I once had a student who absolutely refused to divide using the long division algorithm. Instead, she used a combination of multiplication and addition, which took forever and drove her dad crazy.
While she ended up using great quantities of paper every day, she was frequently able to find the correct answer and explain her thinking.
While hers was certainly not the most efficient strategy, all those sums and products showed what even the long division algorithm doesn’t: that she understood division and how it connected to other operations, like addition and multiplication.
This story stands out as a perfect example of why it’s important to honor all students’ math strategies—even if those strategies are inefficient or confusing or simply aren’t the way we would do it. Because, it turns out, student-generated strategies are the secret sauce when it comes to developing math concepts.
What do you mean by “honor”?
First, let’s go over what it means to honor students’ math strategies. As the story above suggests, students might not be ready to use the strategies presented to them. They might make better sense of a problem using their own methods.
We “honor” where students are coming from by acknowledging their thinking and sense-making. Yes, even if those strategies lead them down the wrong path or to the wrong answer.
This might mean having students walk us through what steps they took or simply acknowledging that we understand what they did to try to solve the problem. You don’t have to do much to let students know that their thinking is valid and to reap the benefits of honoring their thinking.
Empower students to come up with their own math strategies.
For so long, math instruction has been a top-down affair. Teachers tell students the concept they will learn, make the connections to other concepts for them, teach the required algorithm, and correct students when they don’t follow that algorithm to the letter. It’s almost like we don’t trust students to do math by themselves.
Not engaging students to make meaning of concepts for themselves only adds to the uneasy feeling that many students already have that math is incomprehensible and out of their reach. Instead, empower students to come up with their own approaches to solving problems through encouragement and validation.
Try phrases like, “I can see what you did there” or “That’s a good strategy; I hadn’t thought of that one.”
Student strategies show what students know—and don’t.
One way we can measure concept development is to look at the strategies students choose to use and how effective they are at using them. For example, if students count up to add 10 to 7, they might get to the right answer of 17, but their choice of strategy reveals that they don’t understand what happens when 10 is added to a one-digit number.
Use student strategies and explanations to get at what students know about a given concept and how well students can use math terminology.
Try phrases like, “That shows me you really understand … ” and “Do you think you could explain what you did to friend?”
Give students time to think and self-correct.
Similarly, taking the time to have students show you their strategy might mean that they notice, mid-explanation, that their strategy didn’t work out the way they thought it would. Instead of focusing on wrong answers, give them a chance to see their own mistakes by having them narrate how they reached their answer.
Ask questions like, “Does that step make sense?” or “Can you think of another way to solve this problem?” When students figure out what they did wrong for themselves, say, “Nice self-correction! You saw where you made a mistake, which is a hard thing to do.”
Piggyback off of their strategy to teach another approach.
Of course, honoring strategies doesn’t mean you can’t teach a different strategy or even correct erroneous thinking. I like to have students show me how they solved a problem and then show them how I might have solved it. I am careful to avoid saying that one way is better, but I do emphasize that both methods are possible approaches.
Ask students, “Can you see how we got the same answer?” Challenge them to try your strategy next or to evaluate which strategy is more efficient.
How do you encourage students to come up with their own math strategies in your classroom? What kind of prompts do you use to get your students talking about how they solve math problems? Come and share in our WeAreTeachers HELPLINE group on Facebook.