# A Powerful Rethinking of Your Math Classroom

We look at strategies you can reset this year—adjusting your testing regimen, tackling math anxiety, encouraging critical thinking, and fostering a mistake-friendly environment.

The beginning of school is a great time for teachers—both veteran and early career—to consider ways they can improve upon their classroom practices over the next year. This may be especially important for math teachers, who often spend the early days of the school year confronting math anxiety, convincing students that they are indeed “math people,” and coming up with engaging practices to guide students to find pleasure in math challenges. Innovating in these areas has the potential to yield big benefits for students all year long.

To help you re-imagine some of your teaching practices this year, we’ve pulled together a collection of big and small strategies aimed at:

- helping students adopt a more productive, curious mindset when they approach math,
- engaging students as soon as they walk through the door,
- rethinking how you handle testing,
- fostering a mistake-friendly environment,
- incorporating humanities-style discussions into math, and
- building what Peter Liljedahl calls a “thinking classroom.”

### Tackle Negative Math Mindsets

“Most of my work as a math teacher isn’t even math,” former middle school math teacher José Vilson has said. “It’s helping students believe that they can also do math.”

Striking a similar note, middle school math coordinator Alessandra King writes that supporting students—particularly students from marginalized backgrounds—to develop a “positive mathematical identity” is crucial to fostering a sense of belonging in a larger math community, boosting “their willingness and ability to engage” in challenging work.

King recommends spending the first few days of the school year spotlighting mathematicians from the past who reflect the makeup of your classroom. In her all-girls school, for example, she has students read and respond to a play about the achievements and struggles of Maria Agnesi, Sofya Kovalevskaya, and Emmy Noether, three historical female mathematicians. You can also hang up posters of famous Black or Latino mathematicians—such as Euphemia Haynes, the first African American woman to earn a PhD in math, or Robert Luis Santos, a Latino statistician and director of the U.S. Census Bureau—and devote lessons to discussing their achievements and backgrounds.

To get students thinking about—and challenging—their own math identity, educator Rolanda Baldwin suggests asking students early in the year to write a “math autobiography.” They might respond to questions like: “How do you feel about math? How did your relationship with math change over time?” To draw students closer together—and make them realize that many students, of all backgrounds, struggle—have them share their responses in groups, or with the entire class.

Sometimes, our best intentions can go awry. Rachel Fuhrman, a former special education math teacher, notes that teachers can sometimes dampen a student’s math identity by wheeling out phrases that might seem helpful but can actually demotivate students, like: “This is so easy.” Framing something as “easy,” she writes, can leave students feeling uncomfortable or afraid of asking crucial, clarifying questions.

### Engage Students the Moment They Enter the Classroom

To set a playful tone and lower the stakes of the work so students can effectively experiment and collaborate, high school math teacher Lorenzo Robinson suggests starting off class with fun, challenging brain teasers.

For example, ask students to draw a cross on a sheet of paper (you should draw one on the board as a point of reference). Ask students to draw two straight lines that will segment or cut the cross into pieces. The goal is to cut the cross in a way that produces the most pieces.

You could also try math riddles: Ask students to imagine they have two coins that total 30 cents. Tell them that one of the coins is not a nickel, and ask them to figure out what the two coins are.

Robinson finds that brain teasers like these can get students primed for problem solving and critical thinking, without even realizing it.

### Lower the Stakes of Testing

Big tests—centered around units or near the end of a grading period—are staples of many math classrooms. But that doesn’t mean they have to be the only opportunity for students to show what they know. One way to lower the stakes for students and give them more opportunities to practice, while providing yourself more of an opportunity to both teach content and check for understanding is to give short assessments on a regular schedule.

Math coordinator Steven Goldman’s school switched from tests to checkpoints. These short assessments include a mix of current and past topics—retrieval practice is a research-backed way to support greater learning, after all—with some repetition for the most important skills students need to know. The checkpoints are given every two weeks and are not formally graded, but mistakes are noted and teachers leave feedback to guide student revisions. The change, Goldman writes, resulted in a big reduction in student stress levels—something research shows is a benefit of frequent practice tests, alongside a boost in long-term retention. Because the checkpoints happen all the time, Goldman and his colleagues don’t have to “go through all kinds of contortions to finish a unit before a break or on a Thursday so that we could give the test at the right time.”

