1. DEVELOPING PROBLEM SOLVING IN THE MATH CLASSROOM

2. 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?

3. 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

4. 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

5. 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!

6. Implementation, Continued  Wednesday  Help students identify a plausible strategy and begin to solve the problem Let’s make a table… Number of people 12345 Number of handshakes 013610 1234 111

7. 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.

8. 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.

9. 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.

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

11. 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

12. References  Math Forum at Drexel University  Including:  http://mathforum.org/pow/teacher/guide.pdf http://mathforum.org/pow/teacher/guide.pdf  http://mathforum.org/pow/teacher/whatarethepows.pdf http://mathforum.org/pow/teacher/whatarethepows.pdf  Writing to Develop Understanding: The Math Forum @ Drexel's PoWs Suzanne Alejandre, The Math Forum, Drexel University Originally published on pages 33-36 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 Forum @ 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. 35-36 (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 2000. 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