When it comes to teaching math, a debate has persisted for decades: How, and to what degree, should algorithms be a focus of learning math?
The step-by-step procedures are among the most debated topics in math education, and are in some ways at the very heart of the instructional disagreements known as the “math wars.” Some educators feel that students tend to learn the algorithms by rote memorization, rather than really understanding what they represent—even as other researchers insist that learning these steps isn’t anathema to having a strong understanding of key math concepts.
Now, that issue is coming up again with the release of a new book.
Pamela Weber Harris, a former high school math teacher contends in a new volume, Developing Mathematical Reasoning: Avoiding the Trap of Algorithms, that too many classrooms focus too much on memorization. Students and teachers should approach mathematics more focused on reasoning and less on memorization, she argues.
“An algorithm is that generalized procedure that really is kind of opaque. It’s kind of hard to see why it works. The meaning is kind of behind the scenes,” Harris said. “You do a bunch of steps, you’re not even really sure what happened, but voila, all of a sudden you have an answer.
Researchers in the math world argue that algorithms are a useful tool to help students learn math, but need to be coupled with a focus on conceptual understanding.
“The reason [students] do need to memorize math facts: They need to free up their minds to see the more interesting things about math,” said Bethany Rittle-Johnson, a professor of psychology and human development at Vanderbilt University in Tennessee, who has studied how students gain skills and learn key concepts in mathematics.
Is there a balance between the two perspectives?
What are algorithms? Do schools rely on them too much?
Algorithms are defined as the step-by-step procedures students learn to solve problems. Students learn a whole host of algorithms in the elementary grades, such as “regrouping” when adding or subtracting numbers, or the steps they take to solve long division.
But algorithms show up in more advanced topics, too, like the “FOIL” method learned in Algebra 1 for multiplying binomials, such as (x +2) * (3x - 6): You start by multiplying with the first terms, then the outer terms, then the inner terms, and finally the last terms.
The act of knowing algorithms plays an important role in procedural fluency—recognizing a problem, flexibly choosing a strategy to solve it, and executing it—but it is often confused with memorization, said Latrenda Knighten, the president of the National Council of Teachers of Mathematics.
Knighten differentiated algorithms from memorization by their use. Algorithms allow students to solve problems efficiently, and practicing them ultimately leads to knowing them by heart. But that shouldn’t mean they don’t have a strong grasp of the mathematics underlying the steps.
“Students will memorize something just from over time, but at the same time, they still need to be able to understand what those things mean,” said Knighten.
For Harris, too many schools lean into the steps without the conceptual backing.
"[This is how] we teach kids,” she said: “Hey, in order to solve an addition problem, you’re going to line numbers up and you’re going to add the smallest digits first and then the next ones, and you might have to carry over some stuff or regroup.”
Harris gave the example of 99 plus 67, which she says students would “dutifully” follow the specific procedure.
But students who have a broader sense of how numbers work can use a simpler method. They could pull one number from 67 into the 99 and then add simply 100 and 66, she pointed out.
“We can do that with all mathematics. That’s how mathematics was created—it was mathematicians using those relationships,” Harris said.
And people who understand relationships between numbers can then develop strategies.
“You’re involved in the relationships, the magnitudes, that means the size of numbers. But the whole time you’re involved in the structures, the whole time you’re reasoning logically,” Harris said.
“If I’m a teacher, however, who’s never done that kind of mental action, I might hear what I’m suggesting as, ‘Oh OK, here’s another method I need to tell kids how to do.’ In other words, they turn it into memorizing steps and mimicking procedures,” she said. “What we don’t need is to practice algorithms over and over and over, because what that gets us is maybe good at those algorithms. ... In reality, doing math means doing the mental actions that mathematicians do.”
Recognizing patterns and regularities helps develop conceptual understanding
For Rittle-Johnson, who has studied the relationship between conceptual and procedural understanding in math, the two goals can actually build on one another—rather than working at cross purposes.
One way to intertwine the ideas: To have students think about how algorithms are a representation of patterns and rules that they can learn to recognize.
“Students should really be generating explanations, trying to make sense of things, and [looking for patterns is] just a set of instructional methods and tools that, in general, can support and connect procedural fluency and conceptual understanding,” said Rittle-Johnson.
One way Harris likes to do this is through an instructional routine called problem strings, an idea that comes from the Netherlands.
“It’s a structured series of problems that basically high doses kids with those patterns,” Harris said. “Mathematicians could get a low dose of patterns, put them together and kind of create some math. Most of us need a higher dose of those patterns. ... For example, we did 99 plus 67, how could you do something like 49 plus 37? Could you think about 37 plus 50 and then back up one?
“It’s those kinds of generalizations that not only get us the strategy that we call adding over—add too much and adjust back—but it also gets us place value and rounding. All of that keeps kids in reasoning land.”
