For each of the three research lessons, we now provide a table showing the key points made in the final comments, in the order that Nakamura made them. For each lesson, we focus on one or two of these comments in more detail in order to illuminate the kinds of remarks he made.
Research Lesson 1
This research lesson focused on decimals and aimed at enriching students’ number sense by considering the number 2.8 in two ways: as a point on a number line and as a result of addition and subtraction calculations (e.g., 3 − 0.2, 2 + 0.8, etc.). In the introduction to the lesson, the teacher also asked students to think about the number 280 in different ways and then asked the students to consider the relationship between 2.8 and 280.
In the Post-lesson Discussion of this research lesson, seven final comments were made by Nakamura (see Table 2), and we focus below on one of these that we feel is particularly useful in illustrating Nakamura’s approach to making final comments.
Thinking about the order in which students’ representations are shared (Comment 1.2 in Table 2)
In response to the prompt “Let’s express 2.8 using words, expressions and number lines,” the students offered “2 + 0.8,” “3 − 0.2,” and “0.1 × 28” (see Fig. 1).
In the Post-lesson Discussion, Nakamura focused on the order in which the teacher shared the students’ representations when comparing and discussing. First, the teacher called on Yumi (all student names here are pseudonyms), who thought of the number line when considering 2.8. The teacher then called on Yosuke, who thought of the expression “2 + 0.8.” Finally, the teacher asked that the number line and expressions be expressed in words. Next, the teacher examined the idea of “3 − 0.2.” First, the teacher called on Kaori, who explained the idea with words. Specifically, Kaori said that “2.8 has dropped by 2 from 3” and “the number has dropped 0.2.” The teacher then asked, “What expression could represent this?” and the student answered “3 − 0.2.” The teacher then let the student draw a number line.
Nakamura said that it is important to clarify what the student was thinking when she said, “the number has dropped 0.2,” and the number line was a useful tool to clarify this idea. Nakamura pointed out that because 0.2 in the words “the number has dropped 0.2” signifies two gradations on a number line, in order to make the student’s idea clearer, the teacher could have taken up the idea of going down two gradations, rather than bringing in the idea of using an expression. In other words, if the teacher had deepened the meaning using Kaori’s idea, Nakamura thought that the order of words, number lines, and expressions was good.
Nakamura pointed out that expressions, number lines, and words were not examined in this order, since the teacher did not begin with expressions in this lesson. For example, he referred to the following hypothetical lesson outline: The teacher takes up the expression 1 + 1 + 0.8 that a student provides and then has students represent the expression in words. Learning to start with expressions leads to an important activity of interpreting expressions in mathematics.
Research Lesson 2
This research lesson focused on a unit of subtraction involving regrouping and aimed at thinking about methods of calculation of 13 − 9. In this lesson, three main methods were presented and examined. Students explained these methods using words, figures, and expressions.
In the Post-lesson Discussion, nine final comments were provided (see Table 3), and we focus on one of these.
The value of the subtraction–addition method (Comment 2.6 in Table 3)
Two methods were examined to obtain the answer to 13 − 9. The first was a method of partitioning 9 into 3 and 6, subtracting 3 from 13, then subtracting 6 from 10 (subtraction–subtraction method). The other method was to partition 13 into 10 and 3, then to subtract 9 from 10, and add 3 to the 1 obtained (subtraction–addition method) (See Fig. 2 and Seino and Foster 2019).
Nakamura stated that the point of today’s lesson was how students make sense of the value of the subtraction–addition method. He also pointed out that the two terms used by students in the lesson, “taking all at once” and “10,” were key terms, while composing and decomposing of 10 (i.e., partitioning) was a fundamental idea. Furthermore, in the case of 13 − 9, when it was said to take 9 from 10, students knew where 9 was, but when it was said to take 6 from 10, the students did not know where the 6 was coming from. The number 6 was newly constructed. In the case of the subtraction–addition method, when calculating 14 − 9, 1 was added to 4. When calculating 15 − 9, 1 was added to 5. In other words, the answer was obtained by adding 1 to the number in the units column of the minuend. Nakamura pointed out that when this becomes apparent to students, it allows them to sense the value of the subtraction–addition method over the subtraction–subtraction method.
