by NCTM President J. Michael Shaughnessy

*NCTM Summing Up*, March 2011

In the February President’s
message, I addressed the issue of alternative pathways for our secondary
mathematics students as they make the transition from high school mathematics
into post-secondary mathematics in colleges, community colleges, and
universities. In that column, I posed several questions that catalyzed my
reflections on the need for alternatives to the current predominant pathway
available to our secondary students—the pathway that leads to college calculus.
Among the questions that I posed were the following: “What can we do to provide
students with relevant, coherent mathematical options on their pathway through
high school and as they move into college in the 21st century?” and “Is the ‘layer
cake’ of algebra-dominated mathematics that pervades U.S. secondary schools
still relevant?"

These questions have
subsequently prompted me to reflect not just on transition issues, but on the
entire secondary mathematics experience that many, if not most, of our high
school students undergo in this country. I received a large number of responses
to that column, and that input has provided some added motivation for this
message. In this new column, I want to make a case for integrating the
mathematical content areas throughout our students’ secondary mathematics
experience.

In my view, the “layer cake”
approach to high school mathematics that currently dominates so many secondary
school mathematics programs—built on course sequences such as algebra I,
geometry, algebra II, or algebra I, algebra II, geometry—is an outmoded
approach in a 21st-century educational system. There are a number of reasons
why I believe that at this point in our history an integrated approach would be
an improvement over the “layer cake” approach. Among them are some important
interconnected challenges that we face: we need to (1) lay the groundwork for
more mathematics options in the transition from high school to college; (2)
understand the vision and approach of the Common Core State Standards for
Mathematics, which have been adopted by so many states of and (3) reflect on
what it means to be internationally benchmarked. Let’s consider each of these
in turn.

If we are truly going to
build viable options for our high school students to make the transition into
college mathematics by a path that is different from the path to calculus, we
need to lay the foundations for those alternative transition paths throughout
high school mathematics. We must not reduce other possible paths to just an
add-on course in the fourth year of a high school experience. Students can—and
should—have opportunities to learn content in both geometry and data analysis
and statistics while they are learning algebraic skills and algebraic
representations of mathematical concepts. Statistics relies on both symbolic
algebra and functional algebra to represent measures of center, spread, and
association. The geometry of graphic arts depends on linear algebra and
matrices in computer representations. Exposure to relevant applications of
algebra integrated with statistics and geometry *throughout* a high school student’s learning of mathematics will
help instill more meaning and sense making in his or her algebra experience and
lay a foundation for transition options to college mathematics.

The Common Core State
Standards for Mathematics (CCSSM) can be thought of as an unprecedented
opportunity for rethinking potential pathways through K–12 mathematics,
particularly pathways through secondary mathematics. The secondary level
standards in the Common Core are presented in a way that actually invites the
integration and interweaving of algebra, geometry, and data/statistics
throughout the first three years of high school. In fact, the two sample
pathways through secondary mathematics that the appendixes to CCSSM present
provide approaches involving an integration of mathematical content, with one
of those sample pathways offering an approach that is more heavily integrated
than the other.

In
recent years, a hot topic in the news has been the mathematics performance of
students in the United States as compared with that of students in other
countries. With U.S. students placing below the middle on many of these
international comparisons, we have heard continual calls from policymakers and
in the media for mathematics learning in the United States to be
internationally benchmarked. However, it’s not clear just what people mean when
they use the term “internationally benchmarked.” One possible approach to
benchmarking is to focus on the mathematics pathways through which students
learn and experience mathematics. If we take this approach, it is currently
impossible to benchmark mathematics learning in the United States in
international comparisons because it makes no sense to internationally
benchmark a country that takes a “layer cake” approach to its mathematics while
90 percent of the rest of the world teaches mathematics by using an integrated
approach. Our country’s approach to mathematics is the exception when compared
with most of the rest of the world. If we want to be accurately benchmarked
internationally, we will need to take an integrated approach, especially in our
secondary mathematics curriculum.

I
can already hear the arguments against taking an integrated approach to
secondary mathematics. I’ve heard the excuses many times throughout my career:
“But we've just adopted a new curriculum—we can't change again now!"
"We can't do that—the colleges won't accept our students coming in with an
integrated curriculum—the colleges won’t know what to do with them!"
"We can’t switch—we have no money to buy new books." And so on.

The
states that have adopted the Common Core State Standards have three years to
implement them. The two Assessment Consortia are now beginning their work in
developing and piloting the assessment instruments that will be put in place in
2014.

Students
need to see mathematics as an integrated whole, with connections across the
content domains, and they need to experience some of the applications and uses
of mathematics *before *they transition
to college. And the United States will never show well in international
comparisons of mathematics performance as long as other countries have an
integrated mathematics, and we take a “layer cake” approach. In this country,
we have an unprecedented opportunity over the next few years to integrate the
content of our secondary mathematics, and we should do everything we can to
make the most of that opportunity.