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September 05, 2018

Overall, mathematics standards in the United States are far stronger today than they were in 2010, when Fordham conducted its last fifty-state review. And much of that improvement is due to the Common Core math standards, which earned a rating of A- in our 2010 report and a score of 9 out of 10 in our most recent review. In general, the states with the strongest math standards are the ones that have built on the Common Core, modified it in minor ways, or independently drafted separate standards that mirror its pacing and organization.

So why are today’s standards better than the math standards of a decade ago? Here are four strengths that our expert mathematics reviewers found in state math standards in 2018.

**1. ****Stronger focus on arithmetic in grades K–5**

Because it is the foundation for much of the mathematics that students will encounter in higher grades, experts agree that arithmetic should be the primary focus of math instruction in grades K–5. Yet in 2010, the biggest problem we identified in state math standards was that arithmetic wasn’t a sufficient priority. As mathematicians Steven Wilson and Gabrielle Martino lamented at the time:

*Many states include solid arithmetic standards, but these are buried among a multitude of distracting and less important content... By failing to clearly prioritize this essential content, states fail to ensure that it gets the attention it deserves. Only a few states either explicitly or implicitly set arithmetic as a top priority. More often, states devote fewer than 30 percent of their standards in crucial elementary grades to arithmetic.*

Thanks in large part to the Common Core, that is no longer true. To the contrary, a clear focus on arithmetic is now evident in most states’ K–5 math standards. For example, most states’ standards begin with a clear focus on counting, whole numbers, and place value. And most also expect students to know their single-digit addition, subtraction, multiplication, and division facts—and to be proficient with the standard algorithms for these operations, as well as strategies related to place value and the properties of operations (usually by the end of third or fourth grade, depending on the operation and the expectation). Finally, most states systematically develop a strong understanding of fractions and decimals.

To be clear, topics such as geometry and measurement, the representation of data, and algebraic reasoning are also included in most states’ elementary standards. However, in strong standards these topics are connected to number and operations—enhancing rather than diluting the focus on arithmetic.

**2. More coherent treatment of proportionality and linearity in middle school**

The study of fractions is closely tied to proportional relationships and reasoning (i.e., rates and ratios). And such reasoning, in turn, provides students with a platform for understanding slopes and linear relationships (e.g., y=mx+b), which are a key foundation for algebra. Thus, the sequence and pacing of these topics is critical to helping students move from elementary to middle to high school mathematics.

In recent years, the treatment of all of these topics has improved in many states. For example, in most states that used the Common Core as a starting point, ratios and proportional relationships is a main topic in grades six and seven, slope is developed in grade seven, and linear equations are an important part of grade eight, where they are both analyzed and used to describe linear relationships for bivariate data.

**3. Appropriate balance between conceptual understanding, procedural fluency, and application**

In the past, math experts quarreled over the relative importance of students’ conceptual understanding, procedural fluency (or ability to compute quickly and accurately), and ability to apply what they have learned. Yet, as the 2008 National Math Advisory Panel noted in its final report:

*To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills. Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. *

Thankfully, judging from their current math standards, most states have embraced the importance of each of these capacities and the implicit compromise represented by the quote. For example, the introduction to the Common Core states that “mathematical understanding and procedural skill are equally important” while also asking students to “make sense of problems and persevere in solving them.” This tripartite mission is also evident in the standards themselves. For example, most states now ask students to explain their reasoning, in addition to performing computations and solving problems. And in addition to standards about formal mathematical proof and carrying out mathematical procedures accurately, most states’ high school frameworks now include modeling, which links classroom math and statistics to everyday life, work, and decision-making.

**4. Better organization and teacher supports**

Well-organized math standards do at least two things. First, they provide an account of key themes for each grade level or course, as well as a list of major benchmarks to ensure that instruction is appropriately focused. Second, they are organized in a mathematically coherent way that highlights how mathematical topics fit together within a grade or course and how they are connected to prior and future work.

The Common Core math standards are a clear example of well-organized standards. For example, prior to the content standards for each grade level (K–8), there is an introduction describing the focus for the grade and a bulleted list of critical topics. Similarly, each high school domain (or area of math) includes a narrative introduction, followed by the individual standards for each of the clusters in that domain. In general, the organization of the Common Core into domains and clusters provides teachers and other stakeholders with conceptual cues about the connections between individual standards and the intended learning progressions within and across grade levels. And helpfully, states such as California and Massachusetts have extended these positive features to high school courses, a step other states should also consider.

In addition to content standards, most states have also adopted practice or process standards, reflecting the broad consensus among math experts that there are certain “mathematical habits of mind” that educators at all levels should seek to develop in students. For example, the Common Core includes eight “Standards for Mathematical Practice,” abbreviated versions of which are listed in the introduction to each grade (K–8) and high school domain. And again, states such as Massachusetts have helpfully expanded on this approach by articulating particular expectations for each of three grade spans: pre-K–5, 6–8, and 9–12.

Finally, most states now include a mathematical glossary in their standards, as well as other resources and links. The form and content of these are too diverse to summarize here, but many are likely to be useful for teachers. For example, a number of states have developed “vertical alignment charts” that describe the desired progressions for particular topics across grades, and there is a “coherence map” for the Common Core that shows connections across both topics and grades.

***

As others have noted, strong math standards are just the beginning. To implement them well, policymakers, curriculum developers, principals, and, above all else, teachers must understand *why* they are strong so that textbooks, professional development, pedagogy, and practice reflect the same shared vision of mathematical excellence.

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