About the Mathematics: Making Sense of the Commutative Property or Multiplication
Pressing for details in understanding the commutative property of multiplication.
Overview
a x b = b x a. It appears simple enough to explain and comprehend. It leads to ideas of efficiency and flexibility. Asking students what’s the same or different about each expression can be quite revealing when one presses for details (Kazemi & Stipek, 2001; Webb, et al., 2014, Bishop, 2021). Whether or not the multiplicative context is asymmetrical or symmetrical influences how students approach solving the task (Carpenter, et al., 2015) or even how the commutative property is understood (Baroody, et al., 2003). Understanding the units each quantity in the expression represents and how those units transform is required to determine a product. This is all shaped by the context in which those quantities are anchored, and plays deeply on how the commutative property is understood (Brickwedde, 2024). This piece looks at some of the complexities that students need to explore in developing a depth of understanding of this key algebraic property.
Context
A third grade intervention group was working on grounding their understanding of multiplication and the various strategies that could be used to solve such problems. This was in addition to instruction that already had taken place within the larger classroom. The students were asked to work with both manipulatives and number models to solve and record their work. The context was an asymmetrical, equal grouping context.
There were three cookies. Each cookie had four chocolate chips in each cookie. How many chocolate chips were in all the cookies?
The students were already familiar with the ‘groups of’ language to visualize and articulate what the context was asking. The teacher probed students to connect the context to the manipulatives and the numbers. Mathematical vocabulary was reinforced. Students were then sent off to solve the problem and then were called back to share with their peers the different approaches to answering the question of how many chocolate chips in all of the cookies. There was a typical range of additive strategies of counting all by ones, skip counting, adding two fours to make eight and then counting on, etc. In the public sharing of this work, one student shared how it would also work to add four threes particularly by combining each pair of threes into sixes, thus making the problem 6 + 6. The student, when probed, knew that this was the commutative property. The teacher wrote on the screen 3 x 4 = 4 x 3.
It was at this point when the conversation was about to move on that I asked permission to pose some queries with the students. First, I asked which of the recorded number expressions on the screen matched the ‘three cookies with four chocolate chips?’ Which solution matched the ‘four cookies with three chocolate chip cookies?’ Students were asked what was the same and different about each expression. Students focused on the fact that the product (vocabulary used by one of the students) was the same. What was different between the two models was initially elusive to the students. At this point, the teacher drew students’ attention back to the manipulative model. Each expression was modeled. I pressed students beyond, yes, both models had twelve chocolate chips, to focus on the number of cookies. If I was looking for the most amount of cookies, which model, and therefore expression, would I prefer? In that context, three cookies with four chocolate chips is very different from four cookies with three chocolate chips in each.
The asymmetrical nature of the context renders the commutative property more difficult to perceive (Baroody, et al., 2003). One needs to have had enough experience with exploring and trusting that reversing the numbers will, in fact, render the same product. That requires the abstract concept of lifting the quantities and the respective units out of the context, disconnecting the units from the quantities, and commuting those elements to resolve to the same solution/product. That’s very abstract. As an example used by Hurst & Hurrell illustrates, (2016) taking two doses of medication six times a day has different ramifications with taking six doses twice a day of the same medication. Both result in a total of 12 doses a day but the latter may result in a trip to the emergency room. Context still matters.
An attribute of multiplication as an operation is that multiplication is unit transforming. The context the students worked with involved cookies (3), chocolate chips per cookie (4), resulting in individual chocolate chips in total (12). The product does not reflect on the cookies at all. Many curriculums do not have students attend to the units or to how the units transform to create the product. Thus when students begin to work with rates and ratios in middle school, work proportionally, and even encounter unit analysis in high school science classes, they struggle as they have never had to focus on this aspect of their reasoning. Attending to the details of the units is critical for elementary students to begin to grapple with as they work with multiplication and division. A focus on the units is particularly helpful in comprehending the difference between the two types of division. Conceptualizing division as missing factor multiplication tasks is dependent on understanding the units involved in the context.
Figure One
Symmetrical contexts are easier to conceptualize the commutative property. Array (discrete objects in even rows and columns) and area models (contiguous objects in rows and columns) are the two symmetrical representations students encounter. The two models below provide a visual. Whether you see the area model as 6 columns with 4 squares in each column, or as 4 rows with 6 squares in each column is a matter of perspective. The symmetry allows one to easily choose. There is no mathematical social convention that determines a preference. That is very different from asymmetrical packaging.
