Have you ever heard someone say, “It’s exactly the same
question, just with different numbers”? Sometimes a
teacher might say this about a practice test or a set of
exercises in a textbook. The implication is that the details
of the numbers make no substantive difference to the
problem. In mathematics task design, careful thought
is not always given to the particular numbers used in
questions; in some circumstances, these might even be
randomly computer-generated. However, in Japanese
mathematics teaching, the numbers used in questions
(and the details in general) are taken extremely seriously.
In this article, we will explain why the details of tasks
matter so much. We will show across four mathematics
topics with different ages of students how the detailed
choices of examples can be critical for students to learn
what is intended

Colin Foster (6064118)

Tatsuhiko Seino (7844201)

English

Loughborough University Institutional Repository

2 Mathematics in School, November 2019      The MA website www.m-a.org.ukHave you ever heard someone say, “It’s exactly the same question, just with different numbers”? Sometimes a teacher might say this about a practice test or a set of exercises in a textbook. The implication is that the details of the numbers make no substantive difference to the problem. In mathematics task design, careful thought is not always given to the particular numbers used in questions; in some circumstances, these might even be randomly computer-generated. However, in Japanese mathematics teaching, the numbers used in questions (and the details in general) are taken extremely seriously. In this article, we will explain why the details of tasks matter so much. We will show across four mathematics topics with different ages of students how the detailed choices of examples can be critical for students to learn what is intended.Subtraction with RegroupingIn Grade 1 (age 6–7), Japanese students learn subtraction with regrouping, which is typically (2-digit) – (1-digit) across 10, where the minuend is less than 20 and the subtrahend is less than 10. When designing an introductory example, a Japanese task designer, rather than picking the numbers in a haphazard fashion, would consider the 36 possible calculations that satisfy this condition (Fig. 1). Which calculation would you choose for the introduction of subtraction with regrouping?We will compare the numbers used by all six textbooks approved by the Ministry of Education, Culture, Sports, Science and Technology in 2015, extracts from which are shown in Figure 2. Even without being able to read Japanese, you can see that four of the books used 13 – 9 and the other two went with 12 – 9. These were the only examples used. Why might this be?The reason relates to the four main methods for sub-traction with regrouping, which are illustrated below for the example of 13 – 9:1. Counting down from 13: 12, 11, 10, 9, 8, 7, 6, 5, 4, so it’s 4.2. Counting up from 9: 10, 11, 12, 13, so it’s 4.3. Subtraction – subtraction: (13 – 3) – 6 = 10 – 6 = 4.4. Subtraction – addition: (10 – 9) + 3 = 1 + 3 = 4.Methods 3 and 4 are the more sophisticated methods, which the books wish to emphasize. Method 4 is particularly powerful, as it works well even for larger numbers, and may involve less cognitive load than method 3. In method 3, it is necessary to partition the subtrahend (9) according to the minuend (13); in the example above, 9 must be partitioned into 3 and 6, and then the 3 is subtracted from the 13 and the remaining 6 from the 10. This process may involve higher cognitive load for students than method 4, where it is the minuend (13) that is partitioned into 10 and 3.The reason for choosing 13 – 9 and 12 – 9 is that these problems are highly suited to method 4. To subtract 9 from 10, we need to see 13 as ‘10 and 3’, and students have, in their lessons immediately before this, learned to decompose numbers into ‘10 and something’. Method 4 is usually dominant when solving a problem such as 13 – 9, because 9 is so close to 10 (Doig et al., 2011, p. 192). Later on in the books, when the intention is to focus students on method 3 instead, the numerical choices made by the same books are different (Fig. 3). Method 