Virginia Mathematics Teacher Spring 2017

variety of strategies but mostly of meanings that emerges, shows promise for algebra teaching and learning. These emerging meanings are significant, because they offer different entry paths into the tasks of solving algebraic equations and do not restrict to a single view of how this can be done. As well, this variety of meanings for what solving an algebraic equation can be offers significant reinvestment opportunities for pushing further the understanding of algebraic equation solving in mathematics teaching. Issues of conditional equations, of deconstructing an equation regarding operations done on a number, of maintaining the balance, of finding equivalent equations, of seeing an equation as a system of equations, and so forth, offered varied ways of conceiving an equation and of solving it. It opened various paths of understanding. I have not offered details about the discussions that ensued each sharing of strategies, those being contextual and solvers’ understanding related. But, suffice to say that discussions about and explanations of the strategies took an important part of the session: only about 6 tasks were given/ solved in 2 hours! Hence, most of the work in the session revolved around explaining, justifying, contrasting and exploring the strength, meaning and relevance of each strategy developed to solve the equations. In addition, it is through the mental algebra activity that all this emerged, and not through the explicit teaching of these strategies: strategies and meanings became relevant in the need for solving the equations and these meanings were directly connected to those equations. This makes the activity more about the exploration of mathematical ways of meaning, and less about the teaching of explicit strategies for solving. Of course, outcomes will probably vary from one group to other, sometimes offering more, sometimes less. But, in all, it is in the practice of finding ways of solving that these meanings emerged, and their discussion, and it is through this practice that mathematical experiences were enriched and enlarged in relation to ways of solving algebraic equations. Obviously, this is just one example, and much more is to be explored along these lines with (future) secondary mathematics teachers (and with

other mathematical topics!). However, already this emerging variety of meanings through mental algebra shows important promise for enriching algebraic experiences of future teachers, and possibly their students!

Figure 1 . Mental process of preservice teachers

References Arcavi, A. (1994). Symbol sense: informal sense- making in formal mathematics. For the learning of Mathematics , 14 (3), 24-35. Bednarz, N. (2001). Didactique des mathématiques et formation des enseignants. Canadian Journal of Science, Mathematics and Technology Education , 1 (1), 61- 80. Bednarz, N. & Janvier, B. (1992). L’enseignement de l’algèbre au secondaire. Proc. Did. des math. et formation des enseignants (pp. 21- 40). ENS-Marrakech. Davis, R.B. (1975). Cognitive processes involved in solving simple algebraic equations. Journal of Children’s Mathematical Behavior , 1 (3), 7-35. Filloy, E. & Rojano, T. (1989). Solving equations: the transition from arithmetic to algebra. For the Learning of Mathematics , 9 (2), 19- 25. Nathan, M.J. & Koedinger, K.R. (2000). Teachers’ and researchers’ beliefs about the development of algebraic reasoning.

Virginia Mathematics Teacher vol. 43, no. 2

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