DEVELOPING PROBLEM SOLVING
IN THE MATH CLASSROOM
The Handshake Problem
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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?
Problem Solving Strategies
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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
Why?
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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
One Format for Implementation
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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!
Implementation, Continued
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Wednesday
 Help
students identify a plausible strategy and begin
to solve the problem
Let’s make a table…
Number of
people
1
2
3
4
5
Number of
handshakes
0
1
3
6
10
1
2
1
3
1
4
1
Implementation, Continued
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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.
What is expected?
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Not solely an answer but an explanation.
The students must justify their answers:
 Show
ALL work
 Prove that the answer is correct
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This forces students to explain their answers and
write about math – again using higher order
thinking skills.
Who can use this?
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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.
Where to find ideas?

Math Forum
 http://mathforum.org
(you have to pay for this one,
though)
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Math Counts Program
 https://mathcounts.org/Page.aspx?pid=355
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Google!
Math Blogs
What’s the right answer anyway?
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(n2-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
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And for 150 people there are:
 150
x 150 = 22,500
 22,500 – 150 = 22,350
 22,350 ÷ 2 = 11,250 handshakes
References
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Math Forum at Drexel University
Including:
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http://mathforum.org/pow/teacher/guide.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.
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.
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.
Paul Agranoff