To teach math effectively, California must focus on deep, conceptual learning

3 elementary students

Third graders discuss possible ways to solve a new math problem.

Credit: Allison Shelley for American Education

Fierce wars continue to rage around math instruction, but there are many practical changes we should make for mathematics students upon which most of us can probably agree, that could transform their ability to achieve. 

A promising new initiative for California that we have both been involved with tackles two of the most pressing flaws of traditional math instruction with elegant solutions that should be appealing to many, no matter which camp they occupy in the debates. Ask any teacher of math what they wish they did not have to deal with, and they will tell you the excessive amount of content they need to teach, which leads to the second problem — the shallow coverage of hundreds of methods that students do not learn in meaningful ways.

U.S. math textbooks are massive and heavy tomes. By contrast, math textbooks in Japan and China are small and slim. The reason for this is that U.S. curriculum repeats content every year. In China and Japan, content is taught less frequently but more deeply and conceptually. As teachers in the U.S. are forced to “cover” an extensive amount of content in every year of school, students only gain a shallow experience of mathematical methods and rules.

The second problem, linked to the first, is that students are taught hundreds of methods as though they are all equally important, without experiencing the more foundational concepts deeply and conceptually. Some concepts are much more central than others because they link to other areas of content, and they deserve to be learned deeply, over multiple lessons, through applied tasks that relate to students’ lives.  An example of a central concept in grade four is “factors and groups.” Instead of learning about these through short questions and answers, students can learn them through rich tasks in which they are more deeply engaged, as can be seen here.

Students can learn all foundational concepts, such as fractions or functions, by drawing, building and learning about them through real-world examples. Every important idea in mathematics can be learned visually, physically and conceptually, including algebra and calculus. Instead, most students work through pages of numerical calculations, absent of any connection to the world, and spend hours of algebra class manipulating X’s on a page.

A solution to both of these problems is to teach the “big ideas in mathematics” for every grade, as set out in the California Mathematics Framework,  such as “being flexible within 10” (kindergarten) or “unit rates in the world” (grade seven), making sure that for each of the eight or so big ideas in every grade, students have a deep and rich experience of their underlying concepts: by drawing them, building them and talking about them. Even if it is only these eight or so ideas that are experienced in this way each year, they will serve as a foundation for everything else students learn as they progress.

Many California school districts are now waiting for funding to be devoted to the training of teachers to move to the approaches set out in the framework. But in Kern County, leaders have been sharing these ideas for the past three years. Semitropic Elementary school, which serves mainly Latinx, English learners and socioeconomically disadvantaged students, is one example of a school that has moved to the approach of the framework. In the 2018-19 school year, before Covid-19 and the implementation of the new framework, only 5.6% of Semitropic students met or exceeded standards on math Smarter Balanced tests in grades 3-8, with less than 5% in grades four and five, and no students in grades 6 or 8. After the leaders in Kern County supported teachers in learning and implementing the ideas of the framework, through a series of professional development sessions to build capacity, with classroom demonstration lessons to model the new strategies, in action with their students, and coaching to meet teachers where they were, proficiency levels shot up, increasing to 16.3% overall, with the fourth grade showing the most significant increase, to 36.8%. There is more work to be done in this and other districts, but the demonstrable positive changes already unfolding are impressive.

What changed in the classrooms of the schools in Kern County? The teachers focused on big ideas, such as “being flexible within 10” which starts in kindergarten and extends through the elementary grades. Instead of students learning 10 as a fixed number that they use to calculate, they now spend time learning how 10 is made up, and all the ways they can make 10. A powerful strategy teachers started to use was “number talks,” in which teachers pose a number problem and collect the different ways students approach the problem, representing them visually. They also started using richer, deeper tasks, encouraging students to discuss ideas and learn with visuals and manipulatives. The superintendent and county math coaches were thrilled with the high levels of engagement they saw in the classrooms, as well as the significant changes in state test scores.

There are several problems with the systems of mathematics education in many states, and proposed solutions often spark disagreement. But perhaps we should all agree on one thing: Students need to learn important mathematical concepts deeply and well. They should not be working through sets of procedural questions that mean nothing to them, but rather should experience rich applied mathematics that inspires them, helps them learn effectively, and shows them that mathematics is important to their lives.


Jo Boaler is a Stanford professor and author of “Math-ish: Finding Creativity, Diversity & Meaning in Mathematics.” She was one of the writers of California’s new mathematics framework.

Cole Sampson is the administrator of professional learning for the Kern County Superintendent of Schools Office.

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