Why should students learn math through problem solving?
One of our goals as teachers is to help students become better at solving problems, and to think of solving problems as a common, even exciting and engaging process. Another associated goal is to recognize and discuss with students the problem solving strategies that they employ in their work. Applying these techniques allows students to build their understanding of mathematical concepts while increasing the level of their confidence.
If you’ve ever heard a student say “I can’t do this,” or “I don’t know what to do,” then you’ve probably encountered the student’s perception that there is one correct procedure that should be followed when solving a particular problem and that he or she is not confident about what that procedure should be. Finding examples that allow students to explore different strategies and perhaps even find different solutions will assist them in building their skills.
How will these resources help?
The resources presented in the collection can assist teachers in building problem solving acumen. Resources include video clips from actual classrooms in which students work to solve problems, articles that relate helpful techniques, detailed descriptions of several strategies students can use when learning to solve problems, and suggestions of courses that help teachers to advance problem-solving with their classes.
What do the standards expect?
The Common Core State Standards for Mathematics includes a set of eight Standards for Mathematical Practice that “describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.” The first Practice Standard is “Make sense of problems and persevere in solving them.” Students build these skills by having many opportunities to solve problems. By providing problems that allow students to engage in a variety of methods, to decide whether or not their efforts are leading to a solution, and to compare results with other students, teachers help students begin to “make sense” of problems and also to begin to develop perseverance.
Other Practice Standards (as well as the NCTM Process Standards) will be drawn into the problem-solving process as students gain confidence in their abilities. They will make sense not just of the problems they encounter, but also of the underlying mathematical concepts and connections. They will learn to choose and use appropriate tools and representations. Vocabulary develop can occur naturally as a result of collaboration and communication.
Spend some time with these resources, watch the videos, choose some strategies to try with your students, and enjoy the “Aha moments” you share as they develop their skills and deepen their understanding of mathematical concepts.
This document for teachers provides four activities to develop students' ability to understand and interpret problems. These strategies help students deepen their focus and improve their problem-solving skills. The document includes both Problem Solving goals and Communication goals, as well as sample activities and specific examples related to the Wooden Legs Problem of the Week from the Math Forum. A copy of the complete problem, the scenario (with the question removed) and student handouts for applying the problem-solving strategies are also provided.
This packet of information for teachers is designed to be used with the Wooden Legs Problem of the Week from the Math Forum collection for grades 3 to 5. The sample problem provides an exemplar of the type of problem teachers can use to promote students' mathematical thinking and engage their interest. The key concepts in this problem include pattern identification and multiples. The packet contains the sample problem, standards alignment to the Common Core State Standards, possible solutions, teaching suggestions, a print-ready copy of the problem, and a problem-specific scoring rubric.
This page from Suzanne Alejandre's Math Forum blog has links to nine short videos (5 minutes or less) of Ms. Alejandre implementing the "Wooden Legs" Problem of the Week with a fifth grade class. The "Notice/Wonder" strategy is used to introduce the problem, and additional materials describe other problem solving strategies. The blogpost describes the goals of the lesson and also includes links to the teacher materials including the problem, solution, sample student answers, a scoring rubric, and teaching suggestions. Suggested browsers are Chrome, Firefox, or Safari. (It's been reported to us that when using IE9 (PC) the videos do not display.)
This page from the Math Forum blog of Suzanne Alejandre links to a one-minute video during which Max Ray acts out the "Charlie's Gumball" Problem of the Week. The page includes links to PDFs of the scenario, the PoW Packet with the solution and teaching suggestions, a scoring rubric and the "Notice/Wonder" strategy handout that can be used to introduce the problem.
This article discusses an inquiry practice that encourages students to develop their mathematical thinking through discussion. The author describes examples of students taking ownership of their learning when they are encouraged to justify their positions through sound mathematical reasoning. This practice builds a foundation for formal proofs.
This page of videos is designed to showcase classrooms in which the NCTM Process Standards are evident. Scroll to video #45, and select the "VoD" box to view the 20-minute Cranberry Estimation video. Second-graders in Massachusetts estimate the number of scoops of cranberries that will fit in a jar. They report, graph, and discuss group estimates with the class as the concepts of range, mode, and median emerge. Observe as the teacher leads the class in developing strategies, learning about new concepts, and sharing and organizing results.
This 86-page practice guide (pdf) provides educators with five specific, evidence-based recommendations for improving students' mathematical problem solving in grades 4 through 8 by incorporating such activities into regular instruction. The guide contains detailed suggestions and strategies for carrying out each recommendation, including potential roadblocks with possible approaches for overcoming them. It concludes by suggesting a four-step process for incorporating the recommendations into a lesson. The guide includes an extensive list of references.
This course shows teachers how to develop effective ways to use the five National Council of Teachers of Mathematics Process Standards to teach mathematics to all students at the K-2 level. The course materials are freely available online and include information, support materials, activities, video clips, and a forum for communicating with other participants nationwide. The course is designed as a professional development resource for pre-service and in-service elementary school teachers; teacher-leaders in schools and districts; professional developers; and other professionals involved in mathematics education. The materials may be used on your own, in a study group, or as a facilitated online course for graduate credit which is offered at a reasonable cost.
This course shows teachers how to develop effective ways to use the five National Council of Teachers of Mathematics Process Standards to teach mathematics to all students at the grades 3-5 level. The course materials are freely available online and include information, support materials, activities, video clips, and a forum for communicating with other participants nationwide. The course is designed as a professional development resource for pre-service and in-service elementary school teachers; teacher-leaders in schools and districts; professional developers; and other professionals involved in mathematics education. The materials may be used on your own, in a study group, or as a facilitated online course for graduate credit which is offered at a reasonable cost.
This page of videos is designed to showcase classrooms in which the NCTM Process Standards are evident. Scroll to video #48, Problem Solving, and select the "VoD" box to view this half-hour video. It includes 13 classroom excerpts from lessons that illustrate students investigating and learning mathematics through problem solving. Teachers share their approaches and observations.
This article from the NCTM journal Teaching Children Mathematics, begins with a video (transcript also provided) of a conversation between a teacher and a student. The article presents several teachers’ responses to the student as he attempts to solve a problem. Teachers’ responses demonstrated four categories of reasoning that are discussed and evaluated as possible strategies when working with students.
This article from the NCTM journal Teaching Children Mathematics, describes a year-long professional development project involving the integration of problem solving into the third-grade mathematics curriculum and implications for learning and teaching elementary mathematics. The article includes four examples of student work.
Rich problems invite a variety of solution strategies. They provide an effective vehicle for accommodating a range of levels and learning styles. In this six-week online course participants will explore strategies for using problem solving to build skills and concepts for struggling students and to provide challenge for those who soar. The course is available from the Math Forum for a reasonable fee.
After completing this online course, current subscribers of the Math Forum’s Problems of the Week (PoWs) will be familiar with all the resources associated with their PoW membership access. They will be able to make an informed decision about how to start implementing PoWs with their students, and they will have an idea of further steps to try when they are ready. Both PoW membership and this course are available from the Math Forum for a reasonable fee.
This course will provide teachers an opportunity to deepen their understanding of topics and student learning in geometry and measurement and the problem solving process. Teachers will discuss their own solutions to problems about perimeter, area, volume, and properties of triangles. Exploration will include the use of interactive applets and dynamic geometry software. Teachers will analyze student work to discover strengths and weaknesses in order to make more effective instructional decisions. The course is available from the Math Forum for a reasonable fee.