Doing Math vs. Understanding Math
A key instructional shift called for by the Common Core State Standards for Mathematics is the dual emphasis on conceptual understanding and procedural fluency. NCTM draws a connection between these two areas in its position paper on procedural fluency (as I did in my post, Procedural Fluency: More Than Memorizing Math Facts):
Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving.
NCTM also cites research suggesting that "once students have memorized and practiced procedures that they do not understand, they have less motivation to understand their meaning or the reasoning behind them." An implication of this research is that "students need to have a deep and flexible knowledge of a variety of procedures, along with an ability to make critical judgments about which procedures or strategies are appropriate for use in particular situations."
Consider, for example, this 6th grade problem from the Port Angeles School District website:
In a bowling tournament, Elton scored 188, 212, 214, 196, and 200 in the first five games. In order to qualify for the semifinal round, he must average at least 205 for the six games. What is the least he can bowl in his final game to qualify?
Most students will solve this problem as follows:Step 1: Find the minimum total score for six games (205 x 6 = 1230)Step 2: Find Elton's total score for the first five games (188 + 212 + 214 + 196 + 200 = 1010)Step 3: Find the minimum score required for the sixth game (1230 - 1010 = 220)
I, on the other hand, solved this problem by comparing Elton's scores for each of the first five games with the minimum average score of 205. I then computed the net deficit (amount below 205) or surplus (amount above 205). Here's a visual representation of my thinking:
And since the average after five games is 15 less than 205, I knew the score for the sixth game would need to be at least 15 more than 205, or 220.
Would I say that my approach is better than the first approach? No, but I would say it reflects conceptual understanding of averaging. I would also say it reflects procedural fluency, which NCTM defines as the ability to apply procedures accurately, efficiently, and flexibly. In contrast, it's possible for students to get the right answer using the first method without even knowing what "mean" means. It's also possible for students to follow a procedure without having procedural fluency.
The lesson here for us as math educators is that we need to shift the emphasis from answer-getting to the problem-solving process. We also need to model for students--and encourage them to pursue--multiple solution strategies rather than prescribe a standard procedure.
In essence, we need to make the most of opportunities to deepen students' conceptual grasp of math and build their procedural fluency. We need to help students understand math rather than just do math.
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