As a result of relatively poor showings on international tests of Math and Science, there has been a clamor to improve instruction. The most well known of these tests is commonly referred to as TIMSS (Trends in International Math and Science Study). This isn’t a new phenomenon, the National Council of Teachers of Math (NCTM) made this claim back in 1989 when they first published the Standards and it is one of the underlying reasons for the No Child Left Behind legislation.
In their book, The Teaching Gap James Hiebert and James Stigler describe the typical lesson design in the United States, Germany and Japan. They did a video study of the three cultures to see what they could find out about how math is taught. They studied eighth grade classrooms. What they discovered is that there is a significant difference in the way Math is taught. In the United States, the typical lesson today remains very much today like the same lesson design you probably experienced as a student.
One of the most exciting aspects of TIMSS is the opportunity to learn more about teaching in other countries. Some preliminary findings from the TIMSS video study into which we will want to inquire further are the following:
The structure of U.S. mathematics lessons is similar to lesson structure in Germany, but different from that in Japan.
The study reports that eighth-grade lessons in Germany and the U.S. emphasize acquisition of skills in lessons that follow this pattern:
1. Teacher instructs students in a concept or skill.
2. Teacher solves example problems with class.
3. Students practice on their own while the teacher assists individual students. [1, p. 42]
In contrast, the emphasis in Japan is on understanding concepts, and typical lessons could be described as follows:
1. Teacher poses complex thought-provoking problem.
2. Students struggle with the problem.
3. Various students present ideas or solutions to the class.
4. Class discusses the various solution methods.
5. The teacher summarizes the class' conclusions.
6. Students practice similar problems. [1, p. 42]
The expectations in our lessons differ from those in both Japan and Germany.
As mentioned above, U.S. lessons concentrate on skill acquisition, with 95 percent of the lessons including practice on procedures--in contrast to 42 percent of the Japanese lessons. But many Japanese students practice skills in paid tutoring sessions after school. In contrast, 44 percent of the Japanese lessons assign problems in which students have to invent new solutions or procedures that require them to think and reason. This finding was bolstered by the analysis of lesson summaries done by a group of mathematics professors in which they found little mathematical reasoning expected in the U.S. lessons. [see 1, chapter 3]
The structure of U.S. mathematics lessons is similar to lesson structure in Germany, but different from that in Japan.
The study reports that eighth-grade lessons in Germany and the U.S. emphasize acquisition of skills in lessons that follow this pattern:
1. Teacher instructs students in a concept or skill.
2. Teacher solves example problems with class.
3. Students practice on their own while the teacher assists individual students. [1, p. 42]
In contrast, the emphasis in Japan is on understanding concepts, and typical lessons could be described as follows:
1. Teacher poses complex thought-provoking problem.
2. Students struggle with the problem.
3. Various students present ideas or solutions to the class.
4. Class discusses the various solution methods.
5. The teacher summarizes the class' conclusions.
6. Students practice similar problems. [1, p. 42]
The expectations in our lessons differ from those in both Japan and Germany.
As mentioned above, U.S. lessons concentrate on skill acquisition, with 95 percent of the lessons including practice on procedures--in contrast to 42 percent of the Japanese lessons. But many Japanese students practice skills in paid tutoring sessions after school. In contrast, 44 percent of the Japanese lessons assign problems in which students have to invent new solutions or procedures that require them to think and reason. This finding was bolstered by the analysis of lesson summaries done by a group of mathematics professors in which they found little mathematical reasoning expected in the U.S. lessons. [see 1, chapter 3]
Students are taught a skill. More and more this is being done through the use of manipulatives in the elementary schools, but not always. Then they practice the skill by doing a series of similar computations. Then if they show they can do the skill in this isolated format they get to work on applied problems in which this skill is imbedded into a story problem. Take for instance the concept of perimeter. Students are told what skill they are going to learn. They are then shown a shape, usually a rectangle and shown how to calculate the perimeter. Often this is described as the act of ‘adding all around’. They then practice this by calculating the perimeter of a series of rectangles, often this task is done in isolation with little or no talking allowed. If they are able to do this they will get a problem such as the following:
We need to put a border around a bulletin board that measures 5 feet across the bottom and 3 feet on the side. How much border should we buy?
Contrast this with a typical Japanese lesson plan for the same topic. Here students are presented with the problem first.
We want to put ribbon around the edge of this poster board. How much ribbon do we need?
Notice the problem does not give the students the measurements; they must do this for themselves. It also does not use the word perimeter. That vocabulary will be developed within the lesson itself.
Students would begin working on the problem in a problem solving workshop environment (which means working in small groups with tools such as poster board, ribbon, rulers, tape measures and reference books available). As students work on the problem key vocabulary is introduced in the context of the problem. Students must figure out what they must do in order to solve the task. After some time the class is called together to share strategies so far. The class has the opportunity to hear each other’s strategies rather than the one the teacher demonstrated in the typical American lesson plan. The teacher can correct any misconceptions and offer advice at this time and then the students return to the problem to continue their work on this or similar problems.
