V.5 #3 Mathematics - Toward Developing Conceptual Understanding: Four Approaches to Teaching and Lea
Having a deep understanding of multi-digit multiplication is a major student expectation in the upper elementary grades (CCSSO, 2010). This understanding goes beyond procedural fluency where students merely know (or have memorized) a series of steps to reach a solution. If students see a multiplication problem, such as 26 x 48, they should be able to solve it using multiple representations, be able to write accompanying word problems to illustrate its meaning, and engage in discourse where they reason through strategies used to approach the problem. How students engage in these tasks can indicate their level of conceptual understanding.
To work toward conceptual understanding, it is important for teachers to represent mathematical ideas in multiple ways. In doing so, teachers expand students' knowledge of core ideas to help them see connections among the ideas (Shih, Speer, & Babbitt, 2011). Students must demonstrate understanding in ways which are meaningful to them. These could include using drawings or models and presenting findings orally or in writing. These different modalities increase the likelihood that teachers reach every student as students can be more personally connected to the material. Few students, especially those with learning difficulties, can retain and use new connections after one encounter with the material. They will need frequent experiences with the new connection for it to become part of their conceptual repertoire. Explicit modeling of strategies, sustained practice, and planned reviews have often been found to enhance mathematics performance in children with learning difficulties (Hudson & Miller, 2006 as cited in Shih, Speer, & Babbitt, 2011). Providing students with different models can be done in a large group format and then retaught in small groups as necessary to help with conceptual understanding of number sense.
In addition to the traditional multiplication algorithm, this article explores three alternative models that can be taught over time and with sustained practice in order to develop understanding of multiplication: area model, partial products algorithm, and lattice method. Students can learn both concepts and procedures through these alternative models. The models can also assist students with understanding (making sense of) the more traditional algorithm.
Traditional Multiplication Algorithm Many teachers grew up with the traditional algorithm where if they see a problem such as 26 x 48 they might solve it as follows:
26 X 48 208 1040 1248
This traditional algorithm is certainly valuable for efficiency, but not as an entry point to multiplication. Students need time to develop an understanding of the parts of the algorithm so that they can comprehend their significance. Only teaching students this traditional way may cause them to follow a procedure without conceptual understanding. For example, what does the "carried" 4 on top of the problem really represent? Students might see the 4 and think it only represents that amount. However, it really represents "4 tens" which is 40. Furthermore, students are either taught to "bring down a 0" or leave it out and just shift their answers on line 2 over to the left one space (so instead of 1040 it would be 104 with a blank under the 8). But why? It is for place value because students are really multiplying by 40 and not 4 to get the answer for line 2. However, if students do not comprehend that, they are not developing conceptual understanding.
The area model (Figure 1) is a more visual approach which provides a geometric representation of the distributive property of multiplication. Students conceptualize a multiplication problem as finding the area of a rectangle (length x width). The lengths and widths are split into tens and ones. Using the example 26 x 48, the area model would be represented as:
Figure 1. Area model for multiplication.
In this model, students determine the area of the entire rectangle by finding the areas of the four smaller rectangles, in this case, and then adding them together (800+240+160+48=1248). Students could add areas of the smaller rectangles in any order which showcases the associative property of addition. It might be helpful at first to use graph paper so students can see the divisions of the tens and ones more accurately (drawn to scale). Additionally, teachers can use folded paper to delineate the smaller rectangles which aids kinesthetic and visual learners.
Partial Products Algorithm Similar to the area model, the partial products algorithm model also highlights the importance of the distributive property, place value, and addition. Given the same problem 26 x 48, the model would be represented as follows:
26 x 48 48 (8x6) 160 (8x20) 240 (40x6) 800 (40x20) 1248
Comparing the area and partial products models, students can see where the four partial products come from in the area model. This model moves more toward the traditional algorithm except that students write out all four partial products on separate lines as opposed to combining the information into two lines with the traditional model which took into account "carrying" (regrouping) numbers. This model can be used as a precursor to the traditional algorithm because students will be able to see what the "carried" 4 and "carried" 2 (see traditional algorithm model) signify; they also write answers in a vertical format. For this model, initially it would be helpful if students used ruled paper rotated 90° to write/solve problems since it better showcases the place values of the numbers. By doing this, when adding the numbers, they are lined up in the correct places as the columns on the paper represent the different place values.
