Insights from U.S. Lessons

The fourth grade U.S. lesson introduced below also sheds light on how deep questioning on diagrams can elicit students’ conceptual understanding. Very often, teachers in U.S. classrooms tend to ask procedural questions (e.g., What was your strategy? Flow did you get the answer?) These types of questions may only elicit students’ description of procedures rather than deep explanations of the targeted concepts. Additionally, research has reported that some teachers ask initial deep questions but often lack the ability to ask the follow-up questions which would push students far enough to achieve deep learning.

In the following episode, the teacher, DJ, taught a multiplicative comparison task, “D/ picked 7 apples. Teacher Kelly picked 4 times as many apples. How many apples did Teacher Kelly pick?” The teacher first drew 7 apples in a circle. He then asked students how they could represent Teacher Kelly’s apples using a picture, a diagram, or a model. One student suggested drawing 4 big circles with 7 apples in each. The teacher then asked:

T: S1,1 want you to call on someone now, and I want them to tell us why you chose to have 4 big circles with 7 apples inside each one.

SI: Can I say myself?

T: No. I want someone who paid attention to this lesson to tell us. ... S2: You do that because you said that she has 4 times as much and it would be difficult to draw all those in just one big circle.

T: Well, we could draw them all in one big circle, but you said a key word here. She has 4 times as many apples (underlines 4 times). What does that mean that teacher Kelly has 4 times as many apples?

S3: Uh, it means that she has 7 more but 4 times 7 more...

T: What do you mean, she has 7 more but 4 times 7 more?

S3: Like, she has 7 more and then she has another 7 more, so it’s like.... T: Call on someone to help you out to clarify your thinking.

S4: Can I give an example?

T: Please.

S4: Don’t you see how you have 7 apples?

T: I do see I have 7 apples. That’s my favorite number.

S4: You just add on 4 more like. They’re saying like you’re adding on 4 more bags of 7 apples.

T: Well, why does it mean that I have to add on 4 more bags of 7 apples? S4: Because it says 4 times.

T: Okay, because it says 4 times, but why do the bags have to have 7?

S4: Because of the number that you already have, that’s like the...

T: Ah, because of the number I have already. I picked 7, and then we said that Teacher Kelly picked 4 times as many apples I did. You can’t just pick a number out of the sky and say that I’m going to do 4 times 27. Okay, because it’s 4 times as many apples as I already picked. So, she has to have 4 groups, with that same number inside of it. So now, who can tell us an equation that can represent how many apples Teacher Kelly picked?

S5: 7 times 4 equals...

In the above conversation, the teacher asked a set of deep questions that required students to explain of the meaning of “4 times.” I paraphrase those questions below:

• • Why [did you choose]to have 4 big circles with 7 apples inside each one...?
• • What does that mean that Teacher Kelly has 4 times as many apples?
• • What do you mean, she has 7 [apples] more but 4 times 7 more?
• • Well, what does it mean that I have to add on 4 more bags of 7 apples?
• • Okay, because it says 4 times, but why do the bags have to have 7 [apples]?

The first four questions from the above list flowed quite organically from student responses. Indeed, students were actively engaged in the verbalization of thinking about why drawing 4 groups of 7 apples was needed. As seen from the conversation, the two concepts “4 groups of 7 apples” and “4 times” were repeatedly alternated. It is possible that this led some students to make an implicit connection between “4 times” and “4 groups of 7.”

As acknowledged, this teacher’s set of follow-up questions demonstrated an effort to promote conceptual understanding. Perhaps the above discussion could have been more focused on creating explicit connections between “4 times” and “4 groups of the same thing.” More precisely, “Teacher Kelly picked 4 times as many as apples as DJ” means that we would view DJ’s 7 apples as one group, and Teacher Kelly’s apples then contained 4 groups of the same thing. In the above excerpt, it was the teacher who pointed out that “4 times” meant “So she has to have 4 groups.” The teacher could have prompted students to explicitly verbalize this key point. Lacking this level of abstraction can lead to a barrier for future learning. For instance, later in this lesson, the class worked on a similar problem that compared two people’s height: “Mibsam is 2-feet tall⁷. Teacher Chris is 3 times as tall as Mibsam. How tall is teacher Chris?” Some students quickly wrote a multiplication equation (2x3 = 6)^s to find the answer. However, when the teacher suggested using a drawing to show the meaning of “3 times,” several students refused to accept the idea that three stacks of “2-feet” would represent a height that was 3 times as tall. This challenge reveals the importance of understanding the concept of “times” in a solid manner during the teaching of multiplicative comparisons so that students can transfer that knowledge to solving new problems.

