Tuesday, January 28, 2014

Math B-Day: Folding Papers and The Mathematics Behind It

Today in class our professor Henk were unable to present until mid-afternoon, and to keep us busy, he threw us a math problem that we have to solve in groups. This assignment opened me to another insight of how Mathematics (and the Education in general) is much more developed, cared about and well, cool, here in Netherlands than in Indonesia. 

Mathematics B-Day
If you are thinking this is somehow related to birthdays, then our thoughts run on the same track *highfive*

But this is not in any way correlated to someone’s birthday though. Mathematics B-Day (or Wiskunde B-Dag) is an annual mathematics competition organized by The Freudenthal Institute of Utrecht University. It proposes very open-ended problems meant to be solved by a team of 3 or 4 students. As opposite to the timed test often associated with math/science contest, this one in particular can take all day. In the end, they have to write the report paper which in turn will be assessed on national level.

Further investigation on my part reveals that apparently the Dutch split Math courses in upper secondary education into two categories: Math A and Math B. Students make their choice based on majors they plan to take in college; Math A is mean for students preparing for social sciences, while Math B is meant for student interested to take mathematics and natural sciences. Somewhere along the way, they require the assessment to include process skill and timed tests are no longer fit. That’s why Mathematics B-Day is developed. 

Most participating schools use the result of the test as part of the final grade, although the task itself is not necessarily related to the Math-B curriculum. The best papers are then sent in and ranked by teachers in four regional groups. The final best papers are then ranked by national jury.

The Distinction: Why I think this is great
In Indonesia we have OSN (National Science Olympiad). This is probably the most prestigious science competition I can think of. Some high-profile universities hold their own Science competition, mostly a way to steal start in finding new talents. Those are also very popular. Recently there was the PISA – a worldwide test conducted by EOCD to measure 15 yo students’ performance on mathematics, science and readings. 

Mathematics B-Day itself offers problem that put a lot of emphasis in problem solving, conjecturing and proving. Students are challenged to show process skills in developing strategies, making conjectures and try to prove or reject them, logical reasoning, critically reviewing models and adjust them.

I think this is what makes them jump out of the page, at least in my book. In term of not related to the curriculum, it is closely related to Olympiad. But in term of problem solving and open-ended problem, it is closely related to the PISA. But the ‘challenging students to make conjecture and proving it’ part, there is no contest alike in Indonesia, not that I am aware of. Maybe there is one for university students, but not children in compulsory education. I think it is a good idea to start this so early. Logical thinking and mathematical reasoning is important even if you don’t study or work in mathematics or science related field. By offering them an interesting yet challenging problems, coupled with children’ natural enthusiasm and curiosity, this can help to nourish scientific mindset in children. 

A Brief Overview
Just as a quick glance to what the problems in Math B-Day are like, this is the problems we worked on today. 

This is the problem from Math-B-Day 2012, and yep, it is a strip of paper. We are supposed to fold that. Sounds boring, isn’t it, but in mathematics, we call it beautiful. How from such a simple thing, complicated yet captivating problems can arise. Just wait for it. 

Oh, and they also give a very cute and fitting name for the task: (Cr)easy. It is not easy, let me tell ya, but it is so damn worth it. By the time you finish this problem you will be very, very, in awe. But it does give you a lot of creasy strips of paper to clean to, though.

The basic idea is that different way in folding the paper results in different pattern once the paper is unfolded.  We are provided a strip of paper. To mark the start of the strip, we put a mark on the left side of the paper, facing us.

Suppose we fold it to the left, and then to the right, and to the right one more time. Then unfold the paper.  This is what we are going to get.


 By tracing the paper from the starting mark towards the end, we obtain a path where we turn right, right, left, left, right, left, left. It is easy to see that there is a causal relation between the way we fold the paper and the resulted paths. But to discover the mathematics in it, that is the challenge.

The way to fold the paper is called the recipe, and it is denoted by lower case l and r. So in this case the recipe is lrr. The resulted path is called the walking pattern, and it is denoted by uppercase L and R. Therefore the path we got in this is RRLLRLL. Three-letter recipe results in 7-letter walking pattern. Imagine the walking pattern of 8 or 9-letter recipe. Can you predict how many bends you will get if you fold it 10 times? Can you predict to which direction a particular bend will be? Can you guess the change in the walking pattern if you change only 1 letter in the recipe?

Turns out, you can.

The task itself is organized in two parts: Part A and Part B. Part A is the exploration, where we are guided to do several exploratory questions regarding the recipe and the accompanying walking pattern. Lots, lots of folding and students screaming “LEFT, no, RIGHT, wait no, LEFT”. Part B is where we analyze our finding, do research, make conjectures and test it. 

It is mentally exhausting, yet satisfying. I believe it also burns calories because I felt mighty hungry after that. Try it. 

This is the download links for the problem: 

This is the applet that can help you to test your conjecture:

This is the link if you are interested in the previous years' assignment:

More on Utrecht University and Math B-Day(or Wiskunde B-Dag) :

It is not available in English though. I really want to know the winners and read their paper. But it is probably in Dutch.

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