P ROBLEM S OLVING – T HE H EART AND S OUL OF M ATHEMATICS David McKillop, Consultant Pearson Education Canada

AGENDA 8:30…Advance Organizers: Soduko Puzzle and Teacher Self-Evaluation Introduction to the Day Let’s Do Some Problem Solving What is a problem anyway? Problem Solving and Mathematical Proficiency Problem Solving and the Manitoba Report Card Teaching For, Via, and About Problem Solving Personal Reflections 10:00…Break 10:20…Teaching For and Via Problem Solving: The Case of Addition and Subtraction Personal Reflections 12:00…Lunch 1:00…Teaching About Problem Solving Polya’s Model The Braid Model Understanding-the-Problem Strategies 2:30…Break 2:45…Issues in Evaluation of Problem Solving Reflections 3:30…Farewells!

Teacher Self-Evaluation Regarding Problem Solving 1. I make the applications of any concept I teach the centre of most of the activities I plan for my students I make time for, and prepare lesson plans to address, problem-solving processes Some of my lessons have students solving problems that are not directly related to the concepts I have to teach I believe that my students can solve many addition and subtraction problems before they receive any instruction One of the greatest difficulties my students have with problem solving is reading the problems If asked, my students would say that mathematics and problem solving are the same thing I, personally, really like the challenge of solving problems After I finish a unit, my students would not wonder, “When am I ever going to use this?”

F RED HAD A BALANCE OF $427 IN HIS BANK ACCOUNT. A FTER HE MADE ANOTHER DEPOSIT THE BALANCE WAS $705. H OW MUCH WAS THE DEPOSIT HE MADE ? $427 + ? = $705 $705 − $427 = ?

T HE G RADE 4 STUDENTS AT L OUIS R IEL E LEMENTARY HAVE RAISED $600. T HEY DECIDE TO SPEND SOME OF IT ON A CLASS TRIP TO THE S CIENCE M USEUM. T HE ADMISSION PRICE TO THE MUSEUM IS $3 FOR EACH STUDENT, WITH FREE ADMISSION FOR THE TEACHERS. T HE BUS TO GET THEM THERE WILL COST $175. T HE NUMBER OF STUDENTS IN EACH CLASS IS FOUND IN THE TABLE BELOW. H OW MUCH MONEY WILL THE STUDENTS HAVE LEFT AFTER THE TRIP ?

Each different letter represents a different digit. Find the missing digits that will make this addition true. A D D C B B 3 2

Groups of campers were going to an island. On the first day 10 went over and 2 came back. On the second day, 12 went over and 3 came back. If this pattern of going over and coming back continues, how many would be on the island at the end of a week? How many would have left?

DAYCampers OverCampers Back TOTALS − 35 = 77

Jodi’s piggy bank contains nickels, dimes, and quarters. How many different ways can Jodi give her friend Jason 80 cents from this bank?

A student starts with the number 12, adds 3, gets an answer, adds 3 to this answer, gets another answer, adds 3 again, and continues adding 3s to answers until he has added 3 fifty times. Without actually adding all these 3s, predict what the ones digit of the final answer will be.

W HAT IS CONSIDERED A PROBLEM IN MATHEMATICS ?

O NE DEFINITION … A problem is defined as any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific “correct” solution method. (Hiebert et al., Educational Researcher, 1997)

E XAMINE THE 6 P ROBLEMS Y OU D ID Which problems were “problems” to you by this definition?

W HAT F ACTORS C ONTRIBUTE TO A Q UESTION B EING A P ROBLEM, OR N OT ? Prior experiences Developmental level of the individual

I S THIS A PROBLEM ? Mrs. Davison has 8 bags of candies to give out on Halloween. There are 24 candies in each bag. How many candies does she have to give out?

I S THIS A PROBLEM ? Jason bought three bottles of pop that each cost $2.99. He also bought a chocolate bar for $1.65, and two bags of chips that cost $3.49 each. How much did Jason spend?

Five hundred people are lined up outside the Apple Store by 7 am. Since the store doesn’t open for an hour, three of the staff at the store decide to distribute an iPhone brochure to every 2 nd person in line, an iPad brochure to every 5 th person in line, and a gift certificate to every 7 th person in line. If they all start distributing at the head of the line, how many people in line get all three things? I S THIS A PROBLEM ?

P ERHAPS A PROBLEM IS LIKE BEAUTY – IT ’ S IN THE “ EYE OF THE BEHOLDER ”.

W HAT IS CONSIDERED A PROBLEM IN MATHEMATICS E DUCATION ?

I N THE S CHOOL M ATH C LASS Routine Problems such as a Translation Word Problems Non-routine Problems such as Multi-step Translation Word Problems, Multi- concept Problems, Routine Problems before they are routine, Logic/Puzzle Problems

M ATHEMATICAL P ROFICIENCY