Summary: Teaching Multiplicative Inverse (Comparison Model) through TEPS

The U.S. and the Chinese lessons in this section together illustrate TEPS for teaching multiplicative comparison problems. Worked examples that are situated in real-world contexts can be modeled using semi-concrete representations (e.g., manipulatives, bar models/tape diagrams), which leads to numerical solutions. This process employs the concreteness fading method. Note that students should be engaged in the modeling process, and their generated representations can serve as a platform for classroom discussions. Meanwhile, deep questions can be appropriately initiated and then harnessed to their fullest extent through follow-up questioning. I want to emphasize that when teaching the model of multiplicative comparison, the concept of “times” should be explicitly linked to the equal groups model. Once this concept is grasped, a meaningful understanding of multiplicative inverses (e.g., Bigger Smaller = Multiple; Multiple x Smaller = Bigger; Bigger Multiple = Smaller) can be developed based on the multiplicative comparison model.

Notes

• 1 Chinese G2 students’ performance on the other items in the pretest was much worse.
• 2 Our scoring for correctness focused only on the answer. However, based on the meaning of multiplication, the correct solution should be 3 x 9 because Geoff’s amount contains three groups of $9.
• 3 See statistical data and more detailed analysis about the types of errors in Li et al. (2016).
• 4 In the course of my personal communications with mathematicians like Dr. Jim Lewis (University of Nebraska-Lincoln) and Rogers Howe (Yale University), they have expressed agreement with the importance of using a single, consistent meaning for multiplication.
• 5 In Chinese multiplication Koujuc, the variable (first number) is always restricted to be no larger than the fixed (second) number. In this sense, it handles only half the multiplication table, and commutativity is necessary to augment it. For instance, in the Koujue about 7, “two seven fourteen” refers to both 2x7 = 14 and 7 x 2 = 14. In addition, the first number in 7’s Koujue does not go above 7. In other words, the Koujue for 8 x 7 and 9x7 are left for the Koujue about 8 and 9, respectively.
• 6 As mentioned earlier, the new Chinese standards defined the basic meaning of multiplication in both directions: a groups of b is written as either ay, b or b x a. This is a confusing point that is different from the U.S. convention of multiplication: a groups of b is written only as ax b.
• 7 In the lesson interview, the teacher recognized that 2-feet is too short for a student.
• 8 Based on the U.S. convention of multiplication, it should be 3 x 2 = 6 as we have 3 groups of 2-feet here.

References

Beckmann, S., & Izsak, A. (2015). Two perspectives on proportional relationships: extending complementary origins of multiplication in terms of quantities. Journal for Research in Mathematics Education, 46, 17-38.

Cai, J., & Moyer, J. C. (2008). Developing algebraic thinking in earlier grades: Some insights from international comparative studies. In С. E. Greene & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 169-182). Reston, VA: NCTM.

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NEE Heinemann.

Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39, 111-129.

Huang, R. and Li, Y. (2017), Teaching and learning mathematics through variation. Confucian heritage meets western theories. Rotterdam, The Netherlands: Sense.

Li, X., Hassler, R., & Ding, M. (2016, April). Elementary students’ understanding of inverse relations in the U.S. and China. Paper presented at 2016 American Education Research Association (AERA) Annual Conference, Washington, DC.

Ma, L. (1999). Knowing and teaching elementary mathematics: Understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

National Governors Association Center for Best Practices & Council of Chief State- School Officers (NGAC & CCSSO). (2010). Common Core State standards for mathematics. Washington, DC: Authors

Schwartz, J. E. (2008). Elementary mathematics pedagogical content knowledge: Powerful ideas for teachers. Boston, MA: Allyn & Bacon.

Singer, F. M., Ellerton, N., & Cai, J. (2015). Mathematical problem posing: From research to effective practice. New York, NY: Springer.

4 Properties of Addition