In math classes, finding the right solution to a problem might demonstrate procedural knowledge, but it’s just as important to get students to show how they got the right answer to a problem and to articulate the underlying concepts and knowledge they used to solve it.
That’s not easy. Asking students to explain the why and how of their mathematical solutions often results in cursory, unsatisfying answers where not much is revealed. One way to get students to produce more insightful responses is to focus on creating more effective prompts, argue University of Connecticut education professor Tutita M. Casa and colleagues in a recent article for the National Council of Teachers of Mathematics (NCTM). The researchers collaborated with a cohort of K–12 teachers to rethink approaches that lead to better mathematical thinking and reasoning from students in all grade levels.
Their findings suggest that teachers move away from prompts that discourage students from showing multiple solution paths—or prompts that give students permission to regurgitate their computational steps without any deeper reflection. Instead, they suggest a wide range of prompt formats that promote more writing, reasoning, and even arguing and debate. According to the professors, these carefully constructed questions not only illuminate the students’ thinking but allow teachers to “clarify goals for student learning, elicit evidence of their learning, interpret students’ work, and act on these conclusions to guide their subsequent teaching.”
In other words, a simple change to your questions may yield much more productive answers.
However, Casa and her colleagues note, effective prompts and useful conversations about math often don’t happen in classrooms due to the demand of high-stakes testing and mandated curriculum. Teachers can increase opportunities to engage in higher-order thinking, writing, and conversation by making adjustments to existing materials—for example, by adding callout boxes on the side of workbook pages or slightly tweaking the problem-solving sections at the end of a chapter.
And small adjustments to existing prompts—making them more open-ended, for example—can slow things down and resist the pressure to rush students to the “metaphorical finish line defined by correct solution pathways and answers,” Casa and her colleagues write. “Doing so is akin to spoiling the best parts of a movie for a good friend who has yet to watch it.”
Here are three approaches Casa and her colleagues shared to help teachers effectively identify, adapt, and create better prompts to assign to students.
Let Students Do the Heavy Lifting
While directions like “Use a number line to demonstrate…” or “Use a table to prove…” within a prompt can often be framed as suggestions, students usually interpret them as directives. As a result, Casa and her colleagues write, students are often following someone else’s lead—the teacher’s or the curriculum writer’s, most often—instead of doing the hard work of thinking deeply about the problem and creating their own paths toward a solution and learning, for example, why a tool like a number line is truly useful.
Instead of directing students to “use a drawing” to add two sets of fractions, for example, teachers can “strip away” hints and rewrite prompts that are more open-ended: “Describe an efficient way to add these two sets of fractions” or “Show how to mathematically represent the median of [example data sets],” the researchers suggest.
These simple tweaks can get students to surface gaps in their knowledge, help teachers assess levels of conceptual fluency, and inform future instruction.
To dissuade students from simple, single-path responses—the researchers note that “some students initially did not write down as much as their teachers desired when they implemented these approaches”—a few of your prompts might ask students to provide two or three different ways to solve a problem, accompanying their solutions with clear arguments about why they are all accurate.
Get Students to Move Beyond Mere Computation
Although many word prompts in math class often ask students to “explain” their work, this is too often interpreted by students to mean they should just record the computations they made to get an answer. It makes sense—most prompts don’t make explicit requests for more.
To get better insights about a student’s understanding of a given concept, Casa and her colleagues suggest using phrases that elicit deeper thinking. For example, instead of asking students to calculate the volume of a box of crackers and explain their solution, try asking them to “teach a friend about the meaning of each component of the formula used to calculate the volume of your box of crackers.” Other phrases you might try: “Tell a friend how you solved…” or “Describe the meaning of…” or “Explain the patterns you noticed…”
These prompts should get students writing or talking at some depth and give teachers “a window into the extent to which [students] realize the underlying mathematical concepts.” For example, the researchers write, “a middle schooler who states that they found an equivalent fraction by multiplying by a representation of 1 rather than writing that they ‘multiplied by a fraction with the same top and bottom number’ has demonstrated a more advanced understanding of the concept behind the identity property of multiplication.”
Sometimes asking once isn’t enough, however. Students who regularly provide only superficial answers—or mystifying answers—might warrant a follow-up conversation to determine the true depth of their conceptual understanding.
Ask Students to Defend Their Work (or Debate the Work of Others)
Prompts that ask students to assess the validity of their solutions or the solutions of their classmates can help increase their confidence, open their eyes to multiple solution paths, remind them that errors are common in mathematics, and position them as “mathematical thinkers and writers.” The researchers add that these sorts of prompts can help the entire class address common misconceptions and errors.
For example, one style of prompt may ask students to determine if the work of another student, or a fictional student that you invent, is correct. Instead of asking the class to find the absolute value of 30, put a little twist on things: “Alicia and Aloys are debating the solution to |30|. Alicia thinks the answer is –30, and Aloys says it is 30. Write an email to the pair to convince them who is correct, and why.” Another style of prompt might ask students to choose a side and defend their position. For example: “Jaylin thinks that the expected value of rolling a die is 3.5. Alvyn thinks that is impossible and says the expected value is 3. Whom do you agree with, and why?”
Meanwhile, prompts that include phrases like “How do you know?” can establish an expectation within the classroom of mathematical writing and argumentation and get students expounding on how they approached a solution in a way that allows teachers to “assess their sophistication,” the researchers write. An example prompt might look like this: “Aneesha thinks the angles they drew in their art project are congruent. Gavin states that one is greater than the others. Convince them who is right by sharing how you know.”
Ideally, teachers will use a variety of these styles—not only because they will help stimulate different types of thinking and reasoning, but also because varying your prompts forces students to read very closely and think on their feet. By mixing up whether students are analyzing and ultimately attempting to defend or rebut correct solutions, incorrect solutions, or misconceptions, the researchers argue, you ensure that students are “responsible for their mathematical thinking” and won’t disengage from the work at hand.