Making Mathematics Meaningful

In between Christmas shopping (aagh!), making plum puddings, organising tradesmen to fix things around the house one doesn’t get time to do during term time, I have read the November issue of Educational Leadership – the journal of ASCD (Association for Supervision and Curriculum Development) which is themed around mathematics education.

One of the articles  – How Mathematics Counts by Lynn Arthur Steen – asserts, through a focus on fractions and algebra, that high stakes testing that emphasises procedures is alienating students (and their teachers!) from understanding the mathematics and thus makes further involvement in higher mathematics learning unavailable to them. She touches on a number of my ‘pet issues’ as can be seen in the following quotes:

“These examples illustrate two very different aspects of mathematics that apply throughout the discipline. On the one hand is calculation; on the other, interpretation. The one reasons with numbers to produce an answer; the other reasons about numbers to produce understanding. Generally, school mathematics focuses on the former.”

“For mathematics to make sense to students as something other than a purely mental exercise, teachers need to focus on the interplay of numbers and words, especially on expressing quantitative relationships in meaningful sentence”

“Experience shows that many students fail to master important mathematical topics. What’s missing from traditional instruction is sufficient emphasis on three important ingredients: communication, connections, and contexts.”

“One of the common criticisms of school mathematics is that it focuses too narrowly on procedures (algorithms) at the expense of understanding. This is a special problem in relation to fractions and algebra because both represent a level of abstraction that is significantly higher than simple integer arithmetic. As the level of abstraction increases, algorithms proliferate and their links to meaning fade. Why do you invert and multiply? Why is (a + b)2 ≠ a2 + b2? The reasons are obvious if you understand what the symbols mean, but they are mysterious if you do not. Understandably, this apparent disjuncture of procedures from meaning leaves many students thoroughly confused. The recent increase in standardized testing has aggravated this problem because even those teachers who want to avoid this trap find that they cannot. So long as procedures predominate on high-stakes tests, procedures will preoccupy both teachers and students.”

Another article – What’s Right About Looking at What’s Wrong by Deborah Schifter – opens with the paragraph: “To teach mathematics for conceptual understanding, we need to treat it primarily as a realm of ideas to be investigated rather than a set of facts, procedures, and definitions to be used. To implement the former approach, teachers must have a deep understanding of content as well as the skill to implement concept-based pedagogy. And these greater demands on teachers, in turn, require well-thought-out forms of professional development.” Another one of my hobby horses – that mathematics is a set of ideas or concepts and needs to be taught as such to develop real understanding within our students. The article then goes on to describe a 5th grade lesson on multiplying two multi-digit numbers, something that is not commonplace in secondary classrooms but the methodology employed is one I admire very much and intend to use more in my own classes in 2008…”For many decades, mathematics has been taught the same way: The teacher demonstrates procedures for getting correct answers and then monitors students as they practice those procedures on a set of similar problems. Why did Ms. Sweeney ask her students, who already knew one efficient way to multiply 36 × 17, to find alternative strategies to do it? Why, at the end of class, did she ask a student to present a strategy that produced an incorrect result? And why did she ask the rest of the class to examine his strategy for homework?

When we view Ms. Sweeney’s behavior from an alternative perspective, it becomes comprehensible. She acted on the belief that mathematics is much more than a set of discrete facts, definitions, and procedures to memorize and recall on demand. In her view, mathematics is an interconnected body of ideas to explore. To do mathematics is to test, debate, and revise or replace those ideas. Thus, the work of her class went beyond merely finding the answer to 36 × 17; it became an investigation of mathematical relationships.”

There are a number of other articles that are available to read online here. In particular, I think that the following two, in addition to those mentioned above, are most worthwhile to read over the upcoming break:

From Arithmetic to Algebra

Leanne R. Ketterlin-Geller, Kathleen Jungjohann, David J. Chard and Scott Baker

A taste of algebra in elementary school helps students develop abstract reasoning.

Attitude Adjustments

Lesa M. Covington Clarkson, Gay Fawcett, Elaine Shannon-Smith and Nancy T. Goldman

From using graphing calculators to reinforcing math at the zoo, educators share ways to boost the confidence of mathematics students.

Be inspired for the new year ahead !!


About Linda

I have been involved in secondary mathematics education in Victoria, Australia for over 25 years.
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2 Responses to Making Mathematics Meaningful

  1. Silver Lennox says:

    I find myself in strong support of the two articles presented here, particularly that by Deborah Schifter. As a Mathematical Methods and Further Maths student, I’ve always found that by connecting concepts and exploring the mechanics of a formula, one goes beyond knowing it, understanding it thoroughly. Forget mnemonics and repetition; that information which I understand deeply (like the ol’ unit circle and quadratic formula) is recallable 10 times out of 10.

    As one would expect, the problem with a focus on information rather than communication, highlighted by Lynn Arthur Steen, doesn’t solely apply to mathematics. In Chemistry last year, I found myself lagging behind the pace of teaching with the rest of the class, for the first time. We were being left to learn from the text book and consult the teacher should we have difficulty. The inefficiency in this method would seem to be in the poor communication of subject material, as the class was being taught by a textbook first and a person second.

    This blog is neat. Keep writing and I’ll keep reading.

  2. Linda says:

    Thanks for the comments and your insights.
    As you can probably tell, I am passionate about the teaching of mathematics for understanding.
    I hope the academic year has started well for you.

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