Virginia Mathematics Teacher Spring 2017

Exploring the Solving of Algebraic Equations Through Mental Algebra Jérôme Proulx

For numerous years, mathematics teacher educators have attempted to find ways to enrich their future mathematics teachers’ understandings of mathematics. This has often been done with the intention of deepening and making future teachers more mathematically flexible, particularly in view of their future interactions with students’ own understandings of mathematics. Because doing mental mathematics is well recognized as an occasion for promoting meaning making and for enlarging one’s repertoire of ways of solving (see e.g., Reys & Nohda, 1994; Schoen & Zweng, 1986), it appears as an interesting approach to explore for attaining this precise goal with future mathematics teachers. I therefore report in this short article on a study undertook with future secondary mathematics teachers, where we asked them to solve usual algebraic equations of the form A x +B=C, A x +B=C x +D, A x /B=C/D, A x 2 +B x +C=0 without paper or pencil or any other material aids in a restricted period of time (about 15 seconds). The activity was organized in the following way: (1) an equation was offered in writing to the group; (2) future teachers solved the equation mentally; (3) at the teacher educator’s signal they wrote their answer on a piece of paper; (4) answers and strategies were orally shared with the group; (5) the cycle restarted for another equation. As the activity unfolded, diverse strategies were shared for solving algebraic equations. Through those strategies emerged an interesting variety of meanings (implicitly or not) about what solving an algebraic equation can represent. In what follows, I outline this variety of strategies and meanings, in order to illustrate how doing mental algebra can represent an occasion for the enrichment of future teacher’ mathematical experiences. Meaning 1 : Solving an algebraic equation is … finding the value(s) that satisfy, make true, the equality Underneath this meaning is the notion of a

conditional equality, where it is not only the idea of finding the answers/values that make the equation true, but also the fact that the equality can be true or untrue. When future teachers were given 5 x +6+4 x +3=–1+9 x to solve, some rapidly asserted that there was no solution, because one can rapidly see 9 x on both sides of the equation as well as the fact that the remaining numbers on each sides do not equate. It thus led to the conclusion that there was no number that could satisfy this given equation, since no x , whatever it could be, could succeed in making different numbers equal. This strategy is related to what is often termed “global reading” of the equation (Bednarz & Janvier, 1992), that requires consideration of the equation as a whole prior to entering in algebraic manipulations, or what Arcavi (1994) calls a priori inspection of symbols, which is a sensitivity to analyze algebraic expressions before making a decision about their solution 1 . Another strategy future teachers engaged in was one of “solving followed by validation”. When having to solve x 2 –4=5, one future teacher rapidly transformed it into x 2 =9, obtaining 3 as an answer. However, because he knows being in a mental mathematics context and is aware that his answers in this context are often rapidly enunciated and can lack precision, he decided to verify his answer. By mentally verifying if (3) 2 =9, he realized that (–3) 2 also gives 9 and then readjusted his solution. This manner of solving the equation gets close to the idea not only of finding one value that makes the equation true, but also of finding all values that make it true. Meaning 2 : Solving an algebraic equation is … deconstructing the operations applied to an unknown number This meaning requires reading the equation 1 Arcavi gives the example of (2x+3)/(4x+6)=2, which has no solution because whatever the value of x, the numerator is worth half the denominator, making futile undergoing additional steps.

Virginia Mathematics Teacher vol. 43, no. 2

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