Introduction
Artists and mathematicians have a lot more in common than we might initially think. Painters spend long, tedious hours in their studios making a mess, stepping back from their canvas to assess and reassess, and brushing over mistakes. They embrace the process of creation because art isn’t simply about producing a painting. It is a complex endeavor that serves countless purposes. Math is no different. Solving mathematical problems involves processes that are complicated, frustrating, and profoundly rewarding.
The idea that the process of solving a mathematical problem is just as important as getting the right answer is the animating principle behind so-called “new math,” formalized in Common Core. It’s no secret that Common Core is controversial. Its critics argue that it is unnecessarily cumbersome and time consuming. Certainly the voices of concerned parents and teachers should carry weight with any administration deciding whether to adopt a new method of teaching. Yet focusing on the process instead of the right answer has plenty of value, supported by years of research on teaching approaches.
When we as educators focus on helping students get the right answer, we are invariably encouraging them to learn by rote. But for years, studies have indicated that rote learning ultimately hinders students. Rote learning helps them memorize the mechanics of mathematical systems, but fails to teach them how to think and reason mathematically.
This consequence becomes evident when students are asked to solve problems in different contexts. For example, a student who memorized a method for solving equivalent fractions may not recognize that they are working with equivalent fractions when baking a pie at home. Emphasizing the process of solving problems gives students time to develop mathematical reasoning, and it is mathematical reasoning that will help them perform better on tests and apply their education in everyday situations.
It’s important that students learn how to explain and justify their decisions, both inside and outside the classroom. Emphasizing process forces them to think carefully about why they are solving a problem in a particular way. It gives them choice and the opportunity to defend their choice. Perhaps one student chooses a very efficient method, while another chooses a more elegant one. Why did the first student prefer efficiency and the second elegance? Should efficiency always be prioritized? Why or why not? These are the kinds of questions that occasion mathematical dialogue and push students to articulate and justify their ideas.
Obviously, they cannot have these conversations without first memorizing certain facts and formulas. But if that’s where students’ mathematical education ends, they will be poorly prepared for the future.
Problem solving is not just a math skill, it is a life skill that is fostered by process-driven learning. Throughout their education and their adult lives, students will encounter ambiguous and vexing problems of all sorts—mathematical, conceptual, personal, political, to name a few. Answers are rarely straightforward in life, and we all get stuck at some point while trying to solve problems. Knowing how to approach an issue from various angles requires the kind of creativity and imagination people need, to solve not only their own problems, but also problems on a larger scale—for their family, neighborhood, employer, the world.
When we fail to find solutions to a problem, many of us simply quit searching for them and settle for ignorance or an inadequate solution. To persevere through failure and keep working toward solutions, students need to develop grit. Every math teacher has heard students beg to be told the right formula, the missing step, or even the correct answer when they feel frustrated. But if a student develops grit, they will cultivate the intellectual autonomy and emotional maturity to control their frustration and think their way through a difficult problem when they get stuck. An education that conditions children to depend on others for answers and be ruled by their emotions does not serve them in the long run. This is the rationale behind refocusing math instruction on the process of problem solving rather than just getting the right answer.