Sunday / July 21

Seven Tips for Developing Better Problem Solvers

Thinking is hard. Teaching students to think is really hard! Modern mathematics standards and curriculum often put an emphasis on mathematical problem solving—frequently described as what you do when you don’t immediately know what to do—which requires a great deal of student thinking.

Teachers know that teaching students to problem-solve is so much more involved and takes a great deal more skill on the part of the teacher, than simply modeling the steps of a procedure to arrive at a correct answer.

Students have to think about the context, consider what is being asked and make sense of the problem, determine a possible solution pathway, figure out how to document their thinking, and determine if their solution makes sense. For students, problem solving is aided by discussion, exposing oneself to new ideas or options, and answering thoughtfully posed questions and prompts. In other words, problem solving (and thinking) is developed not delivered.

The right answers do matter in math, but when learning math, the right answer isn’t the only thing that matters. There may be an occasional trick for getting a right answer but there certainly are no tricks for getting humans to think! In my learning and experiences, I have found that some approaches go a long way toward developing better problem solvers. Here are 7 of them.

#1 Help your students recognize what problem solving and thinking really mean

Students who have been fed a consistent diet of being shown exactly how to perform mathematical procedures avoid or rush through problem solving that requires them to think, because they haven’t always been well conditioned to know how to think. It can be new, intimidating, and frustrating at first, if they feel insecure. The process of thinking is personal and plays out differently for different individuals as they consider what they already know, work through a problem, have new experiences, are exposed to others’ thinking about problems, and then refine their own thinking. Be honest with your students that learning to problem solve may feel uncomfortable at times, but reassure them that your job is to teach them how to think and support them in doing it.

#2 Avoid proceduralizing problem solving

There is no set of procedures that will reliably solve any problem. Procedures can be useful tools in the problem solving toolbox, but they are not the toolbox itself.  Be careful not to invest instructional time in a “fool proof” method for problem solving. There isn’t one! Instead, ensure students have ample opportunity to analyze, represent, and discuss problems. As teachers, highlight the approaches that work and help students see why they worked. Then, as a class, you can see if the approaches that worked will work again with a new problem.

#3 Build students’ thinking habits

Though problem solving and thinking don’t follow a neat process, you can explicitly teach students problem solving habits of mind. You must first give students the opportunity to work a problem on their own or in groups while monitoring what they do. Discuss their reasoning and representations. AFTER they’ve had a chance to explore and discuss, you can model your own thinking—being careful not to insinuate it’s the only way to think—by

  1. Showing them how you make sense of the problem
  2. Showing how you would represent the problem
  3. Articulating the questions you ask yourself while problem solving
  4. Sharing how you check if it is a reasonable answer
  5. Relating all this back to your students’ own explorations to help them make connections to their own thinking processes

#4 Represent

There is so much to say about representations when it comes to thinking and problem solving. Encourage students to represent how they think best. Connect representations to the problem and one student’s representation to another. Show your own ideas about how to represent a problem. Be careful not to restrict what or how students represent but do spotlight efficient and reasonable representations.

#5 Use tools to aid in thinking

Tools help students, but they can’t do the thinking for the students. Tools are used for visualizing the math and for better accuracy. Pattern blocks and fraction tiles are good for building a model of a situation. Calculators can shift the brain’s burden from calculation to problem solving. We want to make tools available, encourage students to make choices about when to use them, and ensure there is no shame in choosing to use them.

#6 Make it a regular routine

Problem-Solving Friday is a problem! It insinuates that problem solving is done in isolation from “regular math” or after math is learned. Math and problem solving are inseparable. More importantly, it takes a long time to get good at problem solving. Regular problem-solving practice, daily or at least routinely in the week, provides that opportunity. This isn’t to say that an abundance of problems each day necessarily make better problem solvers. Routine problem solving can focus on just one problem! You can do more problem solving with fewer problems by focusing on many of the things shared here. Plus, know that problem solving goes beyond word problems. A good, open question is a problem (e.g., How can you show ¾ in three different ways? What are two problems that would be good to solve with compensation?). It can (and should) be a core component of every lesson. You can find dozens of problem-solving routines and prompts in my new book Daily Routines to Jump-Start Problem Solving.

#7 Practice problem solving without students knowing it

In the age of standards-based instruction, everything done in a classroom is aligned to a standard and understandably so. Playing games focused on number concepts, operations, and basic facts is a good idea. But don’t forget that classic logic games have a place too! These games help students develop strategic thinking, reasoning, anticipation, and perseverance. So, provide classic logic games (and puzzles) like Dots and Boxes or Othello as an occasional option for center and independent work.

You have an opportunity to reframe problem solving – if not this year, then certainly for next year. Look back at these tips and think about which match up best with who you are and how you are growing as a teacher. As you work with your students, acknowledge that problem solving (and thinking) is hard but worthwhile and necessary. Assure them that you are there to help them learn to think for themselves—instead of thinking just like you do! Let them know that you have strategies you can show them, but the goal is that they will learn to make sense of problems on their own. Help them see that getting the right answer isn’t all that matters.

Written by

John SanGiovanni is a nationally recognized leader in mathematics education. John works as a Coordinator of Mathematics in Howard County, Maryland leading mathematics curriculum design, professional learning, assessment, and digital learning. He also works to develop new mathematics leaders at McDaniel College in Maryland. John is an author of more than two dozen books about teaching and learning mathematics and consults nationally for curriculum development and professional learning. He is active in professional organizations recently serving on the Board of Directors for the National Council of Teachers of Mathematics and currently on the Board of Directors for the National Council of Supervisors of Mathematics. John’s most recent book is Daily Routines to Jump-Start Problem Solving (Corwin, 2023).

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