Teaching the Associative Property of Multiplication
Imagine you are a third- or fourth-grade teacher and you are aware that students in elementary school should learn the big idea of the AP of multiplication, (a x b) x c = a x (b x c). How may this big idea be introduced to students, first informally and then formally? To what extent may you prompt your students to understand this big idea at the elementary sehool level? To proceed, you may consider the following questions:
- 1 What examples will you use to teach the AP of multiplication ?
- 2 What representations will you use during teaching?
- 3 What deep questions will you ask during teaching?
Readers may have thought of using an example like (6x5)x2 = 6x (5 x 2) to show that it does not matter whether you multiply the first two numbers first or the latter two numbers first. However, the above instance of the AP could be presented to students in different ways. Based on my observations, one could directly present a computational task like 6x5x2 and ask students to apply the AP to solve it using two different calculations. Alternatively, one could start with a concrete context like the Mango problem (see Section 5.2), which can be solved with two solutions. A comparison between the two solutions will generate the desired instance of the AP. Using different representations might indicate whether the teacher’s goal is to apply or to make sense of (or illustrate) the property.
I argue that during initial learning, it is critical to help students make sense of the AP through concrete contexts before applying this property for computation. Deep questions should be asked to focus students’ attention to the meaning of each solution step within a story situation. Of course, concrete contexts should be gradually faded out so students ean learn the essential features of the property. As such, it is important to ask questions that elieit students’ articulation about the features of the AP equation.
The AP can be applied in various computational contexts to make computation easier (see Section 5.2). Strategies such as adding a “0” when multiplying by tens (8 x 90 = 8 x 9 with a 0 at the end), breaking apart a factor to multiply (e.g., 8x15 = 8x5x3) and doubling while halving ( 8 x 3 = 4 x 6) all utilize the AP. Teachers should make use of representations and ask deep questions to help students understand the undergirding property behind these strategies whenever appropriate.
In my project, there were both similarities and differences between the U.S. and Chinese lessons in terms of the context used to teach the AP. For instance, in both countries, there were relevant lessons that focused on problem solving (e.g., two-step word problems) and computational strategies (e.g., multiplying by tens). Teachers’ approaches to these tasks, however, were markedly different. In order to glean the cross-cultural insights, I will arrange the Chinese and U.S. lessons according to whether the intended goal was an illustration or application of the AP. I will first discuss lessons that illustrate the AP through problem-solving tasks (word problems with the structure of “groups of groups of objects”). Through these lessons, I will discuss how concrete contexts ean be used for sense-making. Next, I will discuss lessons that apply the AP via computation strategies. My focus is on how teachers may help students make sense of these strategies and gain an understanding about the undergirding ideas. For both types of lessons, examples about informal and/or formal teaching will be presented.
Insights from Problem-Solving Lessons: Illustrating the AP
Chinese Grade 3 Lessons: Informal Teaching
In the Chinese textbook, the AP is formally introduced in G4 using a worked example with the problem structure of “6 grades of 5 classes of 23 students.” Prior to G4, Chinese students learned to solve word problems with die same structure (e.g., 6 bags of 5 balls of ¥23). In addition to the similarities in the problem structure, the G3 and G4 lessons both emphasized sense-making, although the G4 lesson went further to formally reveal the AP. In this sense, I consider the G3 lesson to be a precursor to the G4 lesson.
Two G3 teachers (T1 and T2) taught the same lesson using the aforementioned worked example (also see Figure 5.6). This is a two-step word problem with three given conditions:
Figure 5.6 Worked example used in the G3 lessons. This G3 textbook image was taken from the Sun & Wang (2014, v.2, p.ll). Permission of citation was granted. © Jiang Su Phoenix Education Publishing.
a There are 5 Ping-Pong balls per bag. This is a condition hidden in the picture;
b Each Ping-Pong costs ¥2. This is a condition indicated by the price
c One needs to buy 6 bags. This condition is indicated by the picture and also included in the question statement, “How much does it cost for 6 bags?”
