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Developing mathematics teacher knowledge: the paradidactic infrastructure of “open lesson” in Japan

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Abstract

In this paper, we first present a theoretical approach to study mathematics teacher knowledge and the conditions for developing it, which is firmly rooted in a systemic approach to didactic phenomena at large, namely the anthropological theory of the didactic. Then, a case of open lesson is presented and analysed, using this theoretical approach, to show how the format of open lesson contributes to the construction and diffusion of didactic knowledge in the community of mathematics teachers in Japan. The basic idea of this format is that teachers from other schools are invited to observe a class taught by a teacher then participate in a discussion session with him on the details of the lesson. For the case study, we analyse the lesson plan prepared for an open lesson, the observed lesson and the teachers’ discussion. As a result, an open lesson session has been described as a specific form of post-didactic practice related directly to an actual observed lesson, and aiming specifically at elaborating the theoretical aspects of teacher’s didactic practice in the lesson and beyond it.

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Notes

  1. The classroom is here to be construed as the site of the teacher’s central activity, that of teaching in the sense of supporting students’ learning through direct interaction with them.

  2. Similar formats exist in some East-Asian countries (see e.g. Shen et al. 2007, Quan 1992 for China), but virtually non-existent in many other countries.

  3. One of definitions given in an encyclopaedia on Japanese education is “a study whose object is a lesson carried out in school. It includes a clinical study and a study forming its foundation whose objective is to improve the lesson and to develop teacher’s practical abilities” (Hosoya et al. 1990, p. 62).

  4. One can get an impression of the activities of this organisation at the website http://www.lessonstudygroup.net/.

  5. “‘Mathematical values’ are the values supporting problem solving in mathematics. Mainly the following three values are considered: ‘simplify something which is in a complex state’, ‘clarify something which is in an ambiguous state’, and ‘generalise something specific’” (Isono 2010, p.1). Notice also that Nunokawa (2005, p.333), who was collaborating with this teacher, relates the Japanese notion of mathematical values to the Anglophone notion of socio-mathematical norm.

  6. The Japanese term shiki is translated here into “formula”. But in fact shiki is an expression containing only number symbols, operation symbols, eventually empty boxes or letters: e.g. “34 – 16 =”, “16 + □ = 34”. Quite often, shiki, not only the solution, is required to be written by students, as they solve word problems in elementary school.

  7. Here, the “scene” refers to the context of a word problem.

  8. The “unit” refers to the textbook used in this class.

  9. At this age, it is uncommon to attend a private mathematics class (juku) outside of school time, but it occurs.

  10. See footnote 7.

  11. In the professional literature for Japanese teachers, one can find some descriptions of the practice and theory of lesson study (e.g. Inagaki and Sato 1996). As an “open lesson” can be seen as similar to parts of lesson study, one may say that it is not completely without an explicit theoretical block. But the practices of lesson study or open lesson developed in Japan without such a literature and could continue to exist without it.

  12. See footnote 5 for the exact meaning of this term.

  13. “M” means module which is a time unit of 30 minutes. In this school, two modules are used for each mathematics lesson.

  14. Empty box is used to show an unknown number in Japanese elementary school.

  15. English translation could be “Sum and difference problem”. When the sum and difference of the cardinalities of two sets are known, find the cardinality of each set. For example, “in a class, there are 41 students, and there are 9 more boys than girls. Find the number of girls and the number of boys. In junior high school, this is usually solved by the system of equations while in primary school, more informal techniques are used.

  16. The problem with ordinal numbers for which just adding the given numbers leads to an incorrect answer: for example “some children are standing in a line. Taro is the 5th from the front and the 7th from the back. How many children are there as a whole?”

  17. This term is explained in the “Findings” section.

  18. Term “band” means the thick segment which is often used in place of a segment diagram.

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Acknowledgments

We would like to thank Mr. Masato Isono and Attached Elementary School of Joetsu University of Education who generously provided us with materials from an open lesson. We also thank the editor and the anonymous reviewers for fruitful comments on an earlier version of this article. This project was partially supported by JSPS KAKENHI (21830043) to the first author, and a JSPS senior invitation fellowship to the second author.

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Correspondence to Takeshi Miyakawa.

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Takeshi Miyakawa and Carl Winsløw equally contributed to this paper.

Appendices

Appendix (lesson plan translated into English)

2nd grade Mathematics Activity Plan

Sukkiri [Clarity] by Drawing!: Addition and Subtraction

Teacher: Masato Isono

“Preparing pupils to live in human society” in this activity

Through interaction with others, pupils build a sense of mathematical value,Footnote 12 and based on that they act in society by constructing and modifying their thinking.

About the activity

Goal

By drawing the problem situation which calls for addition or subtraction, understand the situation and the relation between numbers and use it for solving the problem.

Reasons for this activity

The activity is to solve word problems by drawing the relation between numbers included in the problem statement, and to improve the way of drawing.

There are three reasons for this activity.

The first reason is that drawing the situation given in the word problem helps pupils to understand the problem situation. Pupils who are weak in word problems cannot often understand the problem situation after reading the problem statement once. Drawing at the same time as reading the statement allows them to organise numerical values, to reconstruct the problem situation by means of their own drawings or words, and therefore, to understand it.

The second reason is that drawing the situation leads pupils to find out the relation between the numbers involved, and their structure. For example, in the situation of “There are two apples on each plate. There are three plates. How many apples are there as a whole?”, pupils will be able to conceive the structure of multiplication—that there are three pairs of two apples—after drawing the plates.

