Look at its surface! The first time you see a ball on that surface, you know an average score.

2. In the same way, if I am counting to 20 in the classroom, when one person comes to me at the end and says “20, 20… I’m out,” I know the total count. If, instead, I ask a few students what they have noticed after several repetitions of counting to 20, many will mention a pattern of counting to 20 that I didn’t explicitly mention. But these students might all mean different things. This kind of “pattern recognition” in learning requires the development of new kinds of knowledge that we don’t normally give students.

3. Our ability to make sense of this information is often so impaired by our language that we just don’t understand each other.

4. Even if students get a good grasp on things like the relative weight of numbers, and we understand the meaning of a ratio, we often don’t make the connection between those two knowledge. When I asked a few students what they had missed in their education, they often simply said: “I missed the lesson.” It was as if they were asking: Why can’t you show these two items of information together?

5. When you are talking about the number 15 that you just learned, and you begin saying, “But, I haven’t really learned that!” you have just made a mistake by stating that you don’t know the answer.

6. Remember that most math problems that start with “20” are really “20, 20…” We don’t really know that we’re doing 20, 20, 20, 20, 20, 20,… before we start. It’s a mistake to make a general statement like “10 is 20” or even similar statements that assume that the problem will come up 20 times for the same answer. In many cases, we are really learning to solve the problem, and so the student is really confusing the question by making it general.

7. Most of the time, the students that talk a lot about the importance of solving problems are the ones who have been taught the most about these problems. We need a lesson in the value of using common sense when making life-or-death decisions.

8. When we have problems with what we have been taught, like a number of the problems that you read about, our instinct is to ignore the problem. You will see very few problems that are so complex that the student has to

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