Creating a Classroom of Mathematics Problem Solvers

1. Grades PreK-2 Creating a Classroom of Mathematics Problem Solvers

2. Grace Ben Jonah Tricia Mahreah Ruby Bain Family This is our family at the summit of Mt. Ellinor in Olympic National Park. It was a great family adventure. We saw many adults turn around before reaching the summit, but we pressed on and enjoyed the summit with a few mountain goats.

3. 1998 graduate of Martin Luther College with a double major in education and music. 2004 earned post-baccalaureate Wisconsin state teaching license. Ten-time classroom supervisor of student teachers. 2010 graduate of Martin Luther College with a Master of Science in Education degree—instruction emphasis. My Background

4. The importance of Mathematics& Defining Problem Solving

5. TIMSS • International studies; comparing eighth graders • TIMSS = Trends in International Mathematics and Science Study • 1995: U.S. ranked 28th out of 41 countries • 1999: U.S. ranked 19th out of 34 countries • 2003: U.S. ranked 15th out of 45 countries • 2007: U.S. ranked 9th out of 47

6. Why is math important for our country? Why is math important for our church? Why is math important for our students?

7. Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008) “The eminence, safety, and well-being of nations have been entwined for centuries with the ability of their people to deal with sophisticated quantitative ideas. Leading societies have commanded mathematical skills that have brought them advantages in medicine and health, in technology and commerce, in navigation and exploration, in defense and finance, and in the ability to understand past failures and to forecast future developments.” (p. xi) National Mathematics Advisory Panel

8. Is this Problem Solving? • In the numeral 185, what does the 8 mean? • Jorge has 12 tickets. William has 15 tickets. How many more tickets does William have? • 14 – 6 = ______ • Which shape has four sides? • (Students are lined up.) Which student is the tallest? • Erin has 2 cookies. MiKayla has twice as many cookies. How many cookies does MiKayla have? • (There is are several shapes shown at once.) How many triangles are there? • Which animal has more legs, a pig or a chicken?

9. NCSM (National Council of Supervisors of Mathematics) NCTM (National Council of Teachers of Mathematics) “the process of applying previously acquired knowledge to new and unfamiliar situations” “problem solving means engaging in a task for which the solution method is not known in advance” Definitions of problem solving

10. Meir Ben-Hur Joan M. Kenney “Problem solving requires analysis, heuristics, and reasoning toward self-defined goals” a process that involves such actions as modeling, formulating, transforming, manipulating, inferring, and communicating Features of Problem Solving

11. Teaching Problem Solving

12. Key Words Strategies Process by George Pólya Teacher’s Role Overview

13. Key Words

14. What words tell a person to add? What words tell a person to subtract? What words do you notice are particularly troublesome to your students?

15. “There are three bears in the living room. There are two more bears in the kitchen. How many bears are in the house altogether?” An Example of Teaching Key words

16. Undermines real problem solving Make problem solving a mechanical process which makes students prone to errors Understanding the language of mathematics is important. “There are five bears in the house. Some bears are in the living room. Two more bears are in the kitchen. How many bears are in the living room?” Ben-Hur, M. (2006). Concept-rich mathematics instruction: Building a strong foundation for reasoning and problem solving. Alexandria, VA: Association for Supervision and Curriculum Development. Xin, Y. P. (2008). The effect of schema-based instruction in solving mathematics word problems: An emphasis on prealgebra conceptualization of multiplicative relations. Journal for Research in Mathematics Education, 39(5), 526-551. What does research say?

17. Word Wall Math Word Dictionary (When they can write.) Vocabulary Cards (Emphasize examples & nonexamples. Do together. Use pictures.) Semantic Feature Analysis Literature Strategies for Math Words

18. Strategies Instruction

19. One Way Another Way • Teacher models the strategy. • Students work on problems using that strategy. • Teacher models the strategy. • Students work on problems which may or may not use the modeled strategy. Ways to Introduce

20. Types of Strategies • Make a Model or Diagram • Make a Table or List • Look for Patterns • Use an Equation or Formula • Consider a Simpler Case • Guess and Check/Test • Work Backward • Act it Out • Others?

