NCTM

Gratuitous Irrelevance

While I may be somewhat critical of the lack of new information in Foundations for Success, I do agree with the findings and recommendations of the National Math Advisory Panel. Which is why I’m so troubled by this article by David Thornburg. The basis for Thornburg’s argument is this:

Recent pronouncements from Washington regarding math education have suggested that pedagogical points of view don’t matter in the teaching of mathematics. For example: “There is no basis in research for favoring teacher-based or student-centered instruction,” Dr. Larry R. Faulkner, the chairman of the panel, said at a briefing last Wednesday. “People may retain their strongly held philosophical inclinations, but the research does not show that either is better than the other.”

Thornburg goes on to cite two “counterexamples” to refute this claim, both from “Rising Above the Gathering Storm“:

  1. Statewide specialty high schools (e.g., IMSA ), and
  2. Inquiry-driven project-based learning.

This is a wonderful example of the misconception of inquiry as being something totally student-centered, with little or no teacher input. Granted, part of the ownership for this misconception lies with the math education community – we do not often enough discuss the concept of inquiry using the word “inquiry.” Instead, we use terms like “problem solving,” “reasoning and proof,” or “connections.”1 It is the stubborn insistence of some educators that math is math and science is science and never the two shall meet.

The science education community, on the other hand, gets it. They understand inquiry.2 It’s part of their standards. In fact, the National Science Teachers Association (NSTA) describes scientific inquiry as

a powerful way of understanding science content. Students learn how to ask questions and use evidence to answer them. In the process of learning the strategies of scientific inquiry, students learn to conduct an investigation and collect evidence from a variety of sources, develop an explanation from the data, and communicate and defend their conclusions.3

Sounds to me a lot like the process standards:
Reasoning and Proof:

Instructional programs from prekindergarten through grade 12 should enable all students to–

  • recognize reasoning and proof as fundamental aspects of mathematics;
  • make and investigate mathematical conjectures;
  • develop and evaluate mathematical arguments and proofs;
  • select and use various types of reasoning and methods of proof.

Problem Solving:

Instructional programs from prekindergarten through grade 12 should enable all students to–

  • build new mathematical knowledge through problem solving;
  • solve problems that arise in mathematics and in other contexts;
  • apply and adapt a variety of appropriate strategies to solve problems;
  • monitor and reflect on the process of mathematical problem solving.

Connections:

Instructional programs from prekindergarten through grade 12 should enable all students to–

  • recognize and use connections among mathematical ideas;
  • understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
  • recognize and apply mathematics in contexts outside of mathematics.

    Not to mention the communication and representation standards.

    But this isn’t the issue at hand; the real issue is how to teach it.If we approach this from a logical perspective, then we understand that students will not develop these skills of scientific inquiry without some direction from the teacher. Inquiry is developed along a continuum, beginning with structured or directed inquiry, moving to the broad category of guided inquiry, and finally – often after much support and scaffolding – to open or student-initiated inquiry. One can also think of this in terms of the Gradual Release of Responsibility model for literacy instruction.

    In other words, Thornburg’s argument is entirely irrelevant. His counterexamples fail miserably to disprove the findings of the panel with regard to student-centered v. teacher-directed instruction. What we know is that a balance of both is critical so that students have the opportunity to develop a solid conceptual foundation of school mathematics.

    —
    1See the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics, with particular attention to the process standards. [back]

    2Some science resources and organizations that discuss inquiry:

    3From the NSTA Position Paper on Inquiry. [back]