Math Exercise: Multiple Approaches to Problem-Solving

For example, if the problem was to fi nd the answer to 8 × 6, students may suggest three options: memorizing the multiplication table for 6, knowing that 8 × 5 = 40 and adding another 8 to equal 48, or adding a column of six 8s. Allowing students to personally choose among approaches all confi rmed as correct and to support their choice will increase their comfort levels. Th is process also builds math logic, intuition, and reasoning skills that extend into other academic subjects and real-life problem solving.

Another example might be to ask students how to fi nd out which fraction is greater: 2/5 or 3/7? Encourage students to draw diagrams or to use any math tools in the room (e.g., manipulatives, rulers, graph paper). Th e answers are likely to match the learning strengths of the students. Explorers may use three manipulatives that are each 1/7 of the same whole and compare the size to two 1/5 pieces. Map Readers may draw two equally sized circles or rectangles on graph paper, divide one into fi ve parts and the other into seven parts, color two and three sections of the respective shapes, and then compare the two colored regions.

Students who have mastered a higher conceptual level of equivalent fractions may find common denominators. Other students who understand that fractions represent division may divide the numerator by the denominator and fi nd which quotient is larger. Other options include making two number lines so students can fairly accurately divide one number line into seven sections and another into fi ve sections. Students who are comfortable with estimation may evaluate which of the two fractions is closer to one whole.

With the large number of options, and a problem in which an exact answer is not required, students come to realize that if they can’t remember a particular rule, they can create their own system of comparison. Th is approach also reinforces for students the benefi t of knowing supporting concepts so they don’t get stuck because they can’t remember the algorithm—a memorized procedure they can reproduce but don’t necessarily understand. Th e important message with multiple-approach problems is that participation is not limited to the students who are faster or always correct, because you emphasize the value of the diff erent ways of approaching the problem, not just the solution. If a student devises an appropriate method to reach a solution but makes an arithmetic error, he or she can still be recognized for the accuracy of the reasoning. You can take this method, demonstrate how it works perfectly when the subtraction or addition is corrected, and prove it by using the method to solve a similar problem with diff erent numbers. Th e student who suggested this method will feel the dopamine reward of a correct approach because he or she realizes that that approach can generate lots of correct answers. Th e student has discovered a concept or creative idea that belongs to him or her and is a useful tool.