Instructional Strategies That Support Mathematical Problem Solving Janis FreckmannBeth SchefelkerMilwaukee Public Schools National Council of Teachers of Mathematics (NCTM) Annual Meeting Atlanta, April Based upon work supported by the National Science Foundation Grant No. EHR

Engaging in Problem Solving What do students need to do to become better problem solvers?

Good Problem Solvers Become aware of what they are doing and frequently monitor, or self-assess, their progress or adjust their strategies as they encounter and solve problems. Principles & Standards (NCTM, 2000), p. 52

Principles & Standards The ability to read,write, listen, think, and communicate about problems will develop and deepen students' understanding of mathematics. NCTM, 2000, p. 194

Principles and Standards Communication Standard Organize and consolidate their mathematical thinking through communication Communicate their mathematical thinking coherently and clearly with peers, teachers, and others Analyze and evaluate the mathematical thinking and strategies of others Use the language of mathematics to express mathematical ideas precisely

Session Goals Experience the Think-A-Loud process through the context of a math problem. Understand ways the Think-A-Loud strategies support student talking, thinking, and writing mathematics.

Why Use Think-A-Louds? T o model and demonstrate the usually hidden mental processes that enable learners to be successful. Stephens & Brown (2005), p. 49

Think-A-Loud A Problem Solving Strategy for Mathematics Understanding the Task Development of Concept and Context Working on the Problem Thinking about the Solution

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Think-A-Loud Understanding the Task Read the problem (whole group, pairs, or independently) Visualize the situation Restate the problem (not focusing on the answer) Connect to real-life situations

Think-A-Loud Development of Concept and Context Develop vocabulary specific to the mathematical content Clarify vocabulary related to the context of the problem Develop the mathematical idea embedded in the problem Connect the mathematical ideas to previous work

Think-A-Loud Working on the Problem Discuss various approaches for entry into the problem (partner,whole class) Redefine the question in the problem Solve the problem independently or with a partner Explain your work or your partner’s work

Think-A-Loud Thinking about the Solution Relate connections between the answer and the problem Share student work samples; (1)discuss the mathematics, (2) the approaches to the problem and (3) the student reasoning.

Why Use Think-A-Louds? Thinking aloud is one of the most powerful strategies in a teacher’s repertoire. Its purpose is to model and demonstrate the usually hidden mental processes that enable learners to be successful. As a teacher’s thinking becomes explicit, it helps students understand processes used to construct meaning & solve problems. Students then use their own Think-A-Louds. Stephens & Brown (2005), p. 49

Adapted from : National Research Council. (2001). Adding it up: Helping children learn mathematics.

Reflective Habits of Mind “Metacognition” Before we go on, are we sure we understand this problem? Do we have a plan? Are we making progress or should we reconsider what we are doing? Why do we think this is true? Principles & Standards (NCTM, 2000), p. 54

Lessons Learned: Teacher #1 “Think-A-Louds are a great for helping students organize their thinking. It helps every child have a place from which to start their work. The steps the students find the most useful are finding the important information and restating what the problem is asking.” “In using Think-A-Louds, I have the opportunity to delve into my students’ thinking.”

Lessons Learned: Teacher #2 Students have been able to easily solve what used to be difficult problems. More students participate during math lessons. Students also seem to demonstrate more confidence and a willingness to work on any problem.

Engaging in Problem Solving What do teachers do to promote problem solving in their classrooms? Ask questions. Share ideas for entry points. Don’t tell too much. Provide discussion time.

Thank you! Janis Freckmann Beth Schefelker

Putting the Strategy into Practice Phase 1 Teacher Models Structure Students Engage in Reasoning Phase 2 Pairs “Try Out” Thinking Aloud Phase 3 Students Think Aloud Independently

Effective Problem Solvers Make sure they understand the problem:  Carefully read.  Ask questions until they understand. Develop a plan. Constantly monitor & adjust their work. If not on the right track, they consider alternatives and do not hesitate to take a completely different approach. Principles & Standards (NCTM, 2000), p. 54

Extras The following three slides provide sample questions teachers can use as they work with students: To promote problem solving. To help students who get stuck. To check student progress.

To promote problem solving… Tell the problem in your own words. Close your eyes and explain what you see happening in the problem. What do you need to find out? What information do you have? Describe a strategy you are going to use? Explain how your answer will connect to the problem.

To help students who get stuck… How would you describe the problem in your own words? Explain the facts you know are in the problem. Draw a picture of the problem. How is your partner going to solve the problem? Let’s change the numbers and then try to solve the problem.

To check student progress… Why does your answer makes sense? Explain your work to your partner. Explain your partner’s work to another. Share your thinking on your method. Compare how you solved the problem to how _____ solved the problem. Is there a more efficient strategy? Would this way work with other numbers? Does anyone have the same answer but another way to explain it?

Student Work Teacher Focus  Compare how the task changed.  Draw inferences about the teacher’s instruction. Student Focus  Describe the student’s progress in his/her written responses (e.g., clarity, detail, organization/structure).

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