Fresno State math instructor Howie Hua suggests lowering the stake in your classroom by allowing students to discuss a test before starting to work on it. Hua recommends having students put their pencils on the floor so they can focus on their discussion. Hand out the test and give students five minutes to discuss strategies they can use to solve the problems.

### Good Mistakes, Better Mistakes

Mistakes are bound to happen in any math classroom, and how you respond to them throughout the year can make a world of difference. Making light of your own mistakes is a good first gambit, but research suggests a more advanced approach: Giving students space to make mistakes—and opportunities to analyze and discuss them with peers—can better encode information in their brains than simply providing them with the correct answer.

To use mistakes as building blocks to better solutions, math teacher Emma Chiapetta uses an ingenious, small-group activity that asks students to identify and reflect on common mistakes, and then explain the rationale behind them to their peers.

Here’s how it works: Randomly separate students into groups and assign each group to a board to generate a problem and solve it incorrectly. Groups rotate so they’re looking at a problem and an incorrect solution, and have to identify the error and solve the problem correctly. After another rotation—so students are looking at a problem and both the incorrect and correct solutions—they explain to the class the mistake made by the first group and the correct solution provided by the second. The activity, Chiapetta writes, helps students think about the same content from various perspectives, which can lead to deeper understanding.

Not all mistakes are created equal, and they often conceal thoughtful, underlying work. Former math teacher Colin Seale asks students to reflect on which wrong answer to a problem is “more right.” He suggests offering students two equations that are both incorrect—perhaps one is wrong conceptually and the other computationally. Seale notes that this exercise gets students to tease out nuances around the skills and concepts they’re learning, while also correcting their approach to similar problems moving forward.

### Bring Humanities Strategies to Math Class

The discussion, analysis of reasoning, and argumentation that happen in humanities classrooms can be extremely useful in math classrooms to help students slow down and think through the work they’re doing, says middle school math teacher Connell Cloyd.

He does this in his classroom by posting four incorrectly solved math problems around the room and having students rotate around each problem in groups to discuss the error and write down (in complete sentences) a claim and supporting evidence to show why they believe the error occurred. As they rotate around, students read the arguments of peers and either support an argument or refute it with new evidence. This is like adding techniques from debate to Chiappetta’s strategy.

Math journals can also inject more writing and reasoning into your classroom. Former math teacher Nell McAnelly prompts students to reflect on concepts they’re having trouble with: They work through a problem and then write about the strategy they used to solve it, or informally journal about other approaches they could have used. Doing so, she writes, gives students a chance to “synthesize learning and address unanswered questions.”

### Driving Deep, Critical Thinking

A traditional math classroom, where the teacher demonstrates a skill numerous times before students take the reins, can inhibit higher order thinking and result in students who “mimic” teachers rather than develop their own strategies to solve complex problems, says Peter Liljedahl, a researcher and professor of mathematics at Simon Fraser University.

Instead of starting lessons with direct instruction, Liljedahl says, give students novel “thinking tasks” to work on in groups. These are problem-solving activities and mental puzzles that should get students in the mindset of challenging themselves. Work groups should not be chosen based on ability or students’ preferences—Liljedahl’s research shows that students are more likely to contribute in randomized groups. Having students stand while they engage in this collaborative, messy thinking—as Chiappetta does in her mistake-analysis exercise—is another way to engage them.

Instead of using notebooks to compute, Liljedahl calls for groups to work at vertical, non-permanent surfaces, such as whiteboards, blackboards, or windows—surfaces that he says promote more risk-taking because students have the freedom to quickly erase false starts without feeling committed. As students work in groups, teachers bounce around the room and avoid directly answering questions such as “Is this right?” that circumvent student thinking and instead make suggestions that lead to further independent thinking.

Because the goal of this approach is to get students to develop perseverance, curiosity, and collaboration, Liljedahl suggests evaluating them in a way that prioritizes these competencies. He developed formative assessments that focus on informing students “about where they are and where they’re going in their learning.” These include observations, check-for-understanding questions, and unmarked quizzes. Summative assessments, meanwhile, should focus less on end products and more on the process of learning through both group and individual work.

Shifting to this thinking classroom model requires a fundamental shift in how you run your classroom, but it could result in large rewards throughout the year.