7 Concrete 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 and would often stumble and become frustrated when they encountered situations where the tricks didn’t work or they forgot exactly what to do.

That’s why math education has transitioned in recent years to focus on teaching deep conceptual understanding rather than encouraging students to rely 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 ease away from teaching tips and tricks so our students have the opportunity to become true 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. Using a carefully considered scope and sequence to structure your school year is the first step in avoiding the pitfalls of math tips and tricks.

During my last few years of teaching, my district used the Carnegie Learning High School Math Solution for our Algebra 1 and Geometry classes. For the first time, I saw how much the scope and sequence really matter. Watching my Algebra 1 students pull from their Module 1 experiences in Module 5 to make sense of quadratic functions was a lightbulb moment–for all of us!

It’s even better if your scope and sequence leaves room to interweave spiral reviews so students can solidify their base knowledge and see the connections between previous and future learning. Because my Algebra 1 students could pull from their foundational knowledge and saw the concepts revisited throughout the year, they didn’t need to rely on shortcuts or tips and 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

Although many students (and teachers!) love math shortcuts because they lead to quick “success,” having a toolbox packed with critical thinking skills and problem-solving strategies for students to pull from is so much more valuable. These skills will serve your students in various situations, whether they’re in advanced math classes or have to think critically about real-world problems.

One way to help students develop their critical thinking and problem-solving skills is to assign high-order math tasks in your classroom. When working on these rich tasks, they can think about what they already know and test out different ways to complete the task until they identify one that works. In the process, your students fill their toolbox with problem-solving strategies and critical thinking skills, eliminating 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 how to develop and deeply understand 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 always amazed at the deep conversations I heard around the room as they completed tables of cell divisions and eventually used those observations to understand why explicit and recursive formulas work.

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.

When students visualize math concepts, they can more easily see patterns and make connections that might not be immediately apparent 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. In that case, they’re more likely to be able to interpret the formula of a quadratic function and apply that conceptual knowledge to different scenarios.

4. Manipulatives and Hands-On Learning

Another way to eliminate the need for tips and tricks (“A negative times a negative is a positive,” anyone?) is with manipulatives such as 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 the paper and pencil note-taking in favor of digging into algebra tiles occasionally! 

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. And while, yes, connecting with your students is one of the best ways to increase engagement, here we’re talking about making mathematical connections.

Teach your students to look for the interconnectedness of mathematical concepts, so they see how ideas fit together and build on one another, and 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 retained much more information when they saw the connections between concepts and were able to apply their knowledge and skills in new situations that I never expected.

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.

Now your students will be more likely to remain engaged and acquire conceptual knowledge that can be generalized across various 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, but if 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!