Teaching mathematical concepts these days includes more than asking students to memorize a process. Instead, says Sybilla Beckmann, it requires teachers have a depth of knowledge to show students multiple ways to find a solution.
As one of the authors of the Common Core mathematics standards, Beckmann also is intimately aware of the decades of research that backs up the grade-level requirements. Starting early in elementary school, mathematical processes build upon each other, she says, as students develop a deeper understanding of the whys and hows behind a solution.
This is a radical change from the rote approach of previous generations.
"We have a long history of very procedural approaches to mathematics, even through middle and high school," says Beckmann, a mathematics professor in both the College of Education and the UGA Franklin College of Arts and Sciences. But knowing how to make a calculation doesn't give you insight into how to apply that method—you need to know how to calculate and what kinds of problems those calculations solve.
For example, the Common Core standard for multiplying multi-digit numbers lays the groundwork for multiplying polynomials. "So you're laying the foundations of understanding these more advanced ideas," she adds. "When you start looking into the guts of why these concepts work, it's really powerful mathematical ideas."
This also means that mathematics teachers must have a greater knowledge of the process. Ideally, a teacher can help a student find a solution that better aligns with their thinking, rather than enforcing rote methods to come up with an answer.
Beckmann, who is also the author of a math textbook for elementary teachers, teaches courses for prospective elementary and middle grades teachers. She notes the Department of Mathematics and Science Education's course in multiplicative reasoning developed by professor Andrew Izsák may be one of the only courses of its kind for secondary teachers that delves into this deeper understanding. Also, current research by Beckmann with Izsák and fellow assistant professor Laine Bradshaw looks at how future middle grades and secondary teachers reason about multiplicative relationships.
"If you want to be a teacher and you want to understand how these ideas are going to develop over a longer period of time, and how you might want to guide children to really think through the reasoning, that requires a level of technical knowledge that's way beyond what an engineer needs," Beckmann says. "That's very technical knowledge."
Ideas build upon each other, and sometimes it's not obvious how they are connected. And the way students in elementary school make calculations is not how you calculate later as an adult. But in the end, it all fits together.
"Math is supposed to make sense," she says. "It's not just this thing you apply by rote. There's a rationale behind it. And it's not just a superficial logic, there's a very coherent, clear logic. It's much more powerful in terms of problem solving."