Mathematics is a playground for learning critical thinking, a currency whose value prevails far beyond the utility of math as a content-laden discipline.

While specific math subject matter is indeed relevant for science majors and engineers, the ability to problem solve, the know-how of critical thinkers, is a priceless advantage in any field, job or occupation.

In other words, the key asset of mathematics learning isn’t just the information. Ideally, the legacy of the learners, and their task, is to leverage a course’s content and unique critical mass of opportunities to develop a disruptive imagination — a tool, sharpened by their own hands, to contribute to changing the world in some meaningful way.

The purposeful shaping of disruptive imaginations is a specific educational goal wherever dynamicity in problem solving is essential. As Harvard University summer courses get started, Brian Mandell uses the term to warn master’s students: the shelf life of the information they acquire at the Kennedy School of Government will expire by the time they graduate in a year’s time, yet the true value of their training lies elsewhere. So it is with mathematics and science.

In large part, the worth of learning mathematics lies in the subject’s ability to teach us how to reason soundly, strategically and innovatively in any context. But wait, you might say, this is not what I learned in my mathematics classes. You may have a sense that memorization and borrowed logic might have hijacked any possibility of making sense of algebra or even calculus, classes in which successful application of seemingly God-given formulas are often the measure of successful performance, even in standardized tests.

Indeed, we often teach mathematical facts and rules, not how to develop mathematical know-how. The latter is tricky to teach, and even trickier to measure, something we Americans have lately become almost compulsively inclined to do.

The stakes are high, however. If standardized tests and final exams fail to measure sense-making ability and know-how, a post-college corporate job will measure exactly that. As we increase college enrollment and channel tuition toward nonacademic commodities, we are likely increasing the number of graduates who cannot manage to problem solve usefully without corporate hand-holding — at best. To many students, a college degree represents an automatic entitlement to a job they may not be qualified to do.

As Karl Taro Greenfeld puts forth in his recent article for The Atlantic, when one’s teenage child can perfectly conjugate a Spanish irregular verb without knowing what the verb means (since that won’t be on the test), a high performance score in a difficult Spanish test is hardly representative of meaningful, usable knowledge. So it is in math — and in science.

But it is not always the students’ fault. As a society almost blindly compelled to improve performance and graduation rates, we err on the side of factualizing knowledge and teaching it prescriptively, not adaptively. We end up delivering ready-made questions and answers, diverting attention from the essential mental gymnastics that connect the two. And we do so for many “justifiable” reasons: efficiency, budget cuts, the institutional rewards of research over teaching and pressure from high-tuition payers entitled to a grade based on “effort” rather than true learning.

In a time when the Kahn Academy and YouTube prolifically supply facts and calculational procedures on demand, the legacy of a mathematics classroom — of any classroom — lies in providing a playground to develop students’ problem-solving abilities, to grow the kind of disruptive imagination essential for creating new knowledge.

So, as teachers, we must learn to teach adaptively, not prescriptively: we must tactfully expose students’ reasoning in the classroom and use our expertise to meta-problem solve on their reasoning work.

As academic leaders, we should become more serious about rewarding good teaching. We must also critically rethink our prolific hard-and-fast measures for assessing knowledge and learning.

And as Americans, we must continue to seek out and defend institutional and political frameworks that allow us to carry out these tasks.

Guadalupe Lozano, a University of Arizona mathematician and educator, works on measuring conceptual understanding, the public face of mathematics, and writing reform calculus textbooks with the Harvard Consortium. Email her at