This document summarizes a presentation on problem solving persistence given by Dr. Mary Pat Sjostrom. It discusses having students work on problems for an extended period of time rather than focusing on quick solutions. The presentation asks teachers to work on sample problems throughout. It describes how mathematicians often work for a long time on problems and how students equate success with speed. The presentation emphasizes having students make sense of problems, plan solutions, check answers, and persevere even when they are confused or solutions are not quick. It provides examples of problems worked on for multiple days and encourages reflection on the problem solving process.

1. Persistence
in Problem Solving
Dr. Mary Pat Sjostrom
msjostro@chaminade.edu
Chaminade University of Honolulu
NCTM 2012 Annual Conference


2. Note to students
This presentation is based on problem investigations with
students in Math Methods for Elementary Teachers and
Secondary Math Methods.
When it was presented at the Annual Conference of NCTM in
2012, teachers were asked to actually work on a problem
periodically throughout the presentation, to give them a small
taste of the students’ experience.

3. Problem: Tiling a Floor
(Work on this for 3 minutes alone)
I want to tile a rectangular floor
with congruent square tiles.
Blue tiles will form the border and
white tiles will cover the interior.
Is it possible to use the same
number of blue tiles as white tiles?

4. Mathematicians
often work hours,
days, or even years
on a single
problem..
Students often
equate excellence
in mathematics
with speed in
solving problems.
If they cannot find
an answer quickly,
they cannot or will
not persist.

5. 1. Make sense of problems and
persevere in solving them.
CCSS – Mathematics
Mathematical Practices
Based on NCTM Process
Standards and NRC
Strands for Mathematical
Proficiency
2. Reason abstractly and
quantitatively.
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express regularity in
repeated reasoning.

6. Make sense of problems and persevere in
solving them.
 Mathematically proficient students start by explaining
to themselves the meaning of a problem and looking for
entry points to its solution.
 They analyze givens, constraints, relationships, and
goals.
 They make conjectures about the form and meaning of
the solution and plan a solution pathway rather than
simply jumping into a solution attempt.

7. Make sense of problems and persevere in
solving them.
 They consider analogous problems, and try special
cases and simpler forms of the original problem in
order to gain insight into its solution.
 They monitor and evaluate their progress
 and change course if necessary.

8. Make sense of problems and persevere in
solving them.
 Mathematically proficient students check their answers
to problems using a different method.
 They continually ask themselves, “Does this make
sense?”
 They can understand the approaches of others to
solving complex problems and identify
correspondences between different approaches.

9. Problem: Tiling a Floor
Work on this for 2 more minutes.
I want to tile a rectangular floor
with congruent square tiles.
Blue tiles will form the border and
white tiles will cover the interior.
If it is possible to use the same number of blue tiles
as white tiles, what area can be covered?
Is there more than one solution?

10. Pierre de Fermat (1601 – 1665)
 French lawyer and
mathematician
 Wrote this theorem in
the margin of a book
 Said he had a proof, but
there wasn’t room to
write it
n
a+n=n
b c
Fermat’s Last Theorem
If n is an integer greater
than 2, there are no
positive integers for
a, b, and c that will
satisfy this equation.

11. Proof?
 Mathematicians worked on this
unsuccessfully for 350 years!
 In the 1980s and 1990s the British
mathematician Andrew Wiles devoted much
of his career to proving Fermat's Last
Theorem.

12. Success?
 Wiles worked for more than 7 years to prove
Fermat’s Last Theorem. His work built on the work
of many other mathematicians.
 In 1993, he claimed to have solved the problem.
 Then other mathematicians found an error in his
work.
 Wiles went back to work, and a year later published
a proof which is now accepted by the mathematics
community.

13. 

14. Problem: Tiling a Floor
(Work on this for 2 more minutes, then talk to your neighbor –
but please don’t ruin your neighbor’s experience!)
I want to tile a rectangular floor
with congruent square tiles.
Blue tiles will form the border and
white tiles will cover the interior.
If it is possible to use the same number of blue tiles
as white tiles, what area can be covered?
Is there a maximum area? How do you know?

