1. Using a Children’s Thinking Approach to Change Prospective Teachers’ Beliefs and Efficacy of Elementary Mathematics AERA Paper Session Sarah Hough, Ph.D. David Pratt, Ph.D. Spring 2006 David Feikes, Ph.D.

2. The research for this paper was supported by the National Science Foundation, DUE 0341217. The views expressed in this paper are those of the authors and do not necessary reflect those of NSF. Focus on How Children Learn Mathematics Mathematical Content Courses for Elementary Teachers

3. Outline  Defining the Problem  Theoretical Perspective  Phase 1 Results  Phase 2 Results  Examples of Supplement  Discussion

4. Problem  Elementary Teachers need more than an understanding of mathematical content knowledge.  Many Preservice Teachers come from traditional mathematics classrooms where procedural knowledge is revered over conceptual understanding and investigation (Ma, 1999)

5. Problem cont.  Underlying beliefs about mathematics Beliefs about mathematics determine future teaching beliefs about mathematics resistant to change (Hill, Rowan, Ball, 2005)

6. Theoretical Perspective  Mathematical Knowledge Necessary for Teaching  Children’s Thinking Approach  Self-Efficacy

7. MKNT - Articulating the “why’s” of procedures and concepts - Interpreting student solutions - Encouraging multiple solution paths to problems 1) Mathematics Content Knowledge-- a textbook understanding of mathematics. 2) Pedagogical Content Knowledge-- how to teach mathematics. Mathematical Knowledge Necessary for Teaching (MKNT) includes: (Hill & Ball, 2004)

8. Children’s Thinking Approach Teachers’ greatest source of knowledge is from the students’ themselves (Empson & Junk, 2004). Concentrating on understanding children’s thinking may help teachers develop a broad and deeper understanding of mathematics (Sowder, et.al., 1998).

9. Efficacy  Bandura: “people’s beliefs about their capabilities to produce designated levels of performance and exercise control over events that affect their lives”  Efficacy has been shown to lead to: Greater levels of planning Enthusiasm Committed to the profession

10. Summary Mathematical knowledge necessary for teaching (MKNT) can be developed through knowledge of children’s mathematical thinking. This new knowledge can help people preparing for teaching children to change their beliefs/efficacy and as a result change the way that they teach in the future.

11. Phase I: Qualitative Results  Open-ended question—How is the mathematics in this course different or similar to the mathematics in previous courses?  Analysis—Constant comparative method (Strauss, 1980)

12. Qualitative Results  Why’s of Mathematics  Multiple Solutions to Problems  Less Rule Oriented  Not One Right Answer  Children’s Thinking Important  Problem Solving  Discovering Patterns

13. The “Whys” of Mathematics  Now we have to think why we are doing what we are doing, and know how to explain it where as before we just did the math.

14. Multiple Solutions  It is very different from the mathematics I did in high school. We are now learning about multiple ways to solve a problem and real life applications of the math we are learning. We did neither of those in high school.

15. Less Rule Oriented  I graduated in 1970 so I’m stretching my memory but I seem to remember it being more rules oriented and less explanation oriented.

16. Not One Right Answer  The math is the same in this course as in different courses because it is dealing with numbers. However the class is different because the final answer is not he most important part of the problem. By far the greatest thing to learn is the understanding.

17. Children’s Thinking Important  The math isn't high school or college level math. The math in this course requires more thinking and attempting to understand how children understand the concepts.

18. Problem Solving  This class is totally different from the last mathematics I took. The other mathematics was strictly learn this method—memorize these equations- whereas this is learning a number of methods and developing thinking strategies and understanding them.

19. Discovering Patterns  This is like no other course. The professor has made it our class. We started out the course with multiple story problems, discussing which different technique each of us has used to solve our problem. At the end of this section, we named each "rule" that helped us figure out our solution. Understand them first, then put a name with it.

