Informal Teaching of the CP of Addition in G1 and G2
U.S. videos in my project also contained opportunities to informally revisit the CP. Those opportunities occurred in G1 and G2 lessons on fact families (inverse relations). This was another situation where semi-concrete representations such as cube trains were used to generate related addition facts. Below is an episode from the G2 classroom (also see Figure 2.3, left). In this interaction, the teacher began by asking a student to determine the related fact for 7+1 based on her cube train.
T: I don’t have another addition fact family up here. Can somebody show me what can you do with these cubes for me to be able to write another addition fact up here using these same exact cubes? I’m not going to change them or take them apart. Who can show me what we can do to make another fact family using these cubes? SI, do you think you can? What would you do? You can’t take them apart.
SI: (Suggested 3 + 5 = 8)
T: Okay SI, I’m gonna hold you up for a second. I love what you’re thinking. She said she wants to write another addition fact but we’re sticking with 7, 1, and 8. So I’m not going to change the colors. I’m not going to add more, take any apart. So, what can I do to still stick with the same family without changing things. S2?
S2: 1 + 7
T: So, what can I do with this to show, this shows 7 + 1. What can I do? S2: Switch it around (S2 went to board and switched the cubes)
T: There you go. Thank you S2. (To class) So S2 changed it around. She didn’t take any away. She didn’t break any apart. She just switched them. And you already told me, S2, what’s the addition fact that would go here then.
S2: 1 + 7
T: Good she’s goi+ng to make it 1 + 7 = 8
In the above episode, the teacher asked students to use the cube train to come up with the related fact. She emphasized that they could not take any cubes away or break any apart. By switching the cube train around, the class was successful in generating 1+7 = 8 based on the number sentence 7+1 = 8, which may have informally allowed students to revisit the CP as well. Note how in this instance, the teacher relies on the nature of her semi-concrete representation to guide the student toward the observation. To develop deeper understanding of the related number facts, the teacher could have asked further questions to prompt comparisons among these number sentences. This may contribute further to students’ understanding of the CP.
There were additional observations from the U.S. lessons that may be worthy of attention. Although these observations go beyond the scope of this book, they call for future exploration on how children may be better supported in learning the basic properties. Given that students in my project demonstrate greater challenges in grasping the basic properties of operations than inverse relations, I believe the following observations shed light on improving classroom teaching from a new angle.
The Metaphor: Turn-around Facts. In most U.S. lessons, the teacher and students identified the CP as a “turn-around fact.” Even some textbooks introduced the CP as the “turn-around property.” This metaphor is most likely related to hands-on activities that involve rotating manipulatives like cube trains. Although this metaphor can be a powerful tool for visualizing the structural properties of the CP, there are also drawbacks. First, there may be some disconnect between what students know about the “turn-around facts” and the mathematical term “commutative property” to be encountered in later learning. According to Sfard’s (1991) dual nature of a mathematical concept, turn-around focuses on the “process” side of a concept that is operational. However, the statement of CP (e.g., a + b = b + a) focuses on the “object” side of the concept that is structural. Furthermore, the idea of the turn-around has been used to encompass ideas that do not fall under the CP. For instance, a G1 classroom in my project identified both the pair 3 + 7 = 10, 7 + 3 = 10 and the pair 10 - 3 = 7, 10 - 7 = 3 as turn-around tacts. Although the numbers in the latter pair also turn around, they are not related by the CP. In fact, such a use of “turn-around tacts” may cause student misconceptions about the CP.
Illustrating the CP: The Change of Meanings. In a few classroom instances, there were flaws in teachers’ representation uses when illustrating the CP. Instead of switching around the two parts of the same whole, the teachers changed the original quantities. Consider, for example, a G3 teacher who used “2 red and 6 yellow” counters. Instead of switching these two parts (6 yellow and 2 red), she represented 6 red and 2 yellow counters. Even though this managed to generate related number sentences (2 + 6 = 8 and 6 + 2 = 8), this representation accidentally changed the quantities in the whole (e.g., 2 red and 2 yellow are not the same). As such, this representation does not lend itself to meaning making. Note that this example was directly taken by the teacher from the textbook. It seems that the U.S. textbook authors could have better designed the tasks to support student learning.
Any Which Way Rule. The aforementioned any which way rule (see Section 4.2) appeared in several U.S. lessons (and textbooks). For instance, after introducing the CP, a G2 teacher gave an example of adding three numbers (3 + 2 + 1= 6, 2+1+3 = 6, 1+2 + 3 = 6, and 2 + 3 + 1 = 6) to serve as an illustration of this property. In fact, we videotaped several lessons titled “adding three numbers” in G1 and G2 classrooms. As explained earlier, the any which way rule is a combined application of the CP and AP, and likely a source of conflation between the two. This conflation is held by undergraduate students as well (Larsen, 2010; Zaslavsky & Peled, 1996), which is a sign of how a lack of clear understanding of these properties can be pervasive throughout a student’s mathematical career.
Separation of Formal and Informal Lessons. Across U.S. lessons, I noticed a disconnect between formal and informal teaching of CP. Both existed, but neither tended to reference the other. For instance, there were often specific, hour-long lessons devoted to CP which were followed by separate lessons on inverse relations (e.g., fact families); yet, the turn-around facts occurring in the fact family lessons were rarely compared to draw students’ attention to the CP. As seen from Chinese lessons (e.g., the swimming pool example), it did not take up much class time to invite students to compare and verbalize the observed features related to the CP. This separation of connected ideas into definitional and procedural lessons (as opposed to compressing the relevant ideas into one lesson) may explain why many U.S. teachers complained that their textbooks contained more topics than could be reasonably completed during the school year. This may also explain the reports that U.S. textbooks are usually three times as thick as Chinese textbooks (Li et al., 2008; Ding & Li, 2010). In addition, I noticed a lack of vertical coherence among the U.S. lessons on this topic. For instance, the CP was basically taught the same way across grades. In contrast, the Chinese lessons informally introduced CP in G1 and then reinforced this idea through checking in G2, all building up to the formal instruction of the CP in G4 during which deep mathematical thinking skills such as proofs and generalization were stressed.
Summary: Teaching the CP of Addition through TEPS
In this section, I described the insights and observations of Chinese and U.S. lessons on the CP of addition. Together, these lessons suggest how to develop students’ understanding of the CP through TEPS. When students initially learn this property, a teacher may start with a real-world context which can be modeled using semi-concrete representations, leading to two-way solutions. This pair of addition number sentences can be used as a base to learn the CP. To do this, comparison and follow-up questions should be asked. For instance, a teacher should ask how a pair of number sentences are the same and different, why switching the two addends does not matter in the results, and so on. To make sense of these questions, students’ attention should be drawn to the part-whole structure that is embedded in either the semi-concrete representations or the concrete contexts. In later lessons where the CP is applied, teachers should continuously ask deep questions to bring students’ attention to the underlying concept of the CP. For instance, they could ask why a strategy worked (e.g., checking, finding related facts). Students may even be challenged to explore this statement at a more abstract level. For instance, is the statement really true since it only resulted from limited examples? Can one find a counterexample to disprove it? An integration of representation usage and deep questions will likely contribute to students’ deep understanding of the CP and broader algebraic thinking.