Saturday, December 29, 2007

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.


  1. I'm in my sixth year of teaching--always in upper elementary grades. I love figuring out how to build my students' conceptual understanding and also how to build real mathematical thinking and problem solving skills.

    I've always had lots of control and freedom in my math instruction. Some concepts are much, much more foundational than others and I think it essential that teachers be given the freedom and trust to pace things and review things and even create curriculum tools as they see fit. Some concepts my kids learn best from daily review "calendar math" type instruction over the course of a month. Other things we dive into deeply for a couple days. Often it's a combination of repetition and inquiry. I really enjoy teaching children to be readers and writers, but at the end of the day, there's nothing I love more than teaching math.

  2. Tricia,

    Franki and I have been having conversations about whether or not teaching/learning math is different from teaching/learning reading. Case in point, we generally have all levels of readers in our reading workshop, and that seems to be an effective format for reading instruction. However, at my school we are doing some flexible grouping across the grade level in math. I feel like a traitor to my ideals, but dang it, it seems to be working. For math.

    What do you think?

  3. I am an adjunct Associate Professor of Mathematics at Rider University, active as a substitute teacher and mentor in high schools, and a retired professor of physics from Rutgers University. I have taken extensive notes from my experiences and given them to my protégés. Recently I collected them into a book. I suggest that your library purchase the book for the benefit of students, parents, and teachers.

    I just wrote a book, "Teaching and Helping Students Think and Do Better". This is available on, ISBN 978-1-4196-7435-8. May I suggest that you order a copy for the library? The readers will be very pleased!

    The reviews are superb. Students, teachers, and professors who have looked at the book give it the highest rating.

    Typical comments that I hear are things like this: "Hi, Dr. Aranoff!" said a girl, "I got a 100 on the test! I am so happy! Thank you so much!"

    Thank you very much.

  4. Tricia,
    Any suggestions on fun math books for third graders? Thank you!

    Have a great New Year's!

  5. Tricia,
    Have you seen "The Number Devil"?
    My "mathy" son loved it and read it over and over. I think it was about 3rd or 4th grade when he first discovered it. In any case, it is a keeper!

    Thanks for creating good math teachers. We so need them!

  6. I am curious about the "inappropriate things" that are not to be learned. Does this mean things that are "too advanced?" When you have a kid, like I do, who finds math incredibly fun, it is so tempting to show him all the neat things you can do with Mystery Numbers X and Y, and how numbers can go down below zero, and that sort of thing, and before you know it, you are trying to remember how to do quadratic equations (and failing, in my case). It's much the same as parents of kids who take to reading giving them increasingly complex books, far beyond the "grade level" they are supposed to be reading at. But parents aren't encouraged to do this sort of thing in math...

  7. Hi Charlotte,
    I interpret "inappropriate things" to be misconceptions. In order to make things easy to learn, teachers often make statements that don't hold true. For example, when adding two numbers together you always get a bigger number. While this is true for young children adding whole numbers, it does not hold true when negative numbers are involved.

  8. Hi Tricia,

    I like the idea of calling kids "mathematicians." I call my students "writers" and "readers" and it is really the same concept.

    I often wonder how things would have been different for me if I'd been taught math in the creative, interesting ways that have been used on my kids. I was always reasonably competent in math but I always hated it. It's a shame because there's a lot to love. (I did like trig. I'm just weird that way.)

  9. As a mother who is on her knees, praying for success in Algebra II for my daughter, I am so interested in this post. My family qualifies for that T-shirt that says, "English Major--You do the Math."