1. Using Mixed Methods to Explore How Research on Children’s Mathematical Thinking Influences Prospective Teachers’ Beliefs and Efficacy Sarah Hough: University of California, Santa Barbara David Pratt: Purdue University North Central AERA: New York, NY March 2008

2. Purpose of this presentation Share results of a mixed model/mixed methods evaluation study Illustrate how the use of mixed research is particularly appropriate in this setting Funding Source: NSF CCLI Grants DUE-0341217 & DUE-0126882 The views expressed in this presentation are those of the authors and do not necessary reflect those of NSF.

3. The Project Being Studied Connecting Mathematics for Elementary Teachers—a supplement based on a Children’s Thinking Approach for use with a traditional text in a typical content course for prospective elementary teachers. Children’s Thinking Approach uses the research based ways in which children think mathematically to teach content.

4. Mathematical knowledge necessary for teaching is fundamentally different than just knowing mathematics content or knowing how to implement pedagogical strategies in the classroom (though it encompasses both). MKNT includes an ability to: Problem that CMET addresses ∙ Articulate the whys of procedures and concepts at a level appropriate to the particular grade taught. ∙ Understand, analyze and use in classroom discussion, unusual methods of solving a problem given by a student ∙ Facilitate student work as they explore multiple solution paths to problems

5. In a typical content course… Chapter 1Problem Solving Chapter 2Sets Chapter 3Whole Numbers Chapter 4Number Theory Chapter 5Integers Chapter 6Rational Numbers – Fractions Chapter 7Decimals, Percents, and Real Numbers Chapter 8Geometry Chapter 9More Geometry Chapter 10Measurement Chapter 11Statistics/Data Analysis Chapter 12Probability Chapter 13Algebraic Reasoning

6. Children’s Thinking Approach The elementary mathematics topics are taught through a focus on how children think about them:  Children’s invented algorithms  Children’s problem solving strategies  Common errors and misconceptions held by children  Children’s development in relationship to content Rather than as content to be transmitted to students using pedagogical approaches learned later.

7. Measurement as a Concept In elementary school, measurement has traditionally been presented as procedures and skills. However, a more careful analysis indicates that measurement is a concept. Teaching measurement is more than teaching the procedures for measuring, it is also helping children understand the concept of measurement.

8. Over seventy-five percent of the fourth grade children missed this question. Most children who missed this question answered 8 or 6. Why 6? Example of children’s thinking problem 1996 National Assessment of Educational Progress, (NAEP)

9. BIG QUESTION Does use of CMET foster Mathematical Knowledge Necessary for Teaching? Problems :  MKNT had not been operationalized at this level— so cannot measure it  Can we expect CMET to effect ideas about teaching when it doesn’t directly teach pedagogy?

10. Evaluation Study In what ways does a course that focuses on children’s mathematical thinking affect pre- service teachers understandings about mathematics? Its teaching? (What?) How does such a course compare to one that does not in terms of the effect on pre-service teachers’ mathematical knowledge for teaching? (How Much?) Used Mixed Model/Mixed Methods to explore:

11. Mixing Models PHASE ONE—Qualitative purpose: Explore the kinds of mathematical understandings gained from course. PHASE TWO—Quantitative purpose: To measure gains in understandings from course. PHASE THREE--Quantitative purpose: To compare outcomes using a control group in a different setting.

12. Mixing Methods PHASE ONE—Qualitative purpose Qualitative data Qual/Quant data (open-ended interviews)(questionnaire) Analyzed qualitativelyAnalyzed qualitatively and then (constant comparative analysis) quantified

13. Samples from questionnaire What is the last regular mathematics course that you took? Did you enjoy this course? Did you do well in this course? How is the mathematics that you are doing in this course the same/different from the type of mathematics you have done previously? Using one of the following areas you learned about (sets, whole numbers, number theory, integers, rational numbers) please give one or two specific examples of what you have learned about how children think about or learn certain topics in mathematics.

14. Sample questions from interview Asked students to elaborate on their answers Do you think the way you learned mathematics in this class will influence your future teaching of any of these mathematics topics to children? How?

15. Results Phase I In addition to building robust understandings of how children think about and do certain mathematics topics prospective teachers began to re-examine fundamental notions of doing and teaching mathematics. The way this course has been taught has made me look at math from a different perspective. In years past, math=rules, not necessarily logic or reason, just rules. In this course it has been made obvious that there are reasons decimals do what they do, and while the reason backs up the rules, the rule has a history. If students get “the why”, their knowledge base has expanded and they can apply that knowledge elsewhere

16. Results Phase I Mathematics as less rule oriented I seem to remember my previous math class as being more rules oriented and less explanation oriented. There aren't so many "set in stone" rules. You have to think of how a child sees math and these problems, so that was new and exciting for me

17. Results Phase I Multiple Solution Paths in Mathematics 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.

18. Results Phase I Why’s of Mathematics Because of this class I know now why I am doing some of the procedures that I am doing, instead of just doing it a certain way because my teacher said to. The math we did in high school did not take into consideration of the “whys” math uses, typically it was “this is how it is done”, no explanation. This course emphasizes that math is a sense making process.

