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.


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]


    Foundations for Success

    File this under, “Stuff I meant to post last week and didn’t.”

    The National Mathematics Advisory Panel released its final report last week, titled Foundations for Success. You can find the report and sub-reports in a variety of file formats at the NMAP home page.

    The report is heavily grounded in “high-quality” research and includes six key elements that I would summarize as follows:

    1. Curriculum Focal Points;
    2. How students learn;
    3. Teacher content knowledge and pedagogical content knowledge;
    4. Quality first instruction;
    5. Quality, focused assessment; and
    6. Education research.

    Remember, these are my summaries, not theirs. I would suggest looking at the report if you’re at all interested in what the group had to say, as they are much more verbose than I.

    Starting Something New

    (cross-posted at Looking for r)

    I like my blog.  It gives me an opportunity to talk about things related to math education that are important to me.  But sometimes I’ve got more to say, and it’s not always about math, and sometimes it’s not even about education.

    Enter the new blog.  I’m excited about this, and hopefully it will keep my creativity flowing a bit more evenly.  Look around and comment.  I’m looking forward to this!  Of course, if you prefer the linearity of the old blog, it will still be around.  I’ve got a lot more to say about math education…

    Making Change: Part 1

    I’ve been doing some thinking about the last post, regarding the need for change.  I was (intentionally) vague in stating how we need to change, although I had planned to address that issue later.  What I realize now is that this is way more than one post; here is part one of [who knows how many?].

    Sometimes the systems in which we work are broken and need fixing.  One of these systems that deserves our attention is the options and support available to students who are below grade level in mathematics, particularly at the secondary level.  The reality is that all teachers should know more about the Response to Intervention (RtI) model besides the fact that such a model exists.

    How do we provide for these students?  We must begin by understanding their individual needs.  Diagnostic assessment data provides one perspective and can help identify potential students .  Another important factor is teacher input; this provides a critical perspective about students who would or would not benefit from some type of intervention.  The system within a school or district needs to support the transition process and provide the opportunity for teachers – those who know the students at the individual level – to provide input into the intervention placement process.

    Once we are aware of individual student needs, we need to approach the numbers of students – however daunting or discouraging – with a positive attitude.  Schools and districts need to find ways to accommodate the needs of these students.  The key to this system is increased time and intensity – “remediation” in the traditional sense will not work, because we are obligated to help students who are below grade level get on a trajectory that will get them to grade level proficiency (hopefully before they leave our buildings). 

    The task, then, is to find a way to increase the time that these students are spending in mathematics – often through “double-dose” courses – with resources that are not changing and sometimes decreasing.  In most cases, this process involves a significant change in the way that schools and school districts do business.  However, this change to the existing system is necessary to facilitate further changes that will benefit the students whom we serve.

    (Next time: Changing the way we think about instruction and instructional resources for students below grade level.)