Meaningful mathematics: using balances for problem solving
Using balances to represent equations forces students to find their own meaning in mathematical problems.
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Mathematics instruction is most effective when students experience ideas in a setting that is potentially meaningful to them — where they are encouraged to give meaning to their experiences rather than follow set procedures. Students should experience mathematics in a variety of settings that encourage sense making. Typically, students are taught to solve linear equations by applying a set of rules derived from the field properties of real numbers. This approach has had limited success because students do not understand the abstract mathematical structure and just memorize rules.
One setting that has proven effective in helping students conceptualize mathematical relationships is a balance scale. Balancing is a bodily experience and thus natural, and so students can easily give meaning to tasks presented in a balance format. Unlike symbolically presented exercises, balance tasks must be interpreted and do not lend themselves to mechanical responses and thus encourage meaning making. Students are asked to think of weights being placed on the two sides of the balance scale to make it balance. Since no operation signs are provided, students must decide what operation to perform.
Students solve a balance problem by first determining the numbers that go in the boxes (the weights on the balance scale) and then writing an algebraic equation that symbolizes their actions. In this way, equations come to have meaning and students can draw on the balance imagery as needed in solving algebraic equations expressed symbolically. It is not enough for students to know a procedure for solving equations, they need to understand what an equation is expressing. The balance format allows students to gain a deeper meaning for algebra and be more successful.
The balance problem symbolized by the first diagram below can help students build meaning for subtraction of fractions. The relationship pictured here can be symbolized by the equation 52/3 + n = 121/3. The equation 3n + 5 = 89 symbolizes the relationship implied by the second balance. Thinking about the pan balances helps students understand the meaning of the equals sign. Thus, using balances, students can learn to solve algebraic equations meaningfully.
Using balances with your class
After a brief discussion of balances and perhaps a demonstration of a balance scale, present a balance task on the overhead or chalkboard. Have students solve it and explain their thinking. Then arrange the students in pairs and give each pair one balance sheet to complete.
The students, working in pairs, are to discuss the problems and, if possible, come to an agreement on how to fill the boxes. This collaborative setting gives each student an opportunity to think through relationships and develop methods that make sense to him or her. When two students disagree, there is rich potential for learning. You may want to have two or three balance sheets available on a given day. When a pair has completed one page, you could give them a second one to consider. It is also wise to have a challenge page (more challenging balance tasks) for students who finish quickly and have developed good strategies. During this time, the goal is not to complete all the problems but to be actively engaged in thinking about the tasks — understanding, not speed, is the goal.
Following the collaborative activity, bring the students together for a whole-class discussion of the problems. Have students present their solution methods to the class. Providing an opportunity for students to explain and justify their reasoning is an important component of mathematics learning. Encourage students to question presenters when they do not understand or if they disagree. These whole-class discussions promote individual learning, develop an intellectual community in the classroom, and provide you with valuable information about the student’s thinking.
