Sunday, November 01, 2009

My Hat Is Still Off To You

The below article is from the opinion section of today's Newsday. I learned to teach from Professor Posamentier, being a student of his in my undergraduate, graduate and post graduate studies. He was a remarkable man then and still is. While I don't always agree with everything he writes (and I have a couple of little differences with this article as well), I know he always speaks the truth and gets to many of the problems in math education today. For those out there who hate my teaching style, blame him. I saw him at a conference a few years ago and realized that I have taken his methods and expanded on them for myself.

I publish the article here because Newsday has just restricted its online service to subscribers only.

Alfred S. Posamentier is professor of mathematics education and former dean of the School of Education at the City College of the City University of New York. His latest book is "Mathematical Amazements and Surprises."

The results of the most recent National Assessment of Educational Progress that show no gain in fourth- and eighth-grade mathematics achievement in New York are quite disconcerting, especially since the state's own standardized testing has been measuring improvement.

Naturally, some educators' immediate defensive reaction was to question the reliability and consistency of the national test. But we have to assume that test writers expect this reaction to poor results and will have taken every possible precaution to protect themselves.

If that's true, then were the New York State tests less complex? Were the tests progressively easier from one year to the next? Cutoff scores for passing have been lowered over the past few years, but Education Department officials in Albany say that the difficulty of passing has remained constant.

We may never know precisely what the gap in scores shows; this would require a complicated study. But we do know that there are plenty of other reasons to be concerned about the teaching of mathematics in this state.

College professors of science and other fields relying on mathematical competence have been extremely disappointed in recent years with their incoming students' preparation in this regard. What is done at the pre-college level - in all grades, not just fourth and eighth - is crucial because it sets the stage for future success in mathematics, not just in college, but in fields like science, engineering and technology.

If we knew exactly where this problem lies, we could solve it immediately; however there are many contributing factors.

We might well begin with the perception of mathematics competence in our society at large. Mathematics is the only subject in the school curriculum in which adults are comfortable - or even proud - to admit to having been unsuccessful during their school days. This declaration is often followed by a claim of continuing ignorance in the subject and confusion about the way math is currently being taught.

Take parents who receive their child, coming home from school with two test results - a 70 on an English test and a 70 on a math test. Typically, the parent will be aghast at the English result. The reaction to the math result will be relief that the child at least passed. Then the parents will praise the child for this achievement - often stating that they themselves didn't do any better when they were in school.

Where does that leave the expectations for the child? High for English, and low for mathematics. Research shows that such expectation is an extremely important factor in a student's achievement.

Let's take this a step further. Elementary school teachers, who are of course a subset of the general population, are responsible for teaching a variety of subjects in their classrooms. Reflecting the societal discomfort, many do not harbor a strong love for mathematics. These teachers could well exhibit their lack of enthusiasm in the classroom, even unconsciously, especially as compared with subjects they truly love, such as, say, history, art or music.

Too many elementary students are not being provided with the enthusiastic motivation that might catapult them toward a love of the subject. Then add the emphasis at the state and federal levels on standardized testing, which leads to "teaching to the tests." This does not support more genuine learning or enthusiasm in either students or teachers.

We would expect better outcomes in middle and high school, where mathematics is taught by math specialists. But here as well, instruction is often compromised not just by testing, but by less-than-properly prepared teachers.

Over the past decade New York State - particularly in the large urban centers and some suburbs - has suffered from severe shortages of math teachers. With schools' absolute need to cover all required classes, some teachers who have passed through alternative certification programs - essentially a summer crash course - have been assigned to teach the subject with as little as one year of college mathematics, with the obligation to take additional mathematics courses as they are teaching to make up for their lack of preparation.

A fair number of these teachers leave the profession within the first two years, forcing the cycle to continue. Just think how many children might have been shortchanged by a teacher whose knowledge of the subject goes barely beyond that of the textbook being used in the classroom? It is as unthinkable as an English teacher offering a unit on Shakespeare having read only "Hamlet."