Although one can multiply the three numbers (5, 2, and 6) in any order to obtain the answer of 60, both Chinese teachers focused their students’ attention to the relevant conditions (or knowns) that could help them to solve the problem. The main point was to first group the items as either “6 bags with 5 balls per bag” or “5 balls costing ¥2 per ball.” Note that “6 bags and ¥2 per ball” are distantly connected and thus hard to put together. Interestingly, Chinese students in both classes came up with a hypothetical method that enabled the combination of 6 and 2 which will be elaborated upon later. Below are the main activities that occurred during both G3 classes where the teachers stressed representational connections through deep questioning.
- • Verbalize the known and unknown quantities
- • Solve the problem in two ways with meaning-making
- • Compare the two solutions for similarities and differences 1
- 1 Verbalize the known and unknown quantities. Both teachers projected the above textbook task (see Figure 5.6) to the board. They then asked students what the given conditions were and what question they needed to solve. Students in both classes were able to identify the pieces of information including the hidden condition that there were five balls per bag. Both teachers then requested that students completely verbalize the word problem based on the words and pictures. T1 pushed this step further by listing the three conditions on the board: “Each Ping-Pong ball is ¥2,” “6 bags of Ping-Pong balls,” and “Each bag has 5 balls” (see three rectangles in Figure 5.7, left).
Figure 5.7 Board depicting the given conditions and thinking methods in one G3 lesson. Redrawn by Anjie Yang (left).
2 Solve the problem in wo ways with meaning-making. Students in both classes were then asked to solve the problem. While both classes came up with two solutions, the two teachers took slightly different approaches to ensure that students were deriving meaning from the problems. T1 asked the students to verbalize their thinking methods before solving the problem while T2 asked students to explain their solution steps afterwards. Below are elaborations.
T1 directly began her lesson with the worked example and she asked her students to share their solution plans (or thinking methods Й.Щ.КК) with the whole class before generating their own number sentences. Overall, die students shared two types of plans: One where they began by finding how many balls there were in total and another where they began by finding how much one bag costed. Below is an example episode:
SI: First use “6 bags of Ping-Pong balls” and “each bag has 5 balls” to find out how many Ping-Pong we have in all.
T1: Okay, you want to use these two conditions first. (Draws a line to link the two connections and writes down “how many balls are in all” on the board, see Figure 5.7). Then?
SI: Then we can use the total number of Ping-Pong balls and that each Ping-Pong ball costs ¥2 to find out how much it costs for 6 bags of Ping-Pong balls.
Tl: Great. This is his thinking method. Do you guys understand?
Tl: Okay, first based on these two (references the bags and the number of balls per bag). And then use what we found to solve the last question. Very good. Any other thinking method? ...
S2: We can also use “Each Ping-Pong ball is ¥2” and “Each bag has 5 balls” to first find out how much it costs for one bag.
Tl: (Draws a line to link the two connections and writes down “how much does one bag cost?” on the board, see Figure 5.7). Okay, how much does one bag cost? And, then?
S2: Then we use the price we found out to multiply by 6.
Tl: Because how many bags [do we have]?
S2: 6 bags.
Tl: So, we can find out how much it costs for 6 bags of Ping-Pong balls, right? Okay, sit down please. This is the second method. It is also very good.
In the above episode, students articulated their solution plans by reasoning upon the relevant conditions. This led to students’ generated solutions, which were documented by Tl as methods A and B. Students were also able to explain the numerical solutions to the story situation in a relatively complete manner, which was likely due to them having verbalized their solution plans. Below are the two solutions recorded by Tl in class:
- 5 x 6 = 30 (balls) 5 x2 = 10 (¥)
- 30 x 2 = 60 (¥) 10 x 6 = 60 (¥)
Alternatively, T2 reviewed the subskills (choosing two relevant conditions among the three) prior to this example task. She then asked students to directly solve the problem on their own without discussing any solution plans. The two solutions that were generated in this class were nearly identical to those in Tl’s class. The only difference observed was using 2x5 = 10 in Method B. Given that multiplication is defined two ways in China, this alternative was permitted.
