As the new school year shifts into high gear, we?re pleased to bring you the second installment of our two-part series on math problem solving.
Teachers today must prepare their students for the increasingly complex world in which we all live. As this month?s article points out, this brave new world demands ?meaningful learning? with a focus on problem-solving. To raise a new generation of willing and able problem-solvers, teachers may need to strengthen their own problem-solving skills. We?re pleased to offer you insights from the creators of the Making Sense of Problem Solving series. This exciting new series offers a supplementary problem-solving curriculum and professional development for grades K?8.
We hope this issue brings you the strategies, tips, and resources you need to tackle this challenging aspect of teaching. The potential benefits are huge?both for us and for generations to come. 'Til next time,
by Cathy Brown, Kathleen Barta, and Jacqueline Cooke
Now that school is back in full swing and you have an idea of your students? range of mathematics abilities, you may be asking yourself, ?What are the most effective instructional strategies and practices that I can use to reach all of my students??
Linda Darling-Hammond (2008) provides solid answers to that question. She analyzes new developments in cognitive research in view of the breathtaking expansion of knowledge and technical information that characterizes the world for which we are preparing our students. She states:
These new demands (of changing information, technologies, jobs, and social conditions) cannot be met through passive, rote-oriented learning focused on basic skills and memorization of disconnected facts. Higher-order goals demand what some analysts have called ?meaningful learning??that is, learning that enables critical thinking, flexible problem solving, and transfer of skills and use of knowledge in new situations.
As Darling-Hammond states, one way to foster meaningful learning is to provide instruction in flexible problem solving.
THE IMPORTANCE OF PROFESSIONAL DEVELOPMENT
The instructional strategies required to stimulate the kind of meaningful learning that can be achieved through problem solving are much more complex than direct instruction. Even with the support of good curricular materials and good assessments, teaching for understanding is demanding. Among other things, the teacher needs a solid grounding in mathematics to be able to recognize the mathematical potential in what students say and to guide instruction so that students engage with and learn fundamentally important mathematics. The need for solid mathematical knowledge should not be underestimated.
In addition, teachers need to be able to structure group work and student discourse so that productive conversations take place that are centered on meaningful mathematics.
Teachers can be supported in learning these strategies by a combination of professional development, a well-chosen problem solving curriculum, and a focused professional learning community.
In Powerful Learning: What We Know About Teaching for Understanding, Alan Schoenfeld (2008) outlines four elements of effective professional development:
Establishing clear goals
Setting aside time for teachers to reflect on their work
Maintaining continuity of focus rather than having a series of divergent workshops
Providing opportunities for teachers to immediately try out content-specific strategies and tools, and to continue to refine those strategies with a group of colleagues in a learning community
INSTRUCTIONAL STRATEGIES THAT PROMOTE SUCCESSFUL PROBLEM SOLVING
A number of principles characterize successful problem-solving instruction. These include:
Creating a safe environment
Engaging students with rich tasks
Differentiating for all students
Encouraging focused student discourse
Using multiple representations including manipulatives
Modeling different solutions, and focused discussion of concepts
Create a Safe Environment
Students learn best when they are actively engaged in the lesson. Working with rich tasks means they may experience disequilibrium at times. This can be uncomfortable. To ensure that students don't shut down even in uncomfortable situations, here are some things to consider.
First, a student needs to feel comfortable in the classroom when he or she is asking a question, sharing a solution method, and possibly sharing errors in thinking or solutions. In other words, a student needs to feel safe. When students take the risk to share their thinking, encourage the rest of the class to look for what is right in the ideas presented. Often, looking for the "right" in one child's thinking is just the piece another student needs to springboard toward more complete understanding. If ideas are presented as contributions to the class as a whole, mistakes are not seen as failures, but as opportunities for everyone to learn and grow.
Engage Students with ?Rich Tasks?
Rich tasks, or "good problems," are characterized by a number of qualities, including:
Being accessible to a wide range of students
Having an intriguing context for the problem solvers, and/or having intriguing mathematical ideas that emerge from the problem
Providing for differentiated responses
Having potential to broaden students? understanding of mathematics content
Before using the task with students, it is helpful to solve the problem yourself in as many ways as possible. Teachers need to be aware of the mathematics embedded in the task?as well as common misconceptions?so that they can engage students in discussions that will deepen mathematical understanding.
Effective reading strategies are helpful for math problem solvers. Discussions about the vocabulary and the relationship of the context of the problem to the real-life experiences of students can help students make sense of the problem.
It can also be helpful, when exploring a ?rich problem situation,? to remove the question and ask the students themselves to pose possible questions. This opportunity engages students not only in effective reading, but also in analysis of the problem situation. The teacher can help students settle on a question that will deepen conceptual understanding.