The drawing of numbers which appear in the problem also leads pupils to realise the advantage of taking 10 objects as a group. For example, when drawing 14 candies, drawing a group of 10 candies and then 4 candies will make it easier to count them. This idea of taking 10 objects as a group will be used later in the written calculation of addition and subtraction.

The third reason is that whole class reflection on a drawing leads pupils to formulate their sense of mathematical value and to see the importance of it in the classroom. The reflections could involve, for example, that if two objects to be added have different size, it can be advantageous to show the difference in the drawing; if the objects are scattered in the drawing, it may clarify the situation to pull them together; if there are many objects, it helps to represent them by grouping some of them.

The pupils’ sense of mathematical value will be built up in a process of producing a clear representation of a mathematical phenomenon in the problem solving situation, and in a process of realising the advantage of representing something in a general way. Through such experiences, one can create an ambience in the classroom which allows pupils to reflect using their sense of mathematical value.

Activity plan and summary of previous activities

<First trimester> 34 MFootnote 13 as a whole.

Phase 1 (10 M)

Realise that a drawing showing the situation of a word problem makes the problem easier to understand

Phase 2 (10 M)

Realise the difference between a drawing showing the situation of a word problem exactly, and a drawing showing the relation between numbers

Phase 3 (6 M)

In the word problems of addition and subtraction, be able to make a drawing showing the relation between numbers

Phase 4 (8 M) Today’s lesson: 27th and 28th M

In the word problem using a box (□),Footnote 14 be able to solve the problem by means of a drawing showing the relation between numbers

In Phase 1, pupils learned that looking at a drawing showing the situation of a word problem makes it easier for them to see the situation and to get a solution than if they just read the problem formulation,.

In Phase 2, the activity involved mainly word problems for which a formula is not easily established, such as wasazan Footnote 15 and problems related to order.Footnote 16 To obtain the right answers in these problems, it is not enough to just add or subtract one by one the numbers appearing in the problem statement. Through the experience of solving this kind of word problems, pupils knew that there are some cases where it is not helpful to establish a formula as the first step of solving a problem. And, in the second half of Phase 2, more abstraction in the drawings was observed as pupils started drawing something similar to a segment diagram.Footnote 17

In Phase 3, pupils started realising that the drawing which is most helpful for solving a word problems is not always one on which the situation is precisely drawn, but often it is one showing the relation between numbers. In addition, they realised that a drawing with a box (□) or a question mark (?) tells them what to do and is, therefore, helpful for problem solving.

About this lesson

Goal

Be able to find the structure of addition and subtraction by drawing the relation between numbers given in word problem, and use it as a rationale for establishing a formula.

Principle ideas for the development of lesson

In this lesson, pupils try to solve a word problem in which the expression “as a whole” is used, while the operation to be done is subtraction. Therefore, the lesson will be developed by focusing on the fact that there are some cases where “as a whole” does not imply the use of addition.

In the first half of the lesson, individual work comes first, and then group work. First of all, I will let pupils make a drawing that gives clarity individually and allows them to establish a formula. Then, I set up an occasion for each pupil to modify their thinking by means of a group discussion with their own individual drawings. After this, in order to communicate the idea of the group to others in the classroom, pupils prepare a common drawing which is more appropriate for the communication, and then present it to the class.

In the second half of the lesson, pupils make a drawing and solve a new word problem by borrowing the ideas presented by other groups or by clarifying the idea of their own group.

Development of lesson: 27th and 28th of 34 M (65 min)

Time

No.: pupils’ activity

•: expected behaviour of pupil

Teacher’s support

5

1. Grasp the task

Task There were 16 people in a bus. Later, some people got on this bus. Now as a whole, the number of people is 34. How many people got on later on?

• Grasp the task by imagining the problem situation

• Try to make a drawing in order to understand the problem situation

When introducing this problem, do not show the underlined sentence and make pupils guess what kind of sentence will be there

15

2. Make a drawing individually and accomplish the task

• Based on the expression “as a whole”, find an answer by addition

• Establish a formula using subtraction

• Make a drawing of the whole and the part

For pupils who get stuck with drawing the situation, support them to make a concrete drawing

25

3. Make a drawing in a group, accomplish the task, and present it

• By drawing a bus and the people, make the problem situation easy to understand

• Abstract the number of people and represent it by “O” or a bandFootnote 18

• Identify the situation of subtraction on the basis of the relation between part and whole

• By using a question mark or an empty box, identify that the situation requires to find a part, so it is about subtraction, and solve the problem

• Realise that there is a case where subtraction is used even when the expression “as a whole” (implying addition) appears.

Distribute the folder including drawings made in previous activities

Make a common drawing of the group on a magnetic sheet

At the time of presentation, make pupils talk about what becomes clear by making a drawing

At the time of presentation, group the magnetic sheets which are based on the same idea

Write on the board the idea of the group (who made the drawing) that makes clarity

20

4. Accomplish a task by modifying the drawing of their own group or by borrowing the ideas of other groups

Task There were some people in a bus. Because 18 people got on this bus later, the number of people is 37 as a whole. How many people were there at the beginning?

• Get stuck with a problem, because of being unable to draw the number of people who were in the bus at the beginning

• Make a drawing with a question mark or an empty box for an unknown number

• Establish a formula on the basis of a drawing, and then find a solution

 
  1. Discussion topic: through the pupils’ actual behaviour, reflect on “preparing pupils to live in human society” as a goal for a mathematics lesson

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Miyakawa, T., Winsløw, C. Developing mathematics teacher knowledge: the paradidactic infrastructure of “open lesson” in Japan. J Math Teacher Educ 16, 185–209 (2013). https://doi.org/10.1007/s10857-013-9236-5

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