Insights from U.S. Lessons
U.S. lessons contributed further insights for teaching multiplicative inverses, particularly in representation uses. In the Chinese lesson discussed above, even though students were prompted to verbalize varied quantitative relationships, there were no transitional tools between the story context and the numerical solutions. As such, the semi-concrete representations occurring in the U.S. lessons (e.g., cube manipulatives, bar models, charts) may be used to support students who need extra help processing the targeted relationships. Additionally, these U.S. lessons lend opportunities to consider how the Chinese lesson approaches may be incorporated to maximize student learning. Below presents two G3 teachers’ (T1 and T2) relevant lessons.
T1 started her lesson with a review problem: “Luisa has 8 eggs. She needs to put them into 4 equal groups.” She first asked students to fold a piece of paper into four equal parts and then to distribute eight cube manipulatives onto it. She then drew a bar model with counters (the bar was separated into four parts with two counters in each part). The folded papers with cubes may have allowed students to form a more concrete understanding of the bar model. This, in turn, set up the class for a new lesson where two types of division problems were discussed: (a) 20 treats shared with 5 dogs, and (b) 20 treats with 5 treats for each dog. After reading each example problem, the teacher guided the class in completing the manipulative model and then the bar model provided by the textbook (see Figure 3.2, left). The teacher
Figure 3.2 Multiple representations in U.S. third-grade classrooms. Teacher 1: Redrawn by Mohen Li. Teacher 2: Redrawn by Anjie Yang.
also asked students to compare both models to identify the similarities and the differences, which was very similar to the comparisons commonly seen in Chinese classrooms. These rich representations (paper folding, cubes, counters, bar model) in Tl’s class had the potential to augment students’ conceptual understanding of the division problems. One potential way for the teacher to go further would have been to ask students to write numerical equations as opposed to filling in the textbook blanks with numerical answers (see Figure 3.2). This would help link students’ understanding established from rich concrete representations to abstract solutions.
T2’s lesson also offered insights into representation use (see Figure
3.2, right). The objective of this lesson was to teach one type of division. The teacher started the lesson review with a multiplication problem: “A robot has 4 hands. Each hand has 6 fingers. How many fingers does the robot have altogether?” She first guided the class in analyzing the known and the unknown quantities using a “multiplication and division chart” (see Figure 3.2, right). Three quantities—“number of groups,” “number of each group,” and “product”—were recorded on the chart along with the corresponding equation that described the problem. This is similar to the Chinese teacher’s emphasis on the quantitative relationship sentence, “number of groups x number in each group = total.” After the multiplication problem (4 x 6 = ?) was solved, the teacher asked students to change it into a division problem which, in turn, brought up the new example task for discussion. Again, expecting students to change a problem to its inverse is a method common to Chinese lessons but rare in the observed U.S. lessons. Similar to her discussion about the multiplication problem, T2 used the multiplication and division chart to discuss the known and unknown quantities and the corresponding equations. In retrospect, I think the chart was an effective representational tool that neatly summarized the quantitative relationships and provided an opportunity for comparison between the two problems.
To determine the computational answers for both the multiplication and division sentences (4 x 6 = ?, 24 -e- 4 = ?), T2’s class discussed varied computational strategies including tallies, which took a significant chunk of class time. This is a different observation from the aforementioned parallel Chinese lesson, where the majority of class time was devoted to quantitative relationships rather than computational methods. One common complaint that I hear from U.S. teachers is that their students lack the mastery necessary to perform calculations quickly. Since time is at a premium in classrooms, teachers often have to choose between either engaging in lengthy calculation techniques or examining mathematical concepts. I suggest teachers consider the benefits of Chinese multiplication Koujue to bridge this divide. As has been demonstrated in several instances, Koujue has the benefit of simultaneously being a memorization tool and a conceptual justification, meaning that teachers who use Koujue don’t have to make the difficult choice between calculations and comprehension.
With regard to deep questions, the teachers in both classrooms asked questions that had the potential to contribute to student connectionmaking. This can be seen from the above descriptions (e.g., Tl’s asking students to compare the counters and the bar models, T2’s asking students to change a multiplication problem to a division problem). To prompt students’ thinking even further, both teachers could have requested more explicit comparisons. For instance, the two “dog and treats” problems used by T1 shared a similar context but had different structures. The teacher could have asked students to compare the two “20 -r 5 = 4” statements since the meaning of each was different (one was to find the number of groups while the other was to find the group size). Similarly, T2 could have explicitly guided her class to compare the two “robot’s fingers” problems. The multiplication and division chart had already documented the two problems and solutions. The teacher could have built off the chart by asking students to compare the similarities and differences between the two problems, as well as having them verbalize when multiplication or division could be used to solve problems.
Additionally, there was a moment in Tl’s lesson when the class noticed “5 groups of 4 counters” (problem 1) and “4 groups of 5 counters” (problem 2) from the drawing for each division problem. Such observations could have been emphasized to make inverse relations more explicit through deep questioning. For instance, after solving each division problem, the teacher could have asked students, “Without counting the drawing, how do we know the answer 4 treats (problem 1) or 4 dogs (problem 2) is correct?” One way to check is to put the answers back into the problem situation and to figure out the total based on multiplicative thinking (5 groups of 4 is 20; 4 groups of 5 is 20).
Summary: Teaching Multiplicative Inverse (Equal-Groups Model) through TEPS
The Chinese and the U.S. lessons on the equal groups model provide insights for teaching multiplicative inverses through TEPS. For representation uses, elements from the U.S. and the Chinese lessons can be combined to demonstrate an application of the concreteness fading method: Starting from a real-world context, using semi-concrete representations (e.g., bar model, cube manipulatives, counters) to depict the problem situation, and then solving the problem numerically. In addition, folding these number sentences back into the concrete context can serve to build further connections between the concrete and the abstract. For deep questions, I suggest that teachers stress the meaning of operations (Why is multiplication used? Why is division used?) and the quantitative relationships. I also suggest that teachers ask questions to foster explicit comparisons between different problem situations, representations, and numerical solutions. An integration of representation uses and deep questions can enable the deep discussion of a worked example. This can, in turn, aid in the development of the mental schema that students require for new learning.