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

W HAT IS THE GOAL OF THE DAY ? To examine and discuss problems and problem solving: WHAT? WHERE? WHEN? WHY? HOW?

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?”

E ACH OF THE SIX G RADE 5 CLASSES IN R ED R IVER E LEMENTARY HAS 26 STUDENTS. H OW MANY GRADE 5 STUDENTS ARE THERE ALTOGETHER ?

Fred has a balance of $427 in his bank account. His friend Carlos has three times as much as Fred in his bank account. Together Fred and Carlos want to buy a new TV that costs $1950. If they both use all the money in their bank accounts, how much more money will they have to get to buy that TV?

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

The average age of a group of teachers and students is 20. The average age of the teachers is 35. The average age of the students is 15. What is the ratio of teachers to students? Express your answer as a fraction in simplest form.

Without calculating the answer, predict what the ones digit of the answer to 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?

P ERHAPS A PROBLEM IS LIKE BEAUTY – IT ’ S IN THE “ EYES 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 (typical story problems associated with a concept) 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

W HAT IS PROBLEM SOLVING ? Considered the most complex of all intellectual functions, problem solving has been defined as higher- order cognitive process that requires the modulation and control of more routine or fundamental skills.

I S P ROBLEM S OLVING Y OUR G OAL IN T EACHING M ATHEMATICS ?

O FTEN O UR D ILEMMA Maintaining a balance between the development of concepts and skills AND the modulation and control of these concepts and skills through problem solving

A P OSSIBLE S OLUTION TO O UR D ILEMMA Teaching For Problem Solving Teaching Via Problem Solving Teaching About Problem Solving