To learn math, students must build a mental toolbox of facts and procedures needed for different problems.
But students who can recall these foundational facts in isolation often struggle to use them flexibly to solve , known as procedural fluency.
鈥淢athematics is not just normalizing procedures and implementing them when somebody tells you to use that procedure. Mathematics is solving problems,鈥� said Bethany Rittle-Johnson, a professor of psychology and human development at Peabody College in Vanderbilt University, who studies math instruction. 鈥淭o solve problems, we have to figure out what strategy to use when鈥攁nd that tends to get too little attention.鈥�
In a series of ongoing experiments, Rittle-Johnson and her colleagues find students develop better procedural fluency when they get opportunities to compare and contrast problem-solving approaches and justify the approaches they use in different situations. While some students may develop this skill on their own, most need explicit instruction, she found.
Rittle-Johnson spoke with 澳门跑狗论坛 about how teachers can use such comparisons to help students develop a deeper understanding of math. This interview has been edited for space and clarity.
For more on the best research-based strategies on improving math instruction, see 澳门跑狗论坛鈥檚 new math mini-course.
How often do teachers talk to students about multiple strategies, and how to select them, in math problem-solving?
Students in the [United States] are very rarely doing rich contextual problems. Even more rarely, they鈥檙e being asked to compare strategies to solve them. I don鈥檛 hear teachers talk about [using different strategies] a lot, and textbooks tend to do a pretty bad job of explaining it.
For example, in Algebra 1, solving systems of equations, there are many standard solutions strategies that are taught in separate chapters and textbooks, ... but I see shockingly little time spent having students think and compare and choose which strategy to use. In one study where teachers were trained [to compare math strategies], only about 20 percent did in the classroom鈥攁nd only about 5 percent of teachers who [did not receive training.]
Sometimes I hear teachers say, 鈥淲ell, multiple strategies, that鈥檚 great for my high-end learners, but I don鈥檛 want to show that to my struggling learners. 鈥� So maybe multiple strategies is the ideal, but I鈥檓 not going to get to it because I鈥檓 tight on time and my kids are behind.鈥� But we hear from struggling learners that they really appreciate the multiple strategies and we see that it helps them, too, across the grade bands and across contexts.
How can teachers decide when to bring in and compare different strategies while introducing a new math concept?
We find comparisons can be useful in all different phases of instruction.
It can be helpful for kids to have had some time to think about one strategy before they think about multiple strategies, maybe at most a lesson. But the risk is in general, if you wait too long, kids just get attached to one strategy. You run the risk of kids becoming really attached to one strategy, and then they become more resistant to wanting to think about and use multiple strategies.
What does this sort of comparison look like in the classroom?
One best practice is to have the steps of the different strategies written out. It can be kids鈥� strategies that they wrote on the board. It can be projecting strategies from textbooks or your solutions, but one thing we know is: Make sure both strategies are visible so that kids don鈥檛 have to remember. Then we ask kids to think about similarities and differences and think about, when is each a good strategy?
Sometimes we have students compare correct and incorrect strategies and explain the concepts that make the correct strategy correct. Just because you teach kids correct ways of doing things, that doesn鈥檛 mean the incorrect strategies disappear. Students really need help thinking and reasoning through why those are wrong.
What are the more common struggles for teachers to teach multiple strategies?
The No. 1 barrier we face is time. Teachers just feel they鈥檙e under so much pressure to cover so much content that they feel like they can鈥檛 take the time to do this, and that they see the value and the payoff in it. It does pay off for what is assessed [in standardized math tests], but it鈥檚 not directly assessed, and so that makes teachers nervous.
Also, sometimes teachers really don鈥檛 like to say this way is better than this other way. Even though mathematicians would say, 鈥測eah, this way is clearly better in this context, and this other way is clearly better in that context,鈥� ... sometimes teachers feel uncomfortable that they鈥檙e making a value judgment.
But the evidence is really clear that it鈥檚 helpful to show correct and incorrect examples and talk through them.
