NYC HOLD Honest Open
Logical Debate on math reform
Are
our school’s math programs adequate?
Experimental mathematics programs and their
consequences
opening
remarks
Do NCTM Standards-Based programs prepare
students for calculus?
Professor of Mathematics
The
In the current
debate on mathematics curriculum reform there are many points of agreement, the
most important of which is that traditional methods and materials have proven
inadequate for conveying mathematics to students and must be modified in order
to achieve that goal. Furthermore, most participants in the discussion would
agree that students should succeed at and enjoy mathematics and should
understand the mathematical methods they are using. Unfortunately, the definition of success in
mathematics, and even what is meant by ‘understanding’ of mathematics, is
itself a point of contention. The question to be addressed herein is much more
focused. What is the effect of proposed
modifications of K-8 curricula on children’s preparation for the mathematics
they will confront in college should they pursue a program of study with a
calculus requirement.
I will address
this question by drawing on thirty years’ experience teaching all levels of the
calculus sequence at the
There is an odd
perception is some of discussions of mathematics reform that concern with
calculus preparation is somehow elitist. In fact, a large percentage of
students are involved, because many majors and programs, including science,
engineering, medicine, architecture, computer science, business, and secondary
math education require one or more semesters of calculus. Indeed, in Fall,
2000, over 462,000 students in four year colleges enrolled in first semester
calculus, compared to a total of 1,156,000 students who began freshman year at
those institutions.
The relation
between NCTM Standards-based curricula
and college mathematics was addressed in the draft document of the report
issued in Spring, 2001 by a commission
of experts convened by Schools Chancellor Harold Levy and chaired by CUNY
Chancellor Matthew Goldstein, as follows:
“ Whenever an emphasis is placed on ensuring that applications are made to
‘real world’ situations …less emphasis is placed on arithmetical or
mathematical ideas and the formal, abstract contextual settings sought
particularly by students who will go on to become scientists, engineers,
mathematicians, computer scientists, physicians, and educators of mathematics.”
“Despite their many strengths, the
NCTM standard do not contain the rigor, algorithmic approach, formal methods,
and logical reasoning which are required of this small[1]
but critically important portion of the population.” (emphasis added)
The above
passage, omitted from the final commission report, expresses the profound
concern of experts that the NCTM standards offer a vision of mathematics
instructions that doesn’t include
appropriate preparation for college mathematics. What is the basis for such a
disturbing suggestion?
Let’s start by
examining a typical page from the TERC teachers’ manual, a page that purports
to demonstrate examples of children’s invented solutions to multiplication and
division problems. Note the ubiquity of children’s English language description
of their mathematical activities, in accordance with TERC doctrine advocating
that children tell a story about what they are doing. What is not obvious at
first glance is in fact quite disturbing: the problems have been carefully
chosen to be amenable to nonstandard solution methods. Indeed, the showcased
children’s solution techniques would not work if applied to randomly chosen
problems. The TERC solutions may be useful as pedagogy, but they don’t provide
a consistent procedure for actually solving math problems. For real world math,
in college or anywhere else, the TERC methods work only sometimes, and that’s
simply not good enough.
At this point I
want you to contrast the pages from the TERC manual with the mathematics that
students will experience in college. The page of mathematics on the screen is
taken from a nicely written first semester calculus final exam paper. What you see contains the solution to one
and a half questions out of a total of ten that appeared on a recent final exam
lasting two hours and twenty minutes. I mention these details to give a sense
of the quantity of mathematics that students must set to paper. What about the
quality?
The most blatant
contrast between the college exam paper and the students’ TERC solutions is
that there is no English language explanation on the calculus final paper. At first glance, one might think that the
calculus final, unlike the TERC example, doesn’t tell a story. Quite the
opposite is true: the calculus problem solution tells a complex and elegant
story, but condenses the narration into a universal language of mathematics
that is written in symbols rather than in words. English language commentary,
were it included, would be irrelevant and confusing, to say nothing of tripling
the amount of writing. The language of mathematics is symbols; the language of
TERC problem solving is English. It is
this dichotomy that lies at the heart of mathematics professors’ concerns that
TERC and similar programs provide an inadequate base for moving from the study
of arithmetic to the study of algebra and calculus.
Does the absence
of ordinary language on the calculus final suggest that the student was
performing by “blind rote” or without proper understanding? Of course not! Her
solution is beautifully organized. She understood exactly what needed to be
done and when to do it. She
analyzed a real life problem;
represented that problem by a diagram;
selected variables and wrote algebra
expressions to convert the problem to
symbolic representation;
selected the appropriate methods for
transforming the symbolic representation to a solution; and
carried out the transformation
correctly to solve the problem.