Some curricula, such as Everyday Mathematics (UCSMP, 2007), teaches students the lattice method (Figure 2). In this approach, students: 1) Write the numbers they are multiplying along the top and side of the grid; 2) Multiply the single digits on the top by the single digits on the side to fill in the squares. (If the product is a single digit answer, make sure to include 0 in the tens place as shown in Figure 2); and 3) Add diagonally to find their answer. When adding, students may have to regroup ("carry") double digit sums to the next place.
Figure 2. Lattice multiplication method.
In this example, the answer to 26 x 48 is 1248 as the numbers along the left outside of the grid show. It is important for students to understand that they add on a diagonal to take into account place value. For this model, it is particularly helpful to use graph paper or paper with blank lattice models on it, especially for those with visual and/or spatial problems who need a template to position the numbers. This model can be more challenging visually and spatially, but also beneficial because students are working with single digit multiplication which is the building block for more complex problems. If using this model, make sure to explain it to parents during back to school night, conference times, or in classroom newsletters. From experience, this model is most foreign to them and the one they are less likely to be able to help their children with without some type of explanation of how it works.
Importance of Multiple Representations
This article showcases four models to teach/learn multiplication to develop conceptual understanding which includes properties of multiplication, place value, and addition. It is important to utilize multiple representations to target different style learners. Once students are exposed to the varied models, they can choose which one(s) fit their personal learning styles. Although the problem presented throughout this article is 2 digit by 2 digit, the models can be used for any number of digits but the diagrams will be adjusted to suit each problem (e.g., 3 digit by 2 digit problems will have six partial products and, thus, six smaller rectangles if using the area model).
Recently, I have had so many of my undergraduate and graduate preservice teacher candidates have "aha" moments when learning these different models at later stages in their lives. "Why weren't we taught with these models?" many exclaim. They finally understood the reasoning behind "bring down a zero," for example. Students do not always have to use each model, but by exposing them to varied models, they can see connections between the numbers and their significance. Since the operation of multiplication builds on ideas about place value, addition, and skip counting, aids in the understanding of fractions and percents, and lays a foundation for future work with ratios and similarity in general (Otto, Caldwell, Lubinski, & Hancock, 2011) gaining a solid foundation and conceptual understanding of multiplication is a necessity.
Council of Chief State School Officers [CCSSO]. (2010). Common Core State Standards for Mathematics. Washington, DC: CCSSO.
Hudson, P., & Miller, S. P. (2006). Designing and implementing mathematics instruction for students with diverse learning needs. Boston, MA: Pearson.
Otto, A. D., Caldwell, J. H., Lubinski, C. A., & Hancock, S. W. (2011). Developing essential understanding of multiplication and division: Grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
Shih, J., Speer, W. R., & Babbitt, B. C. (2011). Instruction: Yesterday, I learned to add; today I forgot. In F. Fennell (Ed.), Achieving fluency: Special education and mathematics (pp. 59-83). Reston, VA: National Council of Teachers of Mathematics.
University of Chicago School Mathematics Project [UCSMP] (2007). Everyday mathematics 3rd ed.). New York, NY: Wright Group/McGraw-Hill.
Joan Gujarati, Ed.D., is an Assistant Professor in the Department of Curriculum and Instruction at Manhattanville College in Purchase, New York where she teaches the childhood mathematics methods courses. She is a former elementary school teacher and Math Teacher Leader. Dr. Gujarati's research interests include early childhood and elementary mathematics education, teacher beliefs and identity, teacher quality, effectiveness, and retention, and curriculum development. Dr. Gujarati can be reached at Joan.Gujarati@mville.edu