Math’s Groundhog Day

In 2018, Ana Redmond, a software developer and parent, described children who could recite math facts but could not say what they meant. The section below adds what current research recommends about teaching understanding and facts together, and what families and teachers can do.

When are we going to teach math understanding, and not just knowing?

I keep reliving this conversation over and over again with different moms. One mom, who now runs an after school center for kids, said when her daughter was young she was behind in math. So, she took her daughter to one of the math centers that make kids do worksheets in a very strict progression. She was excited to see her daughter solve 3+6 one day on the worksheet.

Another day, she was driving with her daughter in the car, and asked her daughter a situational question. There are 6 cars in this parking lot, and 3 on the road, how many in total. She did not expect the answer she got: “I don’t know.” The mom eventually realized that her very smart daughter had memorized the math symbols and answers without understanding their meaning at all.

I met another mom, a very highly qualified Russian immigrant with advanced science degrees, at an event at which I was talking about math learning by doing. She described the time her son was in 3rd grade, and desperate and frustrated with math. The teacher required them to do math in a minute. And, if they weren’t able to, a note came home, to make sure the parents made him do so repeatedly. After several frustrating attempts at math memorization during which her son screamed in frustration, the mom took him out of school, gave up her job and started homeschooling.

Recently, I met another concerned mom and dad whose daughter is struggling with math. Her daughter is in 2nd grade and having trouble with math facts at school. She is frustrated. Her daughter is supposed to work on her math facts at home. She says teaching her daughter math is like Groundhog Day. Her daughter memorizes the math facts, and then the next day they start with the same facts all over again when she forgets them (just like Groundhog Day).

6 years ago, that was also my story. I started building math games for my daughter when she started falling behind in math in kindergarten. She was getting extremely frustrated and agitated when it was time to do math. Yet, she was an advanced reader, reading at 2nd grade level. It was not that she was incapable. The problem was how math is taught to young kids.

Memorizing Math Facts Versus Understanding Meaning

Adults seem to believe that if kids memorize enough math facts kids will figure out the meaning of the symbols without being told what it is. It is true that some kids do figure it out on their own. Many kids either don’t understand the meaning of the math or don’t do so till much later, by when they have already been branded as poor in math.

Jo Boaler, Professor of Mathematics learning at Stanford writes in this article:

“The irony of the emphasis on speed is that some of our world’s leading mathematicians are not fast at math. Laurent Schwartz — who won math’s highest award, the Fields medal, in 1950 — wrote in his autobiography that he was a slow thinker in math, who believed he was ‘stupid’ until he realized that ‘what is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant.’”

Math is not computation. Math is patterns and symbols that have meaning. Math is a means of communicating complex ideas.

Teaching Math Concepts Through Understanding

As a software developer, I also recognize the irony of forcing our children to learn math facts. I never use memorized math facts for calculations ever. I couldn’t even if I wanted. I don’t remember them. But, I constantly use math. I convert the real world to math. I convert requirements into equations and procedures that the computer then runs. The important part of math is understanding how to represent a real world problem using mathematical symbols. How math connects to everyday things. What math symbols mean.

That’s the math I taught my daughter. I built her games that explained concepts simply. She understood the meaning of the symbols by solving simulated problems. The understanding she gained built her number sense. I never again asked her to memorize facts. But, once she understood the meaning of the symbols, the memorization happened automatically. If she forgot the math fact, as she sometimes did, she just went back to basic principle to figure it out again. She developed a strong foundation in math along with her confidence in her ability to figure it out. Now, she is one of the best students in math in her 6th grade class.

Albert Einstein famously said:

“Any fool can know, the point is to understand.”

Then, why do we keep teaching math facts (knowing) instead of math concepts (understanding)? When will we all stop reliving Math’s Groundhog Day?

Previously published in Medium and The Seattle TimesFurther Reading

1. AllKindOfMinds – Understanding Math Concepts
2. Edutopia – 6 Ways to Help Students Understand Math
3. TED Talks – Roger Antonsen: Using Math To Understand The World

Conceptual understanding in math: what it means and how it fits with learning math facts

Conceptual understanding in math means knowing why a calculation works and how its parts connect, for example what 3 + 6 describes in a parking lot with six cars and three more arriving, and not only that the answer is nine. As Ana Redmond explains above, a child who has memorized answers can look successful on a worksheet and still be unable to use them. This section looks at what research recommends about building understanding, where practice with math facts fits, and what parents and teachers can do.

What the research recommends

The What Works Clearinghouse, part of the U.S. Department of Education’s Institute of Education Sciences, published a practice guide in 2021, Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades. It makes six recommendations, each rated as supported by strong evidence. Five of them concern understanding: instruction that builds understanding of mathematical ideas in a systematic way, clear mathematical language that students learn to use, a well-chosen set of concrete and semi-concrete representations, the number line as a tool for teaching concepts and procedures, and deliberate instruction in word problems. The sixth recommends regularly including timed activities as one way to build fluency.

That list speaks to both sides of the argument in Redmond’s essay. The guide gives most of its weight to understanding, which supports Redmond’s concern. It also lists practice aimed at fluency, which means the research does not support dropping practice with math facts, and it describes timing as one way to build fluency, not the centerpiece of instruction. The guide covers interventions for elementary students who are struggling, so its findings apply most directly to that group, and it should not be read as a ruling on every classroom.

How to read the essay’s evidence

Redmond’s account of the family’s own math games is one family’s experience, written by a parent who built math games and observed the result. It is a useful description of what understanding can look like at home, and it does not show that games or a ban on memorizing facts will work for other children. The quotation from Stanford professor Jo Boaler, from a Scientific American article that Redmond links, argues that speed is a poor measure of mathematical ability. Readers can weigh that alongside the practice guide, which treats timed activities as acceptable when they are one part of a program that also teaches meaning.

Why it matters now

The National Assessment of Educational Progress gives a current picture. In the 2024 mathematics results, the grade 4 average score was up 2 points from 2022 and down 3 points from 2019. Thirty-nine percent of fourth graders performed at or above the NAEP Proficient level, and 24 percent performed below NAEP Basic, a share that was 5 points higher than in 2019. These scores do not say which teaching methods caused the change, but they show that many students still need stronger foundations.

What parents and teachers can do

• Ask a child to explain a problem in words or a drawing before asking for the answer, and ask about the same problem in a real situation, as the mother in Redmond’s account did in the car.
• Use a number line or objects to show what an operation does, and move to symbols when the child can describe the picture.
• Teach the words, such as total, difference and equal groups, and ask the child to use them.
• Keep fact practice short and regular, after the meaning is in place, and let a child work out a forgotten fact from first principles instead of treating it as a failure.
• Teachers and leaders can check that a curriculum pairs each set of facts with a representation, includes word problems, and gives students time to explain their thinking.

For more on classroom approaches, see edCircuit’s Teaching for Understanding in Math.

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