That’s quite a
bit of work in ten minutes, but that’s what’s required in calculus. Symbolic
representation and complex symbol manipulation are an absolute prerequisite to
success in college mathematics.
Let’s consider a
number of specific ideas of curricular reform that are discussed in the
mathematics education literature and compare their implementation in TERC with
what actually goes on in college mathematics.
1. Mathematics
should be taught using stories, pictures, and symbols. In TERC, nearly the
entire emphasis is on stories and pictures. That’s useful for pedagogy, but at
some point there has to be a transition from concrete to abstract symbolic
representations of problems. In calculus, stories and pictures play a small
part, whereas about 99 percent of typical student work involves symbol
manipulation.
2. In TERC,
children use blocks, pictures, and fraction strips to model the operations of
arithmetic. That’s fine for motivation; most teachers use such methods to one
degree or another. However, it’s much harder to represent five sevenths by
cutting up a fraction strip than it is to represent one third or one half, the
typical easy fractions of the TERC curriculum. That curriculum shows pervasive
bias toward choosing only the simplest examples, and seems to ignore the fact
that “unfriendly” fractions and numbers are the norm rather than the exception.
The benefits of both modeling and
visualization are limited. Indeed, excessive reliance on these techniques can
be dangerously counterproductive. Any effective curriculum must employ a clear
strategy for helping students graduate from reliance on mathematical
crutches. After all, when children study
algebra in high school and calculus in college, they need to work with x, y,
and z. If they have been trained to rely on models and pictures, children who
encounter variables will be in serious trouble: there’s no way to draw pictures
of x, y, and z.
3. In TERC,
children invent their own procedures for solving problems. As noted above,
these procedures work only for limited classes of problems. In contrast, real
mathematics requires problem solution methods that work both efficiently and
with perfect consistency. That’s why the standard algorithms were developed.
They organize the basic operations of arithmetic in the most efficient manner
possible for solving complex problems.
4. TERC students
are not taught the standard algorithms for either multiplication or division.
Why, in fact, are these “rote procedures” necessary? Aren’t they a waste of
time if they can be carried out easily by pushing buttons on a calculator? Or
so goes the claim of a significant portion of the mathematics education
community.
This argument is
both shallow and specious. As Alan
Siegel has said, mathematics is layered and deep. There is a continuum of
thought and logic that runs from elementary school arithmetic to high school
algebra to college calculus. The algorithms of arithmetic accomplish much more
than obtaining the right answer. They establish patterns of algebraic thought
and execution that reappear repeatedly throughout the precalculus
curriculum and as such are critical to students’ mathematical development. For
example, if students don’t learn to multiply 37 times 22 using the standard
algorithm division algorithm, they will encounter great difficulty when a minor
variant of that algorithm is used to solve the related problem
3x + 7 times 2 x + 2 in high
school algebra. There’s a direct
connection. Similarly, long division matters, not because children need the
answer, but rather because they need hands-on experience with the logic
included in the method. A child who hasn’t become proficient with the standard
long division algorithm will be totally lost when the problem 
transforms to the polynomial division problem 
in high school. In
turn, that polynomial long division algorithm is needed at three distinct
places in the college calculus curriculum. The instructor will assume that
students know it cold: those who don’t will be lost.
The reputations
of standard American algorithms, especially the one for long division, are at
times impugned by math educators’ assertions that different algorithms are used
in different countries. This statement arises from a misperception that the
term “algorithm” refers to the format, or tableau, that is used to present the
solution. In fact, the algorithm is the sequence of logical steps involved in
the solution. All countries use the same
division algorithm. The only variations are minor differences in the number of
intermediate steps that are recorded and in their placement on the division
tableau. Students who learn to do long division in Russia and the United
States, where the division tableaux are in fact rather different, can easily
make the transition from one tableau to the other because both utilize
precisely the same division algorithm, whose use has been universal (or at
least planet-wide) for centuries.
5. TERC
profoundly de-emphasizes computation. Overall, the total number of
computational examples provided with K-5 TERC student sheets is modest. Of
course, teachers can supplement these sheets with classroom activities and
games. Of critical concern, however, is the student sheets’ bias toward the
selection of easy problems. Indeed, in the entire set of published materials,
the number of examples that require students to perform any of the following
single-digit multiplications:
6 x 6, 6 x 7, 6
x 8, 6 x 9, 7 x 7, 7 x 8, 7 x 9, 8 x 8, 8 x 9, and 9 x 9
either as standalone problems or as part of a multi-digit problem, is less than twenty. By a huge margin,
that’s much too little practice to prepare children for the increasing demands
of symbol manipulation they will face as they progress through algebra on their
path toward calculus.
