Make Sense of Problems and Persevere in Solving Them
Part 7 of 8, Strategies for Integrating the Mathematical Practices into Instruction
By Dr. Michele Douglass
This mathematical practice involves the ways students can explain to themselves the meaning of a problem and the ways they find to enter into solving it. It might feel like being a detective who is looking for clues or evidence on how to solve a problem. Students proficient with this practice believe they are mathematicians and try several methods to come to a solution.
This blog series will conclude by examining Practice #1 in this post and Practice #6 in the final post. These two practices can be thought of as overarching habits of mind that productive thinkers use as they work with mathematics.
“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.” —Common Core State Standards (CCSS), MP#1: Make Sense of Problems and Persevere in Solving Them
It is always difficult to watch students struggle as they are solving problems. However, developing this practice depends on allowing students to struggle. It’s hard work keeping them motivated to persevere in solving the problem. I find that students are more likely to persevere when classroom environments have these three characteristics: (1) struggling is expected, (2) tasks are not predictable, and (3) mistakes are learning experiences.
From my experience being a guest teacher in hundreds of classrooms and conducting countless classroom observations, all of this is possible when problems are connected to real-world scenarios, students are encouraged to use a variety of tools and strategies, and individual as well as partner or group work takes place. This allows the teacher to check in for clarity of thought process and to scaffold where necessary.
Teachers might ask students the following questions about any math problem:
- What is this problem asking?
- What makes sense to do next to solve this problem?
- How would you describe the problem in your own words?
- What information is given in the problem?
- How could you start this problem?
- What tools or manipulatives might help you solve this problem?
- What have you already tried, and how did that work in finding a solution? What might need to change?
- What might be another way to solve this problem?
- How is _________ way of solving the problem like your way? How is it different?
- How can you check this problem?
As you design lessons or tasks for developing the mathematical process of making sense of problems and persevering in solving them, find learning experiences that require students to engage with conceptual ideas that connect to the underlying procedures. It might be a multistep problem that requires a plan. When the task can be completed using multiple entry points, you are allowing students a greater experience with this practice. You can beef up many tasks to include this practice by asking students to defend or justify their solution or to show the pathway they used to arrive at the solution.
As I am working on specific practices within a lesson, I find it helpful to be asking myself questions about the task as well as the lesson. Here are a few that I use when focusing on building proficiency in Practice #1:
- Did I allow time for students to initiate a plan?
- What questions did I ask students who might be stuck?
- Am I asking students about their plan and if the solution is making sense?
- Am I asking students to defend and justify their solutions?
- How did I differentiate tasks and the problems to challenge some of the advanced students?
- How am I questioning students about the connections they are making to previous problems or prior attempted pathways to a solution?
Rather than looking at specific problems, I am giving you a few links to classroom episodes. I find that these either show how the teacher is supporting students in building this practice or how a student is demonstrating his/her use of this practice.
Example 1: Finding the Average Area http://mathpractices.edc.org/content/choosing-samples#comments
When you read the student dialogue, there are several places that show how the students are making sense of the problem. In line 3, there is a question about which 5 to pick. In line 10, Sam questions if this is making sense as the best way to find a sample and, in line 15, stops to clarify exactly the information they have so far.
Notice how important it is to not only provide students time for dialogue as they are problem solving but also to intentionally listen to this conversation for evidence of a specific practice.
Example 2: Second Grade Lesson – Part 2 http://www.insidemathematics.org/classroom-videos/public-lessons/2nd-grade-math-word-problem-clues/lesson-part-2
As you watch this clip, you will see how the teacher is scaffolding for students how to question a solution. Notice how mistakes are occurring and how the errors are used as learning experiences.
There are several other grade-level videos at this site you might want to view for this same practice. I find all of these videos a wonderful resource to build my own understanding of the mathematical practices.
References and Additional Resources:
Common Core State Standards Initiative Website: http://www.corestandards.org/Math/Practice/
Implementing the Mathematical Practice Standards Website: http://mathpractices.edc.org/
Inside Mathematics Website: http://www.insidemathematics.org/index.php/mathematical-practice-standards
Mathematics Assessment Resource Service Website: http://map.mathshell.org/materials/stds.php
Michele Douglass, Ph.D., is the president of MD School Solutions, Inc., a company that contracts with school districts on content and pedagogy with teachers and leaders. Her experience ranges from math instructor to director of curriculum and instruction at Educational Testing Services. She has authored several math curricula, as well as technology programs and professional development, including NUMBERS® with John Woodward, Ph.D., and Mary Stroh, M.S.