When each solution was shared, T2 followed up with a set of questions to promote student explanations. Example questions included:
- • 5 x 6 = 30 balls. Do you know which two conditions she chose to compose this number sentence?
- • Who can verbalize this completely? ... In her number sentence, she first figured out what based on which conditions?
- • In this method, what did we find based on “each bag has 5 balls” and “each ball is ¥2”? ... Who else can say it again: based on which two conditions to first find what?
In the above list of questions, the teacher stressed the meaning of each solution step and then the whole solution, which was different from Tl, who asked students to explain the whole solution directly. In short, both teachers spent a significant amount of time stressing meaning-making for both solutions. Both teachers’ questions appeared to draw students’ attention back and forth between the number sentences and the conditions given by the story context.
3 Compare the two solutions for similarities and differences. After their classes solved the problem in two ways, both teachers explicitly asked students to compare and contrast their solutions. As discussed throughout this book, promoting explicit comparisons is common in Chinese lessons and can be easily incorporated into U.S. classrooms. In both Chinese lessons, the teachers first guided students to talk about the similarities. Students noticed that methods A and В both contained two steps and that both were multiplication. They also found that ¥60 was the result of both methods. Based on these findings, both teachers revealed that the title of the day’s lesson was, “Using two-step continuous multiplication to solve problems.” Revealing the lesson title after the discussion of the worked example is also a common occurrence in Chinese classrooms, which is different from U.S. practices where teachers tend to announce the objectives at the very beginning of a lesson.
Based on students’ observations of the solution similarities, T1 further prompted, “What if the answer of one method is different from the other? What does it tell us?” This led students to remind her about how one method could be used to check the other. Similarly, T2 prompted her students, “Both are two-step continuous multiplication and the results for both are ¥60. Who can state an equation based on these two methods?” This resulted in the equation of 5 x 6x2=2x5x6. Although this was not an instance of AP, it would have been if not for the two-way definition of multiplication (elaborated upon later).
After the discussion of the solution similarities, both G3 teachers prompted the students for differences as well. T2 asked, “What are the differences? What did you first find out in this method?” Students pointed out that the differences lied in the meaning of the first steps of two solutions. In a similar vein, T1 posed an open-ended question, “Recall our problem-solving process. Do you have anything to say or do you have anything to remind your peers?” This question elicited students’ input such as reading the given conditions carefully and being clear about the meaning of each step. Some students added that one should be cautious with the “unit” in the problem, especially when discussing the result of the first step. One student elaborated upon this observation with an example that one needed to consider whether 5 x 6 = 30 represented 30 balls or ¥30. Based on students’ input, the teacher summarized:
T2: The result of the first step does not directly address the question raised by the word problem. We cannot directly sec the unit from the question statement. So, we should figure out what we solved first based on the relevant conditions. Therefore, when solving these types of problems, the first step is the most important. Only when you know what you must first find out, will you know the unit of your first step. This is a great reminder.
Other students further added that they did not approve of the random guess method—that is, to multiply the three numbers in any order. This reminder was further elaborated upon by the teacher. Below is a relevant episode:
SI: If you don’t know how to solve this problem, don’t blindly guess. For instance, if you don’t know how to solve this problem and you just use the first two conditions (¥2/baIl and 6 bags) to obtain 12 and then multiply 5, you will also get ¥60. However, this may just happen to be correct. You cannot always use this kind of method to solve problems.
T2: What SI said is exactly what I wanted to remind everyone. When solving problems with continuous multiplication, it seems that we can multiply any two of the three given conditions, right? Some students may choose 5x6x2, others choose 5x2x6. Can I directly multiply 2x6? So, I tell you, when we compose the number sentence, we cannot randomly use the numbers. We need to find the relevant conditions. You should clearly know what you will first find out and then list your number sentence. If you multiply 2 and 6, we cannot explain what we find based on the word problem. So, what SI said is very important. Let’s give him a round of applause. We are not randomly multiplying the numbers. We need to know what it finds out.
