For those of you who have been following my writing, you are aware that I teach math in middle school and that I am extremely interested in the most effective ways of teaching the subject. I majored in mathematics and am pursuing the teaching of it as a second career after having retired several years ago. I balance my teaching by writing articles that address the problems with math education in the U.S. You will also know that I was educated in the 50’s and 60’s, and thus obtained my math education via what is called “traditional” math teaching.
Because of my educational background and my beliefs in how math should and should not be taught, I often find myself engaged in the following dialogue, either in person or on the internet:
Someone: The traditional method of teaching math failed many students.
Me: The traditional method seemed to work well for me and many others I know.
Someone: You’re the exception.
Regarding my claim of traditional math working for me and others, I am mindful of the advice given by David Didau (author of “What if Everything you Know About Education is Wrong?”) who points out the following with respect to educational debates:
“If, in the face of contradictory evidence, we make the claim that a particular practice ‘works for me and my students’, then we are in danger of adopting an unfalsifiable position… We can insulate ourselves from logic and reason and instead trust to faith that we know what’s best for our students and who can prove us wrong?”
I will say in my defense, however, that “You’re the exception” is not much of an argument either. It is usually offered with either anecdotal evidence or none at all, and is based on a largely mischaracterized view of what traditional math is and was. I offer here some rebuttals to three of the most popular “you’re the exception” arguments. Such arguments occur at school board meetings, in casual conversations, on the internet, and —disturbingly — in newspapers and on television. You are free to use them at edu-conferences, cocktail parties, or at troll fests on Twitter.
1. If traditional math teaching were effective, the U.S. would be at the top of the world in math.
This argument ignores that in countries doing well on international tests, students learn math mainly via traditional means — and over the past two decades, increasing numbers of students in the U.S. have learned math using the reform-based methods. Reformers are quick to point out that Japan and perhaps other Asian countries actually use reform methods, ignoring the fact that many students are enrolled in “cram schools” (called Juku in Japan) which use the drilling techniques and memorization held in high disdain by reformers.
The argument also fails to consider that traditional math can also be taught poorly. There have always been good and bad teachers, as well as factors other than curriculum and pedagogy that influence the data. In order for such arguments to work, one would have to evaluate how achievement/scores vary when factors such as teaching, socioeconomic levels and other variables are held constant and when pedagogy or curriculum changes.
Studies have been conducted that examine how math is taught in specific areas of North America, as well as looking at the common traits of high-performing systems across the world. One such study indicates that when both conventional and non-conventional (i.e., reform) math are taught by well-trained teachers, students learning under traditional mathematics instruction show much higher achievement than those learning under the reform math methodology.
2. If traditional math worked, the knowledge learned in school would stay with us.
That people do not maintain proficiency in math as they age says less about traditional or reform math than about the way in which a population’s knowledge and skill base is maintained over a lifetime. It is not evidence of failure of traditional math. The results of not using math on a consistent basis can also be seen in a study conducted by OECD. In the study, people from ages 16-65 in over twenty countries, including the U.S., were given the same exam consisting of math computations and word problems. According to the study, “the percentage of U.S. adults between 55 and 65 years old who scored at the highest proficiency level (4/5) …was not significantly different than the international average for this age group.” These findings can be used in tandem with the first argument above since people in the U.S. in the 55 to 65 age group learned math via traditional math teaching—and the differences in proficiencies between the U.S. and other countries is not significant.
3. Traditional math failed to adequately address the realities of educating a large, diverse, and rapidly changing population during decades of technological innovation and social upheaval
This argument relies on the tracking argument, when many minority students (principally African Americans) were placed into lower level math classes in high school through courses such as business math. It is based largely on the following premises: “Most students did not go on in math beyond algebra, if that, and there were more than enough jobs that didn’t even require a high school diploma. Few went to college. Now most students must take advanced math, so opting out is not an option for them like it was for so many in the past.”
First, in light of the tracking of students which prevailed in the past, the traditional method could be said to have failed thousands of students because those students who were sorted into general and vocational tracks weren’t given the chance to take the higher level math classes in the first place — the instructional method had nothing to do with it. Also, I don’t know that most students must take advanced math in order to enter the job market. And I don’t think that everyone needs to take Algebra 2 in order to be viable in the job market.
Secondly, many of today’s students entering high school are very weak with fractions, math facts and general problem solving techniques. Many are counting on their fingers to add and rely on calculators for the simplest of multiplication or division problems. In the days of tracking and weaker graduation requirements, more students entering high school than now had mastery of math facts and procedures including fractions, decimals and percents.
Some blame the “changing demographics” on the decrease in proficiency, but this overlooks variables like poor curriculum and the reform-based approach to math which views memorization “workarounds” as deep understanding. Also frequently overlooked is the fact that students in low income families who make up the “changing demographic” cited in such arguments do not have access to tutoring or learning centers, while students in more affluent areas are not held hostage — dare I say “tracked”? — to poor curricula and dubious pedagogical practices.
The debate over traditional versus reform-based math has been going on for some time—for so long, in fact, that some on the reform side are saying that there’s nothing to discuss, it’s boring, just let teachers teach. I agree that we should let teachers teach, and that parents be given choices of what type of math they want their children to have. That doesn’t appear to be happening any time soon.
I believe that the debate should continue and that there is plenty to discuss. People may choose to use the information I’ve presented here — or persist in ignoring it. I don’t expect that I’ve changed anyone’s mind about anything, but I am always hopeful that there are some exceptions.
I also do not think that I am alone in drawing a distinction between reform and traditional modes of math teaching. While traditional math can be taught properly as well as badly, I believe that poor teaching is inherent in most if not all reform math programs. I base this on having seen good teachers required to follow programs that present content poorly, lack a coherent logical sequence and rely on questionable pedagogies.
I would like to see studies conducted to document how U.S. students who do well in math and science and pursue STEM majors and careers are learning math. The chances are fairly good that such investigations would show that in K-8, many students are getting support at home, from tutors, or from the many learning centers that are springing up all over the U.S. at rapid rates. Since tutors and learning centers (and parents) tend to use traditional methods for teaching math, I doubt that the clientele are exceptions to some ill-defined rule. In my view, as well as the view of many parents and teachers I’ve met, there are few exceptions to the educational damage reform math programs have caused, even when such programs are taught “well.”
Epilogue and Sales Pitch
Finally, if you have enjoyed this piece, there are others of comparable or better quality in my book “Math Education in the U.S.: Still Crazy After All These Years”. Available here.