20. Phase II: Quantitative Results  Knowledge of Children’s Thinking  Beliefs  Efficacy

21. Knowledge of Children’s Thinking Instrument  If children can count, they understand the concept of number.  Initially, children think of multiplication as repeated addition.  Sometimes children understand negatives in one context and not the other.  Some children believe 6/7 = 8/9

22. Knowledge of Children’s Thinking Results Course 1: 4.1/5.0 Course 2: 3.8/5.0n=41 93% of prospective teachers in Course 1 65% of prospective teachers in Course 2 Above 3.5 (agree more than disagree) By the end of both courses, participants had an understanding of the ways in which children think mathematically.

23. Beliefs Instrument “Mathematical skills should be taught before concepts” (negatively worded) “Frequently when doing mathematics one is discovering patterns and making generalizations” “In mathematics, there is always one best way to solve a problem” (negatively worded)

24. Conflicting Views Prior to Course  Prior to taking the course, preservice teachers would agree with what appeared to be conflicting statements. Factor Analysis -Problem solving is an important aspect of mathematics (Investigative View). -Mathematics is mainly about learning rules and formulas (Procedural View).

25. Beliefs Results MANOVA test results indicates a significant move toward a more non- procedural view of math for both courses. 2.83.33.6 ProceduralNon-Procedural Course 1- pre Course 1 post Course 2- post

26. Efficacy –Connection to the Classroom  Self-Reported Data collected at the end of two courses  Efficacy for two areas: Understanding Math and Teaching Math

27. Efficacy Results T-test results indicate significance differences for both areas (p<.01) Course 1: n=30 Course 2: n=24 Strongly disagree 3.24.0 3.1 4.2 Strongly agree Understanding Math

28. Efficacy Results T-test results indicate significance differences for both areas (p<.01) Course 1: n=30 Course 2: n=24 Strongly disagree 2.4 3.7 2.6 4.0 Strongly agree Teaching Math

29. Context Supporting Change in Beliefs  CMET Project Connecting Mathematics for Elementary Teachers (CMET) Supplement used in Content Course Aligned to content typically taught Includes: Content of the Chapter Problems and Exercises Children's Solutions & Discussion of Problems & Exercises Questions for Discussion

30. Context Supporting Change in Beliefs These descriptions are based on current research and include:  how children come to know number  addition as a counting activity  how manipulatives may embody (Tall, 2004) mathematical activity  concept image (Tall & Vinner, 1981) in understanding geometry

31. Context Supporting Change in Beliefs In addition, the CMET supplement contains:  problems and data from the National Assessment of Educational Progress (NAEP)  our own data from problems given to elementary school children  questions for discussion

32. Students Learn Children’s Methods Different ways children intuitively solve ratio and proportion problems  Unit or Unit Rate Method  Scale Factor or Composite Unit  Building Up

33. Learn About Children’s own Algorithms Fiona worked on a word problem that involved regrouping (of 37 pigeons, 19 flew away). She dropped the 7 from the 37 for the time being. She then subtracted 10 from 30. Then she subtracted 9 more. She puzzled for a while about what to do with the 7, now that she had to put it back somewhere. Should she subtract it or add it? I asked her one question: Did those seven pigeons leave or stay? She said they stayed, and added the 7. 37  19 30  10 = 20 20  9 = 11 11 + 7 = 18

34. Conclusion  One way to help preservice teachers construct both mathematical knowledge and the mathematical knowledge necessary for teaching is by focusing on how children learn and think about mathematics. Prospective teachers can use the way children think about math to learn math themselves. Beliefs can be changed to a more non-procedural or investigative view towards math, when using this approach. Efficacy for understanding AND teaching mathematics can be positively affected by using this approach (even thought the course isn’t focused on instruction, just students’ understandings)  Using knowledge of how children learn and think about mathematics will also improve preservice teachers’ future teaching of mathematics to children.

35. Discussion  Motivation Factor We believe preservice teachers will be more motivated to learn mathematics as they see the context of applicability. If they see how children think mathematically and how they will use the mathematics they are learning in their future teaching, then they will be more likely to develop richer and more powerful mathematical understandings.  Next Steps Establish control group sites More specific instruments based on pilot data Involve more sites in pre-post data collection

36. Thank You!