Note that the above discussion is in direct opposition to the any which way rule method frequently included in the U.S. textbooks and classrooms (see Sections 4.3 and 5.3). It seems that whether a teacher suggests that three numbers should be multiplied in order is intrinsically tied to whether the teacher values contextual meanings over computational answers.
Interestingly, in both classes, there were students who used a hypothetical thinking method to argue for the meaning of the “wrong” combination of quantities. For example, one student argued that it would be appropriate to multiply 2x6 (¥2/ball and 6 bags) in the following way: If we assume that there is only one ball per bag, six bags hold a grand total of six balls. In that case, six bags only cost 2x6= 12(¥). However, since there are actually five balls per bag, that initial total cost of ¥12 should be multiplied by 5, which gives 12x5 = 60(¥). Both teachers accepted and praised this method. One teacher stated that she would accept the method as long as its context could be explained in a reasonable way. This interesting argument highlights the importance of emphasizing meaning-making, which potentially promotes students’ thinking. Again, note the contrast to many U.S. classrooms, where a solution like 2 x 6 x 5 = 60 would be praised for deriving a correct answer, and not for deriving sensible meaning. Perhaps the Chinese teachers could have pressed further by pointing out that when one assumes each bag only contains one ball, the meaning of “6” in the story context changed from six bags to six balls. Therefore, when they multiplied 2 by 6, they were multiplying ¥2/ball by 6 balls (the total cost of the balls in hypothetical bags), instead of multiplying ¥2/ ball by six bags (which has no contextual meaning). In other words, this hypothetical thinking method still complies with the idea of multiplying relevant conditions to ensure meaning-making.
Despite this insight about teaching with meaning, the loosened two- way definition of multiplication in China allows the two relevant quantities to be multiplied in ways that do not necessarily generate an instance for AP (e.g., 5 x 6 x 2 = 2 x 5 x 6). In fact, by defining multiplication as one-way (regardless of whether it is the U.S. or the traditional Chinese definition), the two solutions would result in an instance of AP, if done correctly. For instance, if we consider the US convention of multiplication that “д groups of b" should be represented by a x /?, then the above Ping-pong ball problem (six bags of five balls of ¥2) could be solved as follows:
- 6 x 5 = 30 (balls) 5 x 2 = 10 (¥)
- 30 x 2 = 60 (¥) 6 x 10 = 60 (¥)
Or: (6 x 5) x 2 Or: 6 x (5 x 2)
In Method A, six bags of five balls each gives 30 balls and then 30 balls costing ¥2 each gives ¥60, resulting in the solution of (6 x 5) x 2. In Method B, five balls costing ¥2 each and then six bags costing ¥10 each gives ¥60, resulting in the solution of 6 x (5 x 2). A combination of the solutions will generate an instance of the AP: (6 x 5) x 2 = 6 x (5 x 2).
Alternatively, aligned with the Chinese traditional convention of multiplication, “й groups of b" is represented as b x a, the Ping-pong ball problem will be solved as:
- 5 x 6 = 30 (balls) 2 x 5 = 10 (¥)
- 2 x 30 = 60 (¥) 10 x 6 = 60 (¥)
Or: 2 x (5 x 6) Or: (2 x 5) x 6
In this case, a combination of these instances will also generate an instance of the AP: 2 x (5 x 6) = (2 x 5) x 6.
It is my view that the loosened Chinese definition of multiplication results in a missed opportunity to infuse the AP into this type of lesson, though I am aware that discussion about how to define multiplication goes beyond the scope of this book. Despite what I perceive to be a fundamental shortcoming, I believe the Chinese approach that stresses meaning-making through representational connections and deep questioning is insightful. This approach can be taken into U.S. classrooms and incorporated with the U.S. conventions of multiplication.
Properties of Multiplication 145