6. TERC emphasizes fluency; mathematics
requires proficiency. What’s the difference?
When I use the
term proficiency, I mean that a child masters basic arithmetic facts and
responds to questions such as “6 times 7” reflexively, within a second or two
at most. Communication with TERC curriculum developers reveals that they care
relatively little about this type of performance. Indeed, they are on record as
being concerned more with “fluency,” understood as follow. When a child is
asked “6 times 7” he tries to work out a clever method of approaching the
problem, e.g. “3 times 7 equals 21, then double the answer to get 42.” The
amount of time spent is of minor concern: what’s important is the thought
process.
This is a
perspective that I characterize as interesting, at best. In fact, speed is
essential. It’s fine for children to use such strategies, but only as a step
along the way to mastery. Pre-algebra
mathematics requires mastering and working with a foundation of basic
arithmetic facts that can be summoned automatically for the solution of more
complex problems. As the philosopher and mathematician Alfred North Whitehead
shrewdly observed in his Introduction
to Mathematics:
It is a
profoundly erroneous truism repeated by all copybooks, and by eminent people
when they are making speeches, that we should cultivate the habit of thinking
of what we are doing. The precise opposite is the case. Civilization advances
by extending the number of operations that we can perform without thinking
about them. Operations of thought are like cavalry charges in a battle--they
are strictly limited in number, they require fresh horses, and must only be
made at decisive moments.
Whitehead is
saying, among other things, that mathematics problems must be solved by omitting
unnecessary details. In our setting, the mathematics that leads up to calculus
is structured and hierarchical. Students must master and absorb each stage to
the point of automaticity before proceeding to the
next stage: proficiency is essential.
“Fluency” as
advocated by the TERC developers can indeed be desirable. To the extent,
however, that it encourages a consistent preference for a derived solution over
a memorized one, advocacy of fluency at the expense of proficiency is downright
dangerous for students who must gradually develop the patterns of thought and
practice that are essential to the efficiently and accurate solution of complex
multi-step problems.
In closing, I
would like to add a personal note. Frequently discussed in discussions of
curricular reform are the pain and boredom of memorizing multiplication tables.
Indeed, this sort of memorization is more difficult for some children than for
others. But there is no royal road to proficiency in mathematics. My concern is
with the later part of a child’s journey, when he or she arrives in my calculus
class. At that stage, unfortunately, I
encounter a lot of suffering: not the pain of students struggling on a test,
but rather the shock of flunking calculus and being forced to change majors.
All too many students encounter a sudden and rude awakening, one that amounts
to “You can’t be a doctor [an engineer, a chemist, an architect]; sorry, you
don’t have proficiency.”
When I all too
frequently am forced to be the agent of such a message, I share my students’
pain. The sad reality is that their pain could have been avoided had they
studied a balanced K-12 mathematics curriculum that suitably emphasizes algebra
and formal skills.
Curricular reform is needed. Sadly, at least
from the perspective of children who want to pursue mathematics based careers,
current efforts at reform are moving in
precisely the wrong direction, by de-emphasizing the critical role played by
symbol manipulation and algebraic fluency. Instead, curricular reform should
acknowledge the importance of these skills and seek innovative and effective
methods of helping students develop their proficiency in arithmetic and
algebra. Educators and theorists must awaken to the reality that every calculus
course requires its students to enter with a substantial reservoir of formal
skills.
Perhaps the most
misused term in the mathematics education literature is “(conceptual)
understanding.” In fact, there is no such thing as mathematical understanding
without formal skills. A student or teacher who talks about a problem does not
understand the problem until he or she reduces the statement of the problem to
symbols and then finds an economical way to transform those symbols to a
solution. Talking math is not doing math. College level mathematics without
symbols is cocktail party mathematics, no more, no less.
Perhaps the most
puzzling feature of mathematics curriculum reform has been the absence of
meaningful participation by the community of academic mathematicians. It is imperative
that children who are preparing for college mathematics study curricula that
have been devised in meaningful co-operation with college mathematics and
science teachers. This has not been the case. Indeed, the de-emphasis on symbol
manipulation skills in most if not all NCTM Standards-based curricula poses a
clear and present danger, both to American children and to the future American
technology. To help children succeed in mathematics, we need to have curriculum
development and revision that focus on improving the study and development of algebraic
skills rather than throwing them away.