Four Purposes of Math Problems
"OK, now do problems 1-25 on page ____."
I heard this almost every day as a student when my teachers assigned math problems--odds in class, evens for homework--at the end of each section in the book. And many teachers still do this today.
But does this work for students? Not based on what I've seen and heard from them in my 23 years as an educator, including widespread mathematical understanding and skills deficits, and beliefs that "I'm not good at math."
To help students develop competence and confidence in math, teachers should be concerned with the quality of problems they assign rather than the quantity. They should assign problems purposefully rather than randomly or just because they're in the book. With this in mind, here are four viable purposes for assigning math problems to students:
Learn "new" content. I put new in quotes because a lot of content from one year to the next isn't new at all, but rather relates to and/or builds on previous content. Consider, for example, the connections between fractions and ratios, and equivalent fractions and proportions. Or the relationship between algebra and arithmetic. Such a close relationship that algebra is often referred to as "generalized arithmetic," which makes sense, since using the distributive property, combining like terms, applying the laws of exponents, and other rules and procedures are the same for algebraic expressions as they are for arithmetic expressions. The point--regardless of topic--is that math teachers should strive to introduce content through problems that are accessible yet challenging (think "productive struggle") for students rather than through traditional teacher-led lessons. See my post, Learning by Doing...and Grappling, for an awesome video of a fifth-grade teacher doing this for decimal division.
Deepen understanding of previous content. Even when students have demonstrated mastery of a skill, it's possible for them to gain a deeper understanding of the math involved. But for this to happen, teachers must assign the right problems and be prepared to scaffold students' understanding. Here's one such problem that stretches students (many teachers too) including those who've mastered--or memorized--the laws of exponents: Which is greater: 280 + 280 or 2100 ? (I'll give you a chance to play around with this and share your thinking in the comments section before I share mine.)
Reveal and resolve misconceptions. Like all of us, students learn as much or more from their mistakes as anything. So, don't prevent students' mistakes, prepare for them. Assign problems students are likely to mess up, then help them learn from their mistakes so that they don't make the same mistakes again. Here's a seemingly simple problem that exposes and provides an opportunity to clear up a common misconception among students: Indicate > , < , or = : - 32 32 (Again, please share your thoughts on this problem before I share mine.)
Develop procedural fluency. Not to be confused with fact fluency, procedural fluency--as defined by NCTM--builds conceptual understanding, strategic reasoning, and problem solving. It involves applying procedures accurately, but also efficiently and flexibly; and recognizing when one strategy or procedure is more appropriate to apply than another. To help students develop procedural fluency, teachers must therefore assign problems that are conducive to discovering and discussing multiple solution strategies. And once again, this doesn't require elaborate problems. Sometimes it's just a matter of recognizing the learning potential within straightforward problems. Here's a simple problem that generates rich discussion and helps students develop procedural fluency and number sense related to fractions: Find two fractions between 2/5 and 4/6. (Again, seems straightforward, but think about all the ways of approaching it including common denominator, common numerator, converting to decimals, and comparing with a benchmark fraction such as 1/2).
This list isn't meant to be comprehensive, so please suggest additions. But math teachers would do a lot better for students by only assigning a problem if it serves one or more of these purposes than by assigning a problem just because it's in the textbook.
I'm not saying you should avoid using or assigning problems from textbooks. This isn't a referendum on textbooks. It's about making thoughtful, intentional choices that help students learn and like math AND feel good about themselves in the process. And when I ask teachers why they assign students one similar problem after another, many of them don't have an explanation. It's just a default approach. Following the textbook and assigning problems at the end of each section is what math teachers are supposed to do.
Others rationalize this approach by asserting that mastery requires practice, which may be true. But again, quality often matters more than quantity when it comes to practice. If students believe they can't solve a particular problem, what's the point of assigning them 20 more similar problems? And if they can solve a problem in their sleep, why should they do it again and again when they're awake? Besides, meaningful practice in math involves a mix of content rather than traditional assignments limited to one topic. This is often referred to as "distributed practice" or "spiraled practice," and some textbooks (including Saxon Math) take this into account, but many do not. If 30 problems in a textbook section are really worth assigning, spread them out over the course of the year rather than one or two days. Mix in purposeful problems on other topics, and assess and address students' ongoing needs--from remediation to extension to enrichment--across the curriculum.
And finally, there's the longstanding, misguided belief that repetition is important because memorization is a key to proficiency in math, which Jo Boaler shoots down in her article, Memorizers are the Lowest Achievers and Other Common Core Math Surprises.
But what should you assign students instead of problems that promote memorization? Memorable problems, using the above list as a guide.
And check out my Go-To Resources for Common Core Math if your textbook or other curriculum materials lack these types of problems. Teachers I coach have found Illustrative Mathematics to be particularly useful because it not only features great problems but also includes commentary about the purpose of a problem and how teachers can support that purpose in class.
Image by Markasia, provided by Dreamstime license