What if we lived in a world where people knew, used and enjoyed mathematics? What if mathematics made sense to students? What if teachers teaching fractions in Grade 3 could tell the story of how fractions flow naturally from whole numbers in Grade 2?

Until recently, they had to tell that story from incoherent textbooks based on a cacophony of voices from 50 different sets of state standards. Now that 45 states, including Arizona, have adopted the Common Core State Standards in mathematics, teachers can share ideas for conveying the coherence of mathematics across state lines.

Improvement in mathematics education ultimately depends on teachers. If teachers have focused, coherent and rigorous standards to work from, they can portray the way mathematical ideas build over time, for example by connecting addition of fractions to addition of whole numbers.

If teachers can show students the ways in which mathematics is used in science, engineering, and finance, they can reveal to students the beauty of mathematical ideas.

Illustrative Mathematics, at illustrativemathematics.org, gives teachers the tools to do this. Illustrative Mathematics arose out of a promise I made while working on the standards, to provide illustrative problems illuminating the meaning of the standards. But it has grown into much more: a community of 20,000 teachers and mathematicians who explore the standards and comment on grade-appropriate mathematics tasks.

Those who acquire expertise in writing, reviewing and editing math problems can become apprentices in a growing professional community dedicated to the craft of task design. We aim to create a discerning professional community of teachers who will provide the core of expertise for implementing the standards and build pathways to college and career readiness that all students can follow.

And while some of those students may struggle, they will still be on a pathway, rather than lost in a wilderness of disconnected ideas and techniques that have been memorized but poorly understood.

Here at the University of Arizona, the School of Mathematical Sciences is helping teachers reveal mathematics pathways landmarked by the standards. Illustrative Mathematics is just one of the ways we are doing this. We run weekend workshops for local teachers through our teacher induction program. We run national workshops for teachers across the nation. We weave the focus, coherence and rigor of the standards into our teacher preparation classes, and we develop online modules for teachers to learn about the thematic progressions in the standards.

The other day a teacher left the following comment on one of the tasks at illustrativemathematics.org:

“Used this simple task with my eighth-graders enrolled in High School Math II. It was the single best way I have ever reviewed function notation. And they came away absolutely understanding input, output, x, f(x), independent and dependent variables and how all of those terms are connected. It was fabulous.”

It made my day. In fact it is difficult to assess the kind of understanding the standards promote, and there are many challenges ahead. Neither this nor any single mathematics task is a panacea. But with teachers like this one, actively engaged in talking about what works in the classroom and sharing their knowledge with a national community, we can change the way mathematics education works in this country.

William McCallum is a distinguished professor in the University of Arizona’s Department of Mathematics. Contact him at wmc@math.arizona.edu