The Curriculum Void: Devoid of Proof

by Al Cuoco, EDC  

The NPR story, “The Common Core Curriculum Void,” opened with a crucial distinction that both critics and proponents of Common Core often miss: Standards are not the same as curricula. Standards lay out a set of proficiency goals—what students should be able to do and understand. Curricula are vehicles for attaining goals that provide teachers with texts, support materials, and tools to help students meet standards. Just as Google Maps provides several route options to reach a given destination, different curricula offer a range of pathways to proficiency. Common Core is a standards document that several curricular designs can support. The NPR piece made this distinction, clearly and eloquently.

The story went on to feature interviews with educators from around the country, all of whom claimed that it’s impossible—with such a short turnaround to publish Common Core-aligned curricula—to support teachers as they help their students meet the standards. I disagree. The Common Core did not appear out of a clear blue sky. It is the endpoint (some would say midpoint) of the evolution of ideas that have a long history and pedigree. It builds on the work of generations of mathematicians and educators, all devoted to closing the gap between mathematics as a school subject and mathematics as a scientific discipline.

I have experienced this disjuncture firsthand. For four decades, making school mathematics more faithful to the real thing has been a centerpiece of my work. Yet I didn’t always feel this way about mathematics.

When I started teaching high school, I thought that mathematics was an ever-growing body of knowledge. Algebra was about equations, geometry was about space, arithmetic was about numbers; every branch of mathematics was about some particular mathematical objects. Gradually, I began to realize that what my students (some of them, anyway) were really taking away from my classes was a style of work that manifested itself between the lines of our discussions about triangles and polynomials and sample spaces. I began to see my discipline not only as a collection of results and conjectures, but also as a collection of habits of mind.

This became concrete for me when my family and I were building a house at the same time I was researching a problem in number theory. Pounding nails seems nothing like proving theorems, but there is a remarkable similarity between the two projects. It’s not the fact that house-building requires applications of results from elementary mathematics (it does, by the way). Instead, the two projects were alike because of the kinds of thinking they demand. Both theorem-proving and house-building require you to perform thought experiments, to visualize things that don’t (yet) exist, to predict results of experiments that would be impossible to actually carry out, to deal with complexity, and to find similarities among seemingly different phenomena.

For over 20 years, ever since we built that house, I have focused on fostering students’ mathematical ways of thinking in my classes and curriculum writing. I am not alone in infusing mathematical ways of thinking into curricula, and I am convinced that it is one of the most important things students can take away from their mathematics education. For all students—whether they eventually build houses, run businesses, use spreadsheets, or prove theorems—the real utility of mathematics is that it provides them with the intellectual schemata necessary to make sense of a world in which the products of mathematical thinking are increasingly pervasive in almost every walk of life. To cut back to the Core: The Common Core standards, particularly the eight Standards for Mathematical Practice, emphasize these same mathematical ways of thinking. As is the case with my own curriculum, and that of colleagues, the Core elevates these mathematical ways of thinking to the same level of importance as the results of that thinking. This is not a new approach. It did not happen overnight. And there are existing curricula—with mathematical ways of thinking baked in—that can help teachers support their students.

The “Common Core curriculum void” is far from the black hole described by the NPR story. In addition to curricula, a variety of resources are available—for example, the nonprofit Achieve The Core has produced a set of criteria for publishers, the Massachusetts Math-Science Advisory Council discussed a list of benchmarks for curriculum adoption, and Illustrative Mathematics contains many tools that can help adoption committees. To “bridge the void” we need to point educators to the right resources, sending them and their students on an efficient route to mathematical proficiency.

Al Cuoco is a mathematician and researcher at Education Development Center, Inc. (EDC) based in Waltham. A former high school math teacher, he holds a Ph.D. in mathematics and has spent the past 20 years in curriculum development and teacher education. http://mpi.edc.org

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MPs in High School: Algebra as Bookkeeping

Al Cuoco, the director of the Center for Mathematics Education at EDC, has written some essays that show how mathematical practices can be used in high school. The essays explore examples of Common Core’s Standards for Mathematical Practice. We’ll post all the essays here in the coming weeks.

For more information about EDC programs around the Mathematical Practices, visit http://mathpractices.edc.org and http://mpi.edc.org/dmp-hs-sampler.

The second essay is on the use of formal algebra for bookkeeping, and you can use the methods to solve at least one of this site’s “We Got A Problem” problems. (Click on the link to view PDF.)

MPs in High School: Perseverance

We have two different programs at EDC devoted to the Standards for Mathematical Practice., Implementing the Standards for Mathematical Practice (IMPS) and Developing Mathematical Practice in High School (DMP). For more information, see each program’s website:

http://mathpractices.edc.org/

http://mpi.edc.org/dmp-hs-sampler

These programs involve workshops for teachers. Al Cuoco, the director of the Center for Mathematics Education at EDC, has written some essays that show how mathematical practices can be used in high school. The essays were written for these projects, but we think the essays might be useful to others working to implement the Common Core. We’ll post the essays here in the coming weeks.

The first essay is on perseverance. (Click on the link to view PDF.)