Do Students Really Understand the Math Concepts We Teach Them?

Savvas Insights Team

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Ensuring that students truly grasp the math concepts we’re teaching, beyond just arriving at the correct answer, is essential in today’s math classrooms. Genuine understanding empowers students to effectively transfer and apply their knowledge.

Dr. Marian Small, a globally renowned mathematics educator, professional learning consultant, and author of numerous books and teaching resources, believes in focusing on practical methodologies and effective strategies tailored to cultivate mathematical reasoning and unveil authentic student comprehension. These practical methods ensure students not only know how to do the math, but they understand the math.

Dr. Small says there are three reasons that assessing math understanding is critical:

  • Retention: If students understand the math they’re taught, there’s a better chance they will retain and remember the lesson.
  • Transfer: If students are able to retain the math they’re taught, they’ll be able to apply those ideas to future math lessons, as well as in their own lives.
  • Deeper Learning: If students understand the math they’re taught, they’re more likely to be able to think about why and when things happen in the world around them.

“The reason I think understanding is important is because kids remember things that they understand better than things they don't understand. If we can build understanding, we don't have to keep reteaching things, and that's a valuable thing for us as teachers,” she says.

Dr. Small also points out that learning isn't just the students repeating what the teacher tells them. Learning is applying the ideas they’ve been taught in new situations – and that's hard to do without understanding.

“I don't want students to just rehash the things that I said to them,” she added. “I want them to figure out why this is happening and when it’s happening. And that's an important part of mathematics.”

Strategies to Assess Mathematical Understanding

Dr. Small says it’s important that teachers check in with their students to make sure they actually understand the math they’re being taught through strategies that will help them check for those understandings. These grade-level examples show you how to check for understanding through intentional questions that will lead to thoughtful teaching and deeper learning.

When assessing students’ understanding of mathematical concepts, Dr. Small says there are two kinds of questions we can ask:

  • Do-It Questions: Questions that check that students know how to do the math
  • Understand-It Questions: Questions that check that students understand the math

To ask an understand-it question, Dr. Small says to start by choosing a topic and think about what ideas you wish to bring out. Then set questions to ensure those ideas are heard.

Kindergarten and First Grade Math Understanding Strategy

For example, consider early kindergarten and first-grade math. “We certainly want them to be able to count, show me a number, and tell which of two numbers is greater, but I want them to understand some things, not just be able to do those things,” she says.

Elementary math teachers can use this image to assess a kindergarten or first-grade student’s understanding of a math concept.

If we show students this graphic, the do-it question is:
How many are here?

The understand-it question is:
Which do you think is more? Why?

The do-it question is purposefully not specific to any of the symbols because this allows students to choose which symbol in the image they want to count. For example, one student might say that there are six blue stars and another might say there are five yellow stars. This provides an opportunity for rich, engaging mathematical discussions where students get to hear from each other about what they chose and why.

The intention in asking the follow-up understand-it question, according to Dr. Small, is to explicitly and directly help students understand that the yellow is ‘more’ in a certain way; it has more area. But the blue is ‘more’ in a different way because there are more individual pieces.

“I want to ask that question to ensure the kids understand the difference between those two things. Notice that this allows more inclusion because a kid who is thinking yellow isn't dismissed. There's an opportunity to see why that student is right as well,” says Dr. Small.

Asking understand-it questions also opens the door for peer-to-peer sharing of ideas in the classroom.

“What you see is an opportunity for conversation,” says Dr. Small. “One of the things we've been trying to do in mathematics for the last few years is to make math less silent and more conversational because kids learn by talking and by hearing from their peers. So this is an opportunity for conversation.”

Middle School Math Understanding Strategies

For older students in middle school, consider what you really want students to know about what a fraction means.

The do-it question is:
What is two thirds of 12?

The understand-it question is:
Draw a picture that shows why two thirds of 12 is 8.

The student might draw a picture that looks like this:

This image shows a student’s visual explanation of why 8 is two thirds of 12 and it is used to assess the student’s understanding of fractions.

“I would be delighted if a kid drew something like this,” explained Dr. Small. “They knew that since I said ‘thirds,’ there had to be three groups, and I wanted to know how many counters were in two of those groups. If a kid can draw this, I know they understand what's going on.”

For older students, you can also consider circle measurement formulas. You want them to know formulas for circumference and area, but consider what else you want them to understand.

Middle school teachers can use this image to assess a student’s understanding of a math concept. Middle school teachers can use this image to assess a student’s understanding of a math concept.

If you show students the top image to find the areas of rectangles, the do-it question is:
Tell me two rectangles that have an area of 60 square units.

Then you can use both images to discuss the understand-it question:
Why must a 3 x 10 rectangle have the same area as a 5 x 6 one?

The goal is for students to see that if they move the top half next to the bottom half, they get a 3 x 10 rectangle. They don’t necessarily have to know what the area is. But they’ll know it has to be the same as the 5 x 6 area because they used exactly the same pieces and simply shifted them around.

“I want them to understand why things work – that six rows of five is the same as three rows of 10, and that's why they have the same area,” says Dr. Small.

Developed by Dr. Marian Small

Experience Math

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Dr. Small says it’s still important to ask procedural do-it questions to assess math knowledge. But the typical mix of 90 percent do-it questions and only 10 percent understand-it questions isn’t enough. The balance of questions should shift to more understand-it questions, which will benefit students greatly. She also recommends mixing up the order of when you ask do-it and understand-it questions.

“Do I still think teachers should ask the do-it questions? The answer is absolutely yes. I also suggest flip-flopping the questions. I think the understanding questions will actually help with do-it questions. All students, and I really mean all, need understanding questions. Just do-it questions are really not enough,” she says.

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About the Author

Dr. Marian Small

Dr. Marian Small is an internally renowned mathematics educator, professional learning consultant, and author of numerous books and teaching resources. With over 40 years of education experience, Dr. Small has been a professor of mathematics education and previously served as the Dean of Education at the University of New Brunswick in Canada.

Dr. Small is the author of MathUP¸ the number-one math program in Canada, and Experience Math, a new student-centered math program for the United States. She is a regular speaker on K-12 mathematics around the world. Her focus is on intentional teaching through thoughtful questioning to include and extend all students by appropriately differentiating instruction while increasing critical thinking and creativity.

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