An Opportune Time to Consider Integrated Mathematics
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.