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## Problem Solving

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**Problem Solving**Susan Stuart, Nipissing University**How many rectangles are there in this figure?**S. Stuart, Nipissing University**Agenda**What is problem solving? What actions should we expect to see to ensure effective math learning? Why do we focus on problem solving? Does the curriculum reflect this thinking? What types of problem-solving experiences should I plan? S. Stuart, Nipissing University**Problem Solving is….**The action we do when we are doing mathematics S. Stuart, Nipissing University**What is Problem Solving?**Literature Mathematics Problem Solving Writing Reading S. Stuart, Nipissing University**Problem Solving is the specific phrase given to the inquiry**process in mathematics. Polya’s 4-step Model Understand Make a Plan Do Look Back To this we add: Communicate S. Stuart, Nipissing University**Polya: Dynamic, non-linear**Understand Communicate Look Back Plan Do D. Franks S. Stuart, Nipissing University**Calculate**Measure Solve Plan Mess around Explore Sort organize Select Estimate Explain Extend Represent Identify Observe create What actions do you think students will be doing when they are doing Mathematics? S. Stuart, Nipissing University**Learning to solve math problems is the most significant**learning that takes place in any math classroom S. Stuart, Nipissing University**Why?**• It is the process by where we construct understanding of new concepts • We practice and then learn to transfer concepts and skills to new situations • It is a means of stimulating intellectual curiosity • New knowledge is discovered through problem solving S. Stuart, Nipissing University**Let’s look at an example of a lesson in which grade 4**students are given opportunities to develop an understanding of division through a problem-solving situation. S. Stuart, Nipissing University**Problematic Task: Conceptual**• The ants in the story One Hundred Hungry Ants wanted to get somewhere fast. First they grouped themselves in 2 lines of 50, lines with 25 each, lines with 5 each, and finally lines with 10 each. • How many lines would there be for each group? • Talk with a partner about the question and be ready to share your reasoning. S. Stuart, Nipissing University**Problematic Task: Conceptual**Think about 60 of those ants and find out how they could line up evenly. Show your work with concrete materials or pictures. S. Stuart, Nipissing University**You would follow this lesson with practice questions**involving lining up and sharing • Show how a class of 24 would line up in rows of threes. • Show each • 24 people in rows of 4 • 25 people in rows of 5 • 42 people in rows of 7 • Show all the ways 24 could line up evenly. S. Stuart, Nipissing University**The value of teaching through and about problem solving**• Places the students’ attention on ideas and sense-making • Allows each student to enter the learning at their personal level of understanding • Develops mathematical power through • solving problems • reasoning • connecting • representing • communicating S. Stuart, Nipissing University**Value of teaching through and about problem solving…**• Develops the belief in students that they are capable of doing mathematics and that math makes sense • Provides ongoing assessment data that can be used to make instructional decisions, help students succeed, and inform parents • Is a lot of FUN S. Stuart, Nipissing University**Why such a focus on Problem Solving?**The Ontario Curriculum, Mathematics (1997) stresses the importance of teaching students to use the problem solving process to solve problems they encounter inside and outside the classroom and to use the process for learning mathematics. S. Stuart, Nipissing University**Teaching and Problem Solving:3 Perspectives**• Teaching FOR Problem Solving • Traditional • Teaching ABOUT Problem Solving • Problem Solving Processes and Strategies • Teaching THROUGH Problem Solving • Conceptual Development D. Franks S. Stuart, Nipissing University**Types of problem solving situations**• Routine problems • Application problems • Process problems S. Stuart, Nipissing University**ROUTINE PROBLEMS**Typical one- and two-step problems found in textbooks. These are used to enhance understanding of and practise with procedural skills. They allow students opportunity to construct their own algorithms. e.g. A concert is to be held at the local theatre. Seating costs are $14 for the floor seats and $10 for the balcony seats. There are 70 floor seats and 120 balcony seats. Organizers expect a sellout. How much should they expect to collect? S. Stuart, Nipissing University**APPLICATION PROBLEMS**• Might be called environmental problems • Problem situation is real – must be familiar or make sense to the students • Should be interesting and challenging • Students will apply informal (developed from direct experience) as well as formal knowledge (learned in the classroom) to the situation • Teacher may “lose control” of the math – not all answers will be the same S. Stuart, Nipissing University**Janice went to a store, spent half of her money, and spent**$10 more. She went to a second store, spent half of her remaining money, and then spent $10 more. Then she had no money left. How much money did she have in the beginning when she went to the first store. S. Stuart, Nipissing University**PROCESS PROBLEMS**• Sometimes set in mathematics context, in contrast to “real-life” problems • Concentrates on the mathematical thinking processes and the mathematics itself • Several solution methods (strategies are usually available – process problems are used to teach students about these strategies • Allows teacher to introduce a problem-solving model S. Stuart, Nipissing University**Polya’s 4-step Model**• Understand (Getting to know the problem) This involves making sense of the content of the problem, what information is given and what is being asked. • Make a Plan Students should be encouraged to share possible plans and to discuss alternative approaches. Estimation should be encouraged at this stage. • Do Students choose from strategies they have been taught. • Look Back Does the result correspond with the estimation? Were all conditions met and accounted for? Think about “What if…” S. Stuart, Nipissing University**Dramatize or model the situations**Draw a diagram Construct a table or chart Find a pattern Solve a simpler pattern Guess and check Work backwards Consider all possibilities Change your point of view Strategies S. Stuart, Nipissing University**Routine (traditional) Problem**I have 2 pennies, 1 dime and 2 nickels in my pocket. How much money do I have altogether? Process Problem I have pennies, dimes, and nickels in my pocket. If I take three coins out of my pocket, how much money could I have taken? What strategy would you use to solve this? S. Stuart, Nipissing University**Planning**S. Stuart, Nipissing University**Assessment**S. Stuart, Nipissing University**ESSENTIAL CHARACTERISTICS OF PROBLEM-SOLVING MATHEMATICS**LESSONSThese characteristics describe qualities that are essential for rich, problem-solving focused lessons that attend to the development of student understanding and appreciation of mathematics.1. Students are presented with problems that encourage their mathematical thinking and reasoning.2. Emphasis is placed on having students explain their thinking, both verbally and in writing.3. Students work cooperatively in small groups to maximize opportunities for explaining their ideas4. Manipulative materials are used whenever possible to bring meaning to abstract ideas.5. Different areas of the math curriculum are integrated rather than isolated by specific skills or topics.6. Students' misconceptions and errors are seen as indicators of confusion or partial understanding of math ideas and are accepted as natural to the process of learning.7. Concepts are presented in a variety of ways and, as much as possible, are embedded in contextual settings.8. Rather than explaining a new idea that the students then practice, lessons first engage children in problem-solving activities from which understanding of the math concepts can emerge.9. Calculators are available to students in all lessons and homework assignments.10. Homework is used to further students' understanding and problem-solving abilities, rather than to practice skills. S. Stuart, Nipissing University**ALL MATH LESSONS SHOULD BE PROBLEM-SOLVING LESSONS, EVEN IF**A NEW SKILL OR CONCEPT IS BEING INTRODUCED.Components of many math lessons:Mental math or puzzleOpening problemSmall group interactionIndividual practice or consolidationWhole group discussion at the beginning and at the end of the lesson S. Stuart, Nipissing University