[a] [b]
Symmetrical Context Asymmetrical Context
6 columns of 4 squares per column 4 plates with 6 cookies per plate
4 rows of 6 squares per row 6 plates with 4 cookies per plate
6 x 4 = 4 x 6 4 x 6 = 6 x 4
Figure Two
While the totals in each of the four scenarios results in 24 individual items – squares or cookies, – the asymmetrical context takes longer for students to lift out of the context and abstractly trust that the products will be the same. The units transform in the symmetrical context as well within each respective equation. However, the arrangement of the area model does not change.
Teaching Implications
The packaging of objects in the world around us means that asymmetrical contexts dominate students’ images of how multiplication fits into the mathematical world. Unless students have direct experience with baking or setting up chairs in an auditorium or in other contexts where neat rows and columns are typically used, symmetrical contexts are less prominent images for students to easily draw upon. Yet many curriculums start with area and array models as a means to teach multiplication. The assumption is that if the commutative property is understood in symmetrical contexts it will automatically transfer to asymmetrical ones. Research demonstrates that there is a significant lag time of students transferring the idea of commutativity from one to the other (Baroody et al., 2003).
-
- Begin as early as kindergarten introducing asymmetrical equal grouping contexts. (Carpenter et al., 1993)
- Introduce asymmetrical contexts involving money, rates, and comparison units as early as first grade. (Carpenter et al., 2015)
- Introduce symmetrical area and array context problems as early as first grade. (Carpenter et al., 2015)
- Engage in robust conversations about the commutative property of addition in first and second grades drawing out where and when it will work. (Carpenter et al., 2003)
- In second and third grade, engage in robust conversations about the commutative property of multiplication drawing out where it will work and when it will not. Include how the models in asymmetrical contexts are not the same due to the unit labels attached to each quantity.
- Expect a lag time with students being able to efficiently use the commutative property of multiplication when asymmetrical contexts are being used. (Baroody et al., 2003) Do not assume that using the property in symmetrical contexts will be understood quickly in the other. This is due to the abstract ability to lift the quantities out of the context, separate quantities and the respective units and rearrange the configuration. This is particular to multiplication as an attribute of multiplication as an operation unique from addition as multiplication is unit transforming.
- In public sharing situations, should a student use the commutative property in either asymmetrical or symmetrical contexts, whether intentionally or unintentionally, draw out among students why the property can work, and the efficiency issues around such a decision. It is also important, as an attribute of multiplication is that the operation is unit transforming, how the units transform must also be intentionally discussed. What does the 3 represent? What does the 4 represent? Etc.
The commutative property makes working in addition and multiplication more efficient. Students need a robust understanding of how it works, when it doesn’t work, and how, in asymmetrical contexts in particular, interpretation of the units needs to be attended to. Using the commutative property to count the total pills needed for a prescription, but commuting the quantities without attending to the implications of the units could lead to dire circumstances.
References
Baroody, A.J., Wilkins, J.L.M., & Tiilikainen, S. (2003). The development of children’s understanding of additive commutativity: From protoquantitative concept to general concept? In A.J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ: Lawrence Erlbaum Associates.
Bishop, J.P., (2021) Responsiveness and intellectual work: Features of mathematics classroom discourse related to student achievement, Journal of the Learning Sciences, 30:3, 466-508, DOI: 10.1080/10508406.2021.1922413
Brickwedde, J. (2024), Multiplication and division strategies: An overview of the progression of children’s strategies in multiplication & division, Retrieved from: https://docs.google.com/document/d/163J9vavo7NUZWzzUcI-aAAidYkYhLwxJ/edit?usp=sharing&ouid=104440259025476693639&rtpof=true&sd=true
Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children’s problem-solving processes. Journal for Research in Mathematics Education, 24(5), 428-441.
Carpenter, T.P., Fennema, E., Franke, M.L., Levi,L. & Empson, S.B. (2015). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.
Carpenter, T.P. Franke, M.L. and Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.
Hurst, C. & Hurrell, D. (2016). Investigating Children’s Multiplicative Thinking: Implications for Teaching, European Journal of STEM Education, 1:3, p. 56, doi: http://dx.doi.org/10.20897/lectito.201656
Jacobs, V.R. and Empson, S.B. (2016). Responding to children’s mathematical thinking in the moment: An emerging framework of teaching moves. ZDM Mathematics Education 48:185–197.
Kazemi, E. & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms, The Elementary School Journal. 102 (1) pp. 59-80.
National Council of Teachers of Mathematics. (2014). Principles to actions. National Council of Teachers of Mathematics: Reston, VA.
Webb, N.M., Franke, M.L., Ing, M., Wong, J., Fernandez, C.H., Shin, N., and Turrou, A.C. (2014). Engaging with others’ mathematical ideas: Interrelationships among student participation, teachers’ instructional practices, and learning, International Journal of Educational Research, 63, pp 79-93.