16 1 A Taste of Things to Come We will not understand the psychologi- cal and neurophysiological roots of an important aspect of mathematical prac- tice until we figure out why bubble wrap popping is such an addictive and pleas- ant activity. Still, why is Sudoku popular? I be- lieve the answer is in a rhythm of repeated cycles of operations each of which engages our brains just up to a right and most pleasurable level of intensity. As a student, I experienced the soothing, relaxing effect of car- rying out a recursive algorithm, like long division, or Euclid’s algorithm. Later, in my research work, I felt a similar emotional impact from inductive arguments in finite group theory: you start with a minimal counterexample to the theorem and then simplify it step by step, like removing layers from an onion, until you pinpoint the core contradiction and destroy the counterexample. My teacher Victor Danilovich Mazurov expressed the principle of a “minimal counterexample” using a line from a Russian fairy tale: The oldest brother hid behind the back of the younger one, the younger one hid behind the youngest one, and the youngest brother fell on his knees, raised his hands and pleaded for mercy. In mathematical education, especially at its earlier stages, one of the teacher’s tasks is to give his/her students the opportunity to feel this soothing, comforting effect of a rhythmic repetitive activity. And here I come to the crucial point: why do people love to pop bubble wrap? I would not write this now if the audience of my talk at the WFNMC conference had not immediately agreed with, and ap- proved of, my comparison of the execution of certain types of re- cursive algorithms with bubble wrap popping. I should perhaps ex- plain that the audience included some of the best experts on math- ematical education in the world, especially on advanced and non- standard aspects of mathematics teaching. They definitely knew everything about the so-called “recreational mathematics”, puzzles, brainteasers, and conundrums of every possible kind. Their sup- port allows me to be quite confident in my comparison of Sudoku with bubble wrap popping. In any case, the lady on the train was doing her Sudoku in an immediately recognizable bubble wrap pop- ping rhythm. So, with the authority of the conference on my side, I dare to formulate my thesis: We shall not understand the psychological and neurophysi- ological roots of an important aspect of mathematical prac- tice until we figure out why bubble wrap popping is such

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