In the case of the subtraction–subtraction method, when students explained how to calculate 13 − 9, they often did not explain it as obtaining an answer by subtracting and then subtracting further, but as follows: “I subtract 3 from 9, so I can get 6. Next, I subtract 6 from 10, so I can get 4.” In fact, however, they are considering 9 as 3 and 6. If it is necessary to regroup in 2-digit and 3-digit calculations, the subtraction–subtraction method is difficult for students, because they cannot obtain the answer only by subtracting the number that is most apparent. So, firstly, students solved and mastered problems where the subtraction–addition method is easy to think about: for example, when the subtrahend is 8 or 9. Secondly, students solved some problems where the minuend was close to the subtrahend: for example, 12 − 3 or 17 − 8. This is an easy way to think about the subtraction–subtraction method. However, Nakamura pointed out that by the end of Grade 1, addition and subtraction need to be proficient without using either the subtraction–addition method or the subtraction–subtraction method.
Research Lesson 3
This research lesson covered a unit on fractions with the aim of understanding what ‘1/3 m’ means. In Japan, fractions of a measured quantity, like this, are known as “quantitative fractions.” In this lesson, the following contextual problem of finding treasure was used:
As sign on an island says: There is treasure buried 2 1/3 m from the tree. Use 1 m of tape and look for that treasure.
By solving this problem, it is intended that students make 1/3 m from a 1 m tape.
In the Post-lesson Discussion, nine final comments in total were made by Nakamura, as summarized in Table 4. We highlight two of these comments and describe how they related to incidents from the lesson.
Connecting ‘equally divided’ to prior experiences with decimals (Comment 3.2 in Table 4)
During the introduction to the lesson, the teacher reviewed the unit “dividing fractions” studied in Grade 2. At that time, the teacher had shown the students a tape with a line drawn at the center and had said that the tape was divided into two pieces of the same length by the center line. In the research lesson, a student used the terms “equally divided,” and “bisection.” In response to the student’s remarks, the teacher asked, “What quantity is each of the two equal parts of an original length?” and the student answered “1/2.” The teacher then showed the students a tape with a line drawn at the position of the quarter and asked the same question about 1/4 (Fig. 3).
In the Post-lesson Discussion, Nakamura pointed out that students had used the words “equally divided” and “bisection” in the introduction to the lesson. Then, he asked the participating teachers: When did the students learn the phrase “equally divided?” This question provided teachers with an opportunity to think deeply about the previous knowledge that students had acquired. Nakamura also pointed out decimal numbers and division as learning units in which the phrase “equally divided” is used. When students learn partitive division, they use the term “equally divided.” In the case of decimal numbers, when students learn that “When 1L is divided into 10 equal parts, each part is written as 0.1L,” students use the phrase “equally divided.” Therefore, Nakamura advised as follows: When students say the phrase “equally divided,” the teacher should ask “Where did you learn the phrase ‘equally divided’?” and elicit previous knowledge from students. He explained that such interaction leads students to become aware that fractions are related to division and decimal numbers. Nakamura also took up the remark of one student that “1/3 means dividing, so it is similar to division,” and explained that this student had begun to connect fractions and division. This idea will later be connected with the fraction 2 ÷ 3 = 2/3 that is learned in Grade 5.
Interpreting students’ ideas carefully (Comment 3.5 in Table 4)
After students worked on the ‘treasure’ problem (see above), four pairs of students shared their ideas. Pair A’s idea was written on a small blackboard: “I know 2 m. Because 1/3 m is the length of three 1 m tapes, I arrange three 1 m tapes next to 2 m” (Fig. 4). This was a misunderstanding. Pairs B and C then shared their ideas: “I fold the 1 m tape into three equal pieces. The crease becomes 1/3 m.” Finally, pair D shared their idea: “I need 3 more 1 m lengths of tape” (Fig. 5). This was also a misunderstanding.
The teacher then said that pairs A and D had had similar ideas of a 1 m tape as adding three, while pairs B and C had had similar ideas of equally dividing a 1 m tape into three parts. The teacher then divided the ideas into two pairs (A and D, B and C) on the blackboard.
In the Post-lesson Discussion, Nakamura focused on the episode where the ideas of pairs A, B, C, and D had been divided into two groups, and he discussed why he thought that the students had viewed a 1 m tape as adding three. Like the idea of pair D, from the students’ perspective, 2 m and 1/3 m can be divided into 3 pieces by 1 m (see Fig. 5), comprising two 1 m tapes and one-third. When examining them side by side, each seems to be divided into three. Considering it in this way, 1/3 is the same as 1 m. Therefore, when expressing 2 m and 1/3, pair D assumed that three 1 m tapes were needed. After providing the above interpretation, Nakamura noted that pairs A and D demonstrated difficulties in terms of what is considered to be the unit, ‘1’, and what is considered to be the base quantity, when equally divided.
While understanding students’ ideas is difficult, it is even more difficult to understand why students come up with such ideas. Nakamura pointed out the importance of interpreting students’ ideas politely and properly.