Solid math standards: Necessary...but not sufficient

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Solomon Friedberg

Some authors get paid by the word. I’m so glad that I don’t. In mathematics, my field, a few words can describe a vast terrain. For standards there are four words that all evaluators, all policymakers, and all parents would do well to keep in mind: necessary but not sufficient.

This phrase means what you’d think. Suppose you want to paint a picture. You’ll need some paints, brushes, a canvas to paint on. They are each necessary. But they are certainly not sufficient. You need an idea, and you need skill. The raw ingredients are necessary—you can’t make a painting without them. But they are only the beginning. They are not enough to get the job done. They are not sufficient.

Standards too are necessary but not sufficient. Mathematics standards specify what material is to be taught: In sixth grade, students begin to learn about percentages, and they extend this knowledge in seventh grade. Since so many constructions in math depend on earlier ones (to work with percentages, you need to understand fractions), it really matters that things come in the right order. All sorts of educational decisions get rolled into the standards, and if they are done wrong, few teachers will be able to undo the damage.

But just having good standards is not enough. To make standards work, someone has to think about all the other ingredients that go into schooling. For math, that means communicating the standards in detail to teachers, finding good textbooks and other materials that follow the standards, supporting teachers in using them, designing good evaluation materials to be sure that kids are learning, and finding the expertise and resources to provide extra support for students who need it—such as students who did not learn prior material that reappears as the foundation for a new concept.

All these extra ingredients get mixed in with the standards themselves; they are not the standards, but they are related to them. This is part of why there has been such a vigorous national debate about standards. When a state asserts that kids should learn something in fifth grade, it’s reasonable to ask if they have done so. So testing enters the story (in fact, it is mandated by the U.S. Department of Education). Some policymakers, wanting to spend tax dollars wisely, then impose consequences for schools or teachers whose students don’t test well—though there is nothing in the standards telling them to do so. States might think it’s appropriate to require a certain level of achievement to graduate from high school. Schools and districts might or might not provide effective support for kids who are struggling. These are important educational policy decisions, sometimes made well, sometimes not, but all linked to the standards. Nonetheless, the standards do not tell you to test. They do not tell you to blame or reward. They tell you only what math a typical student needs to learn, at each grade level, to be well-prepared for college and career.

So when we evaluate math standards, we focus on the math content. Are the learning goals structured coherently? Are the most important topics given the most air time? Is there an intelligent balance among skills, understanding, and problem solving? If these elements are flawed, then it is time for changes. But once we have good standards, it is time to focus on something else—implementation.

What does it take to implement good math standards? The most important ingredient is clear: teachers. Teachers—who in each and every class make dozens of choices based on knowledge, observation, and experience—are the key to so much about student learning. They need the support of their communities, they need class sizes that are manageable in schools that are not falling apart, and they need support as teachers of mathematics. That is, math teachers need professional development that is specific to math, and they need the time to discuss student learning and approaches to teaching as a team of math teachers.

These content-specific kinds of support are critical. At the elementary school level, where many teachers are generalists (teaching English, math, science, and social studies), they need a deep knowledge of elementary math. Some of this is very specialized knowledge—for example, how to use student mistakes in ordering decimals to figure out what specific misunderstanding about decimals that student has. School districts should do more to provide cohesive ongoing support about math content. At the middle and high school levels, teachers are now being asked to teach more data and statistics. This requires training. And at all levels, teachers are being asked to do more to develop mathematical thinking, the kind of understanding that will allow kids to use math to handle a completely new problem. This requires sustained and content-based professional development, not a seminar on how to use a smart board or a lesson on kinds of student intelligence that is designed to appeal to all the teachers in a school.

Compared to top countries in math education, we are not doing nearly as much to enable another form of support: teachers supporting each other. Math teachers can form a professional learning community, in which they observe each other, plan lessons together, and discuss important aspects of pedagogy. These activities enable continuous teaching improvement, the sharing of good practices, discussions of the latest research, and on-going professional growth. But to participate in such a professional community, teachers need the time built into their workdays. Most of the countries that do the best on international comparisons provide this time, and so should we.

Good implementation requires real resources. So it is worthwhile to emphasize: Standards are only as good as the teachers, curricular materials, and school systems that implement them. Math can be taught merely as a bunch of arcane rules for solving different types of problems, or as a coherent set of concepts, skills, and reasoning that allow students to analyze today’s problems—and tomorrow’s. Getting to the latter, as envisioned by the best standards, requires people and institutions working together.

With its wide economic impact, investing in implementation is one of the wisest investments in our nation’s future that we can make. Indeed, research demonstrates a strong relationship between students’ math achievement and their subsequent earnings, while there is a clear link at the national level between math performance and GDP. And with data playing such an important role in everything from science to business, quantitative skills are becoming ever more important. For our nation to stay strong, we need to graduate more highly prepared math students than ever. The key to doing so is to invest in implementation.

As my coauthors and I document in Fordham’s new report, The State of State Standards Post-Common Core, the quality of state math standards still varies. But our findings are mostly encouraging—many states now have strong standards, standards that are suitable as the foundation for a high-quality K–12 math curriculum. In these states, further modifications of the standards would have at most a minor effect on student learning, and should not be a priority. Instead, we should remember that good paints alone are not enough to guarantee a good picture. Though good math standards are necessary, they are not sufficient to guarantee that our schools will graduate students who are well-prepared for today’s quantitative world. For the states with high-quality standards, it is time for our educational and political leaders to focus on their implementation.

Solomon Friedberg is James P. McIntyre Professor of Mathematics at Boston College. He is a Fellow of the American Mathematical Society and a member of the U.S. National Commission on Mathematics Instruction of the National Academy of Sciences. He served as leader of the Fordham Institute’s mathematics team in preparing its 2018 report.