CognoMath

  Imagine a student solving problems, one after another, each problem challenging enough to be interesting but not so challenging that the student is discouraged. The problems are sequenced carefully to enable the student to solve each of them.  Enjoying success again and again, the student learns to expect it, gets addicted to it, and is willing to work harder and longer to achieve it.  Being used to solving problems seems to be an important characteristic of successful learning. In the process described above, the student learns and practices problem solving strategies, both general and topic-specific. The student develops stamina and becomes resourceful. These are character traits that are useful—not only for learning mathematics, even not only for learning.  Of course, when we move from one imaginary student to a class of many real students we are faced with questions: Could a math problem, even “difficult enough but not too difficult,” be interesting for every s...

One year in my teaching experience stood out in shaping my professional priorities.  I was teaching math in a charter school. Entering sixth-graders at that school were given a math placement test and then grouped by achievement level into separate classes. That year I got to teach both groups of outliers: the class with the students who did best on the placement test and the lowest scorers’ class. It was such an interesting experience! Never before had I taught at the same time two groups of the same age that were so different academically!  Observing and analyzing the differences in these groups’ behavior changed how I saw the causes of success and failure. Let me tell you about these differences.  How students asked questions The striking difference I observed first was in their questions. My high achievers asked twice - no, ten times - as many questions as my low scorers. Both shy and outgoing kids asked them: the shy ones whispered; the others yelled their questions ...