The document discusses several studies that examine the use of open-ended problems and problem-based learning in mathematics education. It describes how open-ended problems that allow for multiple solutions can improve students' divergent thinking skills compared to multiple choice or algorithm-based questions. Several studies found that problem-based learning approaches enhanced students' self-confidence, organization, attention, and willingness to participate when compared to traditional expository teaching methods. However, the document also notes some challenges in implementing problem-based learning, including the need for regular use and connection to multiple disciplines.

1. Susan Hewett MAED 5040

2. The process of overcoming difficulties that possibly interfere with the attainment of a goal.

3. Originally, I had planned to focus on the difficulties with problem solving However, I narrowed the information down to open ended problems and problem based learning

4. A problem with multiple ways to solve A problem with multiple solutions

5. A scenario is given to students who use reasoning, questioning, and critical thinking to determine a solution (Cerezo, 2004) Small groups receive a scenario with multiple solutions; together they must determine the solution and defend their answer (Belland, 2010) An approach where students apply textbook knowledge to case study situations (Chamberlin & Moon, 2008)

7. 273 7 th grade students Students presented with problem: “ There are 3 jugs, A, B, and C. Find the best way of measuring out a given quantity of water using these jugs. Many looked for and used a pattern within answers

8. A second problem about a circle with an inscribed hexagon was presented. Students were asked to “write as many ideas as …[possible] about the figure” Statements scored based on 3 criteria Students who did not use the same pattern for the first problem did better on the second problem.

9. Students who avoid a pattern are able to develop original ideas in an open ended problem in mathematics

10. 90 4 th graders Compares multiple choice and open ended problems Data analyzed through use of written and verbal responses from students

11. Some students received multiple choice first, some open ended first Problems were based on those that are given on state assessment every year

12. Multiple choice leads to focus on choices, not answer to question in problem Open ended questions do not usually rely on learned algorithms and shortcuts that can apply and usually work Students more likely to solve problem when open ended

13. 14 at risk females in grades 6-8 Attended various schools within the same system Currently using problem based learning in classes Selected because they are at risk, but like using problem based learning

14. Presented math or science situation to solve. Collaboration in small group, followed by group presentation Students willing to participate in group work and in presentation Students interviewed about problem based learning

15. Problem based learning: -enhances self-confidence -leads to better organization -increased student's attention -students better able to process information -real life applications -more willing to participate

16. Small sample size already using problem based learning Selected students liked problem based learning

17. 12 7 th grade students, taught in French Looked at thinking of students As students worked, they were asked to talk through their reasoning and explanations

18. Only a few students solved problem correctly Students preferred informal strategies to formal algebraic ones Main strategy chosen was Trial and Error

19. Most students solved the problem using non-algebraic techniques Strategies used: Estimation and guess and check Trial and error Forward operations Work backwards Write a numerical sentence Write an algebraic equation

20. Small sample size Analyzing thinking is very subjective Presence of researcher as students work problems

21. Not a true research study Presents information comparing problem based learning and model eliciting approach Apply textbook knowledge to real life situations Discusses pros and cons of problem based learning

22. Realistic Leads to creativity Higher order thinking Teacher as facilitator Learn via discovery Self-directed learning Interconnectedness Collaboration Self-assessment

23. Time Don’t focus on single discipline Mathematical models may or may not be used Must be used regularly

24. 164 5 th graders Problems were presented via Internet, but work was done in classroom supervised by homeroom teacher

25. Students were given the following problem: Which of the following numbers: 15, 20, 23, 25 does not belong? Explain why. Open ended problem breaks away from stereotype that there is only one solution The variety of solutions and the reasons were studied

26. 24 6 th grade students Low achieving students Involved after school program Used manipulatives

27. Students worked in groups on given problems Each small group developed their own arguments and justifications Students were eager to share findings Students corrected one another

28. Small sample size Setting Participants

29. 79 7 th grade students Use of computer technology and scaffolding Study not related to math classes Specifically focused on the development of arguments

30. More beneficial to low and average achievers Scaffolding kept groups organized Students stayed focused

31. Involved graduate students and 9 th and 10 th grade students Focused on how to teach problem solving Used individual teacher’s lessons and observations

32. The teachers did not always hear the reasoning of students when an alternative solution was given Teachers were unable to select problems that conveyed what they wanted There must be a connection between different topics At the end of the study, the problems were no longer a means of memorizing and applying a formula, but became a tool for investigation by students.

