7 Strategies to Teach Conceptual Understanding in Math

Please Excuse My Dear Aunt Sally.

Multiplication is repeated addition.

Keep, switch, flip. 

The butterfly method.

These are all examples of math shortcuts, tips, or tricks that many students learn to rely on from an early age. I taught many students throughout my 16 years in the classroom who quickly pulled out these strategies!

But my students couldn’t explain why these tips and tricks work. Sometimes they would struggle and get upset. This happened when they faced situations where the tricks didn't work or they forgot what to do.

That’s why math education has changed recently to focus on teaching students a deep understanding of concepts instead of relying on shortcuts. Educators know that teaching children to deeply understand math leads to the development of problem-solvers and critical thinkers. 

But how can we stop focusing on teaching shortcuts and instead help students become real mathematicians?

Don’t worry; we’ve got a few ideas for you! Check out these seven tips for getting rid of the shortcuts and teaching true conceptual understanding in math.

1. Spiral Practice Through a Well-Thought-Out Scope and Sequence

Mathematics is a body of conceptual knowledge made up of interrelated concepts. It isn’t just a list of disconnected topics to check off a list as students move from grade to grade. Plan your school year carefully to avoid math pitfalls by following a structured scope and sequence.

I used the  Carnegie Learning High School Math Solution for Algebra 1 and Geometry in my last years of teaching. For the first time, I saw how much the scope and sequence really matter. My Algebra 1 students used what they learned in Module 1 to understand quadratic functions in Module 5. It was a lightbulb moment for all of us!

Having a scope and sequence that includes spiral reviews is beneficial. It helps students reinforce basic skills. It also shows them how their current learning connects to past and future lessons.

My Algebra 1 students used their prior knowledge and noticed recurring concepts. This helped them avoid relying on shortcuts or tricks.

A thoughtful scope and sequence incorporating spiral review is key to teaching deep conceptual understanding in math. If we rely on teaching the “easy” shortcuts instead of giving students the time and space to master grade-level skills and see the connections between concepts, they’ll struggle to develop a body of conceptual knowledge that will help them understand more complex ideas in the future.

2. Use High-Order Tasks to Build Critical Thinking Skills

Many students (and teachers!) love math shortcuts for quick “success.” But having a toolbox packed with critical thinking skills and problem-solving strategies is so much more valuable. These skills will serve your students not only in class, but in the real world.

One way to help students develop their critical thinking and problem-solving skills is to assign high-order math tasks in your classroom. Rich tasks help students think about what they already know and test out different methods until they identify one that works. In the process, your students gain skills and strategies that eliminate the need for tips and tricks.

Some of my favorite high-order tasks to use with my Algebra 1 students were in a lesson titled, “Do You Mean: Recursion?” This lesson is filled with activities that encourage students to think critically about arithmetic and geometric sequences and explicit and recursive formulas. They’re even asked to compare the pros and cons of using explicit or recursive formulas, using evidence developed over the last series of lessons!

The fact that there’s no “plug and chug” in this series of high-order tasks meant that my students were constantly using and developing their critical thinking skills and problem-solving strategies. 

I was amazed by the intelligent conversations happening in the room. Students were discussing cell division tables and explaining why explicit and recursive formulas worked!

3. Visual Representations for Better Retrieval

Visual aids are powerful tools for helping students to develop a deep, conceptual understanding of mathematical concepts. I loved supplementing as many lessons as possible with diagrams, graphs, anchor charts, manipulatives, and even high-quality math videos. In doing so, every learner had an entry point into even the most upper-level mathematic concepts.

Visualizing math concepts helps students see patterns and make connections that they may not immediately understand from written or verbal explanations. And when they have a visual cue stored in their brain, it makes retrieving information much more manageable. 

For example, suppose a student can recall that a quadratic function looks like a parabola because they’ve interacted with graphs illustrating a pumpkin catapult or diving into a swimming pool. If that happens, they're more likely to understand and use the formula of a quadratic function in various situations.

4. Manipulatives and Hands-On Learning

Another way to eliminate tips and tricks (“A negative times a negative is a positive,” anyone?) is with manipulatives. I love algebra tiles, counting chips, and even interactive number lines.

And I promise those hands-on materials aren’t just for the younger kids. Your high schoolers won’t mind abandoning note-taking in favor of digging into algebra tiles! 

I’ll never forget using algebra tiles for various purposes with my high schoolers. From watching a student with complex special needs finally understand the meaning and applications of a zero pair to seeing upper-level students suddenly “get” factoring trinomials, each visual and hands-on learning experience was pure magic!

5. Connect Concepts Instead of Teaching Math Shortcuts

Teaching is all about making connections. In this case, we're talking about mathematical connections.

Teach your students to look for the interconnectedness of mathematical concepts. Show them how ideas fit together and build on one another. Watch as they develop a deeper understanding of the underlying concepts. Then, it’s time to kiss the shortcuts goodbye!

For example, the scope and sequence I used encouraged my students to apply their foundational knowledge of concrete geometric investigations and reasoning with shapes to formalize their understanding. Circles were also integrated throughout the course, rather than treating them as isolated geometric figures (as many other curriculums do). 

Watching my Geometry students make connections between circles and angle relationships and complete constructions using arcs was a game changer! They remembered more when they understood how concepts were connected and could use their knowledge in unexpected ways.

6. Help Your Students Make Real-World Connections

Another vital connection that will lead to the elimination of shortcuts, tips, and tricks is between the mathematics your students learn in the classroom and the real-world applications of the concepts.

When you help your students discover these links to the real world, math suddenly loses its abstract nature. It becomes relevant, practical, and motivating.

Your students will stay interested and learn concepts that can be applied in different situations. Here are some examples using real-world scenarios to model integer subtraction that could be used in a 7th-grade class.

7. Don’t Use Math Tips and Tricks—Collaborate!

Most kids love to work in groups, right? It enhances the social aspect of school that many students value. And when structured correctly, these collaborative learning experiences can be the perfect setting for developing deep mathematical understanding.

Collaborating to create their conceptual knowledge is a powerful experience for your students. They may productively struggle, disagree, and even argue a bit, but these experiences are where the magic happens. 

“Allow students to experience and play and notice and wonder,” writes Tina Cardone, author of Nix the Tricks: A Guide to Avoiding Shortcuts That Cut Out Math Concept Development. “They will surprise you! Being a mathematician is not limited to rote memorization…Being a mathematician is about critical thinking, justification, and using tools from past experiences to solve new problems.”

And I can think of no better opportunity to notice, wonder, think critically, and justify those thoughts than when collaborating with peers. It may be hard to give up that “sage on the stage” lecture style (I definitely struggled!), but hearing your students engage in rich, mathematical conversations and watching them abandon the shortcuts in favor of deeply understanding the math is worth it. The feeling is second to none!

Don’t Let Tips and Tricks Take Away the Beauty of Math

Math is a beautiful, creative, and thought-provoking subject that sets the perfect stage for your students to become critical thinkers, problem solvers, and leaders of tomorrow. Don’t let a reliance on math shortcuts, tips, and tricks rob them of that experience!

I hope you’re ready to ditch the tips and tricks in your classroom. I you need more convincing, check out this case study from Muleshoe Independent School District in Texas. They were able to teach their students deep conceptual understanding in math and get rid of the shortcuts—with some great results to show for it!