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A new class of intransitive objects-geometrical and mathematical constructions forming intransitive cycles A ≻ B ≻ C ≻ A (where “≻” means “is preferable to”) are presented. 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 in-transitivity. Classification of the types of situations in which the transitivity axiom does and does not work is presented. 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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All content in this area was uploaded by Alexander Poddiakov on Jun 07, 2021

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... Their invention is not analyzed in this article for the sake of brevity. The objects and general paradigm of these studies are described in (Poddiakov, 2010(Poddiakov, , 2019Poddiakov & Valsiner, 2013). A scientific and popular illustrated description for the wider public is given in . ...

Invention of problem situations and experimental objects to study others’ thinking is a special kind of creativity worthy of scientific interest. The objects are considered in terms of Latour’s actor network theory (as nonhuman actants), cultural psychology (as cultural tools), and Gibson’s theory of affordances (as meta-affordances). A fundamental problem of validity in studies of curiosity and exploration is discussed. The author’s experience of inventions of exploratory objects to be experimented with is analyzed. An inter(trans)-disciplinary insight penetrating and integrating all levels of the work on an exploratory object is described. An example of participants’ revealing of an object’s unexpected faculties (“serendipities”) undesirable from the experimenter’s point of view is given. It is shown that an object designed to study thinking in one sample can be used in another sample with unexpected results. It is argued that automatic generation of problem posing-and-solving situations and exploratory objects, beginning from some levels of their novelty and complexity, is hardly possible because of fundamental limitations of “standard means to generate or score non-standard ends”.

Изобретение, конструирование проблем и задач для других людей – то, что мы делаем в самых разных ситуациях. И в бытовых, когда придумываем загадки, задачи для партнера по игре. И в институционализированных – например, составление огромного количества задачников по разным предметам, задач для экзаменов и олимпиад, разработка психологических тестов, в том числе на мышление, и т.д. При этом в психологии и педагогике нет систематических исследований задачного творчества, несмотря на множество работ по изучению и внедрению задачного конструирования в учебную деятельность школьников.
На открытом семинаре ВШЭ по образованию Александр Поддьяков, профессор Факультета социальных наук ВШЭ, представит анализ авторского опыта разработки оригинальных объектов, проблемных ситуаций и задач в нескольких областях.
1. Создание интерактивных исследовательских объектов типа головоломок для изучения мышления и самостоятельного экспериментирования детей. Они не требуют формулирования задач со стороны взрослого – важно, чем ребенок займется и что обнаружит сам.
2. Создание парадоксальных объектов, находящихся в нетранзитивных отношениях превосходства (отношениях «камень-ножницы-бумага»), и задач на их понимание для взрослых. Например, в известном шоу «Битвы роботов» можно наблюдать парадокс, названный «рободарвинизм»: боевые машины первого типа чаще выигрывают у устройств второго типа, чем проигрывают им. Устройства второго типа чаще выигрывают у устройств третьего типа, но те – у устройств первого типа. Можно ли смоделировать эту закономерность в чистом виде? Да, можно – с помощью соответствующих объектов – и на этой основе конструировать задачи на нетранзитивность.
3. Создание заданий для олимпиады школьников по психологии и для дисциплины «Психология мышления».

Изобретение, конструирование проблем и задач для других людей – мыслительная деятельность, осуществляемая достаточно большим количеством профессионалов и любителей в самых разных областях и ситуациях – от институционализированных (составление огромного числа задачников по разным предметам, задач для экзаменов, олимпиад, разработка психологических тестов, в том числе на мышление, и др.) до бытовых (придумывание загадок, задач для партнера по игре). Для оценки этой деятельности есть свои официальные конкурсы – чемпионаты мира по составлению шахматных задач, международные конкурсы дизайна механических головоломок, кроссвордов и т.д. Создаются и публикуются различные рекомендации по конструированию проблем и задач. При этом нет сколько-нибудь систематизированных психологических работ, посвященных анализу того, как люди создают, конструируют проблемы и задачи для других.
После обзора аспектов, описанных выше, в докладе представлен анализ авторского опыта разработки оригинальных объектов, проблемных ситуаций и задач для изучения мышления. Рассмотрены этапы этой деятельности – от источников возникновения психологического замысла до реализации в технической конструкции (инженерное творчество), в развертывающемся психологическом эксперименте и до социальной коммуникации по поводу этих объектов и задач (представление в журналы, общение с психологами и профессионалами из других областей, делающими затем уже свои профессиональные ходы). Обсуждается динамика и взаимосвязь инсайтов разного уровня: и идущих с нижнего, технического уровня, и верхнего уровня. Вводится новое в психологии мышление понятие - понятие комплексного, или системного инсайта.