3 is preferred when the number in the units column of Why the details matter:  Learning from Japanese Kyouzai Kenkyuu by Tatsuhiko Seino and Colin Foster 11 – 2 11 – 3  12 – 3 11 – 4  12 – 4  13 – 4 11 – 5  12 – 5  13 – 5  14 – 5 11 – 6  12 – 6  13 – 6  14 – 6  15 – 6 11 – 7  12 – 7  13 – 7  14 – 7  15 – 7  16 – 7 11 – 8  12 – 8  13 – 8  14 – 8  15 – 8  16 – 8  17 – 8 11 – 9  12 – 9  13 – 9  14 – 9  15 – 9  16 – 9  17 – 9  18 – 9Fig. 1Mathematics in School, November 2019      The MA website www.m-a.org.uk 3(1 book)(1 book).In addition, Keirinkan (2015), Kyouiku syuppan (2015) and Nihonbunkyou shuppan (2015) first consider the problem of dividing fractions with unit fraction divisor ( 13, 14, 13), before using the non-unit-fraction divisor ( 23, 34, 23). So, again, we ask, why do four of the books use exactly the same calculation? You might like to think about what might be so good about this particular example before reading on.One reason for using 25÷ 34 is that both 25 and 34 can be expressed as decimals, meaning that the answer to25÷ 34 can be obtained from the calculation 0.4 ÷ 0.75 using students’ prior knowledge of decimal division from Grade 5. Also notice that this calculation leads to a recurring decimal answer of 0.5333..., so the goal of the subsequent thinking can be to find a way to answer the question precisely in fractions. Another important feature of 25÷ 34 is that the four numbers used as numerators and denominators are all different, which may help students to generalize from this one example. By contrast, a calculation such as 35÷ 34 would be less helpful in this the minuend is close to the subtrahend number (Doig et al., 2011, p. 192), so examples such as 12 – 3 and 11 – 2 are natural choices. In both cases, examples are chosen to focus students’ thinking on the mathematical method that the teacher is emphasizing at a particular time.Division of FractionsIn Japan, students learn about division of fractions in Grade 6 (age 11–12). Figure 4 shows the examples used by the six textbooks to introduce division of fractions. All six books use the scenario of painting of a board, walls, fences or desks, and the numerical values used are:(4 books) Tokyo shoseki (2015)There are 13 acorns. We used 9. How many acorns are left?Keirinkan (2015)There are 13 persimmons. If we take 9, how many persimmons are left?Gakkou tosho (2015)There are 12 acorns. We used 9 to make the [spinning] tops. How many acorns are left?Kyouiku shuppan (2015)There were 12 bottles of milk. We distributed 9. How many bottles of milk are left? Dainippon tosho (2015)There were 13 people in the park. 9 people left. How many people are left in the park? Nihonbunkyou shuppan (2015)There were 13 persimmons. We took 9. How many persimmons are left?Fig. 225÷ 3458÷ 2335÷ 23Tokyo shoseki (2015)  12 – 3Keirinkan (2015)  13 – 4Gakkou tosho (2015)  11 – 2Kyouiku shuppan (2015)  12 – 3Dainippon tosho (2015)  12 – 3Nihonbunkyou shuppan (2015)  12 – 3Fig. 34 Mathematics in School, November 2019      The MA website www.m-a.org.uk                               instead of the correctSo, keeping all four numbers distinct is extremely useful.respect, since, if students were to write 35× 43, there would be ambiguity over which of the 3s here had come from the numerator of 35 and which from the numerator of 34. This could lead to a faulty generalization of Tokyo shoseki (2015)With 34	dL L of paint, we could paint 25m2 2 of board. What area of board can we paint with 1dL of this paint? Keirinkan (2015)With 13 dL of paint, we can paint 35	m2 2 of wall. Let’s write an expression for the area of wall that we can paint with 1 dL of this paint.Then…We used 23	dL L of paint to paint 35	m2 2 of wall. What area of wall can we paint with 1dL of this paint? Gakkou tosho (2015)We use 34	dL L of yellow paint to paint 25m2 2 of fence. What area of fence can we paint with 1dL of this paint? Kyouiku shuppan (2015)With 14 dL of paint, we can paint 25m2 2 of board. What area of board can we paint with 1 dL of this paint?Then…With 34	dL L of paint, we can paint 25m2 2 of board. What area of board can we paint with 1dL of this paint? Dainippon tosho (2015)With 34	dL L of paint, we can paint 25m2 2 of board. What