cohen of Vassar, that this problem appeared in print as, “Cutting the cheese”, by J.L. Woodbridge, as problem E554 in the 1943 American Math Monthly.]

Back to the visit to the Petrik Lajos School, Márta had prepared a logic lesson for her students, based on some of Raymond Smullyan’s Knights and Knaves problems (in What is the Name of This Book?) She met with us before the lesson and described her students and her concerns that they might be hesitant toward the lesson, as it differently from the typical lecture mode of delivery. She was ready to explain that logic problems really do appear on the standardized exams, which they might find motivating. Also, Petrik Lajos is a dual language school, so the class would be taught in English.

Márta was partially right! The students did ask those classic questions, like, “will this be on the test?” But soon they were captured by the beautiful problems, and quite engaged. Márta’s students seemed to enjoy discussing each problem, and how to resolve them.

That’s part of the point. Students get engrossed by these delightful problems, and become open to genuinely learning the mathematics underneath, and really internalize the content. When skillfully used, an experienced math instructor can select and sequence good problems into an effective and enjoyable math course.

The Hungarian tradition in mathematical problem solving is now in the process of being rolled out to a wider high school audience, complete with educational studies of efficacy.

I liked what I saw at Petrik Lajos, but was curious to visit more Hungarian math classrooms. In later posts, I’ll write about more experiences I had with BSME classrooms, and two other school visits I was able to arrange independently of BSME.