Previously, we discussed the background of the eight Common Core State Standards for Mathematical Practice. Today, we’ll examine the first of these standards: Make sense of problems and persevere in solving them.
In an age of anecdotally short attention spans, learning how to engage deeply with a problem, think laterally, select strategies, and justify solutions has become essential. The reality is that many students expect that “solving a problem” should involve a few short steps taken from content learned within the last two weeks (or even the last two days). Is the problem taking longer than five minutes to complete? Something must be wrong.
In real life, problems don’t come embedded in specific content units with convenient formulas listed on the previous page. In real life, you can’t check the answer in the back of the book; you have to verify your answers. Real-life problems usually take a bit more than five minutes to solve, and most do not have immediately evident solution paths.
How can we teach and assess this practice? We can’t just drop students, cold turkey, into solving complex, multi-step problems. Problem solving and perseverance are learned skills that need to be developed over time. Here are six simple strategies to jumpstart problem-solving skills. All apply to both word problems and decontextualized math problems.
- Ask students to give an initial estimate of a problem’s solution. This prompts a review of the givens, constraints, and the student’s contextual background knowledge, and creates a “gut check” for the final answer.
- Give students several worked solutions to a problem, each utilizing a different strategy. Ask them to analyze and compare the strategies. The worked solutions may be student-generated, in which case, students can advocate for a strategy of their choosing.
- Give students a set of problems drawn from different units, and ask them to map the problems to content areas. This requires metacognitive, lateral thinking and builds recognition of mathematical context.
- Prior to actually beginning the math on a problem, have students write a few sentences about their strategy and what kinds of work they anticipate doing. Verbalizing mathematical thinking can be surprisingly difficult, so students will probably need models and scaffolding at first.
- Assign a complex problem and require students to work on it through noncontiguous time periods. Perhaps they work on it in class for 10 minutes every day for a few days.
- Games! From Mastermind to Rush Hour to Rubik’s Cubes, there are lots of great, easy-to-learn games available that teach sustained, strategic, logical thinking. This is a great strategy for shortened school days or other odd periods of time.
Standardized tests are getting better at assessing problem-solving skills. Smart people are learning how to write tests that can be graded quickly and also begin to tease out student thinking at a deep level. Even as we continue to increase the quality of our standardized tests, though, the best assessment still comes from classroom teachers closely examining their students’ work.
I would argue that this standard is the single most important thing students can get out of our math classrooms. Of course, students who go on to science and engineering jobs will utilize the actual math content we teach. But for non-STEM bound students—80% of high school students, according to the Department of Education!—the ability to corral mental resources, sort out what’s important and what’s not in a problem, review and select old and new strategies, and verify results will be carried with them as a lifelong skill, cutting across disciplines and job categories.
Next week, we’ll look at the second and fourth Mathematical Practice Standards: reasoning and modeling.
How have you used or assessed problem solving? Leave your thoughts in the comments!