# Creating a Generation of Independent Thinkers

An author and former educator explains how teachers can change the way students think about math by giving them the time to reason and productively struggle.

By Brittany Goerig

In the past, math education too often focused on having students memorize multiplication tables and rush through “mad minute” exercises that rewarded finding an answer quickly. For today’s students to be successful in math, though, I believe educators need to emphasize that getting the right answer isn’t necessarily the only goal of math class. Students need to be okay with struggling and not see problem-solving as a negative aspect of understanding a math concept.

The goal of this is not just to prepare them for their next math class, but to prepare them for their future careers. We don’t know what the jobs of the future will be, so we’ve got to make sure our kids are thinkers. According to research by UCLA professor James Stigler, placement tests show that 60 percent of U.S. students who enter community colleges are not qualified to take a college mathematics course and many never graduate for that reason. He cites the American teaching method of memorizing and regurgitating answers, rather than deep understanding of mathematical concepts, as part of the problem. Rather than standing in the front of the class and telling kids the rules of math, we want them to figure it out for themselves in their own way, with teacher support to guide their process as needed. Here are three ways educators can help create a generation of independent thinkers.

Fostering Mathematical Reasoning

We all see math a bit differently, but some ways of thinking are more beneficial than others. It’s an educator’s job to present a problem, allow students to strategize, and use tools to model that strategy for them.

According to research done by Pamela Weber Harris, kids begin their journey to mathematical reasoning with counting strategies and perceiving numbers as 1, 2, 3. They eventually start thinking additively. With practice, a student who’s thinking additively about 8 + 7 is thinking 8 and 2 is 10, 5 more is 15. Or 7 and 7 is 14 and 1 more is 15. Rather than counting on their fingers from 8, they see numbers in chunks.

Once students advance, they are taught to think multiplicatively. Multiplicative thinking is seeing numbers in groups rather than chunks. Someone who’s thinking multiplicatively about 6 x 7 is working through 6 x 6 = 36, plus another group of 6 is 42. Or, 7 x 5 = 35, plus another group of 7 is 42. After multiplicative comes proportional thinking and then function reasoning.

What happens sometimes, though, is that kids get stuck in counting, which makes it hard for them to catch on in middle school when they are required to think proportionally. Creating a community of learners starts with valuing everyone’s abilities and voice. We want kids to be comfortable sharing their answers and strategies.

Open Discussion and Brainstorming

The ultimate goal is to have kids see a problem and be able to determine what strategy is best for those numbers. The problem is that the strategy happens in students’ minds and may not be reflected on their paper. Adding math talks to your classroom allows students to see different strategies that their peers are using. It also allows educators to see what paths students are taking and the ones they haven’t quite grasped.

If I threw out a problem like 7 + 8 in my classroom and one my kids explained, “8 + 2 =10, and 5 more is 15,” that gave me one understanding of their strategy. If, on the other hand, I heard a student say, “I started at 8 and went 9, 10, 11, 12, 13, 14, 15,” I had insight into their instinct to avoid the near-double strategy. To hone in on that skill, I would present a number string that illustrated that strategy. After I give them some practice, I would do another math talk to see if they’d made changes to their path.

The idea of holding math talks is to help students gain number sense by watching and hearing their peers’ thought processes visually modeled and explained. Developing insight into the various ways one problem can be solved allows students to see numbers and decide the best solution for themselves. If done right, math talks don’t place pressure on students to get the right answer; they’re about brainstorming the most efficient path to an answer.

Valuing Productive Struggle

While it’s important for students to be able to recall basic math, educators should persist in teaching the deep relationship between numbers. Instead of just memorizing facts, we want students to be able to visualize models and construct strategies. A surface-level understanding of numbers isn’t going cultivate the confidence students need to independently and creatively solve math problems.

Math classrooms are changing by letting students productively struggle. Rather than encouraging them to get the correct answer quickly, educators are giving students the time they need to reason with mathematics. Most of the time, the teacher isn’t teaching strategy directly. The strategy is a natural outcome of the product. After students experience a few number strings, they are able to pick up on the patterns. From there, they can construct the path to reason about the numbers.

We don’t know what future job markets look like, so we have to make sure the future generation is full of thinkers. We can help them prepare by equipping them with the ability to solve problems independently, without someone telling them what to do every step of the way.

Brittany Goerig is a product manager at hand2mind and lead-author and co-creator of Daily Math Fluency. Before that, she was a K–5 mathematics consultant and an elementary and middle school teacher for 17 years.