Mathematical Practice and Problem Solving: How to Support Your Teachers
By: Ross Brewer, Ph.D., Exemplars President
The Common Core State Standards – Mathematics is divided into two parts: Content Standards, and Standards for Mathematical Practice. A major focus of the Standards for Mathematical Practice is on using problem solving to reinforce important concepts and skills and to demonstrate a student’s mathematical understanding.
To fully implement the Common Core, teachers must have an understanding of what problem solving is, why it is important and how to go about implementing it. For many, the successful teaching of problem solving will require real pedagogical shifts. What do teachers need to know?
To help answer this question and prepare your staff, you might turn to findings in the recent report, Improving Mathematical Problem Solving in Grades 4 Through 8, published in May 2012 under the aegis of the What Works Clearinghouse (NCEE 2012-4055, U.S. Department of Education, available online from the Institute of Education Sciences). This report provides educators with “specific, evidence-based recommendations that address the challenge of improving mathematical problem solving.”
In the Introduction, the panel that authored the report makes the following points:
- Problem solving is important.
“Students who develop proficiency in mathematical problem solving early are better prepared for advanced mathematics and other complex problem-solving tasks.” The panel recommends that problem solving be part of each curricular unit.
- Instruction in problem solving should begin in the earliest grades.
“Problem solving involves reasoning and analysis, argument construction, and the development of innovative strategies. These should be included throughout the curriculum and begin in kindergarten.”
- The teaching of problem solving should not be isolated.
“… instead, it can serve to support and enrich the learning of mathematics concepts and notation.”
- Despite its importance, problem solving is given short shrift in most classrooms.
To address these points and improve the teaching of problem solving, the panel offers five recommendations.
Recommendation 1
Prepare problems and use them in whole-class instruction.
In selecting or creating problems, it is critical that the language used in the problem and the context of the problem are not barriers to a student’s being able to solve the problem. The same is true for a student’s understanding of the mathematical content necessary to solve the problem.
Recommendation 2
Assist students in monitoring and reflecting on the problem-solving process.
“Students learn mathematics and solve problems better when they monitor their thinking and problem-solving steps as they solve problems.”
Recommendation 3
Teach students how to use visual representations.
Students who learn to visually represent the mathematical information in problems prior to writing an equation are more effective at problem solving.
Recommendation 4
Expose students to multiple problem-solving strategies.
Students who are taught multiple strategies approach problems with “greater ease and flexibility.”
Recommendation 5
Help students recognize and articulate mathematical concepts and notation.
When students have a strong understanding of mathematical concepts and notation, they are better able to recognize the mathematics present in the problem, extend their understanding to new problems, and explore various options when solving problems. Building from students’ prior knowledge of mathematical concepts and notation is instrumental in developing problem-solving skills.
The panel also identifies two specific “roadblocks” to implementing these recommendations:
Roadblock 1
“Traditional textbooks often do not provide students rich experiences in problem solving. Textbooks are dominated by sets of problems that are not cognitively demanding …”
Exemplars was started precisely to meet this need — to provide the rich problem-solving tasks that teachers and students lacked in traditional texts.
Roadblock 2
Lack of time/opportunity to do problem solving in the classroom.
The panel notes that in addition to spending time solving problems, research shows that students benefit by studying already solved problems.
Exemplars annotated anchor papers help meet this need.
As president and founder of Exemplars, it is validating to see the fundamental elements of our material affirmed in this rich research-based report. So much of what is discussed is at the core of what Exemplars math material is all about and has been since we began publishing 19 years ago:
- The importance of success with problem solving
- The critical role formative assessment plays in the classroom
- Students’ use of representations in making the link between the problem and the underlying mathematics
- Students’ ability to communicate their thinking
- Students’ application of appropriate mathematical language and notation
- Helping teachers instruct students in mathematical understanding and allowing students to demonstrate that understanding.
We believe all of these factors should play a critical role in instruction, assessment and professional development.
As teachers are asked to implement more problem solving in their classrooms in support of the Common Core Standards for Mathematical Practice, Exemplars math tasks provide a valuable resource. The tasks are also an effective tool for staff development.