Over the last few years, schools and teachers have begun to realize the importance of building students’ background knowledge when it comes to new learning. Research has shown that background knowledge makes learning new material easier and richer for a variety of reasons—increased vocabulary and knowledge in art, history and science bolsters reading comprehension, for example, while greater stores of knowledge in long-term memory eases cognitive load and makes it easier for new knowledge to stick.
The idea that prior knowledge is key to learning—“What you know determines what you see,” as Paul Kirschner wrote more than thirty years ago—is a relatively new one to American education. Most teachers say they never learned about the role of knowledge, long-term memory and working memory in their training.
But now things are beginning to change. Books like Natalie Wexler’s The Knowledge Gap and the work of cognitive scientists like Dan Willingham have spurred national attention to helping increase kids’ knowledge of geography, history and science topics, including ‘knowledge-building’ ELA curricula like Core Knowledge and Wit & Wisdom. Academic texts like the recent Developing Curriculum for Deep Thinking: The Knowledge Revival (a recent Bell Ringer book club pick—you can watch our reader discussions here and here) outline how knowledge works in the brain, and then dissects how educators can help build the “web of knowledge” in students’ minds that leads to analyzing and deep thinking.
Yet the same push to focus on building background knowledge hasn’t happened in math—even though it plays just as important a role in student success. “Everything in math requires background knowledge and knowledge of math language,” said University of Texas at Austin professor and math education expert Sarah Powell.
“We really need to be thoughtful about how all of this knowledge works across grade levels and even into our adult life, so that people actually understand math instead of just all these bits and pieces of math,” she said.
Because math is entirely cumulative—new skills are built upon already mastered ones constantly—background knowledge plays an essential role in everything students do, Powell said, in ways that go beyond the basic math content. Students need knowledge of math vocabulary and strategies. Word problems, which are quite complex, require stores of knowledge in reading and language as well as being able to do the math.
Powell, who spends much of her time training math teachers, said a knowledge-building movement for math looks different than for reading. In a recent interview, she outlined how teachers and schools can think about intentionally building knowledge in math.
Building math knowledge
Math vocabulary is essential background knowledge, and often overlooked.
Though math is made up primarily of numbers, it’s learned through language, Powell said. If students don’t have a handle on math’s extensive vocabulary—kindergarteners are exposed to more than 100 math vocabulary terms in common math curricula, middle schoolers over 500—as well as all the symbolic language of numerals, they will have trouble fully accessing math content.
“Not every math teacher sees themselves as a language teacher or a vocab teacher, but they are,” Powell said.
“If I am talking about a proportion, do I know what a proportion is? Can I use the word ratio to be able to describe the two ratios that are in comparison in this proportion?” she said. “If I am talking about place value, do I know what hundreds is and how that's different from hundredths? There are so many nuances here.”
Math vocabulary shows up in speaking about math ideas in class, but also in reading and writing—especially in story problems, a key indicator used to measure how well students are performing in math. Many math terms have other non-math meanings—think “degree” or “base”—that can be confusing for students, and teachers often have to be explicit with how the math term differs from its other uses.
Students who are learning English, or who have dyslexia or other learning disabilities, might need extra attention to all the words associated with learning math.
Continual practice is what activates needed background knowledge—and students may not be getting enough.
As educators add new concepts and build up student understanding over time, yesterday’s math content becomes today’s background knowledge. But often teachers are working from curriculum pacing guides that move too quickly through new concepts, Powell said, making it challenging to do the kind of review that’s essential knowledge for whatever is in the new lesson.
Turning math content into background knowledge stored in long-term memory takes practice, repetition and time—something math teachers are notoriously short on. To continually activate background knowledge, Powell said, students need well-placed interleaved and distributed or spaced practice to revisit key knowledge multiple times. But a lot of math curricula doesn’t prioritize it.
“We do this chapter within this unit, and then we don't always come back to it later on in the year. Because everything that we learn in math is learned through practice, it’s the practice that helps us continually activate that background knowledge,” Powell said.
