By Bruce Winokur
Mathematics Instructor
Stuyvesant High School
New York City
May 24, 2005
I've been teaching mathematics in New York City public schools since 1969, first at IS 201 in District 5, then at JHS 17 in District 2, and since 1983 at Stuyvesant High School. I'm also the father of a 10 year old daughter who attends NYC District 2 schools and a member of an organization, NYCHOLD (www.nychold.com), dedicated to bringing sanity to mathematics education.
I'm a firm believer in public education, the great equalizer. Sadly, over the past 10 years I've witnessed how badly things can go wrong. I am referring specifically to the constructivist mathematics curricula that abound in our city public schools in general and more specifically in the district where I live, teach and raise my daughter, District 2.
Constructivist curricula, such as TERC and CMP, forsake algorithms, postulates, and theorems (the foundation of mathematics) as well as teacher centered learning. Instead, they have students working among themselves in groups, loosely guided by the teacher in a drawn out attempt to "discover" mathematical truths.
When these types of curricula were introduced in District 2 in the 90's, they were already highly controversial. They had been in use in both California and Texas where parents and educators protested their use in great numbers and with varying effect. In numerous cases, the curricula were replaced by more traditional ones. Never had there been such a controversial attempt at math reform. I recall the "new math" of the 70's and how disastrous a reform attempt that was. Even that failed effort generated significantly less negative publicity than this.
With an eye towards bigger and better things, people in District 2 embraced these controversial curricula. A lot of positive publicity was generated. People in line with the District leadership's initiatives were regularly promoted within the system. The curricula became entrenched. The District leadership spoke in one voice regardless of the well-known failures of these curricula. "Proving" their effectiveness was more important than honestly assessing them. We who either teach or parent District 2 students know of the failures of these curricula. We send our kids to math tutors in record numbers. Intelligent, hard working kids have trouble doing simple math. We who have grown up with an understanding of elementary mathematics find that we can't help our kids; that many of the games they play and homework they do are so convoluted we either can't figure them out or don't see their significance. We're forced to sit by and watch our kids' frustration, both kids who are having great difficulty and kids who are so talented that they're terribly bored with their school mathematics. When we speak to school officials about our frustration we're condescendingly told that we just need to understand what they're doing. The truth is that many of us do understand what they're doing. They're doing irreparable harm to thousands of kids.
In my neighborhood, the Upper East Side of Manhattan, an incredible number of intelligent young students from the fourth grade and up are seeing private math tutors. Many of these are not the type of kids who would normally struggle in arithmetic or elementary algebra. As a result of the way they're taught elementary mathematics, they find themselves unable to do real math. When they're taught math in a more traditional way by their tutors, they invariably find themselves relieved and highly critical of the way they've been taught mathematics.
At Stuyvesant, we have a disproportionate number of freshmen from District 2 taking our introductory algebra course. Most Stuyvesant students have already completed that course before they enter our school. The ratio of District 2 students to Non-District 2 students in those classes is close to twice that same ratio in the freshman class as a whole.
Would you invest all of your money in a speculative and highly controversial venture and then turn a blind eye to major problems that develop in that venture? That is what District 2 has done with our most precious assets when it concerns their mathematics education.
Mathematics is a science that has been developed over thousands of years. It's not just about adding, subtracting, multiplying, dividing and solving the type of everyday problems associated with going to the supermarket. If it were, perhaps these curricula wouldn't be so bad. Beginning at the middle school level and continuing to the high school and college levels it evolves into Euclidean and non Euclidean geometries, algebra, trigonometry, Calculus, and various types of discrete mathematics. Today's high school student is expected to master a significant amount of the aforementioned.
Constructivist mathematics curricula attempt to teach mathematics by having the students "discover" their own methods for solving problems. A great deal of time and energy is spent having students "discover" things such as if you're multiplying 98 x 28 , you could multiply the "friendly number 100" x 28 and from that subtract that extra 2x28. 2x28 can be found by multiplying 2x30 and subtracting 2x2.
This is fine for this problem and in fact is how many good mathematicians would perform this computation in their heads. However, it takes too long and it won't work for calculations such as 34 x 67, 286 x 327, or most others one would need to perform. The purpose of a standard algorithm is to easily and quickly solve a whole class of problems. It generalizes. We can do all problems of this type with the standard multiplication algorithm. In the constructivist curricula, a similarly haphazard way of working with fractions is taught, with similarly disastrous results.
A constructivist would argue that kids just memorize standard algorithms and get no feel for numbers. This is a misconception. When taught properly, a student is first introduced to place value. Then the distributive property is taught. With these principles already established, perhaps some student could discover the standard algorithm or, more likely, a teacher would introduce and drill it in a constructive fashion. Since it easily solves all problems of its type, it is then mastered. With that in place, students wouldn't have to struggle to do simple calculations. Many would truly "discover" the short cuts that the constructivists so value after they were well grounded in both the principles that enable us to multiply numbers and the standard algorithm. After all, the same principles that lead to the standard multiplication algorithm are applied in the short cuts.
What these constructivist curricula fail to recognize is that mathematicians who employ short cuts have first developed a level of mastery of the subject that could only be developed through years of hard work and deep thought. Students who are taught to rely on these shortcuts often crash and burn when faced with the real problems of mathematics.
Mathematics has certain rules. Students must understand these rules in order to succeed in the field. Fundamental principles must be taught. A foundation must be set. Then one can build upon that foundation using the established rules of mathematics. When students are left on their own, or led by a well-meaning teacher who is trained to teach these convoluted and inadequate curricula, not only don't they learn to do simple arithmetic adequately, but they are unable to take the natural step of learning to work with variables, which is where algebra begins. In fact, it's totally unnatural for them. One must understand both the rules of the game, the fundamentals, and the standard algorithms of arithmetic in order to understand how algebra and higher mathematics work. Without these understandings, constructivist mathematics students have a very difficult time understanding, appreciating and doing mathematics at higher levels.
So we're stuck with kids who have trouble with basic arithmetic and find it difficult, if not impossible, to advance much further.
In middle school and high school these students rarely complete the curricula because it takes far too long to "discover" truths that can easily be taught much better and faster by a skilled teacher. Even in the rare cases where students do finish the material, they don't have a sufficient background to succeed at higher mathematics.
I wonder why District 2 hasn't required the Regents exam for their students. Could it be that their curricula haven't adequately prepared students for this minimum competency exam that is so simple that it requires little algebra to succeed? How can that be acceptable?
There is a simple solution to this problem. Courageous parents and educators must unite and demand that mathematical sanity be returned to our district and our city. Politics seems to be the only thing that has an effect on our leadership. Let's let them know that the status quo is no longer acceptable. Let's rescue our kids. It's our responsibility.
You may get some ideas as to how to make a difference at www.nychold.com and www.mathematicallycorrect.com.
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