Another Reason For Problem Solving

1. “Problem Solving: What is it? Why is it important? How to make it like Kool-Aide.”Lenny VerMaasESU #6 Milford NEEmail lvermaas@esu6.orgWeb page http://lvermaas.wikispaces.com/

2. Another Reason For Problem Solving • Ten Things to Consider When Teaching Proof, November 2009 Mathematics Teacher p 251-257 • I should think about proof as a problem-solving activity. • I have come to believe that proof is difficult for many students because it is a problem-solving activity, one that cannot be proceduralized.

3. In contrast to arithmetic and algebra (or at least many students’ earlier mathematical experiences with these), proof is not an activity that can be done routinely. Although the two-column from provides a structure that helps scaffold proof for students, there is little a teacher can do—beyond telling students to write down the “givens”—to turn this activity into something less than problem solving. I am not saying that this is not a bad thing. I only point out that if students have not had sufficient problem-solving experiences before the geometry course, they are likely to find doing proofs an unfamiliar, challenging, and frustrating experience. Again, this is a curricular issue as well.

4. Resources • My web page at lvermaas.wikispaces.com

5. Students… • think all math problems can be done quickly • think that if the problem can not be done quickly they can not do the problem or have not learned the trick • Believe that there is a “math gene” • lack the patience to spend extended time on one problem or look for alternate ways to solve a problem

6. Students… • Think that mathematics is mostly a set of rules and that learning mathematics means memorizing the rules (1996 NAEP) • “when students are seldom given challenging mathematical problems to solve they come to expect that memorizing rather than sense making paves the road to learning mathematics” (p131 Adding it up)

7. Students… • Enter school eager to become competent at mathematics. They have high levels of persistence and eagerness to learn. • One study found first graders rating themselves a 6 on a 1 to 7 scale on their interest in mathematics

8. Students Should… • be expected to justify and explain their mathematical ideas • be given challenging mathematical problems to solve so that they learn to make sense rather than memorize • learn to attribute success to effort rather than ability • see mathematics as both useful and worthwhile

9. Student’s View of Intelligence • “fixed mindset” based on ability • “growth mindset” based on effort • One study identified students and followed them for two years • Those with “growth mindset” steadily increased in math grades over two year while students with the “fixed mind” set decreased • From the Scientific Journal “Child Development” by Carol Dweck

10. Student’s View of Intelligence • It is important that student perceive achievement as a product of effort as opposed to ability. • “Extensive research has shown children who attribute success to a relatively fixed ability are likely to approach new tasks with a performance rather than a learning orientation.” (p171 Adding it up)

11. Cathy Seeley • Cathy Seeley, former president of NCTM • Productive, Structured Struggling • Attitude is more important than aptitude. • Do Math and you can do anything. • Relevant, assessable, engaging, and appropriate for all.

12. Marilyn Burns: 10 Big Math Ideas • Success comes from understanding. • Not steps and tricks to be memorized • Have students explain their reasoning. • Probe children’s thinking • Math class is a time for talk. • Students can learn from each other. • Make writing a part of math learning. • Present math activities in contexts.

13. Support learning with manipulatives. • Let your students push the curriculum. • Choose depth over breadth • The best activities meet the needs of all students. • Confusion is a part of the process. • Encourage different ways of thinking. • http://www2.scholastic.com/browse/article.jsp?id=3596

14. What is Problem Solving

15. Where to Start • Expose students to multilevel problems that take a “long time” • Make the “long problem” a common occurrence in student’s lives • Create an atmosphere that supports risk taking • Remind students that life is a “word problem” • Look at the process in addition to the answer.

16. Problem Selection • Problems that can be solved with a variety of strategies. • Several possible answers • Choose numbers to maximize students opportunity to learn. • Provide a choice for a set of numbers in the problem. • Present problem to the whole class and discuss the meaning of words. • Have students put in their own words what this problem is saying

17. Classroom Instruction Should.. • Engage children in writing their own story problems. • Invite children to talk about their work throughout their problem-solving investigations. • Invite children to examine the structural similarities and differences among problems.