[This corrects the article DOI: 10.3389/fpsyg.2018.01791.].

An experimental study of changes of opinions about intransitivity caused by observation of deterministic intransitive objects was conducted. Aim of the experiment was to investigate influence of adults’ observation of triads of objects (The Intransitive Mobile Assault Towers and The Intransitive Gears) being in intransitive relations of superiority, on changes of their opinions about possibility/impossibility of existence of other "intransitive" objects in different domains. It was shown that opinions about intransitive relations of superiority were domain-specific: the participants thought that objects, being in intransitive relations, are possible in some domains and impossible in other domains (though in reality they are possible in the other domains as well). The transitivity axiom was applied by the participants very selectively, if at all. In many situations it was the right solution. Demonstration of different "intransitive" objects caused different effects. Some of the models provide more opportunities for right generalizations (have more "heuristic power") than others. Both types of models can be used for investigation of understanding of transitivity/intransitivity.

A most interesting area in transitivity / intransitivity relations consists of relations like dominance / subordination, superiority / inferiority, preferences, etc. If A dominates B and B dominates C, must it be so that A dominates C? If A is superior to B, and B is superior to C, must it be so that A is superior to C? What happens if superiority/inferiority (dominance /subordination, etc.) relations form a cycle, an intransitive loop?
Human rationality is often assumed to be based on the logical relation of transitivity. Yet, although transitivity fits relationships between physical objects or human decisions about targets that are independent of one another, it fails to fit the phenomena of systemic and developmental organization. Intransitivity has been shown to be present in various kinds of systems, ranging from the brain to society. In cyclical systems transitivity constitutes a special case of intransitivity. In this chapter, we examine different forms of emergence of intransitivity cycles, fixation of transitive parts in these cycles, and the organization of different levels of reflexivity within the systems. We conclude that reflexivity of cognitive processes— rather than transitivity in specific forms of thought—is the defining criterion of rationality.

With nontransitive dice, you can always pick a dice with a better chance of winning than your opponent. There are well-known sets of three or sets of four nontransitive dice. Here, we explore designing a set of nontransitive dice that allows the player to beat two opponents at the same time. Three-player games have been designed before using seven dice. We introduce an improved three-player game using five dice, exploiting a reversing property of some nontransitive dice.

The contemporary state of the competitive intransitivity hypothesis is considered. Intransitive competition among species occurs when, for example, species A outcompetes species B, B outcompetes C, and C outcompetes A (sometimes written as A > B > C > A). In the first part of the article, a summary of the studies of competitive intransitivity is given. Examples of actually existing intransitive loops, as well as simulation models that provide a theoretical explanation for these processes, are discussed. For competitive intransitivity to emerge, it is necessary (but is it enough?) that the community has sufficient potential diversity, that species interactions are carried out in a relatively stable limited space, and that there is a penalty for the acquisition of competitive ability. In the second part, the competitive intransitivity hypothesis is compared with neutral theory and niche theory. The results are believed to make it possible to form some generalizations, which could stimulate a deeper understanding of the species coexistence phenomenon.

Researchers have long been interested in the emergence of transitive reasoning abilities (e.g., if A>B and B>C, then A>C). Preschool-aged children are found to make transitive inferences. Additionally, nonhuman animals demonstrate parallel abilities, pointing to evolutionary roots of transitive reasoning. The present research examines whether 16-month-old infants can make transitive inferences about other people's preferences. If an agent prefers object-A over B (A>B) and B over C (B>C), infants seem to reason that she also prefers A over C (A>C) (Experiment 1). Experiment 2 provides indirect evidence that a one-directional linear ordering of the three items (A>B>C) may have helped infants to succeed in the task. These and control results present the first piece of evidence that precursors of transitive reasoning cognitive abilities exist in infancy.

The effectiveness of different concrete and pictorial models on students' understanding of the part-whole construct for fractions was investigated. Using interview data from fourth and fifth grade students from three different districts that adopted the Mathematics Trailblazers series, authors identified strengths and limitations of models used. Pattern blocks had limited value in aiding students' construction of mental images for the part-whole model as well as limited value in building meaning for adding and subtracting fractions. A paper fraction chart based on a paper folding model supported students' ability to order fractions with same numerators but was less useful in helping students on estimation tasks. The dot paper model and chips did not support fifth grade students' initial understanding of the algorithm.