1. My Philosophy of Learning and Teaching Mathematics

2. Start with 2 identical sheets of paper. • Tape the long sides of one sheet together to form an open “silo.” • Make another silo from the second sheet, by taping the short sides together. Which silo will hold the most? Explain your reasoning. (Problem adapted from Lappan, et al.)

3. Piaget • At each stage of development, the learner has • a characteristic way of viewing the world and • explaining it to him/herself. • In order to “let go” of a conflicting theory, the • learner must resolve cognitive dissonance. • Vygotsky • Learner’s ideas may be pushed by teachers • and peers, within a “Zone of Proximal Development.” • Role of culture and social interaction in learning.

4. The desire to share knowing with another human being is a fundamental one. It is at heart a desire to make your thoughts known to the other and to learn whether they are understood, even shared— always with the chance that I will mean more than I meant before because of the way the other has understood. Deanna Kuhn, 1996 In a 1994 study of college students (Geil and Moshman) a correct problem solution was given by 75% of the groups, but only 9% of the individuals. In three-eighths of the correctly-responding groups, no individual had initially exhibited the correct response.

5. The principle problem of human memory is not storage, but retrieval.The key to retrieval is organization…material that is organized in terms of a person’s own interests and cognitive structures has the best chance of being accessible in memory. In a study of 12-year-old children, asked to remember pairs of words, those who created their own mediators remembered many more pairs than those with no mediators, or mediators provided to them.

6. I have never seen anybody improve in the art and technique of inquiry by any means other than engaging in inquiry. Jerome Bruner

7. Implications of learning theories for instruction

8. Implications of learning theories for instruction • Design activities which push the learner’s level of • understanding, and confront learner misconceptions.

9. Implications of learning theories for instruction • Design activities which push the learner’s level of • understanding, and confront learner misconceptions. • Design instruction which causes learners to actively • practice the art of inquiry.

10. Implications of learning theories for instruction • Design activities which push the learner’s level of • understanding, and confront learner misconceptions. • Design instruction which causes learners to actively • practice the art of inquiry. • Provide opportunities for spoken and written discourse.

11. The End