What did you expect?

From the ASCD Blog:

“More math, particularly Algebra, in California high schools has yet to pay off. Last week’s most-clicked Smartbrief story reported that enrollment in remedial math courses still remains high in California Universities, leading many to question why high school reforms are not transferring to college-preparedness.”

Are we surprised? This is much less an issue of what we teach than an issue of how it is taught.

Standards are not the problem. Expectations for student learning and the pervasive “sit and get” culture of high school mathematics teaching are the culprits. Our focus needs to extend beyond the curriculum to include sound instructional strategies.


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]

    The Way We Do Business

    I’ve been thinking lately about math in high school.  This is not unusual, by any means, especially for someone in my position.  Nationally, we are struggling with math in high school; some schools, districts, and even states are getting it right, but most are struggling.

    A word if caution: if you’re looking for solutions in this post, you will be disappointed.

    One thing that frustrates me most about the work we (as a nation) have to do at the high school level is the lack of concern about instructional practices.  Most high school math teachers have a solid grounding in the mathematics that they teach.  This is often incorrectly equated to a solid understanding of how to teach that content.  Unfortunately, content knowledge does not imply pedagogical knowledge.  (I would pose the question: is the converse true?  My jury of one is still out on this, although I thought I knew the answer until about 30 seconds ago.)

    So how do we change instruction?  I’m not even going to attempt to answer that today.  What I know for certain is that if we want the change to happen in the future, we have to start now.  That sounds pretty obvious, but here’s why:

    1. Current high school students who are thinking about becoming math teachers are learning how to teach even now.  They watch their teachers.  If all of that student’s teachers are teaching they way that they were taught, then that student is going to someday teach the same way.  The cycle continues because…
    2. Secondary teacher preparation programs at the university level do not do enough to promote a change in instructional practice.  Those that do are often ineffective in reforming students’ attitudes and beliefs about pedagogy that were learned in high school. The result is…
    3. More teachers, teaching the same way they were taught, and influencing the next generation of teachers.

    You’re probably thinking that I’m the world’s biggest pessimist right now.  I really believe that we can change.  I also believe that we will not wholly change the way that high school math is taught in the future unless we start the change now.  In summary,

    • Systems matter – they need fixing sometimes.
    • Resources matter – they need to be of a high quality and aligned to appropriate benchmarks.
    • Instruction matters – it’s the critical third element that we too often overlook.

    Changing the way we do business in high school means taking a close look at all three areas.  Until we do, we aren’t going to have the overall effect that our nation so desperately needs.

    The Whole Truth…

    I had the opportunity to talk about assessment with a group of elementary school teachers earlier this week.  Our discussion focused on scoring, with particular emphasis on the scoring of constructed response items similar to those that appear on many state assessments.  In Colorado, there exists a set of holistic rubrics for this purpose.  They are sound, but require some understanding of what is being measured to be used effectively.  We also talked about preassessment – why and how to preassess kids.  This topic is better suited for a later post.

    Knowing that this presentation was looming on the horizon, I had been on the lookout for assessment-related ideas anywhere I could find them.  So last Saturday, while enjoying a Shrekmarathon with my wife and kids, I found what was looking for.  In the first movie, Shrek overhears Fiona lamenting about her ugliness and thinks she is talking about him.  Dejected (he had just worked up his courage to express his undying love), Shrek walks off into the night.

    I showed this one-minute segment to teachers.  Then I made my point: when it comes to assessing student learning, be sure

    1. you have the whole picture; and
    2. you have the right picture.

    Sometimes, whether observing students working or scoring an assessment, it is easy to miss the big picture or to only get a part of the picture.  If we use assessment data to guide our instruction, but we’re working with an incorrect or incomplete picture, then our instruction will miss the mark. 

    So next time you think assessment, think Shrek – and get the whole picture the first time.