The teacher starts by displaying, either with a poster or using a projector, the picture from the textbook of four different rabbit cages, shown in Figure 2 (it is not uncommon for Japanese elementary students to care for rabbits in several rabbit hutches, so this is a familiar context). The teacher then asks, “Should each cage have the same number of rabbits? Write that idea in your notebook.” Other students count the rabbits and decide that B and C are equally crowded because they look like they are the same size, but they are unsure about D. He first calls on a student who thinks that B and C are the same size. The teacher posts a table with the areas of the four cages (Figure 4).
Neither of these behaviors will serve students in the long run.
Inevitably, someday, every one of your students will encounter problems that they will not have explicitly studied in school and their ability to find a solution will have important consequences for them.
Through this discussion, the lesson enables students to learn new mathematical ideas or procedures. Let's illustrate this with an example from a hypothetical fifth-grade lesson based on the most popular elementary mathematics textbook in Japan.
(This textbook has been translated into English as and is available at Ed
” There is some discussion about the rabbits in cage B, and students decide that just because they are bunched together right now, they probably won't stay that way. ” This last question becomes the key mathematical question of the lesson, and the teacher writes it on the board: “Let's think about how to compare crowdedness.” Students copy this problem in their notebooks while he writes. A would have an area of 48 m2 while C would have an area of 45 m2, so B is more crowded.
Students recognize that cages A and B are the same size, and since cage A has more rabbits (9 vs. The teacher writes that observation on the board: “When two cages are the same size, the one with more rabbits is more crowded.” “What about the others? The teacher gives students a piece of paper with the pictures from Figure 3 to glue in their notebooks and gives them 5 minutes to think about the problem. Idea 4: Divide: (area) ÷ (# of rabbits) = amount of area per rabbit Idea 5: Divide: (# of rabbits) ÷ (area) = number of rabbits per unit area The teacher invites students to explain their ideas to the class, selecting students based on the order above, while he records each idea on the blackboard.In a lesson about problem solving, students might work on a problem and then share with the class how using one of these strategies helped them solve the problem.Other students applaud, the students sit down, and the lesson ends.What do your students do when faced with a math problem they don't know how to solve? At best, they seek help from another student or the teacher.At worst, they shut down, seeing their failure as more evidence that they just aren't good at math.5) During most Japanese lessons, the textbook is closed, but the textbook shows how the authors think the lesson might play out.When the lesson begins, the blackboard is completely empty. Some students notice that some of the cages are different sizes.They studied George Polya's , they began exploring what it would mean to make problem solving “the focus of school mathematics.” And they succeeded.Today, most elementary mathematics lessons in Japan are organized around the solving of one or a very few problems, using an approach known as “teaching through problem solving.” “Teaching through problem solving” needs to be clearly distinguished from “teaching problem solving.” The latter, which is not uncommon in the United States, focuses on teaching certain strategies — guess-and-check, working backwards, drawing a diagram, and others.Early in the year, before students learn a particular skill, the task could be a problem; later, it becomes an exercise, because now they know how to solve it.In Japan, math educators have been thinking about how to develop problem solving for several decades.