Learning intransitivity: from intransitive geometrical objects to “rhizomatic” intransitivity

Learning intransitivity: from intransitive geometrical objects to “rhizomatic” intransitivity.
Research report by Alexander Poddyakov, professor at HSE, Russia, at the conference “Psychology and Technology in Mathematics Education”.
March 18-21, 2019, Moscow, Russia.
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Poddiakov starts by distinguishing two difficult points in understanding intransitivity: first, only some mathematicians know about the intrasitivity of superiority, and second, other people - non-mathematicians - also understand intransitivity in a certain way.
It is usually stated that if A is more than B and B is more than C, then A is also more than C, but this statement is not true for all cases. There are situations in which this principle is violated. There is no psychological and pedagogical tradition of studying correct and incorrect understanding of intransitive relations.
After examining the intransitivity nature in his experimental research, Professor Poddiakov proposes a new class of intransitive objects - geometrical and mathematical constructions forming intransitive cycles A is more than B is more than C is more than A. In contrast to the famous intransitive dice, lotteries, etc., they show deterministic (not probabilistic) intransitive relations. The simplest ones visualize intransitivity that can be understood at a qualitative level and does not require quantitative reasoning. They can be used as manipulatives for learning intransitivity.
Poddiakov also identifies 4 types of complexity of intransitive relations and situations that correspond to them. It is also noted that the experiment participants applied the principle of intransitivity only selectively.
Classification of the types of situations in which the transitivity axiom does and does not apply is outlined. Four levels of complexity of intransitivity are introduced, from simple combinatorial intransitivity to a "rhizomatic" one. A possible version of the main educational message for students in teaching and learning transitivity-intransitivity is presented.
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