It’s time to come together around good math instruction

A growing consensus has been emerging that reading instruction must align to the science of reading by including phonics, decoding, vocabulary acquisition, and background knowledge. This has led to important changes in schools. In contrast, the conversation around how to teach math well doesn’t have the same clarity or urgency. It can, and it should.

Whether you call it the science of math or something else, there are proven and well-researched principles that underpin high-quality math instruction. These have been lifted up by college- and career-ready standards in states and by professional groups like the National Council of Teachers of Mathematics. But they haven’t received the broader attention they deserve.

The nation is in a math crisis, worsened by the pandemic. On the latest Nation’s Report Card, just about a third of U.S. 4th graders were deemed proficient in math, and only about a quarter of 8th graders met the benchmark (National Assessment of Educational Progress, 2022). A subsequent study showed many of the efforts to help kids recover from the disruption to schooling caused by the pandemic have stalled (Lewis & Kuhfeld, 2023). It’s imperative that all kids get access to math instruction that includes three key components: coherence in math instruction, an emphasis on deep conceptual understanding along with procedural skills, and the application of math in school and beyond.

Coherence: Connecting the dots

A big problem in math instruction is that many U.S. schools for too long have been teaching concepts and procedures in isolation. Thankfully, that’s changing, as there is a growing understanding that kids need to learn math in a more connected way, in which concepts build on one another. This involves encouraging kids to see the links between topics, like connecting addition and multiplication in elementary school and ratio and rate to whole number multiplication and division in middle school.

We always smile when we’re visiting classrooms and hear teachers start a sentence with the phrase, “Remember when we learned . . . .” That explicitly cues kids to build on previous knowledge. We also love when teachers begin a math lesson with an activity that gives kids the chance to practice a skill or solidify something they learned.

In addition to explicitly making connections across content areas, good math instruction encourages the use of consistent strategies and models over time. A teacher might introduce a visual model like a tape diagram, a rectangular model used to illustrate number relationships, to help 1st-grade students solve simple addition. Then, in 5th grade, students might use that tape diagram to explain and solve a complex word problem (see Figure 1).

It’s not enough for teachers to be experts in the grade level they teach. They need a solid understanding of the math content that precedes and follows their grade. To accomplish that, we should ensure all classrooms have strong curricula and all teachers have the training and time they need to study the curriculum and collaborate with colleagues within and across grades.

Conceptual understanding is critical

Another core principle of high-quality math instruction is that it must foster deep conceptual understanding of mathematics. Kids need to understand the reason math works and why a math idea is important.

For too long, math instruction focused primarily on rules and procedures and relied on teaching kids tricks to solve problems. That left many students struggling and turned them off the subject and away from pursuing it throughout their studies and beyond. If students understand how a mathematical idea develops, they develop ownership of the mechanism by which it works. For example, understanding how “2 apples + 1 apple = 3 apples” relates to “2 fifths + 1 fifth = 3 fifths” makes addition of fractions much easier to understand.

There are a variety of strategies teachers can use to build conceptual understanding. For example, in many elementary classrooms we work with, teachers ask kids to read or explain a math assignment and then discuss what is happening in the problem. Then they ask students to draw a visual aid or mathematical model that represents the problem, like groupings of objects, a tape diagram, or a number line. After that, they might ask the kids to describe a solution and provide the answer. In each case, students better understand the math concepts at hand.

One way to provide a system for problem solving is to help students progress from concrete understanding to representational understanding to abstract (symbolic) understanding. For example, a young child might need to use blocks to count and regroup to see that three blocks and five blocks makes eight blocks (concrete). Remove the blocks and the child may draw a diagram of three and five (representation). Eventually, the student knows 3+5=8 and can express it symbolically without using supports (abstract). Students ultimately develop something like muscle memory that enables them to move directly to the abstract approach, such as writing the symbolic equation without using objects or a model first. But if that student later encounters an unfamiliar situation, they may go back to the beginning to solve the problem using concrete or representational strategies.

We liken it to driving a car. Before taking driver’s education, you must familiarize yourself with a car. Then, you must go through a checklist of steps before starting a car: Adjust the seat, check the mirrors, press the brake, and turn the key. Actually driving requires more steps. The process eventually becomes so familiar, most people drive “automatically” without thinking about each step. But if you drive an unfamiliar vehicle, reverting to the checklist is a comfort.

By emphasizing conceptual understanding, we don’t mean to imply that procedural skills don’t matter. They do; they’re just not the only thing that matters. Math is built on a series of important rules and procedures. Students need to do these procedures quickly and efficiently. They also must be able to quickly recall associated math facts, which frees up working memory to engage in more complex thinking and problem solving.

We do need to do away with relying on unsupported math tricks to remember how to do procedures, however. Remember that old butterfly method for adding fractions? It was a commonly taught gimmick involving drawing wings around numbers. It probably sounds familiar even if you don’t remember how to do it. That’s the problem with a trick. It’s too easy to forget the reason for it or get mixed up later. The tricks also tend to fail when problems get more complex. We need to focus instead on robust ways to teach kids to learn concepts in a sustainable way.

Applying what you know

Kids also need to know how to apply what they know and can do, for example when calculating grades or batting averages, measuring distances, or making change. Providing students with opportunities to discuss and engage in real-world applications of math provides them with meaning and context for their learning.

A task we’ve seen teachers assign involves asking kids to apply their knowledge of area to determine the area of a room in a given floor plan. The kids then measure the side lengths of rooms in a floor plan to calculate the areas of the rooms. Then they redesign the floor plan while keeping the same area for each room in the house. The class can then discuss the different floor plans and — more importantly — the math involved in coming up with the designs. Such lively classroom discussions about math help cement the learning. It’s also engaging and inspiring to provide kids with examples of how math has been used over the years and across societies. For example, we’ve seen teachers engage kids with a lesson on how NASA scientists have used precise measurements to get astronauts safely to the moon.

Math is fundamental

We’ve come a long way toward understanding what great math instruction looks like, but there is a lot more work to do to ensure its core principles are used across the country. Our kids can’t afford to lose any more ground in math. Policy makers, school leaders, and educators must come together to provide high-quality instruction to all students.

The two of us have been deeply engaged in the science of reading conversation and are very glad that leaders have come together to stand up for the right of every child to learn to read in a research-backed way. It’s well past time we took similar steps to improve math education. Like reading, math is fundamental to students’ success in school, careers, and life. Providing all young people with the learning opportunities they need and deserve in this vital subject ought to be a shared national goal.

References

Lewis, K. & Kuhfeld, M. (2023). Education’s long COVID: 2022-23 achievement data reveal stalled progress toward pandemic recovery. NWEA.

National Assessment of Educational Progress. (2022). Largest score declines in NAEP mathematics at grades 4 and 8 since initial assessments in 1990. National Center for Education Statistics.

This article appears in the April 2024 issue of Kappan, Vol. 105, No. 7, pp. 64-65.