18. Classroom Instruction Should.. • Require children to represent their understanding in several ways. • Make thinking and sense making the cornerstones of the classroom community. • From “Learning to Solve Problems in Primary Grades” p 426-432 Teaching Children Mathematics, March 2008

19. What is Problem Solving • The course of action is not immediately evident. • The solution may be found in several different ways or using various strategies. • More than one answer may be possible.

20. Looking at a Continuum • Exercises • 15 + 27 = ? • Word Problems • There are 15 students in one class and 27 students in another class. How many students are in both classes? • Problem Solving • Investigation

21. A Mathematical InvestigationFrom Teaching Children Mathematics May 2003, p 251 • Has multidimensional content • Is open-ended, with several acceptable solutions • Is an exploration requiring a full period or longer to complete • Is centered on a theme or event • Is often embedded in a focus or driving question

22. In Addition, A Mathematical Investigation Involves Process That Include • Researching outside sources • Collecting data • Collaborating with peers • Using multiple strategies to reach conclusions

23. How is Problem Solving Different than Solving a Word Problem • Traditional word problems. • One solution and one strategy to get to the solution. • Translate the situation into an arithmetic sentence and then solve that sentence. • Real-life problems or Investigations • Not all of the information may be provided • Several paths to arrive at multiple solutions

24. Education is what’s left over when you forget all the things the teachers made you memorize in school Mark Twain

25. Is The Extra Time Required To Teach Problem Solving Worth It? • Research points to the importance of including instructional activities that require higher-order thinking. • In a study of data from the National Assessment of Educational Progress, Wenglinsky (2000) found that students whose teachers emphasize higher-order thinking skills in mathematics outperform their peers by about 40% per grade level.

26. “Teaching with the Brain in Mind” by Eric Jensen. • “The single best way to grow a better brain is to engage in challenging problem solving. Surprisingly, it doesn’t matter to our brains whether we come up with the right answer or not: the neural growth happens because of the process, not because we have found the correct answer.

27. Cathy Seeley • Upside-down teaching • Start with a rich problem • Engage students in dealing with the problem by discussing, comparing, and interacting • Help students connect and notice what they’ve learned • Then assign exercises and homework • Demonstration of upside-down teaching at www.utdanacenter.org/amdm

28. John Dewey progressive education reformer from early 1900’s • Learning does not start and intelligence is not engaged until the learner is confronted with a problematic situation.

29. Your children want to solve problems, and their enjoyment of problems increases when children can solve problems in ways that make sense to them. • Children Who Enjoy Problem Solving, Teaching Children Mathematics, May 2003

30. Young children want to learn from their mistakes, and their enjoyment of problem solving increases when children know that mistakes will be used as stepping stones to learning. • Children Who Enjoy Problem Solving, Teaching Children Mathematics, May 2003

31. Young children want to engage in complex tasks that are completed over extended periods of time, and their enjoyment of problems solving increases when children experience the satisfaction of overcoming a challenge. • Children Who Enjoy Problem Solving, Teaching Children Mathematics, May 2003

32. Young children want to communicate their solutions to problems with others, and their enjoyment of problem solving increases when children can share and discuss their solutions with peers. • Children Who Enjoy Problem Solving, Teaching Children Mathematics, May 2003

33. Enjoyment of Problem Solving increases when students: • Solve problems in ways that make sense to them. • Learn to appreciate mistakes as a necessary and valuable part of the problem-solving process. • Experience the satisfaction of overcoming a challenge. • Share and discuss their solutions with peers. • receive support and encouragement with their development slows or appears to regress.

34. When given the chance to routinely share their solutions with their peers, children: • Find ways to communicate their thoughts. • Invent ways to examine and evaluate their ideas before sharing them in public. • Develop techniques to critique others’ ideas and provide useful feedback. • Develop the capacity to compare different solutions and expand their understanding of the mathematics embedded in problems.