*Mathematical and Analogical Reasoning in Young Learners*says there are. The three kinds of teachers are 1. theoretical, 2. experiential, and 3. intuitive. In reading over the respective descriptions and case studies of actual teachers, the categories began to remind me of those primary school reading groups, you know, the squirrels, bears and rabbits. Everybody knows the squirrels are the best and the rabbits, shall we say, are not, regardless of the attempt to camouflage the differences with a neutral category name.

The researchers asked about twenty teachers of children ranging from kindergarten to third grade the following questions:

1. What doesmathematical reasoningmean to you?

2. Are you familiar with the term,analogical reasoning? What does the term mean to you?

3. How competent at mathematical and analogical reasoning do you consider young children to be?

4. How did you acquire your understanding of mathematical reasoning?

5. How did you acquire your understanding of analogical reasoning?

6. How do you perceive your role in developing the children's mathematical reasoning?

7. How do you attempt to stimulate your children's mathematical reasoning?

8. What differences in reasoning ability do you see in your children?

9. How do you attempt to address these differences?

10. What kinds of mathematical reasoning abilities do you think your children will need to be successful in first and second grade?

11. Do you take these future reasoning needs of your children into account? If so, how do you do it?

12. In what ways do you think the children's out-of-school experiences contribute to the growth of their mathematical reasoning?

The beliefs and practice of the teachers of our youngest students should be examined more often. These teachers are instrumental in laying the academic foundations which may mean the difference between later academic achievement vs academic frustration. According to the authors, theoretical teachers displayed substantial and detailed knowledge of mathematical and analogical reasoning, pedagogy, and cognition. Theoretical teachers believed that children are highly capable and competent. They also considered home experiences and mathematical language to be essential to the the children's development of mathematical and analogical reasoning skills.

On every question, as a group (but not necessarily as individuals) the authors found the experiential teachers and the intuitive teachers to display less knowledge and lower expectations. In summary, and base their practice on the collected knowledge and experience of others enhanced by their own experience. Experiential teachers relied primarily on their own experience and their reflections on that experience without reference to the professional literature. Intuitive teachers were just guessing and hoping; their answers to the research questions “were brief and

**often included jargon**” (emphasis supplied) (page 148). In fact, their responses often included the phrase, “I guess.” Although all groups emphasized the process of doing math over getting answers, the intuitive teachers “did not see a need for direct instruction. This may relate to their lack of attention to conceptual development and their emphasis on learning being “fun” (page 149). (

*In the near future, I will be examining the jargon and buzzwords of education, and how terminology can substitute for clear thinking.*)

I mentioned earlier that I got the impression that the labels, “theoretical,” “experiential” and “intuitive,” were hardly more than euphemisms for good, middling and poor. By the end of the book, the impression was pretty much confirmed.

The knowledge, beliefs, and practices of the teachers in this study can be placed on a continuum from intuitive to experiential to theoretical. Theoretical teachers explained their knowledge and and beliefs by referencing theoretical frameworks, teaching experience, and listening to children. They provided rich examples of practices associated with effective/exemplary teachers. The experiential teachers made decisions based on knowledge that appeared to reflect their experience as opposed to the multiple sources of knowledge used by theoretical teachers. In contrast, the intuitive teachers appeared to make instructional decisions more spontaneously. Although the examples of practice the intuitive teachers provided were not necessarily ineffective, these teachers could not clearly articulate any rationale for the decisions they make regarding their instructional practices (p 167).

Since the book was written by a bunch of education researchers, the high standing of theoretical teachers may mean nothing more than theoretical teachers quote education researchers so education researchers like them. The authors recommend that future research focus on the question of whether more learning occurs in the classrooms of theoretical teachers than in the classrooms of other teachers. Maybe I am a little rankled because I know that most of the here-today-gone-tomorrow education fads that have burdened veteran teachers over the years are usually perpetrated by education researchers, researchers who may have little teaching experience of their own.