area of board can we paint with 1 dL of this paint? Nihonbunkyou shuppan (2015)With 13 dL of paint, we could paint 25m2 2 of desk. What area of desk can we can paint with 1 dL of this paint?Then…  With 23	dL L of paint, we could paint 58	m2 2 of desk. What area of desk can we paint with 1 dL of this paint?Fig. 4ab÷ cd= cb× daab÷ cd= ab× dcMathematics in School, November 2019      The MA website www.m-a.org.uk 5Tokyo shoseki (2015)The perimeter is 16 in the upper left, 14 in the upper right, 16 in the lower left, and 16 in the lower right.Keirinkan (2015)The perimeter is 16 in the upper left, 14 in the upper right, and 16 in the lower drawing.Gakkou tosho (2015)The perimeters are all 20.  Kyouiku shuppan (2015)The perimeters are all 20.Dainippon tosho (2015)The perimeters are all 20.Nihonbunkyou shuppan (2015)The perimeters are all 16.Fig. 56 Mathematics in School, November 2019      The MA website www.m-a.org.ukimportant to compare oblongs with squares. So, we start by creating the square. A side length of 3 is too short, and 6 feels a little too long. An appropriate length is 4 or 5, meaning that the perimeter will be 16 for a 4 × 4 square or 20 for a 5 × 5 square, and these values dictate the perimeters used.Calculating the Area of Compound ShapesIn Grade 4 (age 9–10), students in Japan learn how to calculate the area of rectangles and squares. They also learn how to calculate the area of rectilinear compound figures consisting of juxtaposed rectangles. Figure 6 shows how each textbook presents compound figures.Area and PerimeterIn Grade 4 (age 9–10), when students learn about area, all of the six textbooks introduce this by presenting figures of equal perimeter and asking for which figure the amount of space is largest, and the numbers used are quite similar across the textbooks (Fig. 5). Students may think that the longer the perimeter, the larger the area, and may also think that when the perimeter is the same, the area is the same (Kaji, 1983). These examples highlight this misconception and challenge it. But why the similarity in the numbers chosen? In the case of figures of equal perimeter, we obtain the maximum area when the rectangle is a square. Therefore, it is  Tokyo shoseki (2015) 4 × 3 + 2 × 3 = 18Answer 18cm2 Keirinkan (2015) 2 × 4 + 5 × 3 = 2323cm2Gakkou tosho (2015) Kyouiku shuppan (2015)4 × 6 – 2 × 3 Dainippon tosho (2015) 5 × 6 + 7 × 10 = 100Answer 100cm2Nihonbunkyou shuppan (2015) 3 × 4 + 3 × 6 = 30Answer 30cm2Fig. 6Mathematics in School, November 2019      The MA website www.m-a.org.uk 7shoseki and Gakkou tosho the same – and different from those used in the other textbooks? Again, the answers lie in the different possible methods; in this case, for calculating the area (Fig. 7).When we calculate the area, it is important to understand which edges we need to measure. Keirinkan and Kyouiku shuppan emphasize this point. But, for children, it is The dimensions are presented in various ways: the figure is placed on the grid and the length is read off (Tokyo shoseki and Gakkou tosho); the length is stated alongside the edges of the figure (Dainippon tosho and Nihonbunkyou shuppan); and the figure has neither grid nor stated measurements and the length is measured by the student (Keirinkan and Kyouiku shuppan). Why these differences? And why are the numbers used in Tokyo 1. Vertical split  4 × 3 + 2 × 3 = 18Answer 18cm22. Horizontal split  2 × 3 + 2 × 6 = 18Answer 18cm23. Large rectangle – small rectangle 4 × 6 – 2 × 3 = 18Answer 18cm24. Rearrangement 1 (2 + 4) × 3 = 18Answer 18cm25. Rearrangement 2  2 × (3 + 6) = 18Answer 18cm26. Rearrangement 3 3 × 6 = 18Answer 18cm27. Duplication                                                                                                           4 × (3 + 6) ÷ 2 = 18              Answer 18cm2Fig. 78 Mathematics in School, November 2019      The MA website www.m-a.org.ukdifficult to measure the length of the edges themselves, and the textbooks come to different conclusions about the balance between the difficulty and importance of measurement.The figures and numerical values in the Tokyo shoseki and Kyouiku shuppan books allow all seven methods to be used, whereas the Keirinkan and Dainippon tosho