The difference can be summarized as follows: the American lesson design is part to whole instruction while the Japanese lesson design is whole to part. The best analogy I could think of is if you were trying to teach basketball and you taught it like this –
“Class today we are going to learn about basketball. It is a sport played on a court. Here are the dimensions. (Show diagram). There are 5 players on each team. In order to play this game well you will need to perform many skills well. Today we are going to introduce the skill called dribbling. When you dribble you push the ball towards the floor with one hand, today use your right hand. Push with your fingers and let the ball bounce back to you. Good. Now continue to do that while I come around and check you. Those of you that I told were okay you can try with your left hand. Others come over here; I need to re-teach you this skill. Class, you did very well today. Those of you still having trouble with this skill, I will work with more at recess. Tomorrow we will continue with dribbling. Tomorrow we will try to dribble and move at the same time. In a few short days we will move to passing, and in only a week or two we might try to shoot the ball at the basket.”
This lesson design takes the game of basketball and breaks it up into a series of discrete skills. In doing so it takes a lot of the fun out of the game. The fun is in actually playing. By teaching the game in parts to the whole, we have taken something that kids naturally enjoy doing, and made it so dry that students feel it is boring. Another problem with this lesson design is it is hard for students to put together all the various sub-skills that make up the whole. By teaching in these isolated discrete skills, the basketball player above may be able to dribble, pass, and shoot, but that doesn’t mean that they can actually play basketball. In math terms, just because a student can add, subtract, multiply and divide, doesn’t mean that they can do math – apply those skills to solve a problem. Many adults experience “aha” moments in adulthood when it finally clicks. Through application in their work or adult lives the math actually begins to make sense. They see the connections between the isolated skills that eluded them in school.
In the whole to part lesson design, we would start with the kids actually playing the game. The coach would gather the group to offer constructive criticism of various skills as they were being performed in the context of the game. So if a player was not dribbling correctly, the coach could call the team together and review the skill of dribbling and the players could then practice with the understanding of how dribbling is a skill that helps you play basketball.
To carry this into the math class, we need to provide students with opportunities to solve problems. This lesson design is sometimes called the Problem Solving Workshop and is similar to the Writing Workshop model described by Lucy Calkins and her team at Columbia. The students work in the Problem Solving Workshop to create a product I call a Math Report. Just like other forms of writing, this genre has qualities that define it as “good”. Please see the related FAQs, “What is a Math Report?” and “What is the Problem Solving Workshop?”
A quote from Marilyn Burns in an Interview in ENC Focus
What are your ideas on how to make a classroom, at whatever grade level, a place where students use sense-making as their basic learning strategy?
First of all, in planning instruction a teacher needs to look at the content and say, "What is it that I want my children to understand?" and secondly, "What is it that I want them to be able to do with that understanding?" It is really a question of both concepts and skills.
Teachers first need to have a sense of what the content is, and that's outlined in the five content standards. Then the next thing, the question that I always ask myself, is: "What experiences can I provide the children that would give them a way to start to make sense of this for themselves?" This is where I look at the process standards because they really address what children need to do to learn math-
• Problem solving-What kinds of problems can I present to children that would give them a chance to grapple with important ideas and skills?
• Reasoning and proof-What kinds of situations can I pose to children so that their reasoning is engaged and they have experience giving convincing arguments?
• Communication-How do I involve children in talking and writing to help them communicate what they are studying and learning, and hear the ideas of others?
• Connections-How do I help children see the connections among mathematical ideas rather than seeing concepts as isolated and separate from one another?
• Representation-How do I help children use the symbolism of mathematics to describe their thinking?
When I am planning new experiences for children, I use those five process standards as my guidelines. I say, "Am I providing kids the opportunity to problem solve, reason/give proof, communicate, connect, and represent?"
But there is another thing I think about, and this has been the biggest change for me in the last five to ten years. Teachers often say to me, "Children are supposed to understand, but is it okay to tell them something? How do I teach it?" One criticism that I frequently hear is that in reform math teaching there is no direct instruction.
But, of course, there is direct instruction. I make the distinction this way: I ask myself when teaching something new, "Where is the source of the knowledge for the child-inside or outside?" Mathematical concepts and skills rely on logical structures and learning them calls for making sense of these structures, so the source of learning is internal. But social conventions are also part of learning math, and the source for learning the conventions is external and can be another person, a book, television, or some other source outside the child.
Let's say, for example, that I'm teaching the signs for "greater than" and "less than." You can't figure out the sign-it's just a funny-shaped arrow. So it's appropriate for the teacher to teach this by telling.
But if I want to teach a child how to figure out if five-eighths is more than or less than or equal to a half, the child needs to make sense of these fractions, not merely memorize them or learn a rule, like "cross multiplying," that they might not understand. The source of that understanding is inside the child's head.
To me that is the critical difference about a Standards-based curriculum-it makes the distinction between when you teach by telling and when you teach by giving children experiences to grapple with. The issue is-where is the source of the understanding for the child?
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