A few days ago I posted on MathNotations a comment I had left on Edspresso, a reply to Barry Garelick's essay, Living in a Post-National Math Panel World.
My post was entitled, Balancing the Equation.
Barry Garelick is an analyst for the U.S. Environmental Protection Agency in Washington, D.C. He is a national advisor to NYC HOLD, an education advocacy organization that addresses mathematics education in schools throughout the United States.
I just replied to Mr. Garelick's reply to my original comment. Unfortunately, I received the following automatic response from edspresso:
Your comment submission failed for the following reasons: In an effort to curb malicious comment posting by abusive users, I've enabled a feature that requires a weblog commenter to wait a short amount of time before being able to post again. Please try to post your comment again in a short while. Thanks for your patience.
I assume I will be able to re-post that comment in another day or so, but I thought MathNotations' readers may want to see a preview.
Here is my original comment to Mr. Garelick's post and his recent reply:
"There are also teachers who maintain a truly balanced approach and who, while rejecting the discovery-oriented and textbook-less programs being foisted on schools across the country, are admonished by their administrators to do as they are told."
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.
Mr. Garelick's Reply:
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.
Posted by: Barry Garelick | April 14, 2008 11:06 AM
Here is my latest reply of 4-18-08 which was temporarily rejected:
Dear Mr. Garelick,
"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."
"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..."
I never suggested that direct instruction of powerful models such as rate problems is not desirable, nor do I hold these traditional rate problems in disdain. In fact, in my classroom, I always used the same RATE X TIME = DISTANCE chart that I learned in high school, several decades ago. The emphasis on my blog is on rich open-ended investigations that one normally does not see in most textbooks. Problems that teachers have to search for in most cases. Most of my regular readers know that success with these problems require a strong base of skill and knowledge of traditional algorithms. I've stated this repeatedly.
Consider what happened when the AP Calculus Exam changed dramatically a few years ago. The tried-and-true methods still were needed and still were assessed. But the test changed dramatically from an emphasis on technique to an emphasis on application and deeper understanding of fundamental concepts. The decline in scores was predictable and occurred until textbooks and instruction were revised to address this shift in focus. The most capable students were able to 'generalize' from formulaic problems - they always will. However, the scores on the AB Exam initially reflected that many others could not. It has been my experience that students perform well on non-routine problems when they have that strong base of skill AND experience with many nonroutine problems that require more than a superficial understanding of content. When presented with such problems, students need TIME TO EXPLORE. Direct instruction initially will usually fail in this case. After giving a reasonable amount of time to grapple with the problem and discuss it, the solution is explained clearly and thoroughly, with alternate methods presented if time permits. One cannot rush this process, that's why it is called a process...
That's my definition of 'balance.' There's nothing fuzzy about it. I've never suggested that students be given challenging problems and left entirely to their own devices to invent something out of nothing. I have suggested they need to know their trig values, their trig identities, their differentiation and integration formulas well. After they have demonstrated this KNOWLEDGE, they can be challenged with problems they have never seen before. Exploration does not mean one is "a blind man, searching in the dark for a black cat." (Luv that Escalante quote!). Exploration means that one is PREPARED to explore and then given the OPPORTUNITY to EXPLORE!
Me. Williams and I share many common beliefs about teaching. But he doesn't represent even a majority of math teachers out there. This is why I repeatedly emailed the Panel asking for more frontline teacher representation on the NMP (and, of course, I was politely dismissed). There was not a single high school math teacher represented, not to mention an underrepresentation of research mathematicians.
I appreciate your thoughtful reply, but you attempted to pigeonhole my beliefs. No one yet has been able to do that and I'm afraid you did not succeed either! I welcome an opportunity to discuss this further with you, perhaps in another venue. How about a debate on my blog? We'll probably discover we have far more in common than one might think.
Oh well, nothing has changed. Does anyone out there really get my balanced perspective, other than my faithful readers of course!