## Abstract

This study aims to reveal the teaching script and structure of lesson practice of two seventh-grade Japanese mathematics teachers—a “novice” and “expert”—through comparative analysis of mathematics lessons. Specifically, it aims to clarify how the teachers’ views of teaching as tacit knowledge determine lesson structure and share the same culture in different forms in practice. This comparative analysis shows how the lessons can be described as sharing the same teaching culture in different forms from the following two perspectives: (1) methods of mathematical communication between teacher and students, and (2) approaches to dealing with mathematical concepts.

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## Acknowledgements

This research was supported in part by the Japan Society for the Promotion of Science (JSPS) under a Grant-in-Aid for Scientific Research C (reference no. 15H03477). The author would like to express his gratitude to the JSPS for the assistance that made this research possible. He is also grateful to the principals, teachers, and students of the high schools in Japan for their valuable contributions to this study.

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## Appendix: The process of the mathematics lessons based on the analysis framework

### Appendix: The process of the mathematics lessons based on the analysis framework

Category | Middle school U (public) | Middle school S (private) |
---|---|---|

Novice teacher | Expert teacher | |

Introduction | Segment 1 (T1–S65): Using a picture in the textbook (a world map giving the maximum and minimum temperatures in each city) to set the learning topic | Segment 1 (T1–T99): Preparing to present positive and negative numbers found in newspapers, books, encyclopedias, media, etc. at home |

The teacher had the students open their textbooks and explained that, along with city names and photographs, the world map in the textbook had written on it the maximum and minimum temperatures in April in each city. At T19, the teacher asked, “Let’s have a think. Is there anything we notice from the photos and the temperatures we can see here?” The students answered as follows: | The teacher had set students the task of finding numbers with pluses and minuses on them in the newspaper, and checked this homework as follows: | |

S32: “The upper part of the map seems cold, and the lower seems warm.” [omitted] S37: “Some places you need summer clothes and other ones winter clothes.” As these answers were not what the teacher was looking for, he continued to ask “What else?” while pointing to students to have them give their answers. The closest answer to that sought by the teacher was: | T24 “Yesterday I asked you to bring something if you could. [Y], what have you brought?” [omitted] S26 “Umm, are there minuses in the newspaper?” | |

S43: “There’s a minus next to the minimum temperature in Moscow” | T27 “Yes there are. Minuses, and positive numbers, or pluses. If you had something like that, I said could you please find it in the newspaper and make sure you ask someone in your household before you brought it to class” | |

Upon receiving this answer, the teacher set the learning topic he had planned for the lesson: “Thinking about numbers with minuses.” He had the students write this learning objective in their notebooks | The first two students who were called upon used the projector as follows to present to the class traffic accident death toll numbers in a newspaper article | |

S112: “Pluses and minuses are written here or here. The pluses and minuses written here, the pluses are where this year’s numbers are written” [omitted] | ||

S114: “At the bottom are the number of deaths, and there are two more than last year, and here two less, and here one less” [omitted] | ||

S124 “Are there any questions?” There were no questions from other students |

Development | Segment 2 (T66–T137): Questions and answers about how to say temperatures with minuses, using an image of a thermometer that the teacher prepared (e.g., as temperatures go below zero the number has a minus put on it) | Segment 2 (T100–T161): Having students present, with priority on those who found the positive and negative numbers in the newspaper at home (e.g., the number of traffic accidents in different areas), and questions and answers about this |

The teacher pointed out the picture of the thermometer showing the temperatures in Tokyo and Moscow and had students read the temperatures | The format of the presentations was such that students first presented their work, other students were encouraged to ask questions, then the teacher summarized the content | |

T72: “What temperature is it in Tokyo?” [omitted] T85: “OK, how about Moscow?” | For instance, T128: “About the death toll, it’s good that the number of traffic accidents has decreased in some areas, but there were also some areas where they increased, some unfortunate areas. There were pluses and minuses” | |

The lesson progressed in this fashion, with the students answering each of the teacher’s questions. Part way through, the teacher asked, T94: “OK, I have a question for you. Why is it minus in Moscow? Why is there a minus on it?” | The teacher reminded students that increases and decreases in numbers could be expressed using pluses and minuses, probably with the aim of conceptualizing the topic | |

Upon receiving the answer, S111: “As temperatures go below zero the number gets a minus on it,” the teacher picked up on this, saying, | The next students who were called upon gave their presentation about average stock prices. The teacher then commented as follows: | |

T113: “Tokyo is 6 degrees, which means it’s 6 degrees above zero, right?” | T160: “Yes, rising and falling. Remember that this is another place we can use pluses and minuses” | |

From time to time the teacher drew upon the comments of the students to develop the lesson. (e.g., T114: “In other words, minus 6 means the temperature is above or below zero?”, T115: “Below, isn’t it?”, T116: “Someone gave me the answer. I think it was [0]. Good comment. Nice”) | This presentation also failed to draw any questions from the audience | |

Segment 3 (T138–T270): Working on textbook problems (how to say temperatures with minuses and numbers with minuses) for about 5 min (the teacher walked around looking at each student’s notebook, marking the correct answer and speaking to students), and finally checking the answers together | Segment 3 (T162–T235): Having students present what they found at home based on the textbook content (e.g., the maximum and minimum temperatures in every place in Japan), and questions and answers about this |

After about 5 min, the teacher said, T156: “Alright then, let’s all check the answers together” | Here two presentations on temperature are considered. Although both focused on temperature, the content of the presentations differed. S170: “This shows the maximum and minimum temperatures in every place in Japan. For example, the maximum temperature in Nagoya is 13.9 degrees. And the minimum is 5.2.” [omitted] T173: “Ah, that’s where some minuses were. I’m not seeing any pluses here—do we not need them?” | |

In response to S177: “Sapporo is minus 4.3 degrees,” the teacher repeated, “Yes, Sapporo, good, 4.3, that’s cold. Thank you.” This thanks served not only to praise students but also to encourage them to actively present their answers in class. After this, students’ facial expressions became more relaxed, and more and more of them raised their hands to give their answers | There was also a comment about the numbers next to the maximum and minimum temperatures, S196: “The average,” and the teacher responded as follows: | |

T216: “Temperatures below zero have a minus on them. OK, so what about numbers that are, that, that are smaller than zero? What do you think?” In this way, the topic was shifted from temperatures to numbers | T210: “Whether the temperature is falling, or rising. This is also something we express with pluses and minuses, you see” | |

Given the answer S217: “Temperatures below zero,” the teacher assumed that the students had not understood, and repeated the question like so: | This added a new example for teaching knowledge about expressing positive and negative numbers | |

T219: “Small numbers, temperatures below zero have a minus on them. Temperatures below zero degrees” [omitted] | The next student presentation was slightly different from the ones that had preceded it. T217: “OK, so, now let’s make this the last one for our homework.” [omitted] T221: “Oh, this one is just like the textbook, isn’t it?” S222: “This maximum temperature shows a temperature higher than zero, and for the minimums, up the top, it’s showing temperatures lower than zero, and with a minus, the bigger the number that comes after it is, the lower the temperature, we discovered. Are there any questions?” [omitted] T225: “OK, thank you. Applause please” |

T221: “OK, numbers smaller than zero. What about them?” [omitted] | There were no questions. The teacher continued as follows: | |

T223: “What about numbers smaller than zero” | T226: “They’ve just told us something very important. There was a minus 2—they’re saying the bigger that 2 gets, the colder it is” | |

Not long after this, one student said, S226: “They have a minus,” but no opportunity was given for the rest of the students to check this answer and confirm that they understood it. The teacher continued the lesson as though all students in the class had understood | Next, the teacher engaged in the following exchange with students | |

T229: “If it’s smaller than zero, lower than zero, we show this by putting a minus. Let’s practice this.” The teacher then had students work on practice problems. After these practice problems, the teacher solidified the knowledge of negative numbers as follows: | T228: “If it got colder than minus 2, if it got colder than minus 2, what temperature could it be, [H]?” | |

T259: “Write it in red pen” | S229: “Minus 6” | |

T260: “Numbers smaller than zero are called negative numbers (writes on board)” [omitted] T262: “You’ve got it. Remember this.” [omitted] T265: “What do we call the kinds of numbers you learned about in primary school? The opposite of negative numbers: positive numbers. Positive numbers. This is another important term.” Saying this, the teacher wrote on the board, “Positive numbers: Numbers greater than zero.” The students copied this into their notebooks | T230: “Ah, minus 6? Or?” [omitted] | |

S232: “Minus 4” | ||

T233: “In other words, if we use numbers higher than minus 2, higher than minus 2, like 3 or 4, then it means it is getting colder” | ||

T234: “Do you all understand? It was in your newspapers, your newspapers you brought from home, wasn’t it?” [omitted] T236: “OK, so, while we’re here, is there anyone who had something different to what the groups have presented?” |

Turn | Segment 4 (T271–T327): Playing a game where students quickly guess whether a number is positive or negative by raising their right or left hands. Then, asking about zero (some students put up both hands) and replying (the number zero is neither a positive nor a negative number) | Segment 4 (T236–T431): Having students present the positive and negative numbers they found in the newspaper during the lesson (e.g., minuses and pluses on the calendar or minuses, zero, and pluses in golf), and questions and answers about this |

The teacher checked that all of the students had written this in their books and continued the lesson by saying, T274: “Time to play a game. Whoever thinks the numbers I say are negative numbers raise your left hand. Positive numbers, raise your right hand” | In this section, students search for numbers expressed as pluses and minuses in the newspaper and report on these | |

