Historical studies on the development of mathematical concepts will help mathematics teachers to relate their students" difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians such as Arnauld, Leibniz, Wallis, Euler and d"Alembert. Not only does division by negative numbers pose problems for the number line, but even the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics, we argue for the introduction of negative numbers in education within the context of symbolic operations.
Keywords: negative numbers, number line, obstaclesMy enthusiasm for mathematics was based principally perhaps on my horror of hypocrisy; hypocrisy as I saw it, was my aunt Séraphie, Mme Vignon and their priests. In my opinion hypocrisy wasn"t possible in mathematics and, in my youthful simplicity, I thought the same went for all the sciences in which I had heard it said they were applied. What then when I realized that no one could explain to me how it is that a minus times a minus equals a plus?(From The Life of Henry Brulard, pseudonym for Marie-Henri Beyle, better known as
This paper deals with a sub-class of recreational problems which are solved by a simple memorized rule resulting from an elementary arithmetical or algebraic solution, called proto-algebraic rules. Their recreational aspect is derived from a surprise or trick solution which is not immediately obvious to the subjects involved. Around 1560 many such problems wane from arithmetic and algebra textbooks to reappear in the eighteenth century. Several hypotheses are investigated why popular Renaissance recreational problems lost their appeal. We arrive at the conclusion that the emergence of algebra as a general problem solving method changed the scope of what is considered recreational in mathematics.
SommarioQuesto saggio tratta di una sottoclasse di problemi ricreativi risolti tramite memorizzazione di una semplice regola risultante da una soluzione algebraica o aritmetica, chiamata regola proto-algebraica. L'aspetto ricreativo di questi problemi deriva da una soluzione a sorpresa o da un trucco non immediatamente ovvi ai soggetti coinvolti. Intorno al 1560 svariati problemi di questo tipo sparirono dai manuali di algebra e aritmetica, per riapparire nel diciottesimo secolo. Diverse ipotesi sono vagliate sul perché problemi ricreativi popolari nel Rinascimento persero attrattività, per giungere alla conclusione che l'emergere dell'algebra come metodo generale di risoluzione di problemi cambiò la portata di ciò che era considerato ricreativo in matematica.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.