For example, take place value—something Powell has often heard teachers say students struggle with. “How much are we explicitly working on place value? Because place value is always there, but are we actually really digging into it and making sure that students understand it—not just in September and October, but also in January and March and May and so on?”
Teachers should focus on teaching for retention, not just for pacing, she said. Students won’t be able to activate the background knowledge of things like place value when they need it if they don’t have it.
Solving word problems needs several different kinds of knowledge all working at the same time.
Word problems are unique in that they require crucial background knowledge from multiple areas in order to solve, Powell said. Students need to be able to read the problem and know what the words mean; they need to be able to understand what the problem is asking them for, and be able to filter out irrelevant information; and they need the math skills to solve it.
Weakness in one area can stop a word problem in its tracks. “If you do have difficulty with working memory, if you do have difficulty with reading, or if you have difficulty understanding let’s say place value—that's going to make the word problem solving just so much harder for kids, than if all three of those things are working in balance,” she said.
In her work with teachers and students, Powell has found that the reading and cognitive components are the most difficult hurdles for students—they’re the components that help get the problem set up correctly, and require both domain-specific math knowledge (do students know what the numbers mean and represent, do they know how to do the calculation being asked for), and some general knowledge (how to read and understand the words, and how to organize and translate those words into an equation and answer that makes sense).
Students also need strong executive functioning skills, which help with planning and organization. “Outside of the math domain, these skills are really integral for at least setting up a word problem and then being able to make sure that your answer makes sense,” Powell said.
Then there’s the math. “If there's a word problem that says, ‘Holly buys five dog bones for each of her three dogs. How many dog bones did she buy?’ First I have to understand—she's got these dogs, and each dog has the same number of bones. So there's this grouping structure, this kind of relates to word problem schemas,” Powell said.
“I've got these dogs and each dog has the same number of bones, now I could set up this equation. I have to understand multiplication or understand repeating grouping. So either I have to have multiplication knowledge or addition knowledge to help me solve that problem—and that's a very simple problem. In a lot of word problems, we're talking about multi-digit operations that are involved fractions, decimals. If I don't understand the math and grouping, it's going to be really hard to visualize that each dog has the same number of dog bones.”
Once the problem has been correctly set up, students rarely mess up the computation.
“Teachers always agree that the setting up of the word problem is actually more difficult than the solving,” she said. “If we get to the point where I understand that that problem is three times five, more often than not, kids are going to be able to answer that correctly. The hard thing for kids is getting to the point where they get to that setup of three times five. It’s a combination of your outside-of-math knowledge and your inside-of-math knowledge.”
Educators should pay more attention to all the knowledge that goes into solving a word problem, Powell said.
“The expectations that we have for students are often quite unreasonable,” Powell said. “It doesn't matter if the kid knows five times three is fifteen. They have to be able to read these seventeen words and extract all of that information, and get to the point where they know that they have to end up multiplying five times three.”
My own thoughts: a word on math background knowledge for teachers
Though Sarah Powell and I only discussed the role knowledge plays in student math learning, it’s become clear in my reporting how teachers’ math knowledge—both of the content and how to teach it—plays a central role in student success. I’ve spoken with quite a few elementary school math teachers who feel like they didn’t know enough about math to teach it well. Teachers have also said they learn little or no research on how the brain learns, or teaching techniques supported by research.
There’s some substantial research showing that teachers (a) don’t get the opportunity to learn enough math in their undergraduate and graduate prep programs and (b) don’t feel confident—sometimes downright anxious—in doing math as well as teaching it.
If background knowledge is essential to learning, it must be doubly so for teaching. One of the most important developments might be that universities and colleges recognize the role background knowledge and long-term memory play in teacher learning, too. Then teachers walk into classrooms better prepared, less anxious and more confident, to teach math to students.
Editor’s note: This was first published on the author’s Substack, The Bell Ringer.
