PME Paper Session

1. Developing Preservice Teachers’ Beliefs about Mathematics Using a Children’s Thinking Approach in Content Area Courses PME Paper Session Dave Feikes, Ph.D. Fall 2006 Sarah Hough, Ph.D. David Pratt, Ph.D.

2. Focus on How Children Learn Mathematics Mathematical Content Courses for Elementary Teachers 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.

3. Outline • Defining the Problem • Theoretical Perspective • Methodology • Results • Examples of Supplement • Discussion

4. Problem • Elementary Teachers need more than an understanding of mathematics 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

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

10. Supplemental Text Based on Research 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

11. CMET Supplement cont. 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

12. Preservice Teachers 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

13. 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

14. Methodology • 15 Likert-Scale questionnaire asking participants the extent to which they agreed or disagreed with statements. • Pre and Post test with group using supplemental materials. • Factor Analysis and MONOVA was used to analyze responses.

15. 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)

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

17. Beliefs Results Course 1- pre Course 2- post Procedural 2.8 3.3 3.6 Non-Procedural Course 1 post MANOVA test results indicates a significant move toward a more non-procedural view of math for both courses.

18. 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. • Using knowledge of how children learn and think about mathematics will also improve preservice teachers’ future teaching of mathematics to children.

19. 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 • Use IMAP survey instrument to determine change in beliefs with a control and experimental group.

20. Thank You!