21. Act it out Work backward Students pretend to be each person. They form a line and check to see if it fits the criteria listed in the problem. The final statement puts Ashanti as last. Use an A to represent Ashanti. The second to last statement means that Drew cannot be first. That leaves only Chris, so Chris is first. “Drew, Ashanti, and Chris line up to get off the bus. (Ashanti is the oldest. Drew is the youngest.) The first person is not the youngest. The oldest person is last. Who is first?”

22. Teaching strategies improves mathematics problem solving abilities. Teaching strategies does not improve overall math achievement. Teachers need to avoid teaching strategies as an algorithm. Rickard, A. (2005). Evolution of a teacher’s problem solving instruction: A case study of aligning teaching practice with reform in middle school mathematics. Research in Middle Level Education Online, 29(1), 1-15. Higgins, K. M. (1997). The effect of year-long instruction in mathematical problem solving on middle-school students’ attitudes, beliefs, and abilities. Journal of Experimental Education, 66(1). Jitendra, A., DiPipi, C. M., & Perron-Jones, N. (2002). An exploratory study of schema-based word-problem-solving instruction for middle school students with learning disabilities: An emphasis on conceptual and procedural understanding. Journal of Special Education, 36(1), 23-38. Mastromatteo, M. (1994). Problem solving in mathematics: A classroom research. Teaching and Change, 1(2), 182-189. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145-166. What does Research Say?

23. Process by George PÓlya

24. Understand the Problem Make a Plan Follow/adjust the Plan Look Back The Elements of the Process

25. Define important words. Identify necessary and unnecessary information. Stating what is known and unknown. Determine if other information is needed. Decide if calculations need to be made prior to another calculation. Rephrase the problem. Consider this: pose a problem situation without a question. Understanding the Problem

26. Select a strategy Use what is known to determine how to find a solution The goal would be that students be able to explain, with reasons, why they think their strategy could work. Make a Plan

27. Students carry out the plan they made. Students show the work that they do, and they may be asked to write explanations. Students adjust their plan if they notice something isn’t working or they have determined a better way to solve the problem. Follow/adjust the Plan

28. Very valuable! Check that solution fits problem. Consider strategy choices and their consequences. Create related problem(s) that could be solved the same way. Offer changes to the problem and infer their affect on the solution. Connect to other problems already studied. Make generalizations. Look Back

29. The Teacher’s role

30. Undergeneralizations Ex.: A rectangle must be horizontal. Ex.: There is only one way, or very few ways, to represent 3. Overgeneralizations Ex.: Adding two numbers makes a bigger number Ex.: Misapplication of regrouping Ex.: Address Misconceptions

31. Foster a classroom environment friendly to asking questions. Adjust content to meet student needs. Use a wide variety of activities. Allow time for exploration. Organize and represent concepts in different ways. Pose probing questions to foster meta-cognition. Model meta-cognition. Promote dialogue. The Environment

32. Assessment/Evaluation

33. Creates a framework to help students see the importance of explaining why they are doing what they are doing. Use in groups or individually. Can be used in portfolios to share with parents at conferences or for student self-reflection of progress. Problem Solving Form

34. Teacher notes while observing problem solving Listen for evidence the student seeks information to fully understand the problem. Consider a student’s ability to persevere. Note use of appropriate strategies. Listen to student oral explanations for misconceptions or proper conceptual understanding. Look for algorithmic errors. Anecdotal Records

35. Much the same as anecdotal records, but this may be scored from viewing work. A rubric may be formed using Pólya’s process or specific to the learning goals of the lesson. Share whatever rubric you use with students and make sure they understand it. Provide opportunities for self-evaluation. Checklist or Rubric

36. Questions or Comments?