33. Saturday Graduate students

34. 48 10 th grade females Control and experimental groups were determined on basis of pre-test scores 2 teachers

35. Primary grades usually use expository teaching methods for math Problem solving techniques leads to students integrating the content Problem solving methods have become the “norm” in math classes

36. Experimental group showed larger gains Interesting side note : experimental group was actually a combination of expository and problem-solving Reasons : -students lacked basic math skills -caused students to have difficulty

37. Small sample size Location of school in Pakistan Lack of basic skills prior to study Students hesitant to change how they learn

38. 110 3 rd graders, placed in 2 groups Pre-test 7 weeks Computer software

39. Motivated students All students solve problems

40. Use regularly with non-routine problems Justify explanations Work collaboratively Multiple solutions or methods Time to implement Greater achievement Real life applications

41. Belland, B. (2010). Portraits of middle school students constructing evidence-based arguments during problem-based learning: the impact of computer-based scaffolds. Educational Technology Research & Development , 58 (3), 285-309. doi:10.1007/s11423-009-9139-4. Cerezo, N. (2004). Problem-based learning in the middle school: A research case study of the perceptions of at-risk females. Research in Middle Level Education Online , 27 (1), 20-42. Retrieved from Education Research Complete database. Chamberlin, S. A. , & Moon, S. M. (2008). How does the problem based learning approach compare to the model-eliciting activity approach in mathematics?. International Journal for Mathematics Teaching and Learning , Nov 28 . Retrieved from http://www.cimt.plymouth.ac.uk/journal/chamberlin.pdf Imai, T. (2000). The influence of overcoming fixation in mathematics towards divergent thinking… International Journal of mathematical Education in Science and Technology , 31(2), 187 – 193. Retrieved from Education Research Complete database. Karp, A. (2010). Analyzing and attempting to overcome prospective teachers’ difficulties during problem-solving instruction. Journal of Mathematics Teacher Education , 13 (2), 121-139. doi:10.1007/s10857-009-9127-y. Kazemi, E. (2002). Exploring test performance in mathematics: the questions children’s answers raise. Journal of Mathematical Behavior , 21 (2), 203. Retrieved from Education Research Complete database. Klavir, R., & Hershkovitch, S. (2008). Teaching and evaluating ‘open-ended’ problems. International Journal for Mathematics Teaching and Learning , May 20 . Retrieved from http://www.cimt.plymouth.ac.uk/journal/klavir.pdf

42. Mueller, M., & Masher, C. (2009). Learning to Reason in an Informal Math After-School Program. Mathematics Education Research Journal , 21 (3), 7-35. Retrieved from Education Research Complete database Osta, I., & Labban, S. (2007). Seventh graders' prealgebraic problem solving strategies: geometric, arithmetic, and algebraic interplay. International Journal for Mathematics Teaching and Learning , Nov 28 . Retrieved from http://www.cimt.plymouth.ac.uk/journal/osta.pdf Perveen, K. (2010). Effect Of The Problem-Solving Approach On Academic Achievement Of Students In Mathematics At The Secondary Level. Contemporary Issues in Education Research , 3 (3), 9-13. Retrieved from Education Research Complete database. Schoppek, W., & Tulis, M. (2010). Enhancing Arithmetic and Word-Problem Solving Skills Efficiently by Individualized Computer-Assisted Practice. Journal of Educational Research , 103 (4), 239-252. Retrieved from Education Research Complete database