15. Project: Mathematician at Work
I want my students to gain experience in
 Problem solving
 Extended investigation - persistence
 Log all work and thinking - communication
 Reflection - metacognition

16. Purpose
The purpose of this project is to give you an opportunity to investigate a
problem at length.
The purpose is not to solve the problem quickly; it is not even necessary to
successfully solve the problem.
You are to immerse yourself in the problem over the course of several
days.
Live with it!
Get to know it intimately!
Own it!
Love it!

17. Directions
 Choose a problem.
 Work 15 minutes a day for 5 days.
 Log your work: Write down everything you think or do.
 If you don’t solve it, that’s okay.
 If you solve it, extend the problem.
 Summarize the mathematics.
 Reflect on the process.

18. Rubric

Log shows that student worked on problem for at least 15
minutes a day for 5 days.

Work is clearly shown; student explains thinking and
attempts at solution.

Summary and work show some mathematical
understanding.

Reflection discusses the experience of extended work on a
problem.

19. The Problems

Red Paint: There are 27 small cubes arranged in a 3 by 3 by 3 cube. The
top and sides of the large cube are painted red. How many of the 27 small
cubes have 0 faces painted? 1 face? 2 faces? 3 faces? 4 faces? 5 faces? 6
faces?

Double Your Money: On the first day, Natasha puts a penny in her piggy
bank. On the second day, she puts in another penny, doubling the
amount of money in the bank. On the third day, Natasha puts in 2
pennies (the amount already in the bank), again doubling her money.
Each day the pattern continues: Natasha puts in the number of pennies
needed to double the amount of money in the bank. How long will it take
Natasha to save 500,000 pennies?

20. The Problems

Diophantus Diophantus was a famous Greek mathematician who lived in
Alexandria, Egypt, in the third century, A.D. After he died, someone
described his life in this puzzle: He was a boy for 1/6 of his life. After
1/12 more, he acquired a beard. After another 1/7, he married. In the 5th
year after his marriage, his son was born. The son lived half as many
years as his father. Diophantus died 4 years after his son. How old was
Diophantus when he died?

Sea Sick Suppose a boat is located 30 miles from shore and must get a
passenger to a hospital that is located 60 miles downshore from the boat's
current position. The boat travels at 20 mph, and the ambulance that
meets the boat travels at 50 mph. Where should the ambulance meet the
boat to minimize the amount of time needed to reach the hospital?

Goldbach's Conjecture Every even number greater than 4 can be written
as the sum of two odd prime numbers. Can you find Goldbach pairs for
all even integers between 4 and 100? Which have more than one
Goldbach pair? Can you find any patterns?

21. Problem Solving Strategies

Looked for patterns, drew pictures, made tables and graphs,
tried to find an equation

Explained thinking, wrote about ideas and confusion

Built on previous days’ work

Checked understanding

Looked for different methods of solution

Extended problem

22. Students’ Problem
Solving Strategies

I used Polya's heuristics to help guide me during the problem
solving process.

This process made me really appreciate the technology I had access
to. Graphing my data by hand …would have taken up quite a bit
of time so I opted to use the features in Excel so that I could move
onto my analysis sooner. Using PowerPoint I was also able to
keep all the different graphs from Excel together in a wellorganized fashion. I think that my experience with this problem
solving activity is a great example of using technology as a tool to
allow students (AND teachers) to explore mathematical ideas!!

23. Problem Solving Strategies

I think I also realized the benefit of breaking up a problem into
manageable chunks. I remember being a young student and seeing
a problem like this and being completely overwhelmed by all the
WORDS..

I started by adding the fractions that were given, but that did not
make sense as I progressed through the problem. Then I decided
to draw a timeline and go from there.

I think this is one of the few times that I produced a graph trying to
see a pattern in the problem solving process.

