When I was a student, in every mathematics class I had, the teacher presented problems and explained how to solve them. The teacher would do a sample problem with us, then give us a worksheet full of similar problems to try on our own. Our success depended on how well we remembered the procedure we had been shown. It never occurred to me that there was any other way to learn math. As it turns out, there are places in the world where math instruction is organized is precisely the opposite direction. Students are given time to explore problems without knowing what procedure might be useful in their solution. Then students present their findings to the rest of the class. The teacher comes in at the end to summarize and extend the mathematics that was discovered. (This example is from Japan.)

The juxtaposition in these two approaches is one of the reasons that I’m excited about the classroom videos from The Third International Mathematics and Science Study (TIMSS) 1999 Video Study. Through the web site, more than 50 math and science classroom videos from seven different countries are available, along with translated transcripts, lesson plans, and commentary from the presenting teachers and researchers. Examining these international videos and lesson plans allow us to slow down and examine education in classrooms around the world, seeing it with new eyes since we are comparing across cultures.

The TIMSS Video Study collected and analyzed video from math and science classrooms in Australia, the Czech Republic, Hong Kong, the Netherlands, Switzerland, the United States and Japan. The 1999 study was an expanded version of a similar video study in 1995 conducted in order to gain insight into anticipated achievement differences between students in the United States and other countries, as shown by the TIMSS results. As you might expect, there were differences in how instruction was organized in classrooms in these countries.

For me, the videos of Japanese classrooms provide interesting examples of how math instruction can use elements of problem-posing to guide effective math instruction. Over the past few months, I have been reading about the Japanese approach to teaching math, in which lessons are usually organized around a single problem (or a just a few problems), the solution for which students don’t know in advance. While planning the lesson, the teacher will try to anticipate all of the common approaches to the problem so that she can be strategic in the questions she asks students and in the way solutions are presented after students have worked independently. The general format for instruction is described by Tom McDougal and Akihiko Takahashi in their article, Teaching Mathematics Through Problem Solving:

“[The] lesson would begin with the teacher setting up the context and introducing the problem. Students then work on the problem… while the teacher monitors their progress and notes which students are using which approaches. Then the teacher begins a whole-class discussion… The teacher may call on students to share their ideas, but, instead of ending the lesson there, the teacher will ask students to think about and compare the different ideas — which ideas are incorrect and why, which ideas are correct, which ones are similar to each other, which ones are more efficient or more elegant. Through this discussion, the lesson enables students to learn new mathematical ideas or procedures.”

One of the TIMSS classroom videos is of Mr. Kazu Marata’s 8th grade math class in Kasu City, Japan. The goal of the lesson was, “Make the students able to express the relative size of quantities in inequalities and use inequalities to solve the problems as a procedure.” This lesson shows the basic elements of a problem-solving approach to teaching math and provides an example of the wonderful range of solution strategies that can come from one problem.

This model of teaching math demonstrated in this video is so established that there are specific Japanese words to describe the different elements of instruction (the timestamp links will take you to the appropriate section of a YouTube version of the classroom video).

**Presentation of the problem**

*Hatsumon* (1:14-4:43) refers to the problem setup and key questions asked by the teacher. In this lesson, the teacher explains the setup of the following word problem about two boys making prayer offerings at a temple by presenting a drawing of an offering box and representations of the coins that the boys are offering.

**Individual problem-solving by students**

*Jirikikaiketsu* (4:39-18:20) is the name for students solving, or struggling, with a problem on their own. In this class, students work independently for about 13 minutes. (If a similar format was used in a longer adult education class, this section could be longer to give students time to get to a solution. Students might also work in small groups after some time working independently.)

*Kikan-shido* is the term for teachers moving around the classroom, asking questions and providing guidance to students at their desks. In this lesson, you will see the teacher walking around the classroom, checking in with students and taking notes on his clipboard, presumably to keep track of each student’s solution strategy to decide who he will call to the board later to present. One of the nice things about the TIMSS site is that a transcript of each class is included, with links to each section in the video. One way to watch these videos is by listening for the questions the teacher asks the students. The transcript makes it easy to search through the class video to find examples of questioning. In this class, I notice that the teacher encourages students to continue with different strategies, but doesn’t tell them if they are correct or not.

**Whole-class discussion about methods for solving the problem**

*Neriage* (18:30-31:14) refers to the “polishing” of student ideas and the presentation of student solutions in group discussion such that mathematical connections are made across solution strategies (manipulatives, drawings, diagrams, charts, equations, etc.). In this class, Mr. Murata brings students to the board to present the following solution methods (concrete use of a manipulative supplied by the teacher, a chart listing days and the amount of money each boy has remaining, and a calculation based on the difference in how much money each boy offers each day, followed by more abstract solutions involving algebra). The teacher helps each student present his/her method to the class, ending with a student solution that involves setting up an inequality. The deliberate progression from concrete to abstract strategies prepares the class to understand how an algebraic inequality can be used to understand this problem. One of the things that struck me most about this class is how prepared the teacher is for every student-generated approach that comes up. He has cards prepared in advance to label each approach on the board as they are presented!

