The Common Core State Standards for mathematics incorporate three "key shifts" that are meant to make math education as a whole more effective. These shifts include focusing on fewer topics, increasing the rigor of math curricula and implementing mathematical coherence. Coherence is an idea that's been around for years in math education, but with its addition to the Common Core, more schools and districts are making an effort to incorporate it into lesson plans. But what exactly is mathematical coherence? Here's more on this key shift in math instruction and why it's important:

**What is mathematical coherence?**

In basic terms, mathematical coherence is the idea that all math subjects are connected to one another, and that educators should implement that connectedness into their teaching from grade to grade. According to an article published in the journal American Educator, coherent standards are those that "evolve from particulars to deeper structures inherent in the discipline." In other words, basic mathematical concepts can be built upon and related to one another, resulting in a deeper understanding of more complicated mathematical concepts. The same article describes the state of coherence in the U.S. at the time, saying, "In the United States, mathematics standards are long laundry lists of seemingly unrelated, separate topics." Over the last few years, math standards, especially those in the Common Core, have been working to change math education from teaching those "seemingly unrelated" ideas into connecting each mathematical concept to a previous one.

**Why is coherence important?**

Coherence is an important part of math education because, when done effectively, it helps students understand mathematical ideas more deeply while also making complex subjects easier to understand. For instance, basic math operations like multiplication and division, which are taught in early elementary school, will be used continuously over the course of students' math education. Those skills can be built upon later in elementary school to help students learn how to multiply and divide fractions and decimals. Then they'll be added to again for more complex operations in subjects like algebra and geometry. In that case, a true understanding of those basic concepts coupled with a curriculum that helps students make connections is key to a coherent math education as a whole.

**How can teachers implement coherence?**

The Common Core Standards in math were written to sequence mathematical concepts in a coherent way across grade levels, based on evidence of what has worked in the past. That means that, in states that have adopted the Common Core, schools and school districts are already building more coherent curricula. But there are some ways that individual teachers can implement coherence into their classrooms. One idea is that math teachers at each school or within the same school district can work with one another to create more coherent transitions and progressions between grades, which many math educators are already doing, according to Education Week.

An article from the National Council of Teachers of Mathematics provides a few other ways for teachers to take coherence into their own hands. Most importantly, teachers can assist their students in making sense of math concepts and teach mathematical reasoning. Helping students make sense of why certain mathematical operations work is a key way to a full understanding of what they're doing in class. Teachers can reinforce these concepts by encouraging students ask questions, showing them alternative ways to get the same results and allowing them to defend their answers. This promotes deeper learning, rather than simple memorization, and will help students better understand future math concepts.

Mathematical coherence is an integral part of any effective math curriculum, which is why it's such a large focus of the Common Core. As schools and districts create their own Common Core-aligned curricula, they should consider the progressions and sequences of learning that make the most sense for a coherent understanding of math.