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DEVELOPING PROBLEM SOLVING IN THE MATH CLASSROOM

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The Handshake Problem It’s the holiday season and you are at another Christmas party. You, being mathematically minded of course, begin to wonder how many handshakes would take place if each person in the room shakes everyone else's hand. Suppose there are 50 people in the room – how many handshakes would there be? What if there is 150 people in the room?

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Problem Solving Strategies Estimate Act it out Systematic guessing and checking Solve a simpler problem Divide into cases (solve it for specific situations) Consider extreme cases Draw a picture/diagram Make a list/table/chart Label, assign variables, formulate expressions and equations Look for a pattern Work backwards Use logical reasoning (e.g. show the opposite can't be true) Make a model

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Why? Challenge students Higher order thinking skills Connect mathematics to real life Multiple ways to solve a problem Develop persistence Learn mathematical thinking and habits Provide a variety of opportunities for success

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One Format for Implementation Monday: Introduce the problem Help students sort the important information from extraneous details. Help students identify what is being asked in the problem. What is important in the handshake problem? Counting all handshakes Finding a way to solve this without taking forever – Work hard to be lazy!

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Implementation, Continued Wednesday Help students identify a plausible strategy and begin to solve the problem Let’s make a table… Number of people Number of handshakes

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Implementation, Continued Once the light bulbs start going off, the teacher stops helping with the problem. The problem is due on Friday The flexibility is key – the teacher can explain as much or as little as needed in order to bring the students to the brink of understanding. The student takes the final steps, however, and can experience the success.

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What is expected? Not solely an answer but an explanation. The students must justify their answers: Show ALL work Prove that the answer is correct This forces students to explain their answers and write about math – again using higher order thinking skills.

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Who can use this? Any math teachers The teacher can select problems appropriate to his or her students’ levels. It is essential, however, that the students are challenged – they often surprise themselves and the teacher in the problems they can solve.

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Where to find ideas? Math Forum (you have to pay for this one, though) Math Counts Program https://mathcounts.org/Page.aspx?pid=355 Google! Math Blogs

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What’s the right answer anyway? (n 2 -n)÷2 where n is equal to the number of people in the room. So for 50 people there are: 50 x 50 = 2,500 2,500 – 50 = 2, 450 2,450 ÷ 2 = 1,225 handshakes And for 150 people there are: 150 x 150 = 22,500 22,500 – 150 = 22,350 22,350 ÷ 2 = 11,250 handshakes

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References Math Forum at Drexel University Including: Writing to Develop Understanding: The Math Drexel's PoWs Suzanne Alejandre, The Math Forum, Drexel University Originally published on pages in the December, 2006, issue of the CMC ComMuniCator, the journal of the California Mathematics Council. Copies of an unedited version of the PDF file may be distributed. Writing to Develop Understanding: The Math Drexel's PoWs Problems of the Week Engage Students with Special Needs Annie Fetter, The Math Forum, Drexel University Eisenhower National Clearinghouse Focus, volume 10, pp (2003). Reprinted with permission. Problems of the Week Engage Students with Special Needs The Impact of the Math Forum's Problem(s) of the Week on Students' Mathematical Thinking K. Ann Renninger, Laura Farra, and Claire Feldman-Riordan Swarthmore College, Swarthmore, Pennsylvania Copyright 2000, The Regents of the University of Michigan. Presented at the International Conference of the Learning Sciences Copies may be distributed with the University's permission. The Impact of the Math Forum's Problem(s) of the Week on Students' Mathematical Thinking Paul Agranoff

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