* Bansho* refers to the strategic planning of the blackboard so that all presented solutions will be visible for the final group discussion. In the course of Mr. Murata’s class, the board is slowly filled from left to right, first with a drawing illustrating the problem, then demonstration of student solutions and, finally, a summary and extension by the teacher.

**Summing up by the teacher**

* Matome* (31:15) is the teacher’s summing up and extending of mathematics at the end of the lesson. After the student presentations in this class, the teacher gives a short explanation of inequalities and leads the class into an extension problem that will allow them to apply what they learned about inequalities.

A really nice thing about the TIMSS video site is the ability to read full lesson plans and reflection from the teacher after the filming of the class. In this lesson, Mr. Marata included the following statement in his lesson plan:

Although it is tempting to want to emulate Japanese math teaching specifically, considering how well their students have fared on international texts, the researchers of the TIMSS Video Study determined there was no one best method for teaching math since the highest performing countries in their study used a range of instructional techniques. Stigler and Hiebert, who wrote The Teaching Gap, recommend looking at the teaching in other high performing countries as well. In their words, “It appears that there is not one way to teach effectively, but many.” And as Elizabeth Green, author of How to Build a Better Teacher, wrote: “Of all the lessons Japan has to offer the United States, the most important might be the belief in patience and the possibility of change. Japan, after all, was able to shift a country full of teachers to a new approach.” These videos let us see goals, planning and implementation of instruction based on different ways of thinking about education. The benefit of the TIMSS Video Study web site is that it gives us a range of approaches from different countries to consider, while we work to improve mathematics instruction in our own classrooms.

I’m interested to hear what you think about the TIMSS videos. What compelling examples of classroom instruction do you find in other countries, including the United States? What differences and similarities do you see with the classes you’ve been in? What elements of these different kinds of instruction should we try out?

I think that it is possible to start learning mathematics by following a model provided by a teacher. However, not all problems presents the same level of difficulty. Once a student practice a giving model he should be allowed to attempt to solve a problem of a different level on his own.

Thanks for the comment, Ruben. I definitely think it’s possible to learn math by following a model as well. Clearly, many, many people have learned math that way and have gone on to great success. I wonder about the others who haven’t gone on to great success, or the students who have memorized procedures without understanding what they’re doing, or the students we see in adult education who have mis-memorized the procedures (a negative plus a negative is a positive, anyone?) and don’t have a conceptual understanding that will help correct the procedure.

The other wonderful thing that comes from students working through problems on their own without a model is a proliferation of different strategies for solving the same problem. You can see it in the inequalities lesson above. There are at least 5 different ways to solve the same problem, and students can see connections between the concrete strategy, the table/chart, the algebraic, etc. If teachers model problem solutions, they generally show one way, which tells students that there is one way, a best way, etc. So if there is a new problem presented that students haven’t seen, they might have a tendency to wait to see “the” way to solve it rather than being creative and practicing persistence.

You make an interesting point about how problems can be presented at different levels of difficulty. Could you give an example of a problem you would model and then the problems students would do on their own? Would the problems students do on their own be at the same difficulty level as the modeled problem? More difficult? Easier?

In the end, we need to find strategies for teaching that work. And that might be a mix-and-match collection of strategies that include problem-posing at times and modeling/telling kinds of instruction at other times, depending on what is appropriate for the students and the content.

I would be interested to know if anyone has tried structuring a problem-solving classroom in the way that is described above, with problem presentation, individual problem-solving, general discussion and summary.

I think that your idea is wonderful. However, I now have a question for you to consider. Would you give a student a wrong answer to a problem knowing the right answer in order to avoid or discourage the fact that the teacher is always right?

You mean would I intentionally give a student a wrong answer to a problem? I don’t think so… I might not tell them if they have a wrong answer, but I don’t think I would intentionally lead students astray. If my intention is for them to come to their own conclusions, I think it makes more sense to withhold information than to give them bad info.

On the other hand, I could imagine just for the sake of discussion “acting out” a misconception so that the class can correct me and clear up the misconception for other students. To go back to the negative plus a negative, I could imagine saying to the class something like…

“So, -10 is like a debt of $10. If I owe you $10, that’s like me having negative $10. If I borrow another $10 from you and spend it, that would be another negative $10. I owe it to you, so it’s -10.

(acting out thinking…) “I’m trying to remember the rule. Isn’t it a negative plus a negative is a positive. So, if I have -10 and -10, that’s the same as having +20. So, I have $20! Right?”

I’m thinking of an example where I know people in the room are going to correct me, but some other students might still have the misconception. I would want my “mistake” to be corrected by students immediately.

However, in general, I wouldn’t give students wrong answers. I would just withhold information. Did I respond to the right question?

Best,

Eric

Hi all,

I wanted to share a great radio documentary from American Radio Works on teacher preparation. The documentary mentions the TIMSS video project and specifically references the teaching video on inequalities with Ichiro and his brother.

http://www.americanradioworks.org/documentaries/teaching-teachers/

Highly recommended.

Have you seen this great video by Michael Pershan? Using the TIMMS video project as his inspiration, Michael asks (and answers!), “What if Khan Academy was Made in Japan?” (https://www.youtube.com/watch?v=CHoXRvGTtAQ)