Mathematics Recalculated : Riverside Press Enterprise, Sunday 8 February 1998
by
John de Pillis
Professor of Mathematics, University of California, Riverside
Imagine
teaching a class in navigation even though your students are using instruments
that often produce incorrect readings. Imagine teaching your students to
fly an airplane, all the while, paying scant attention to what working
pilots have to say about your course.
This is pretty much the case with respect to how reform mathematics is being taught to our children --- with the use of error-prone mathematical "navigational devices" such as the hand-held calculator and with scant attention being paid to the "pilots" or working mathematicians (creators and users of mathematics).
To see why this is so, let us first consider
(1) Calculators make real errors. The student should be learning that for any non-zero number A, (1/A) times A = 1. This happens always, not just once in a while. But most commonly available hand calculators contradict this principle, telling you that this result is true only once in a while.
To see this for yourself, ask your calculator to multiply (1/3) by 3. (The answer should be 1.00.) First, produce the number (1/3) on your calculator by entering the four keystrokes, [1],[/],[3],[=]. Now multiply this by 3 with the three additional keystrokes, [*],[3],[=]. Your calculator is likely to tell you that (1/3) times 3 equals 0.9999999, not 1.00.
Worse yet, you will get a different answer if you reverse the order, 3 times (1/3) now equals 1.0000000.
My personal experience shows that most mathematics teachers are surprised to learn of this intrinsic error. Then, after the surprise passes, the error is too often excused or minimized because it is "so small."
Woops! What's happening to critical thinking here? What do we tell the inquisitive student or parent who wants this contradiction explained?
(2) Exclusion of mathematicians. Mathematicians have been ignored in the debate over mathematics reform for years. In fact, Henry Alder, Professor of Mathematics at UC Davis, in a 1995 statement to the California State Board of Education noted that no mathematician from any UC campus, or from Stanford, or from USC was included in the Mathematics Task Force (a body appointed in April 1995 by State superintendent of Public Instruction to address the need to improve the mathematics achievement of California's students).
In spite of the exclusion of mathematicians, many of the right words were used in the formulation of goals from groups such as the National Council of Teachers of Mathematics (NCTM).
Who can argue, after all, against the encouragement of conceptual understanding, critical thinking, and the application of these principles to the solving of real-world problems? What's not to like here? Especially since no one --- but no one --- is advocating the teaching of mathematics through mindless memorization of rules and tricks.
(3) How does practice compare to the promise? Goals are one thing --- methods are another. If we ask how well reform methodology produced mathematical understanding by California students, we can not ignore the fact that their recent test scores are not stellar --- they are cellar! This alone should cause some critical thinking about the effectiveness of current methodology.
As I inspected the textbooks, spoke to educators and sat in on some workshops, these were among the startling and disturbing facts that emerged:
A powerful characteristic of mathematics is the fundamental role of clearly stated, unambiguous definitions.
And what an intellectual bonanza this is! Clarity of terms that do not change their meaning produces solid results that are easier to understand. Clarity of terminology clears the table so that new results (theorems, consequences) can be developed and understood by a process called "proof."
For example, a "prime number," has had the same meaning to all mathematicians for centuries. (A whole number, such as 7, that is divisible only by itself and by one.) With universal agreement of what a prime number is, we then "prove," among other things, that any whole number can be factored as a product of "prime numbers" --- and this factoring can be done in only one way. (Anyone, anywhere, at any time who factors 30 into primes will always produce the same three prime factors, namely, 2, 3, and 5. That is, 30 = 2 x 3 x 5.)
In spite of the importance of this beautiful and powerful tool, clearly stated definitions are either ignored or minimized in our classrooms. For example, no mathematician would have endorsed one officially adopted mathematics textbook which states that a "fraction is a fundamental concept" while it fails to provide any definition whatever.
It should be noted that in almost all other areas, yearn as we may, we are not so fortunate as to have clear, timeless definitions that everyone accepts. (How many people share your definition of "liberty," or "happiness?") Yet, to be understood, we must try to attain the ideal of clarity in the meanings of our words. How ironic, then, that this ideal, so freely available in mathematics, is exactly what is given short shrift in most mathematics class rooms.
For some reason, current textbooks avoid presenting mathematics in a logical, linear manner in favor of glitzy graphics and prose that carry you back and forth, from examples on one page to exercises on another to expanded topics on yet another page.
I asked a teacher how her textbook could help the independent student to push ahead on her own. Simple. Just ask the teacher for the special sequence of pages that related to the topic. Hmmm.
And this from a methodology that says it encourages independent exploration and learning!
This school of thought has it that mathematics must always be put in a real world context. Real experience is thought to give the student a feeling for mathematics. And to some extent, it does. When a student actually measures sides of several triangles then there is meaning to the theorem that the sum of any two sides exceeds the third. What educators are missing, however, is that direct experience is not enough. Not in today's world.
For example, direct experience does not include living in the fourth dimension. Yet it is abstract four-dimensional mathematics that produces the three-dimensional computerized graphics you see every day in film and television.
This is only one example of how mathematics goes beyond "real" world experience to a "fantasy" world of infinities and higher dimensions, only to return with results that actually re-apply themselves to the real world. Mathematical creativity needs more than experience with paper, scissors and tape.
Summing Up: Maybe it is time to listen to the pilots. Here is what some mathematicians and mathematics users have to say: Abigail Thompson, Professor of Mathematics at UC Davis, notes that her daughter "wasn't learning any math" in school. The teachers just didn't know enough about mathematics. To address this deficiency, she launched a UC Davis program called "Starting With Math." According to Professor Thompson, an unexpected "hilarious" consequence of her involvement in strengthening the mathematics curriculum, is the artificial politicizing that follows. Although she happens to be a liberal Democrat, her "fundamentalist" mathematical philosophy has caused her to be labeled as a right-wing conservative.
Professor H. Wu, at UC Berkeley, notes in a lengthy report to the American Mathematical Society (AMS), that "mathematics reform is way overdue," but current reform has defects so great that we would need "yet another reform." And listen to Ken Ross, Professor of Mathematics at the University of Oregon, who, as President of the Mathematical Association of America said, "The reformers have a laudable focus on understanding [but it] has led to some decline in mathematical skill." He continues, "The reform movements need to address this issue."
Indeed it does. This was clear to Michael McKeown, a biology researcher at the Salk Institute and adjunct faculty member of UC San Diego. After recovering from what he saw in the mathematics classroom, he started the anti-reform San Diego based group Mathematically Correct.
If we avoid defensiveness and admit there are failings --- if we consult the pilots, then maybe we can get our kids to fly this mathematics ship right.
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updated and revised 26 February 1998
Other articles by J de Pillis send e-mail ===> jdp@math.ucr.edu