1. A PROBLEM ANALYSIS COURSE FOR MIDDLE GRADES TEACHERS Mary Garner mgarner@kennesaw.edu Department of Mathematics and Statistics Kennesaw State University mgarner@kennesaw.edu

2. Problem Analysis Course: Advanced Perspectives on School Mathematics I Students: Middle grades and secondary mathematics education majors (certified for 7-12) at KSU Emphasis: Algebraic reasoning and skills, number, function – linear, quadratic, exponential

3. Problem Analysis A substance is 99% water. Some water evaporates, leaving a substance that is 98% water. How much of the water has evaporated? a.Solve the problem. b.A numerical approach. Find a solution again this time using no algebra but only concrete, numerical reasoning. c.A diagrammatic approach. Solve the problem yet again, this time using a diagram as your basic reasoning tool. d.Generalizing the problem. Solve the problem again, but this time replace the numerical values 99% and 98% in the statement of the problem with general parameters. e.A functions approach. Solve the problem again, but this time keep the specific numerical value 1% of the drop in water content, and express the proportion of water evaporated as a function of the original proportion of water in the substance. f.Another functions approach. Express the proportion of water evaporated as a function of the absolute amount of water in the substance. From Mathematics for High School Teachers. An Advanced Perspective by Usiskin, Peressini, Marchisotto, Stanley

4. Problem Analysis 1.Approaches 2.Generalization 3.Analysis 4.Extension 5.Concept 6.Connections

5. Problem Analysis 1. Approaches Methods (with explanations for why the method works or doesn’t work): Diagrammatic reasoning Guess and check Algebraic approach Graphical approach Wrong solutions Questions: Does a solution exist? Is the solution unique? Does the solution make sense?

6. Problem Analysis 1. Approaches Example: Tile Problem

7. Problem Analysis 2. Generalization Which solutions methods can be generalized to any input? How? (Opportunity to exercise algebraic skills.) What form does the solution have? Why? What are the characteristics of the solution? (For what inputs does no solution exist or many solutions exist?) Example: Tile Problem

8. Problem Analysis 3. Analysis How does variation in inputs affect solution? How are pairs of inputs related? Can solution be expressed in different ways? (For example, as a function of ratio of inputs.) Example: Tile Problem

9. Problem Analysis 4. Extension Write an extension of the problem. The extension must require the use of mathematical techniques not necessary to solve the original problem. Example: Problem: If Sam can paint a house in 5 hours and Joe can paint a house in 3 hours, how long will it take for them to paint the house together? Extension: Two teams are competing in a pie eating contest – team A and team B. On team A is Sam who can eat a whole pie in 5 minutes. On team B is Joe who can eat a whole pie in 3 minutes. How fast does Sam’s partner and Joe’s partner have to be so that team A wins? How fast do the partner’s have to be so that team B wins?

10. Problem Analysis 5. Concept & 6. Connections Key concepts and techniques underlying the problem and its solution are identified. One particular concept is explored in depth by investigating how the concept is defined, how it is represented, its important properties, applications, connections to other mathematical concepts, and where it appears in the curriculum. Connect college-level concepts to the problem, its concepts and techniques. Connect to other problems in the curriculum. Connect to state curriculum or performance standards.