p.s. thanks, barry!

Thank you very much for your comment. I’ve seen your writing on MathNotations and am glad you wrote.

I am aware that there are people who hold that “problems which are not formulaic” are not well-addressed by teaching students the components of math and algebra delineated in the NMP’s report. Such problems are the so-called “messy” problems that have a range of answers or are open-ended, and so forth. Problems such as the “work” problems and others in math textbooks are held in disdain and thought not to lead to problem solving skills. Proper presentation of the solution of say, work problems, however, opens the door to “rate problems” in general, and which generalize to the solution of a great many problems in engineering and science. In fact, many of the standard so-called “formulaic” problems in algebra and other math classes are widely generalizable and have their purpose as I can attest as one who majored in math and work in a field that requires knowledge of scientific and engineering principles.

Providing students the opportunity to solve non-formulaic problems does not in and of itself prepare them to solve problems. Analytic and procedural skills and knowledge of form, which generalize do in fact provide such preparation. I tend to think the term “balanced approach” is one that is not well defined. I used the term “true balanced approach” in my essay, meaning an opportunity for student-centered instruction (such as discovery) that makes use of prior knowledge, rather than the melange of “just in time” skills, procedures and concepts that some teachers, textbook writers and policy makers seem to think students will discover because they need them to solve a problem.

It is my hope that those teachers who use textbooks that are written topics presented logically, sequentially, with expectation of mastery, and which builds upon concepts, will not be punished for doing so. Vern Williams who I quote in the essay is one of those teachers. He gives students very tough “out of the box” problems that are not in the textbook necessarily, but he makes sure they have the requisite skills and information (which he imparts via instruction) before giving them such problems.

Although now retired, I was one of these educators for the past several decades.I believe the Panel paid lip service to these educators. Mr. Garelick, just what benefit does this report have for this group of math teachers? There are many dedicated professionals who have always balanced the need for ‘correct answers’ with conceptual understanding. Educators who always knew that there must be mastery of essentials before one can move on in mathematics. Educators who continue to find creative ways to satisfy their administration and their personal integrity…

The problem is that it is just not easy to blend skill practice, mastery and rich problem-solving experiences and explorations when one has to essentially create one’s own materials. Particularly when the rewards for going ‘above and beyond’ are purely intrisic in the teaching profession. Experienced math teachers know that computational proficiency is absolutely essential but, when confronted with problems that are not formulaic and require recognition of essential concepts and making connections, many of our students flounder. Yes, it is really hard to do the right thing, isn’t it?

In your opinion how will textbook publishers respond to the Panel’s report? IMO, skills-based texts that neglect exploration and more challenging problem-solving would be just as damaging to this next generation as many of the reform texts have been to the current generation. Perhaps such ‘skills’ texts will not be the response to the Panel’s report from textbook publishers. Perhaps…

But that’s ok, the most dedicated of our profession will compensate for whatever materials they are handed. They’ll continue to write their own and do what’s right, just as they always have.

Dave Marain
MathNotations