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“ A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations, he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.” -George Polya

George Polya the father of problem solving in mathematics education. 1887–1985

Impact of Instructional Approaches Key words Every problem has a given strategy There is one way to solve a given problem Teacher’s Way or the Highway Problem Solving should be nice and neat and quick If you can’t remember the formula there is no hope of solving the problem

The Poster Method Mario has a wall in his room that measures 13 ft long and 8 ½ ft high. He wants to hang his favorite posters on the wall. Each poster measures 3 ft by 2 ft. What is the greatest number of posters that he can hang on the wall so that the posters do not overlap?

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What is a Good Problem-Solving Task? Non-routine Non-routine No prescribed method or memorized rules No prescribed method or memorized rules No perception by students that there is a specific correct solution method. No perception by students that there is a specific correct solution method.

Selecting the Problem Solving Task Does it relate to current instructional focus? Does it match students’ current instructional level? Is it relevant and engaging to students? Does it require students to “stretch” their mathematical reasoning abilities? Involve more than one strand or standard of mathematics? Is there more than one way to solve the problem?

Common Core Math Practices Solve Problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments Model with mathematics Use tools strategically Attend to precision

Emphasize mathematical reasoning and verification and use of math vocabulary Believe that students have the capacity to problem solve Don’t “save” students Train components of the process Build student capacity to problem solve

Promoting and Implementing Effective Discourse  Clarify students’ ideas in a variety of ways  Clarify students’ ideas in a variety of ways restate students ideas as questions have students restate others ideas  Em phasize Reasoning right  Em phasize Reasoning ask follow up questions whether the answer is right or wrong teacher’s role is to understand student’s thinking  Encourage Student Dialog  Encourage Student Dialog provide opportunity to share in small group or with peer

Guiding Questions What are you doing? Why are you doing it? How does that help you? What did you do that helped you understand? Did you find the information that you didn’t need? How did you decide what to do? How did you decide your answer was correct?

Sequence to Build Capacity Poster Method Poster Method with a Write-Up Alternative Method Poster Method– 2x’s Month

Primary Task Write Up Primary Task Write Up

Rationale “Writing in math class supports learning because it requires students to organize, clarify, and reflect on their ideas—all useful processes for making sense of mathematics. In addition, when students write, their papers provide a window into their understandings, their misconceptions, and their feelings about the content they’re learning.” Source: Marilyn Burns, “Writing in Math,” Education Leadership, 62(2) (October 2004), p. 30.

Original Problem: Mr. Robinson is building a new cage for his rabbit, Nicholas. He has 12 feet of fencing to build the rabbit cage. The cage needs to be a rectangle. What are the lengths and widths of the different sizes of cages that he can make with that amount of fencing? Which cage will give Nicholas the largest area to play around in? Revised Problem: Mr. Robinson is building a new cage for his rabbit, Nicholas. He has 12 feet of fencing. What is the largest cage Mr. Robinson can build? Justify your final solution with pictures and words. Revising Word Problems

The animation is already done for you; just copy and paste the slide into your existing presentation. Class Discussion Class Write Up Class Solution Kindergarten Problem Solving Solve with Partner Student WorkRead Together

The animation is already done for you; just copy and paste the slide into your existing presentation. ata Sheet Individual Work Reflect ection Group Work Group Poster Finalize Poster Justification 1 st Grade Problem Solving Circle Discussion: Teacher guided Class Write Up: Shared Writing Visit Explanation & Share Refl Set up D

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Explain the “hint” process to students. Alternative Method Preparation Assign students to small cooperative groups of 3. Distribute the problem and an alternative write-up guide. Have students count off (students number themselves: one, two or three within each group). Have students create a Data Sheet. Explain the “hint” process to students.

Alternative Method - Solving the Problem Students attempt to solve the problem independently (5 minutes). Students record their independent work. Begin initial group work (8-12 minutes). Have students record the initial group work.

Do the Second Rotation Do the First Rotation Students numbered “1” rotate to new groups Begin work with the second group (8-12 minutes) Have students record the second group’s new information. Do the Second Rotation Students numbered “2” rotate to new groups Begin work with the third group (8-12 minutes). Have students record the third group’s new information. The “hint” process is ongoing during group work.

Ask students to complete the “Answer and verification” section. Create the final product independently Process the solution (next day in class). Assess the completed student work (peer, self-, and teacher evaluations).

Alternative Method Set up Problem Review Rubric Small Groups Data sheet Independent Work and Recording (5 Minutes) “Hint” Process (Throughout work time.) Initial Group Work and Recording (10-12 Minutes) 1 st Rotation Continued Group Work and Recording (10-12 Minutes) Answer & Verification (10-12 Minutes) Create Final Product Independently Process the Solution (Next class period, 30 minutes) Self-Assessment (Based on the rubric, How did I do?) 2 nd Rotation Continued Group Work and Recording (10-12 Minutes)

Language Frames for Math Discourse

“Listen to understand not speak to be understood”

Find or create a word problem that meets the criteria discussed.