Have you ever been in a team planning session and heard “I’ve found the most incredible strategy for teaching my students multi-digit multiplication.”?
This is the time of year when upper elementary teachers begin teaching multiplication strategies. However, how do you evaluate the usefulness and appropriateness of a multiplication strategy? Today, I share two considerations to determine whether an alternative strategy is appropriate to use with students.
Some years back, I discovered a strategy called lattice. I planned to use the strategy to help my fourth-grade students with the multiplication of large numbers. I was so excited that I called one of my middle school math friends from college and enthusiastically told her about my discovery. But, I was disappointed when my excitement was met with silence and a scolding.
I was so thrilled about using this strategy with my students who were unsuccessful with the standard algorithm that I forgot to consider how the strategy would help my students in the future. There are many strategies floating around in textbooks and on the web. However, it’s important that we consider the following when selecting an alternative strategy to use.
- Is the strategy mathematical?
- Will the strategy support the learning of operations later in a student’s mathematical career?
Let’s use the lattice method to consider the questions above. Explore the example of the lattice model below. This model shows the solution for 87 x 24. The lattice strategy requires students to multiply each digit on the top by each digit on the right side, place their product into the intersecting box in the array, and add the diagonal “lines” to get the product.
Is this multiplication strategy mathematical?
The Common Core State Standards for Math emphasize a progression of computation strategies be used to help students build an understanding of the more-standard algorithm (Fuson and Beckmann, 2012). This means that the alternative strategies should emphasize place value concepts as related to the computational process and the properties of operations. If the lattice model reinforced these important mathematical understandings, students would leave with the understanding that when the model shows that 7 x 2 = 14, the meaning is actually 7 x 20 = 140. Now, I know what you’re thinking– the standard algorithm presents the same limitation; however, the formatting of the lattice method makes it more challenging for students to easily see the place value of each number.
Will the multiplication strategy support the learning of operations later in a student’s mathematical career?
My middle school math friend contended that this method does not work well with decimals and therefore, in her opinion, would not support students as learning progresses into the middle grades. However, much like the traditional algorithm, students can just total the number of digits on the right side of the decimal in both factors and then count that number of places from the last digit in the product to know where to place the decimal.
So what now?
The major issue here is formatting. The lattice method just does not communicate place value well. A solid understanding of place value is essential to the conceptual development and understanding of decimal multiplication. In fact, if you modify the lattice model to emphasize place value, you get another model– the area model (see the image below).
The area model accomplishes the same goal as lattice and is visually similar. It will also help students multiply with decimals effectively once students have developed the concept of decimal multiplication. The area model also lends itself to a strategy that Algebra teachers use to teach the multiplication of polynomials. For example, students can use a similar model to multiply (x + 2) by (x – 3).
As teachers, we are always on the look-out for ways to better help our students master the curriculum. However, it’s important to make certain that the strategies we choose will support them beyond their time in our classrooms.
Reference: Fuson, K. C., Beckmann, S. (2012). Standard algorithms in the common core state standards. NCSM Journal, pp. 14-30.