Mathematically Inclined

I go back to work in a few short days, so in my free time I am reading like mad for spring courses. My current book selections are all about math. Even though I teach the same math class year after year, it's never exactly the same. Sure, I need to help my students understand how children learn math and how best to teach it, but I seem to approach it differently each go-round. I suppose this keeps me and the material fresh.

I just finished reading The Glass Wall: Why Mathematics Can Seem Difficult, by Frank Smith. A whole lot of circular reasoning at the outset had me quite frustrated and wondering why I was reading, but I did press on and found some really interesting ideas in the end. Chapter 12 begins with this idea.

Is mathematics something we know or something we do? The answer is both--mathematics is part facts and part act. And there is no clear dividing line between the two.

I generally talk about the difference between procedural and conceptual knowledge and the need for children to develop both. I'd like to begin with this question on the first day and use it to frame our conversations throughout the semester.

Smith makes several important points in Chapter 13. They include these ideas.

Many teachers have little choice about the materials they use and the curriculum they follow. Yet their most critical practical concern must always be to the understand the mathematical situation the student is in. The issue is not so much what the teacher (or the textbook) is trying to teach as what the learner is learning.

Being a mathematician is a state of mind rather than a repertoire of skills and knowledge. Becoming a mathematician should be an initiation, an affirmation, an induction into a club that is open to all learners, no matter how limited their experience.

Anyone can learn to understand and enjoy mathematics provided nothing goes wrong. And nothing will go wrong provided four essential conditions are met. They are:

1. The mathematics must be interesting and comprehensible.
2. There's no fear of mathematics.
3. Inappropriate things aren't learned.
4. There's sufficient time.

These are all important ideas I try to convey to my students. I think the most difficult thing for them to accept relates to the notion of time. Classroom teachers have limited amounts of time, but learning can't be forced. What happens when the pacing guide only allows a teacher to devote 2 weeks to teaching fractions, but at the end of these two weeks the students still don't understand? What then? Do teachers move on, or take more time to ensure students understand the material? Overwhelmingly, the response is move on.

One change I ask students to make to develop the notion that all children can do math, is to use the word mathematician on the top of worksheets and tests (instead of name). This is such a small thing, but it really makes a noticeable difference in the attitude of students toward their work in math. I do the same thing in science and social studies, using the titles scientist, historian and geographer.

As I work on my syllabus, I am going to think more about the four conditions and how I can use them in helping students think about essentials of instruction.