Effective Problem Solving in a Tamara Pearson, PhD Assistant Professor Mathematics Department How should we be teaching? There is a shift towards applying mathematical concepts and skills in the context of authentic problems and for the student to understand concepts rather than merely follow a sequence of procedures. -Common Core Georgia Performance Standards Conceptual and Procedural Conceptual Procedural Logical relationships constructed internally and existing in the mind as a part of a network of ideas. of rules, procedures and symbolism used in carrying out routine mathematical tasks. Created by Tamara Pearson, PhD 1
Conceptual and Procedural Conceptual Procedural All mathematical procedures can and should be connected to the conceptual ideas that explain why they work. Engage Activity 7 10 Here are two circular discs. A number is written on the top of each disc. There is another number written on the reverse side of each disc. By tossing the two discs in the air and then adding together the numbers, which land uppermost, I can produce anyone of the following four totals: 11, 12, 16, 17. Work out what numbers are written on the reverse side of each disc. Engage Activity What strategies did you use in order to solve this problems? How can these strategies help you to solve other mathematics problems? Created by Tamara Pearson, PhD 2
Mathematical Practices 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. Mathematical Practices 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. 1. Make sense of problems and persevere in solving them. Explain to themselves the meaning of a problem and look for entry points to its solution Analyze givens, constraints, relationships, and goals Make conjectures about the form and meaning of the solution and plan a solution pathway Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution Monitor and evaluate their progress and change course if necessary Explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends Check their answers to problems using a different method Understand the approaches of others to solving complex problems and identify correspondences between different approaches. Created by Tamara Pearson, PhD 3
3. Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, definitions, and previously established results in constructing arguments Make conjectures and build a logical progression of statements to explore the truth of their conjectures Analyze situations by breaking them into cases, and can recognize and use counterexamples Justify their conclusions, communicate them to others, and respond to the arguments of others Reason inductively about data, making plausible arguments that take into account the context from which the data arose Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is Listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments 6. Attend to precision Communicate precisely to others State the meaning of the symbols they choose, including using the equal sign consistently and appropriately Careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. Examine claims and make explicit use of definitions Problem Solving Problem solving is the cornerstone of school mathematics. Without the ability to solve problems, the usefulness and power of mathematical ideas, knowledge, and skills of severely limited Students who can both develop and carry out a plan to solve a mathematical problem are exhibiting knowledge that is deeper and more useful than simply carrying out a computation. Principles and Standards for School Mathematics National Council for Teachers of Mathematics Created by Tamara Pearson, PhD 4
Three Teaching Methods Teaching For Problem Solving Teaching About Problem Solving Teaching Through Problem Solving Three Teaching Methods Teaching For Problem Solving Teaching About Problem Solving Teaching Through Problem Solving Teaching For Problem Solving This approach can be summarized as teaching a skill so that a student can later problem solve, which follows the format of many textbooks designed with skills taught first. Elementary and Middle School Mathematics John Van de Walle Created by Tamara Pearson, PhD 5
Teaching For Problem Solving Teacher presents the mathematics Students practice the skill Then they study word or story problems involving that skill For example Today we will learn about long division Now complete this worksheet to practice long division Now solve the following word problem using long division What s the problem with this method? Problems with this Method Places student as passive learner Problem solving becomes separate activity Students get used to being told how to do mathematics. Created by Tamara Pearson, PhD 6
Three Teaching Methods Teaching For Problem Solving Teaching About Problem Solving Teaching Through Problem Solving Teaching About Problem Solving This approach involves direct instruction about general problem solving strategies. Fostering Student s Mathematical Power Arthur J. Baroody Teaching About Problem Solving Understand the problem Make a plan for solving the problem Carry out the plan Look back and reflect on the answer in terms of the initial question Adapted from: How to Solve It George Polya Created by Tamara Pearson, PhD 7
Understand the Problem Good problem solvers: l Ask themselves questions l Monitor progress l Re-evaluate choices Teachers can help students talk through this process l Helps confused students to process information l Important mathematical information is brought out Make a plan Enables students to gain insights into mathematical relationships Helps students learn more sophisticated problem solving strategies Make A Plan Try some simple cases Find a helpful diagram Organize systematically Make a table Spot patterns Use the patterns Find a general rule Created by Tamara Pearson, PhD 8
Carry out the Plan Carefully monitor the solution procedure Is there a need to create a new plan? Look Back Check the reasonableness of the answer Does the solution make sense? How do you know it s correct? Look Back - Types of Justifications Numerically In Words Algebraically Using a Diagram Created by Tamara Pearson, PhD 9
Teaching About Problem Solving Understand the problem l How did you restate the problem? Make a plan for solving the problem l What strategy did you decide to use for solving the problem? Carry out the plan l Did your plan work? Look back and reflect on the answer in terms of the initial question l Are there other ways to solve the problem? Multiple Representations Manipulative models Pictures Oral language Real-world situations Written Symbols Three Teaching Methods Teaching For Problem Solving Teaching About Problem Solving Teaching Through Problem Solving Created by Tamara Pearson, PhD 10
Teaching Through Problem Solving Problem solving is not a distinct topic, but a process that should permeate the study of mathematics and provide a context in which concepts and skills are learned. Principles and Standards for School Mathematics National Council for Teachers of Mathematics Teaching Through Problem Solving Develops mathematical power Develops student confidence Provides a context to help students build meaning Allows an entry point for a wide range of students Provides ongoing assessment data Created by Tamara Pearson, PhD 11