Consider the child who comes up with alternative or unconventional ways to solve a problem, do a computation or analyze a concept. A well-prepared teacher can respond comfortably and with confidence. But a teacher without a proper mathematics background would have great trouble adapting to the situation and, worse, could discourage the student and any enthusiasm he might be building for the subject by forcing him to think strictly in the way the textbook presents the material.

The inexperienced teacher is likely to use the exercises in the textbooks, which are mostly drill, whereas a qualified math teacher would quite likely - and at every opportunity - infuse problem-solving strategies into his teaching. Rather than mechanical learning, students would be encouraged to solve a problem, for example, by considering a simpler analogous problem, or testing an extreme condition on a variable to see how it might behave. (I give an example in the accompanying sidebar.)

Another problem for math performance has been a trend to move more substantive subject matter to the lower grades. At that level, the material has to be taught by drilling rather than as concepts. The younger students are generally not mature enough to understand the advanced ideas, although they can learn to mechanically get the answers to problems.

The shift of material, such as some algebra topics, was justified in middle school, where there was a lot of repetition of material from one year to the next and students can handle the broader concepts. But in elementary grades, making math a mechanical exercise, as is too often the case, is not a good way for students to gain a true competence or love of the subject. And ultimately, they all need command of the concepts to have a solid foundation for future study and certainly for any career where math might be involved.


Above all else, as we assess the dilemma of math achievement we must make every effort to ensure that the most important element of mathematics learning - the quality of our math teachers - is at the highest levels.

This means recognizing and financially rewarding good math teachers, which will in turn attract more qualified people. At the same time, we need to work relentlessly to provide support for parents. Schools should offer them programs that show the beauty of mathematics and the nature of the school's instructional program.

This will make the parents not only willing partners in the education of their children, but stronger supporters of mathematics in the community outside of the school.

Mathematics is essential for preparing today's youth for our ever-increasing technological era. It's high time we all take a more positive view toward its teaching and learning.

A more inspired approach to solving a problem

To illustrate how a problem-solving technique that relies on reasoning might be more useful than a textbook approach that uses formulas, we'll use the technique of exploring extreme situations.

PROBLEM: A car is driving along a highway at a constant speed of 55 mph. The driver notices a second car, exactly a half-mile behind. The second car passes the first, exactly one minute later. How fast was the second car traveling, assuming its speed was constant?

The traditional solution is to set up a series of equations, plugging the information we have into a formula or a chart. In this case the formula is: the rate of speed multiplied by the time traveled equals the distance traveled. (A quick example: A car going 60 miles per hour, traveling for two hours, will travel 120 miles; rate x time = distance.)

An alternate approach is to consider extremes. We assume that the first car is going extremely slowly, that is, at 0 mph.

The second car travels half a mile in 1 minute to catch the first car. Under these conditions, the second car would have to travel at a speed of 30 mph.

When the first car is moving at 0 mph, then the second car is traveling 30 mph faster.

So if the first car is traveling at 55 mph, then the second car must be traveling at 85 mph.

- Alfred S. Posamentier


Anonymous said...

thanks for sharing that.

mathematicamama said...

Thanks for the info. I just finished a search for the best price on this book and ordered it. It sounds great; I just hope I can find the time to read it. Maybe over the holiday break.

mathman42 said...

By the way, an excellent analysis though more can be said. I may have read too quickly but many elementary teachers are weak in Math concepts. Did you see the recent article as to how badly NJ students did on a recent projected state wide Algebra test ? Would love to see the urban/ suburban breakdown.

Alan said...

National math test scores continue to be disappointing. This poor trend persists in spite of new texts, standardized tests with attached implied threats, or laptops in the class. At some point, maybe we should admit that math, as it is taught currently and in the recent past, seems irrelevant to a large percentage of grade school kids.

Why blame a sixth grade student or teacher trapped by meaningless lessons? Teachers are frustrated. Students check out.

The missing element is reality. Instead of insisting that students learn another sixteen formulae, we need to involve them in tangible life projects. And the task must be interesting.

Project-oriented math engages kids. It is fun. They have a reason to learn the math they may have ignored in the standard lecture format of a class room.

Alan Cook