by NCTM President J. Michael Shaughnessy

*NCTM Summing Up*, February 2011

For some time I have been concerned about the mathematics
that we are teaching our K–12 students and whether it will prepare them for the
problems and challenges that they will encounter in mathematics, science,
technology, and engineering in the 21^{st} century. Recently, my
concern has reached the red-alert level, especially my concern about how we
prepare secondary students for the transition to college. I constantly ask
myself two questions: (1) Are we really offering our secondary students an
appropriate mathematics experience? (2) What can we do to provide students with
relevant, coherent mathematical options on the pathway throughout high school
and as they move into college? Or to put it another way:
(1) Is the “layer cake” of algebra-dominated mathematics that pervades our U.S.
secondary schools still relevant? (2) Is calculus the be-all and end-all goal
for the preparation of students for a successful transition to college? My
answer is, I think not.

For more than 20 years national organizations and prominent
leaders in mathematics education, including NCTM, have warned that our national
rush to calculus is misguided and not even an appropriate path for many
students. For secondary school mathematics, NCTM recommends a different
approach: “These four years of mathematical study will revolve around a
broadened curriculum that includes extensions of the core topics and for which
calculus is no longer viewed as the capstone experience.” (*Curriculum and Evaluation
Standards for School Mathematics*. NCTM, 1989). According to Lynn Steen, former president of the Mathematical Association of America
(MAA), “It is probably about time that we face facts: Aiming school mathematics
for calculus is not an effective strategy to achieve the goal of improving all
students’ mathematical competence.” (*Mathematics
Teacher* 100 [February 2006]). More recently, the MAA published
the *Curriculum Renewal Across the First Two Years* (CRAFTY) report, which examined the mathematical needs of many client disciplines, such
as biology, chemistry, economics, engineering, physics, and others. CRAFTY
advocates secondary mathematics that facilitates students’ transition from high
school to college by providing (1) a greater emphasis on modeling; (2)
consideration of multivariate topics; (3) an emphasis on computational skills
that are useful in other fields; and (4) a strong foundation in units, scaling,
and dimensional analysis.

Last month, I participated in a panel presentation, *Transition from High School to College:
Should There be an Alternate to Calculus?* at the Joint Mathematics Meetings
conducted by the Mutual Concerns Committee of NCTM and the MAA. This experience
caused me to look even more critically at what many of our students are
experiencing as they move through high school and on to college. I am not the only
one who is concerned. My co-presenters and many of the participants packed into
the room echoed similar thoughts. Both high schools and colleges are operating
under outdated assumptions. Among these are the assumptions that high school
students should take or be prepared to take calculus, and that the path to
calculus needs to be paved with frequent and repetitive overdoses of algebra.

Consider, for example, a typical student’s mathematics
transition path. In high school, a student takes algebra I, algebra II, and
perhaps pre-calculus. In college, this student may be put into Intermediate
algebra, followed by college algebra, and perhaps, yet again, pre-calculus.
This endless sequence of algebra courses is not an uncommon experience for many
students, and the attrition rate along this path is very high. Many students thus
mired in algebra discover they don’t need calculus, and they exit math at the
level of college algebra, never to return. This is an out-of-date, wasteful,
and repetitive transition path for our students. Worse, it does nothing to
improve our students’ disposition toward mathematics. (Read __NCTM President’s Message__, October
2010) When students are confined to this tunnel of
repetitive algebra, they never have opportunities to experience the beauty,
excitement, power, or usefulness of mathematics as called for in the NCTM
Standards (1989, 2000) or suggested by the CRAFTY report from the MAA.

The NCTM/MAA Mutual Concerns panel presented four concrete,
relevant, alternative mathematical transition paths for high schools and
colleges to consider. One path emphasizes quantifying uncertainty and analyzing
numerical trends. Its mathematical foci include data analysis, combinatorics,
probability, and the use of data collection devices, interactive statistical
software, and spreadsheet analyses of numerical trends. A second transition
path concentrates entirely on the development of students’ statistical
thinking, beginning in high school and continuing into the first year of
college. Statistical thinking involves understanding the need for data, the
importance of data production, the omnipresence of variability, and decision
making under uncertainty. This path differs both in purpose and approach from
an AP statistics course. A third path recommends building a transition grounded
in linear algebra. Linear algebra integrates algebra and geometry through
powerful vector methods. It offers an arena in which students can work with important
multivariable problems and provides students with general-purpose matrix methods
that will serve them well in many fields, including mathematics, science,
engineering, computer science, and economics. Finally, a fourth transition path
incorporates a suggestion that an alternative to calculus can be found in
calculus itself—but a vastly different calculus from the traditional calculus
I. This path concentrates on multivariate applications of both calculus and
statistics, because today’s application problems rarely involve single-variable
calculus or univariate statistics. We live in a multivariate world. Therefore,
students’ mathematics experience in preparation for their transition to college
should emphasize multivariate functions, partial derivatives, multivariate data
sets, and analyzing covariance.

The Common Core State Standards provide us with an
opportunity to rethink the sequence of school mathematics, as well as a
challenge to provide exciting new pathways and transitions from high school to
college mathematics. We need to offer students alternative pathways as they make
their transition from secondary school and into colleges. The mathematics paths
that we provide for our students need to prepare them for existing fields that
are changing rapidly, as well as for emerging fields—and for fields that don’t
yet exist. In my view, the current deadly sequence of ever-repetitive and out-of-
touch experiences in algebra—the sequence intended to lead students to a single
variable calculus course—will not accomplish this goal. It is time that we
replace the eternal algebra transition from high school to college with some
viable and exciting 21^{st} century mathematics alternatives.