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The 3 Most Critical Aspects of Teaching Mathematics Going into Next Year

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Math education has a troubled history. It’s a sore spot for many elementary teachers around the country. In fact, it’s a sore spot for many adults. Students across the board struggle, often feeling helpless and buying into the myth that they aren’t a “math person.”

Couple that with a historic year of interrupted schooling and learning loss, and math teachers everywhere are left looking to next school year and wondering, what now?

Setting Students Up for Success

Let me start by saying this: I don’t know your unique situation. How many of your students had the technology needed for remote learning? How often were your students in school for face-to-face instruction? What personal hardships did you and your students go through? Your plans will require creatively attending to your unique challenges, yet there are a few general strategies that may help you answer the question, “what now?”

Looking forward to the next school year, set down things that aren’t essential. Focus on what we know constitutes high-quality instruction. Progressing through math standards one by one is not a reliable recipe to catch students up, especially if they're already months or years behind. However, ensuring that students are thinking mathematically, talking about math, doing math, and—perhaps most importantly—enjoying math sets them up for long-term success closing learning gaps and accelerating learning.

Aspect 1: Create Moments of Disequilibrium.

Let’s start with a problem: What’s missing? Hint: The answer is NOT 6.

Refocusing instructional math norms missing number puzzle

Before I solve the problem, really think for a moment. Put down your phone. Close your email. What could go there other than 6?

What you’re presumably experiencing right now is a moment of disequilibrium. This idea (and the use of that particular word) stems from renowned child development expert Jean Piaget’s theory of cognitive development. A moment of disequilibrium is a moment where you recognize the difference between prediction and reality, and it is a springboard for curiosity and wondering. It motivates us intrinsically to know more or try something different and primes the brain for its greatest potential for new learning.

Ready for the intended answer? These are the gears of a manual transmission. The missing value is “R.” With the right setup and content, however, there are endless ways to adapt a problem like that for the right age and topic. Here’s an example for younger children. How many cubes are in the shape below? The answer is not 8. (There are 7: the back-bottom-left corner is empty.)

Refocusing instructional norms missing cube puzzle

There is research to back up the effect of teaching with disequilibrium in mind. According to a study conducted by Jordan and Brownlee (1981), cited in John Hattie’s Visible Learning for Mathematics (2012), Piagetian programs, which include the idea of students experiencing disequilibrium, have an effect size of 1.28. For reference, Hattie determined that the desired effect size is anything 0.40 or greater.

Aspect 2: Facilitate Physical Experiences

The theory of experiential education revolves around the idea that learning is enhanced when students acquire knowledge through active processes that engage them. One of the many challenges that come with students not being in school physically is how that limits their possible physical experiences. There is ample evidence, even pre–COVID-19, that when students are not physically in school, they lose learning, with the largest effects on math skills. For example, researchers have measured the loss after summer vacation, after school closures due to Hurricane Katrina, or for immigrants who experience interrupted schooling. Our full report on learning loss explores these studies in depth.

For teachers who will be taking advantage of remote and hybrid learning next year, it is important to note that it is still possible to get students engaging in physical experiences—only not with each other face to face. Students can still draw, write, role-play, move, look for math outside of school, and manipulate objects they personally own.

Next year is the time to zero in on how mathematics is about knowing and doing. To effectively solve problems in mathematics, students need to demonstrate both conceptual understanding and procedural fluency, which infers verbal and physical demonstration. Look for ways to make math playful and physical anywhere you can, be it in or away from school:

  • Don’t just count. Dance.
  • Use number lines that students can stand and play on.
  • Play games that involve moving, and discuss the math around scoring or strategies.
  • Measure throws, kicks, and jumps.
  • Look for math in places students can walk, such as building height, plant geometry, or animal features.

Aspect 3: Facilitate Social Interaction

There is one aspect of math that is no different from learning a foreign language or history: in order to learn about it, students must talk about it. Famed education psychologist John Dewey viewed the classroom as a social entity where children can learn and problem solve together as a community. He believed in the process of “discussion, debate, and decision-making” to make this happen.

Similarly, Lev Vygotsky, another giant in education and child development, believed that cognitive development stems from guided learning within the zone of proximal development combined with social interactions as students and their peers co-construct knowledge. The more opportunities teachers provide for students to interact with others, the more viewpoints students will hear. Social interaction not only prompts students to think about their own viewpoints, they also help students clarify or cement their individual thinking. While individual thinking time is critical for students when gathering their own thoughts, social interaction is a required condition for learning.

As much as possible, avoid the trappings of rote fact fluency practice, which are rarely social and can hamper conceptual understanding of mathematics—a necessary element to advance or accelerate math learning. Instead look for problems with low floors and high ceilings that let students try new ideas and compare strategies. An entire class spent struggling over one problem is often more effective than a class spent repetitively solving short, related problems. Mathematical mindset expert Jo Boaler provides examples in a former Heinemann blog.

Looking to Next Year

Earlier in the pandemic, we proposed thinking about learning today as taking place along a spectrum of connectedness. Connected learning is possible when the surrounding culture prioritizes social and emotional well-being, supports professional learning, and includes family and community engagement.

There is much outside of your control, but leaning into powerful and memorable instruction utilizing the three aspects of learning discussed here is one interlocking piece of the connectedness puzzle.


MATH 180 is a proven, flexible solution to not only catch up students in Grades 5–12 who are behind, but also show them the excitement and relevance of mathematics. Find out how to unlock learning for students who struggle with mathematics.