Differentiate for All Students
Effective differentiation is best achieved with a systematic approach to matching content, process, and products to the readiness, interests, and learning modes of each student. Use activities that provide various levels of difficulty but focus on the same essential learning goals, so that each student has a role to play as a valued part of the learning community.
In Differentiation in Practice: A Resource Guide for Differentiating the Curriculum, Grades 5?9, Carol Tomlinson writes, ?The best tasks are those that students find a little too difficult to complete comfortably. Good instruction stretches learners.? (p. 14)
One differentiation technique is to pose a single question or task that students can approach at different levels, with different processes or strategies, and with somewhat different levels of mathematical knowledge. Another method is to pose two or three parallel tasks that are designed to meet the needs of students at different developmental levels, but that target the same big idea and are close enough in context that they can be discussed simultaneously. A third way is to provide series of tasks on a specific topic with progressive levels of challenge. All students participate in warm-up exercises and then progress to a more challenging task. Students who are ready for further challenge engage in solving extension problems.
After your students solve problems, analyze their work and any journal comments. Then adjust your differentiated instruction based on what you learn from that analysis.
Encourage Focused Student Discourse
Encourage students to rely on their own and each other?s thinking. Foster purposeful, focused student-to-student dialogue. Many studies have found that well-focused cooperative learning among peers often stimulates problem solving and reasoning to a greater degree than students achieve when working independently (Teasley, 1995; Webb, 1991).
Use Multiple Representations, Including Manipulatives
Encourage students to visualize solutions, especially with the use of manipulatives, and to use a variety of mathematical representations (as appropriate). Then ask students to compare various representations. For example, ask: ?Where do you see the work that Janyce did (15 ÷ 6 = 2.5) in the sketch that Xavier made??
The use of multiple representations provides opportunities for students to use a variety of strategies and processes that make sense to them. Students learn new ways of approaching problems from other students? representations and strategies.
Model Different Solutions and Have a Focused Discussion of Concepts
Shimizu (2003) describes an instructional model typically used in Japanese classrooms. Lessons are usually organized around multiple solutions to a single problem because of the belief that students learn most effectively when they are engaged in solving a challenging problem. Teachers prepare for lessons by anticipating a wide range of student solutions and methods, so that they are prepared to lead a discussion that draws out and highlights the essential mathematical ideas that students are likely to present.
While students are working, select a few groups that have different solution strategies to share later. When students are finished solving the problem, invite the groups you selected to share their work and explain their thinking. Ask the group with the most concrete solution to share first, so that all students can understand. Let the students with the most abstract solution share last. Ask other students to compare specific parts of the more abstract work to visual or otherwise more concrete solutions.
The good news is this: Virtually all of the available scholarly evidence indicates that teaching for understanding in mathematics?devoting significant amounts of class time to conceptual understanding, problem solving, and reasoning; having students communicate mathematical ideas orally and in writing; having students seek and make connections; and guiding students to reflect on their work?pays significant dividends.
We hope that the ideas presented in this article will be helpful reminders. We teachers have one of the most challenging jobs, but also the privilege of being conduits to the future. Best wishes for a productive and rewarding year!
Darling-Hammond, L., Schoenfeld, A.H., et al. (2008). Powerful learning: What we know about teaching for understanding. Hoboken, NJ: Jossey-Bass.
Shimizu, Y. (2003). Problem solving as a vehicle for teaching mathematics: A Japanese perspective. In F. K. Lester, Jr., R. I. Charles (Eds.), Teaching mathematics through problem solving (pp. 205-214). Reston, VA: NCTM.
Teasley, S. D. (1995). The role of talk in children?s peer collaborations. Developmental Psychology, 31, 207?220.
Tomlinson, C. A., Eidson, C. C. (2003). Differentiation in practice: A resource guide for differentiating curriculum, grades 5?9. Alexandria, VA: ASCD.
Webb, N. (1991). Task-related verbal interaction and mathematics learning in small groups. Journal for Research in Mathematics Education, 22, 366?389.
Teaching math problem solving is a challenge, requiring teachers to master new instructional strategies and have a solid grounding in the math itself. Turn your classroom into a problem-solving environment with Making Sense of Problem Solving, the new professional development series from Teacher to Teacher Publications. The nine books in the series (spanning grades K through 8) provide a wealth of problem-solving strategies as well as sample lessons targeted to the NCTM Curriculum Focal Points.
We believe this comprehensive teacher guide is a resource you won't want to be without. Download these sample pages today!
Teaching with your students' learning styles in mind is an important component of successful instruction. Join us next for an informative article on this topical theme, plus Internet resources and a product special on a new activities book that brings learning styles front and center.
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