books use examples where it is difficult to deform the compound figures into a single rectangle. The method of duplication works only for the Tokyo shoseki, Kyouiku shuppan and Nihonbunkyou shuppan books. So, the choices of these examples reflect the methods that the textbooks wish to prioritize. Indeed, the Tokyo Shoseki book (2011) goes so far as to provide a blank space adjacent to the figure to allow a duplicate shape to be drawn in (Fig. 8).Notice also throughout that area is always expressed as ‘height × width’. This consistent approach to expressing calculations facilitates here the teacher–student and student–student discussions about, and understanding of, the different methods used by different students.ConclusionThe examples used in these four topics show the care that is taken in Japan over the details of the mathematical examples and questions used in textbooks. They demonstrate how the tasks are designed to ensure that student thinking carefully builds from prior knowledge and takes into account how current thinking needs to inform future learning. Considering in detail the possible methods that students might use, and mapping these onto possible examples, allows the task designer to select numbers that will promote or elucidate the use of a particular method. This then enables tasks to be constructed that really help students to focus on the deep features of the mathematics to be learned.AcknowledgementsWe would like to thank Tokyo Shoseki, Gakkou tosho, Dainippon tosho, Keirinkan, Kyouiku shuppan and Nihonbunkyou shuppan for permission to include excerpts from the textbooks approved by the Ministry of Education, Culture, Sports, Science and Technology.We would also like to thank Professor Geoff Wake for helpful comments on an earlier draft of this article.• This work was supported by the Economic and Social Research Council [grant number ES/S014292/1].• This article is based on a talk given at the conference Coherent currriculum design and lesson study in Japanese maths: What can we learn? in London on Friday 12 July 2019, which was organized in conjunction with Collaborative Lesson Research.ReferencesDoig, B., Groves, S. and Fujii, T. 2011 ‘The Critical Role of Task Development in Lesson Study’. In Hart, L. C., Alston, A. S. and Murata, A. (Eds), Lesson Study Research and Practice in Mathematics Education (pp. 181–199), Springer, Dordrecht.Fujii, T., Ishihara, N., Ichikawa, S., Enomoto, A., Ohta, S., Ohtani, M., … Watanabe, K. 2015 Atarashii Sansuu (Mathematics: Grade 1–6), Tokyo Shoseki, Tokyo. Fujii, T., Iitaka, S., Ichikawa, S., Ichikawa, R., Enomoto, A., Ohta,  …Watanabe, K. 2011 Atarashii Sansuu (Mathematics: Grade 1–6), Tokyo Shoseki, Tokyo.Hitotsumatsu, S., Okada, Y., Akai, T., Ikeda, T., Kurosawa, S., Tanaka, H., … Watanabe, K. 2015 Shougakkou Sansuu (Mathematics: Grade 1–6), Gakkou tosho, Tokyo. Kaji, T. 1983 ‘On the Children’s Cognition of Area and Perimeter’, Journal of Japan Society of Mathematical Education, 39, 40, pp. 49–66.Koyama, M., Iida, S., Itoh, T., Iwasaki, H., Iwata, K., Ueda, Y., … Yamada, A. 2015 Shougaku Sansuu (Mathematics: Grade 1–6), Nihonbunkyou shuppan, Osaka. Seki, S., Hashimoto, Y., Abiru, S., Ikeno, M., Itoh, T., Emori, H., … Yamada, M. 2015 Tanoshii Sansuu (Mathematics: Grade 1–6), Dainippon tosho, Tokyo. Shimizu, S., Funakoshi, S., Negami, S., Teragauchi, S., Aoyama, K., Iijima, Y., … Watanabe, M. 2015 Wakuwaku Sansuu (Mathematics: Grade 1–6), Keirinkan, Osaka.  Tsubota, K., Kanemoto, Y., Ohkubo, K., Ohsawa, T., Ohno, K., Kimura, H., … Yoshikawa, S. 2015 Shougaku Sansuu (Mathematics: Grade 1–6), Kyouiku shuppan, Tokyo. Fig. 8Keywords:  Japanese textbooks;   Kyouzai Kenkyuu;   Task design.Author  Tatsuhiko Seino, Associate Professor of Education, Faculty of Education, Tokyo Gakugei University, 4 Chome-1-1 Nukui Kitamachi, Koganei, Tokyo 184-8501, Japan.e-mail: tseino@u-gakugei.ac.jpAuthor  Colin Foster, Reader in Mathematics Education, Mathematics Education Centre, Schofield Building, Loughborough University, Loughborough LE11 3TU.e-mail: c.foster@lboro.ac.ukwebsite: www.foster77.co.uk

Why the details matter: Learning from Japanese Kyouzai kenkyuu

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