24. Summarize the math
Prompts:

What methods did you use? Did you solve it more than one
way?

What was the solution?

Did you extend the problem? If so, how?

If you did not solve the problem, talk about the mathematics
you tried.

What new mathematical insights did you have as a result of
working on this problem?

25. Insights

Diagonals in a 17-sided polygon: “After looking back at my
previous work and experiments I figured and wondered if
each side/point could only connect to an “x” number of
sides. For example…one dot can only connect to 14 of the
other 17 dots because you’re at 1 and the two dots on the side
would not be considered a diagonal…”

26. Insights

Goldbach’s Conjecture: “I noticed many properties of prime
numbers. The first one right away was that prime numbers
do not occur at any kind of regular frequency (as far as I
could tell). The second thing that amazed me was that there
really wasn't a perceivable pattern to the number of prime
pairs as I worked from 6 through 100. No one factor or
property that I noticed could help me predetermine the
amount of prime pairs for the next number I would work
on.”

27. Reflect on the process
Prompts:
 Talk about the experience of working on a
math problem for several days.
 Did you enjoy it?
 Did the problem intrigue you?

28. Student reflections

It is important to look back at your work to see what you
could have possibly missed. Mathematics isn’t always black
and white because the only way you get the answer is
through walking through the grey area.

I would be sitting in class or watching a movie and think of
another way to solve it

29. Student reflections

There were times while doing this project that I did not
realize how much time had passed until someone in my
family interrupted my train of thought to ask me if I needed
anything. I believe that this is the type of interest that I
would like to be able to inspire in my students.

I actually enjoyed working on it. (that's weird)

I found out that if you look at a problem, and think that you
can't do it, it is possible and also fun.

30. Student reflections

Mathematicians must have a lot of headaches also. But they
also have a lot of patience and not much of social life if they
are constantly thinking about math, trying to prove a theorem
or create their own

It felt like a treasure hunt: such much so that I would neglect
getting other tasks done, for the sake of solving the problem.
I think, as teachers, we often forget that students may not
have developed an enjoyment for this type of problem
solving.

31. Student reflections

I learned that thinking skills are activated through an open
ended problem solving activity such as this

I cannot imagine any student who looks forward to
completing their homework assignments of every other
even/odd problem in a book, then leading up to review for a
chapter test. Assigning this type of problem, on the other
hand, has the potential of stirring up excitement in students
because they want to find the answer.

32. Student reflections

I cannot remember a teacher in my whole schooling in
mathematics that said wrong answers were ok as long as you
had some explanation to how you performed it…I have gone
from being very fearful that making mistakes made you
ignorant to making mistakes allows you to learn and helps
you to perform better next time.

I also liked that I was able to write a lot in this assignment
about what was going on in my head, kind of like a math
problem narrative.

33. Student reflections

I felt slightly disappointed once it was solved, however right
when I thought I was close to the solution, I was thrilled.
Math can truly be like sex if done correctly.

34. Problem: Tiling a Floor
Extensions
 How would this problem be
different if there were half as
many blue tiles as white tiles?
 What if there were three times as
many blue tiles as white tiles?

35. Make sense of problems and persevere in
solving them.
A mathematics professor had a party at her home, and one of the
guests was admiring the family photos on the picture wall.
"I see you have three lovely children," the guest commented. "How
old are they?"
"I won't tell you their ages," replied the mathematics professor.
“However, I will tell you that the product of their ages is 72, and the
sum of their ages is the same as my house number."
The guest went outside to look at the house number, returned,
and complained, "You haven't given me enough information to solve
the problem!"
"Oh, there's one more thing," said the professor. "The oldest child
likes strawberry ice cream."
"Thank you," said the guest, and told her the ages of the children.

36. Project Materials
 LiveText:
 Log in at www.livetext.com
 Visitor Pass: 64F9CF64
 Questions or comments:
 Dr. Mary Pat Sjostrom
msjostro@chaminade.edu