Persistence

in Problem Solving

Dr. Mary Pat Sjostrom msjostro@chaminade.edu

Chaminade University of Honolulu

NCTM 2012 Annual Conference

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.

Problem: Tiling a Floor

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?

(Work on this for 3 minutes alone)

 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.

 CCSS – Mathematics

Mathematical Practices

Based on NCTM Process Standards and NRC Strands for Mathematical Proficiency

1. Make sense of problems and persevere in solving them.

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.

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.

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.

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.

Problem: Tiling a Floor

I want to tile a rectangular floor with congruent square tiles. Blue tiles will form the border and white tiles will cover the interior.

Work on this for 2 more minutes.

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?

 

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     

Fermat’s Last Theorem

Pierre de Fermat (1601 – 1665)

an bn c n

If n is an integer greater than 2, there are no positive integers for a, b, and c that will satisfy this equation.

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.

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.

Andrew Wiles (click to play a short videoclip)Andrew Wiles (Click to play a short video)

Problem: Tiling a Floor

I want to tile a rectangular floor with congruent square tiles. Blue tiles will form the border and white tiles will cover the interior.

(Work on this for 2 more minutes, then talk to your neighbor – but please don’t ruin your neighbor’s experience!)

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?

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

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!

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.

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.

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?

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?

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

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 well-organized 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!!

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.

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?

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…”

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.”

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?

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

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. 

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.

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. 

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.

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. 

Problem: Tiling a Floor

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?

Extensions

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.

Make sense of problems and persevere in solving them.

Project Materials

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Questions or comments: Dr. Mary Pat Sjostrom

msjostro@chaminade.edu