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Teaching Mathematics for Elementary Teachers through Problem Solving Martha VanCleave MathFest 2000 UCLA August 5, 2000

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Topics mathematical reasoning & problem solving sets whole numbers; their operations properties and computations (mental, written and electronic) number theory rational numbers as fractions decimals, percent, ratio & proportion integers rational numbers real numbers, functions, graphing, and using algebra.

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Goals And Objectives To review the principles of elementary mathematics To present enriching topics which strengthen an individual's mathematical background To increase the student's interest and confidence in confronting mathematical situations. To develop positive attitudes about mathematics To instill an appreciation for the art and beauty of mathematics To promote active student involvement in the learning of mathematics To encourage individual responsibility for learning

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Current Text Bennett, Jr., A. B., & Nelson, L. T. (1998). Mathematics for Elementary Teachers:A Conceptual Approach, 4th Edition. Boston, MA. The McGraw- Hill Companies, Inc.

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Problem Solving Activities Scoring the problem solving of peers using Oregon’s Mathematics Problem Solving Official Scoring Guide “Menus of Problems” taken from Mathematics Teaching in the Middle Grades Group problem solving

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Oregon’s Problem Solving Official Scoring Guide

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1999 - 2000 Mathematics Problem Solving Official Scoring Guide 1999 - 2000 Official Scoring Guides/Mathematics For use during the 99 - 00 statewide assessment Office of Assessment and Evaluation Oregon Department of Education Wording may be refined based on samples of student work May 1999

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Conceptual Understanding The “what” The student interprets the concepts related to the task and translates them into mathematical ideas The student uses mathematics that fit what is requested in the problem

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Processes and Strategies The “how” of solving the task The student chooses and carries out processes and strategies that can work. The student uses appropriate pictures, models, diagrams, and/or symbols.

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Verification* The “defense” in solving the task The student clearly shows the review of concepts, processes, and calculations used to get the solution. The student shows that the solution answers the question posed. If possible, the entire problem is worked in a second way. * not evaluated in 1999-2000

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Communication The “connecting path” among other dimensions leading to the solution The student clearly shows the path leading to the solution shown with no gaps for the reader to fill in. The student’s work fits all the parts together by using pictures, charts, diagrams, and/or words.

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Accuracy The “correctness” of the solution The final answer is complete, justifiable, and clearly identified. The answer matches what the problem is asking.

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Scoring Problem Solving Four dimensions; conceptual understanding, processes and strategies, verification, and communication are scored on a 6 point scale. Accuracy, scored last, receives a score of 5, 4, or 1. Scoring provides students with specific feedback related to the four dimensions and accuracy of their solutions.

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Scoring the Four Dimensions 6enhanced 5thoroughly developed 4work is complete 3work is partially effective or partially complete 2work is underdeveloped or sketchy 1work is ineffective, minimal, or not evident

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Scoring for Accuracy 5the answer given is mathematically justifiable and supported by the work 4the answer given is adequate, and/or it may contain a minor error, but no additional instruction appears necessary 1the answer given in incorrect, incomplete, or correct but conflicts with the work

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Examples of Scoring

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Student Activities

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Menu of Problems (MP) From Mathematics Teaching in the Middle Grades Provide practice applying problem solving strategies - no time pressures In class discussion of solutions provides: –Immediate feedback –Opportunity for individuals to present solutions –Exposure to alternate methods of solution

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Individual Problem Solving & Scoring In class problem solving creates a “pressure” situation Scoring peer’s work provides –Opportunity to use the scoring guide –Exposure to alternate methods of solution –Feedback Instructor Scoring provides –Feedback on scoring

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Group Problem Solving As a portion of exam –During previous class meeting, does not take away from time allowed for individual work –Encourages cooperative work –Provides opportunity to discuss problem solving strategies Evaluation based on Scoring Guide dimensions

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Outcomes And Advantages

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Focus On Process Students tend to begin the course focused on the product (answer to the problem) rather than the process The four dimensions of the problem solving guide emphasize the importance of the process The problem solving guide provides students with a framework for the process

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Changes in Students Changes in attitude Changes in self-concept Development of problem solving skills Changes in conceptualization of mathematics

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Quotes from Students’ Self-Evaluations I actually started to like it (math). I enjoy math (more) than I ever did. I learned group problem solving skills.

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Quotes from Students’ Writings After completing the worksheet (MP) I felt rather confident about things. It’s a big accomplishment for me to be able to do this (MP) on my own, especially since I usually require that each problem be worked out for me. I’m proud of this piece of work because we solved it very systematic(ally). My group explained it well and there is work shown that makes the problem easy to follow.

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Preparation for Teaching Most graduates from our elementary education program go on to teach in Oregon Oregon uses the Scoring Guide for evaluation of the state performance assessment at grades 5, 8, and 10 Prospective teachers must be prepared to teach their students ALL the skills needed to become proficient problem solvers.

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Future Directions Enhance instruction in using the Scoring Guide Follow-up with inservice teachers –Feedback on preparation –Feedback on students’ problem solving skills Conduct a formal study

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References Bennett, Jr., A. B., & Nelson, L. T. (1998). Mathematics for Elementary Teachers:A Conceptual Approach, 4th Edition. Boston, MA. The McGraw-Hill Companies, Inc. Mathematics Problem Solving Scoring Guide: “Sampler” from the Teacher Support Packets, Oregon Department of Education, www.ode.state.or.us, May 1999 Menu of Problems, Mathematics Teaching in the Middle Grades, National Council of Teachers of Mathematics. Excerpts from student writings and evaluations, Spring 2000.

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