Traditional math practice used to consist of neat rows of problems on a worksheet. Sometimes teachers would set a timer. Students were asked to show their work, but it’s likely that every student’s work looked more or less the same. But times have changed. Educators now know that repetition of math facts and memorization of formulas is no longer sufficient instruction in math. Don’t get us wrong: fact fluency is still important, as is the ability to recall necessary procedures in math. However, we now know that deep conceptual understanding requires much more than repetitive practice.

### What Does This Look Like?

Here’s a scenario. A third grade math class was working on fractions. Students spent the 90 minute block working in partnerships on the following word problem:

*Javier, Jen, and Jayson are siblings. Javier is ⅔ Jen’s age right now, and Jayson is the youngest. In 4 years, Jayson will be half the age of one of his siblings. What are two possible combinations of ages of the three siblings?*

The teacher introduced the word problem, set up partnerships, and let students get to work. They showed their work and solutions on large pieces of construction paper. Most partnerships needed the full 50 minutes allotted to work out their strategy, their solutions, and show their work clearly. Students spent the rest of the math block taking turns presenting their work to the class, and asking and answering questions of their peers. Not only did the students come up with wildly different answers, their answers ranged from very simple to quite complex, and some involved decimals and/or fractions. And not a single partnership approached the problem in exactly the same way.

### Students’ Brains and Math

The students in this hypothetical classroom gained more understanding from that one problem than they would have if they’d spent the same amount of time solving 100 practice problems written in standard form. Plus, they probably had a lot more fun. This is because students’ brains need to engage much more deeply with math concepts in order to fully make sense of those concepts.

The students in the class above had to approach a type of problem they had never seen before. They had to grapple with logic and reasoning, rather than mindlessly follow a fixed series of steps in order to find a solution. For older students, the same principle applies. We can ask students to memorize the formula for finding the volume of a circle (and they should, for efficiency’s sake), but do they know where pi comes from and why it works? These are the details that take up more time in a math classroom, but are so worth the investment.