19. Results Phase I Teach Differently because of the way children approach problems The math taught in this class addresses the challenge of not only understanding how to do the material ourselves, but how children perceive problems and solutions and how we as teachers need to instruct and explain the math to our students. If they [children] are given a problem, they might do it a different way and come up with an answer different from our own. I am starting to understand that math can be taught in different ways from the way I learned it.

20. Efficacy: Results of interview with participants indicated clearly that they felt confident in their own understanding of the mathematics taught as a result of the class and their own efficacy to use what they had learned to teach others. Results Phase I

21. Mixing Methods PHASE TWO—Quantitative purpose Quantitative data collection tools derived from phase one (N=93 prospective teachers ) Analyzed qualitatively Analyzed quantitatively (Descriptive/Exploratory ( MANOVA, t-tests ) Factor Analysis)

22. Mathematics is mainly about learning rules and formulas (q8). An elementary teacher should immediately explain the correct procedure when a child makes a mistake (q5). Children's own methods of problem solving are as important as learning procedures (q2). Frequently when doing mathematics one is discovering patterns and making generalizations (q11). Sample Beliefs Items

23. Results Phase Two Exploratory Factor Analysis

24. Results Phase Two: Descriptives Mathematics is primarily a step-by-step mechanical process. 2.3 (.77)2.6 (.95)3.5 (.84)** Mathematical skills should be taught before concepts. 2.6 (.81)3.0 (.83)*3.0 (.99) Children best learn mathematics through extensive drill and practice. 3.0 (.94)3.7 (1.00)*4.3 (.62) Mathematics is mainly about learning rules and formulas. 3.0 (1.00)3.6 (1.00)**4.0 (.67) An elementary teacher should immediately explain the correct procedure when a child makes a mistake. 2.8 (1.10)3.8 (1.00)**4.3 (.63) A good textbook is more important for helping students learn mathematics 3.0 (1.00)3.8(.85) **4.2 (.83) Items that loaded on factor I only Pre course 1 Post course1 Post course 2

25. Results Phase Two Items loading on factors I and II Pre course 1 Post course1 Post course 2 Children should master the basic facts before doing problem solving. 2.5 (.94)2.7 (1.00)2.7 ( 1.01) Children should be able to figure out for themselves whether an answer is mathematically reasonable. 3.5 (.83)3.6 (.95)3.7 (.64) Frequently when doing mathematics one is discovering patterns and making generalizations. 3.9 (.70)4.0 (.74)4.2 (.40)

26. Results Phase Two Items that loaded on factor II only Pre course 1 Post course1 Post course 2 Children's own methods of problem solving are as important as learning procedures. 4.1 (.77)4.5 (.63)**4.6 (.50) In mathematics there is one correct answer. 3.3 (1.00)3.7 (.99) **3.8 (.98) Children are often creative when solving problems. 4.0 (.64)4.5 (.68) **4.7 (.49) Problem solving is an important aspect of mathematics 4.2 (.50)4.4 (.53)4.3 (.70) In mathematics there is always one best way to solve a problem. 4.0 (.73)4.2 (.70)4.4 (.58) For elementary school children it is not important to understand why a mathematical procedure works. 4.2 (.83)4.3 (.96)4.4 (.94)

27. Results Phase Two Course IPrePost Efficacy for understanding mathematics 3.24.0** Efficacy for teaching mathematics 2.43.7 ** Course II Efficacy for understanding mathematics 3.14.2 ** Efficacy for teaching mathematics 2.64.0 **

28. Mixing Methods PHASE THREE—Quantitative purpose Quantitative data collection (Forced choice questionnaire) Analyzed quantitatively (ANOVAs)

29. Comparison and treatment classes at different Midwestern University Used 7 key items from beliefs scale Ran MANOVAs on TOTAL beliefs scores Found significant differences between control classes and CMET class. No significant differences were found between the two control classes. Post hoc tests showed individual items contributing to the overall significant result: Mathematics is mainly about learning rules and formulas; Children learn best through drill and practice; In mathematics there is only one correct answer.

30. Conclusions Key use of mixed methods in this study was first allowing the sequential use of qual/quan model and methods. Second the use of exploratory factor analytic techniques along with descriptive measurements to better describe both the structure and the strength of notions of doing and teaching mathematics.

31. For more information, please contact: Sarah Hough at sarahh@education.ucsb.edusarahh@education.ucsb.edu Dave Pratt at dpratt@pnc.edudpratt@pnc.edu