Teaching through Problem Solving Kindergarten to Grade 3 Presented by:

Agenda We will be involved in problem-solving activities throughout the day! Welcome and introductions. Why should I teach through problem solving? What are the characteristics of a problem-solving environment? What does problem solving look like in a classroom setting? How can I provide a problem-solving context? How might today’s session connect to my classroom?

Welcome! How are you feeling today and why might you be feeling that way? What are some experiences you have had with problem solving? What might be your hopes for today’s session?

How Many Pockets? Reproduced with permission from CartoonStock.com, “Pickpockets Anonymous,” CartoonStock.com, Catalogue Reference mban1707, www.cartoonstock.com (Accessed January 28, 2008).

Problem Solving with How Many Pockets? How did the activity involve you in problem solving? How might this be a worthwhile activity for students?

Why Teach through Problem Solving? Problem-solving Get-together You have been given one of the benefits of teaching through problem solving. Meet with others that have the same colour paper as you. Read it over and take notes on how you might paraphrase the information for others. Include a classroom example, if possible, to illustrate the benefit. Mingle and share your benefit and your thoughts about it with people who have a different colour paper than you do. Try to hear about all the other benefits.

Why Teach through Problem Solving? Places the focus of attention on ideas and sense making. Develops the belief in students that they are capable and that mathematics makes sense. Develops “mathematical power.” All seven process skills are likely to be involved.

Why Teach through Problem Solving? Provides ongoing assessment data. Allows multiple entry points for students. Engages students. It is a lot of fun … for both teachers and students!

Why Teach through Problem Solving? It is in the program of studies. Where?

Program of Studies page 6 (front matter) PROBLEM SOLVING [PS] Learning through problem solving should be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the type How would you …? or How could you …?, the problem-solving approach is being modelled. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies.

What Is a Problem? A problem is: a situation in which a person is seeking some goal and for which a suitable course of action is not immediately apparent in the context of mathematics, one requiring that mathematical skills, concepts or processes be used to arrive at the goal.

Criteria for Mathematical Problems There is a perplexing situation that the student understands. The student is interested in finding a solution. The student is unable to proceed directly toward a solution. The solution requires use of mathematical ideas. Was the How Many Pockets? problem a problem for you? Why or why not? Answer in terms of the four criteria. Reproduced with permission (pending) from M. Burns, About Teaching Mathematics: A K–8 Resource (Sausalito, CA: Math Solutions Publications, 2000), p. 17.

Teaching through problem solving is not: a unit taught in isolation of other mathematics teacher-directed demonstrating a strategy and having students use that strategy to solve a problem expecting all students to use the same strategy to find a solution involving students in solving numerous problems that follow the same procedures. A problem-based approach does not dictate how a child must think about a problem in order to solve it.

Program of Studies A problem-solving activity must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learnings in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement.

What Do I Do with the Problem-solving Strategies I Teach? Remember these: Act it out? Draw a diagram? Look for a pattern? Use a table? Guess and check? Work backward? Solve a simpler problem? When students present these strategies in class, the teacher can facilitate a discussion about the strategy and present it as one way of many ways to solve the problem. These strategies can be background information for teachers to understand strategies that students may use.

Program of Studies Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. Creating an environment where students openly look for, and engage in, finding a variety of strategies for solving problems empowers students to explore alternatives and develops confident, cognitive mathematical risk takers.

What Does a Problem-solving Environment Look and Sound Like? Classroom Characteristics Jigsaw (part 1) Read Problem-solving Standard for Grades Pre-K–2. In expert groups, discuss and make notes on either: Student involvement: what might you: see students doing hear students saying? OR Teacher involvement: what might you see teachers doing while: preparing for class working with students in class? what might you hear teachers saying?

What Does a Problem-solving Environment Look and Sound Like? Classroom Characteristics Jigsaw (part 2) Form new groups of 4: 2 student experts and 2 teacher experts. Designate at timekeeper. Each person will have 2 minutes to speak. Present the information on which you have become an expert. Experts may build on each others’ information. Student involvement: what might you: see students doing hear students saying? OR Teacher involvement: what might you see teachers doing while: preparing for class working with students in class? what might you hear teachers saying?

What Does a Problem-solving Environment Look and Sound Like? Classroom Characteristics Jigsaw (part 3) Discuss in groups: How might problem solving be the same or different than in your own classroom? What might be some considerations you will take back with you to your classroom?

What Does a Problem-solving Environment Look and Sound Like? Classroom Characteristics (reading) Read Problem-solving Standard for Grades Pre-K–2. In expert groups, discuss and take notes on chart paper on one of the following topics: Student involvement: what might you: see students doing hear students saying? OR Teacher involvement: what might you see teachers doing while: preparing for class working with students in class? what might you hear teachers saying?

