Guide for using “Thinking Systematically”

From APEC HRDWG Wiki

As part of the APEC Education Network (EDNET) projectClassroom Innovations through Lesson Study, a Guide for Planning and Analyzing Mathematics Lessons in Lesson Study was created to help educators learn about Lesson Study so as to improve mathematics education. This page provides an example of using the Guide to review a 6th grade lesson from Japan entitled, "Thinking Systematically", and analyze it using the guidelines presented.

Posing the problem

The beginning of this lesson, (link to video 1) is designed to engage students’ interest. The teacher orients them to the task by asking students to guess what the lesson might be about. He relates mathematics to life outside school by asking questions such as how much students would pay for pens and pencils. This builds the “productive disposition” strand of mathematical proficiency. The segment ends when the teacher poses the problem for the lesson.
To mathematics experts, this problem is a standard algebra problem, with two unknowns and two linear equations. However, these students are too young for algebra and the intention of the lesson is to give them experience of tackling problems systematically. The lesson is not about algebra. The problem has been deliberately chosen to highlight thinking strategies. Every aspect of the problem has been considered in the planning, including the size of the numbers, their numerical relationships, as well as the mathematical structure. In the first part of the lesson, the students focus on the problem to gain experience of the benefits of using a table. This experience is discussed at the end of the lesson.
PROBLEM: You bought some 40 yen pencils and some 70 yen ball-point pens. There were 10 of them in total for 460 yen. How many pencils and ball-point pens did you buy?

Understanding the problem

In the second section, (link to video 2) the teacher begins by ensuring that all students understand the problem. He does this by first asking students to read the problem aloud, and then by asking the class some simple questions related to the problem. In this way, the lesson models Polya’s first phase of problem solving: understand the problem. The teacher then asks students to suggest how the problem can be solved, and he writes important suggestions (“multiples”, “tables”) on the board to emphasize that these are important.
The construction of the table is very carefully managed by the teacher, who stresses the importance of using a labeled table, and shows very strong evidence of lesson planning. A student suggests labeling the rows of the table “pencil” and “ball-point pen”. The teacher implicitly corrects this by bringing out the pre-prepared label “number of pencils” and “number of ball-point pens”. This is an important distinction to make when learning algebra later, because students who believe a letter stands for an object (such as a pencil) rather than a number (such as the number of pencils) make systematic errors when setting up equations.
The teacher has prepared strips of paper with three spaces that hold one number of pencils, the corresponding number of ball-point pens and the total price. So, for example, on one strip of paper the teacher can write 1 (pencil), 9 (pens), 670 (yen). Having 12 strips that hold the 3 quantities, rather than 3 horizontal strips (e.g. one for pencils, one for pens, and one for total) is a subtle point of the lesson design. This emphasizes the vertical relationships between the quantities that are the key to solving the problem (e.g. number of pencils = 10 – number of pens). For this problem, these “vertical relationships” are more important than the “horizontal” relationships. It is also interesting that, at this early stage of the lesson, a student in the class notes that the total price for 1 pencil and 9 pens is 30 yen less than the total price for 0 pencils and 10 pens. This difference of 30 will be the key to solving the problem. However, at this stage the teacher is not ready to incorporate this comment into the flow of the lesson. Perhaps he wants all students to find this for themselves, rather than taking it from one student.
In this lesson, the teacher calls upon many students, and conducts whole class discussion. In this way, he is able to monitor the degree to which the various class members are following the lesson. The extent of engagement of the whole class cannot be judged from the video, but we see the teacher taking steps to monitor this for himself at all points of the lesson.

Individual Work and Class Discussion

In the third section, (link to video 3) students construct the tables in their notebooks and some students write their combinations and total price on the strips placed randomly around the board. In the fourth section, (link to video 4) the calculations are checked, the strips are checked for completeness and the answer is identified.
The teacher then draws the class’s attention to the “confusion” with strips randomly arranged, and two students come to the board to make a systematic table using the pre-prepared labels. The class then discusses why arranging the entries systematically is good. They identify the patterns that are evident and note again the fact that when the number of pencils increases by 1, the total price decreases by 30 yen. The table makes this pattern easy to see. Students in the class suggest the reason for this and the teacher gives two students the opportunity to try to express this relatively complex mathematical pattern clearly in words. Looking at the table, students observe several patterns (e.g. the total price decreases by 30 yen moving from the left and increases by 30 yen moving from the right).
During the class discussion, a student unexpectedly comes to the board and draws a graph showing the number of pens as a function of the number of pencils (i.e. part of the line y = 10 – x). The teacher illustrates what he has done to the class by joining points on a histogram. This very clear explanation is targeted directly to students at Grade 6 level and it give evidence of the teacher being very well prepared mathematically, so that he can help the many students to benefit from the insight of one.

Further Discussion (HaKaSe)

In the fifth section, (link to video 5) the teacher reminds the class of the HA-KA-SE (fast-easy-accurate) principle that they know well, and they discuss whether the table method is indeed fast, easy and accurate. The children decide it is not fast, and would be too slow if the number of pens and pencils was large. This leads for a search for a fast method, and students suggest possibilities, although they all struggle to verbalize these.
Note how the students and teacher write the unclosed expressions on the board. For example, they write (70 – 40) rather than 30. This is a characteristic of Japanese mathematics education which emphasizes relationships.
The suggestions from the class at this point of the lesson have to be handled carefully. Students make suggestions and improve upon each others suggestions. For example, one student explains: “What he wanted to say was 460 – 40 x 10 = 60 and 60 ÷ (70 – 40) = 2 and that (2) is the number of pens bought” Another student explained the meaning of the first expression as “If you buy 10 pencils, the change is 60 yen. Divide by the difference of prices, which is (70- 40) yen”. Later, the teacher summarizes these contributions and explains the logic behind the method of solution by using the table extensively. This part of the lesson very clearly illustrates the development of two aspects of mathematical proficiency: strategic competence and adaptive reasoning, as well as the teachers’ detailed preparation and thinking ahead as to how this can be explained well.
Another student suggests starting at the other end of the table, purchasing 10 pens for 700 yen but with time running short this is not followed up. Wishing the students to practice using their new idea, the teacher then gives students a new problem with the same mathematical structure to solve individually.

Closing the Lesson

At the end of the lesson, (link to video 6) the students are asked to suggest the title of the lesson. In their reflection, they say it will involve tables and mathematical expressions. The teacher ends the lesson by naming it “From tables to mathematical sentences”.
The lesson has shown a strong ‘story line”. It started with a problem that is strong enough to carry the whole lesson. There was a clear period of deepening understanding; the follow up problem closely followed the main idea developed by the students, and the lesson ended with a time where students reflected on their mathematical learning.