The numbers the teacher gave were 3, minus 4, 16th, and 9:00, and the students answered quickly, keeping up with the pace at which the teacher presented the problems. Here there was also a scene in which students guessed whether zero was a positive or negative number | Calendars | |

T302: “Right, how about zero?” [omitted] T304: “Positive number or negative number, or don’t know? Zero – which is it? Zero, which do you think?” | S247: “Minuses and pluses on the calendar” [omitted] | |

The teacher watched the students’ reactions. T306: “OK, right now we’ve got more people saying positive number, don’t we?” [omitted] T309: “Some of you put up both hands. Alright, which could it be?” | T260: “Ah, I see. I suppose you mean how we express the waning of the moon using minuses” | |

After 2 min, the teacher gave the answer. T268: “Actually, the number zero is neither a positive nor a negative number.” T276: “So those who put up both hands were correct.” After revealing this answer to the students, the teacher wrote on the board and explained that, T322: “Zero is neither a positive nor a negative number” | Golf | |

T289: “There are minuses, zeros, and pluses in golf, aren’t there?” [omitted] | ||

T294: “[R], what is this zero? This one.” [Teacher H calls upon a student who likes golf] [omitted] | ||

S301: “It’s a set number that shows the correct number of strokes” |

Cherry blossom news | ||

T392: “So we’ve got global temperatures and weather in the newspaper, don’t we? And next to that we have the cherry blossom news. This is a good opportunity, so I want you all to have a think about this.” [omitted] T410: “Where the cherry blossoms are not yet in full bloom, where they are close to full bloom, have just started blooming, are half in bloom.” [omitted] T423: “I wonder if we can express this as positive and negative numbers too” | ||

The lesson proceeded in this way, with the teacher picking up topics he noticed | ||

Conclusion | Segment 5 (T328–T389): The teacher sums up that pluses and minuses are attached to positive and negative numbers, and explains | Segment 5 (T432–T525): The teacher looks back over the presentations and summarizes the main points |

The teacher tried to transform the scene to deepen the students’ understanding of what was discussed in the lesson. In doing so, he | The lesson was summarized in a question and answer format | |

instilled some discipline as follows: | T440: “What kind of numbers are positive numbers?” [omitted] | |

T328: “So, ready everyone? Those of you who’ve finished writing, sit up straight, show me what you’ve got” | S442: “With pluses, and they are greater than the previous number” [omitted] | |

The teacher also responded to the reaction of one of the students and attempted to ensure all of the students had properly understood. This can be seen in the following quotes: | T445: “And what is our reference point? We learned when we talked about golf” [omitted] | |

T335: “So, negative numbers are written with a minus sign. Since primary school you’ve written positive numbers like 1, 2, 3, 4, 5, just as they are. But, you see, actually, positive numbers are sometimes written with something like the minus sign we write in front of negative numbers” | S447: “Zero” | |

After this, the lesson continued to develop based on the teacher’s explanations | T448: “That’s right, isn’t it? Positive numbers are numbers greater than, greater than zero” [omitted] |

T351: “Negative numbers have a minus on them, so positive numbers have a plus on them. So, remember this too please. [Writes on board]” [omitted] | T505: “Plus is the positive sign, and this is the negative sign.” [Writes “positive sign” on the board below a plus sign, and “negative sign” below a minus sign] | |

Segment 6 (T390–S429): Working on practice problems to check students’ understanding of the knowledge in this lesson. Giving information about the next lesson | The teacher then went on to check the presentations in this way | |

Finally, the teacher practiced some problems in the textbook (Let us think about number lines) and ended the lesson by indicating the learning topic of the next lesson as follows: | T507 “There were some good presentations. [H] told us that as numbers go from minus 2 to minus 4 to minus 7, the temperature is getting colder. Very good” | |

T423: “OK, so what we’re doing next time is, we’re going to put these positive numbers and negative numbers on a number line, like you learned in primary school. Next lesson I want us all to think about number lines together” | As mentioned above, the contrasting concept of zero as a reference point was not discussed here. Discussion of the golf example was the only scene in which this concept could be seen to be applied | |

Segment 6 (T526–S554): Working on practice problems to check students’ understanding of the knowledge in this lesson. Giving information about the next lesson | ||

The students opened their textbooks and started to work on problems about expressing numbers using pluses and minuses. Teacher H decided to leave checking the answers to the next lesson the following day | ||

T548: “Today I think you have all understood very well positive and negative numbers, and in particular pluses and minuses, so I want you to go home and make sure you do your math study and revision” |

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Arani, M.R.S. Shared teaching culture in different forms: a comparison of expert and novice teachers’ practices.
*Educ Res Policy Prac* **16**, 235–255 (2017). https://doi.org/10.1007/s10671-016-9205-8

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DOI: https://doi.org/10.1007/s10671-016-9205-8