What Does a Problem-solving Environment Look and Sound Like? Classroom Characteristics (presentations) As an expert group, designate a presenter. Present the information on which you have become an expert. Student involvement: what might you: see students doing hear students saying? OR Teacher involvement: what might you see teachers doing while: preparing for class working with students in class? what might you hear teachers saying?

What Does a Problem-solving Environment Look and Sound Like? Classroom Characteristics (discussions) Discuss in groups: How might problem solving be the same or different than in your own classroom? What might be some considerations you will take back with you to your classroom?

The Napping House by Audrey Wood

What Could Live in this House? Curriculum Correlation Grade 1 Number, Specific Outcome 4: Represent and describe numbers to 20, concretely, pictorially and symbolically. [C, CN, V] Demonstrate an understanding of addition of numbers with answers to 20 and their corresponding subtraction facts, concretely, pictorially and symbolically, by: using familiar mathematical language to describe additive and subtractive actions creating and solving problems in context that involve addition and subtraction modelling addition and subtraction, using a variety of concrete and visual representations, and recording the process symbolically. [C, CN, ME, PS, R, V]

What Could Live in this House? Curriculum Correlation Grade 2 Number, Specific Outcome 4: Represent and describe numbers to 100, concretely, pictorially and symbolically. [C, CN, V] Number, Specific Outcome 9: Demonstrate an understanding of addition (limited to 1- and 2-digit numerals) with answers to 100 and the corresponding subtraction by: using personal strategies for adding and subtracting with and without the support of manipulatives creating and solving problems that involve addition and subtraction using the commutative property of addition (the order in which numbers are added does not affect the sum) using the associative property of addition (grouping a set of numbers in different ways does not affect the sum) explaining that the order in which numbers are subtracted may affect the difference. [C, CN, ME, PS, R, V]

Problem Solving ~ Personal Strategies 8 12 6 15 11 9

Problem Solving ~ Personal Strategies 20 15 30 35

Problem Solving ~ Personal Strategies 32

Problem Solving I drew a shape with 4 sides. Draw what my shape might look like.

Problem Solving ~ Personal Strategies 1 2 3 4 5 6 7 8 9 Use each digit at least once and form three 3-digit numbers with the sum of 968.

Teaching through Problem Solving— Classroom Examples Classrooms Examples Jigsaw (part 1) In expert groups, read and discuss the assigned article. In relation to the article, discuss the: involvement of students role of the teacher characteristics of problem-solving classrooms criteria for problem solving. There is a perplexing situation that the student understands. The student is interested in finding a solution. The student is unable to proceed directly toward a solution. The solution requires use of mathematical ideas. Reproduced with permission (pending) from M. Burns, About Teaching Mathematics: A K–8 Resource (Sausalito, CA: Math Solutions Publications, 2000), p. 17.

Teaching through Problem Solving— Classroom Examples Classrooms Examples Jigsaw (part 2) Form new groups (home groups) with one expert from each article. Designate a timekeeper. Each group member will have 5 minutes to share their article. Provide an overview of the problem-solving task in your article. Share the: involvement of students role of the teacher characteristics of problem solving classrooms criteria for problem solving.

Teaching through Problem Solving— Classroom Examples Classrooms Examples Jigsaw (part 3) In home groups, discuss what made all of the different problem-solving lessons successful. On chart paper, record the elements common to all lessons that made them successful. Be prepared to share your chart with the whole group.

Problem Solving ~ Personal Strategies 8 12 6 15 11 9

Problem Solving ~ Personal Strategies 20 15 30 35

Problem Solving ~ Personal Strategies 32

Problem Solving I drew a shape with 4 sides. Draw what my shape might look like.

Problem Solving ~ Personal Strategies 1 2 3 4 5 6 7 8 9 Use each digit at least once, and form three 3-digit numbers with the sum of 968.

Examining Problems Look at samples A and B of the mathematics tasks. Each task is presented in a traditional form and an open-ended form. In groups, discuss the following: How do the open-ended forms engage students and involve them in rich problem solving? How might you adapt samples C and D to provide a richer problem-solving context?

Modifying Problems Fold your handout as indicated. In the centre of your handout, there is a problem that would be appropriate for your grade level. Directly below the problem, write a modification to increase the challenge. Directly above the problem, write a modification to decrease the challenge. Directly to the left and the right of the problem provided, write a similar problem that addresses the same mathematics concept. Repeat the same steps above and below the problem. Fold, as indicated, to record possible solutions students might present.

Connecting to Your Classroom Use the next 2 minutes to take notes to yourself about: What would you like to remember about problem solving when you return to your classroom? What might you share with other teachers about problem solving? Share your thoughts and/or take notes with an elbow partner.

Thank You for a Great Day! If you have questions or comments following today’s session, please contact me: