I am always trying to improve my math teaching skills. My goal is for my young learners to truly mathematize their world. So I use a constructivist view of learning—where learners generate knowledge and meaning from an interaction between their experiences and their ideas. I feel that I am pretty solid with presenting real-life math situations in the classroom, but I could improve on helping learners go from their initial, and usually inefficient strategies, to fluent operations. The series of books “Young Mathematicians at Work” by Catherine Twomey Fosnot and Maarten Dolk illustrates what it means to do and learn mathematics. The second volume “Constructing Multiplication and Division” provides strategies to help teachers turn their classrooms into “math workshops that encourage and reflect mathematizing.”
The book starts by describing mathematizing, which is creating a classroom environment that encourages problem solving. Next, “Young Mathematicians at Work” provides open and contextual math questions, and explains the process of turning a classroom into mathematical communities with the teacher on the edge. Then, the authors describe real-life and practical ways to develop multiplication and division strategies by using models. I really appreciated the descriptions of real-life lessons, including the examples of teacher dialogue and of student work.
I focused on “Chapter 6: Algorithms Versus Number Sense” as this chapter elaborates on the transition from a student’s strategy to an algorithm. The first sentence asks you to calculate 76 x 89. Most of us would need paper and pencil. The author challenges you to draw an array 76 x 89 and then within this large array, find the smaller arrays that represent the algorithm you calculated. If it is difficult for you, like it used to be for me, then the algorithm you were taught works “against your conceptual understanding of multiplication.”
Most Chinese teachers break up the algorithm conceptually: 76 x 89 = (70 + 6) x (80 +9) = (6 x 9) + (6 x 80) + (70 x 9) + (70 x 80) = 54 + 480 + 630 + 5,600. “In contrast, 70 percent of American teachers teach the algorithm as a series of procedures and interpret children’s errors as a problem with carrying and lining up.” As a young teacher, I remember how hard it was to teach double-digit multiplication. I had to give pages and pages of homework hoping that eventually the routine would be memorized. Now, I provide my learners with a lot of experiences building arrays to let them develop their own strategies. While refining their strategies, I am usually able to lead the students to understanding the algorithm. The authors note that when mathematicians were asked to solve 76 x 89, only 4 percent used the algorithm. The other mathematicians used a variety of strategies, depending on the number relationships. So why do I feel pressure to teach the algorithm? Probably because my students’ parents need to see their children doing the standard algorithm in order for them to feel their child is learning. So for now I teach my students the algorithm, but I take every opportunity to remind parents that good number sense and estimating skills are more important than doing triple digit multiplication. My favorite quote from this chapter is:
By abandoning the rote teaching of algorithms, we are not asking children to learn less, we are asking them to learn more. We are asking them to mathematize, to think like mathematicians, to look at the numbers before they calculate. To paraphrase Plato, we are asking children to approach mathematics as freemen. Children can and do construct their own strategies, and when they are allowed to make sense of calculations in their own ways, they understand better.
The book is rich with details, experiences, and goals that I value, support, and strive to accomplish as a teacher. I highly recommend this book as well as the others in this series.
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