As prepared in advance for The Fourth Interdisciplinary Scientific Conference on Mathematical Transgressions, March 2019, with reflections on the conference added afterwards. Please note that if this paper is read as a pdf in a web browser, then clicking underlined references will directly open up the link if it is still available on the internet. Complementing supportive and problematic aspects of mathematics to resolve transgressions in long-term sense making David Tall University of Warwick, UK david.tall@warwick.ac.uk In the teaching and learning of mathematics, while it is important to focus on what happens at each stage of development, what matters even more is the cumulative effect of learning over the long-term. As mathematics grows in sophistication, new contexts require new ways of thinking that can act as barriers to progress. Passing through such a barrier may be called a transgression. This presentation focuses on aspects of mathematics that remain consistent over several changes in context and contrasts them with others that cause conflict at any given stage. For instance, how we speak, and write mathematics reveal new insights into making long-term sense of increasingly sophisticated mathematical symbolism in arithmetic and algebra. How the eye tracks a moving object affects how we interpret the notion of variable in the calculus both visually on a number line and symbolically as a variable quantity. Studying successive changes in mathematics and the positive and negative emotional affects leads to an overall framework for long-term development that applies both to historical evolution and to the individual development of different learners. It offers a practical approach in the classroom and a theoretical framework that brings together widely differing interpretations held by mathematicians, educators, curriculum designers, philosophers, psychologists, neuro-physiologists, and even politicians who currently specify the curriculum. 1. Introduction: the notion of transgression This paper has been prepared for a plenary at the 4 th Interdisciplinary Scientific Conference on Mathematical Transgressions where the term ‘transgress’ has the broad meaning of ‘crossing over’ a limiting boundary to a new, previously untenable context. Here I focus on mathematical transgressions in the long-term growth of the individual as more sophisticated mathematics is encountered that requires a reconstruction of earlier knowledge. I will use an analysis of how we humans construct mathematical ideas formulated in my book How Humans Learn to Think Mathematically (Tall, 2013) from birth to the full range of adult thinking which has corresponding links to the historical evolution of mathematical thinking. This will be enhanced by new ideas that have been developed since that book was published, to give a more comprehensive framework for teaching and learning mathematics meaningfully over the long term. It does not see difficulties that students encounter as ‘misconceptions’ that need to be corrected. Instead it seeks fundamental ideas in mathematics that link to the natural operation of the biological brain, to focus on thought processes that are supportive over the longer term and to contrast these explicitly with problematic aspects that require new ways of thinking in new contexts. Supportive connections produce electrochemical changes in the brain that enhance mental activity while problematic links cause conflict that inhibits thinking. The theory has practical implications that seek to make sense for a wide range of classroom teachers and learners.