1. Using a Children’s Thinking Approach to Change Prospective Elementary Teachers’ Efficacy and Beliefs of Mathematics AERA Paper Session Spring 2007, Chicago Sarah Hough, Ph.D. David Pratt, Ph.D. David Feikes, Ph.D. Keith Schwingendorf, Ph.D.

2. Outline Defining the Problem Defining the Problem Context of Study Context of Study Samples from Supplement Samples from Supplement Literature Review Literature Review Results Results Discussion Discussion Future Studies and New Possible Grants Future Studies and New Possible Grants

3. Breaking the Traditional Beliefs about Math Many preservice teachers come from traditional mathematics classrooms where procedural knowledge is revered over conceptual understanding and investigation (Ma, 1999)

4. Math Teachers Need to Know There appears to be a disconnect between math needed to know in elementary school and that which is taught in math content courses in college. Nathan & Koedinger (2000).

5. Beliefs and Knowledge about Math Beliefs about mathematics determine future teaching beliefs about mathematics and are resistant to change Beliefs about mathematics determine future teaching beliefs about mathematics and are resistant to change Many elementary school teachers have limited understanding of the subject matter they are responsible for teaching. Many elementary school teachers have limited understanding of the subject matter they are responsible for teaching. (Ball, 1990; Ma, 1999; Mewborn, 2000)

6. A Special Type of Math Content Course is Needed Increasing the number of math content courses is not effective. The specialized knowledge of mathematics needed is different from the mathematical content contained in most college mathematics courses, which are principally designed for those whose professional uses of mathematics will be in mathematics, science, and other technical fields. Courses that reflect a serious examination of the nature of the mathematics that teachers use in the practice of teaching do have some promise of improving student performance. National Research Council (2001).

7. 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. Mathematical Content Course Supplement for Elementary Teachers Focus on How Children Learn Mathematics

8. Context Supporting Change in Beliefs The CMET supplement contains: problems and data from the National Assessment of Educational Progress (NAEP) problems and data from the National Assessment of Educational Progress (NAEP) our own data from problems given to elementary school children our own data from problems given to elementary school children questions for discussion questions for discussion - Supplement to existing textbook - Aligned to content typically taught

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

10. Qualitative Results from Phase 1 of Study What is the Focus of the CMET materials? Understanding Mathematics Conceptually Understanding Mathematics Conceptually Multiple Solutions to Problems Multiple Solutions to Problems Less Rule Oriented Less Rule Oriented Focus on Process, not final answer Focus on Process, not final answer Children’s Thinking Important Children’s Thinking Important Problem Solving Emphasized Problem Solving Emphasized Discovering Patterns Discovering Patterns

11. Math as a Process “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 of how the solution was reached.” “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 of how the solution was reached.”

12. 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. 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.

13. 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. 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.

14. Literature Review Mathematical Knowledge Necessary for Teaching Mathematical Knowledge Necessary for Teaching Children’s Thinking Approach Children’s Thinking Approach Self-Efficacy for Understanding and Teaching Math Self-Efficacy for Understanding and Teaching Math

15. 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)

16. 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).

17. Efficacy Bandura: “people’s beliefs about their capabilities to produce designated levels of performance and exercise control over events that affect their lives” 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: Efficacy has been shown to lead to: Greater levels of planning Greater levels of planning Increased enthusiasm Increased enthusiasm Being more committed to the profession Being more committed to the profession

18. Efficacy and Instruction Several studies have indicated a consistent relationship between teacher efficacy and classroom instructional strategies (Wertheim & Leyser, 2002) as well as willingness to embrace reform strategies (Hami, Czerniak, & Lumpe, 1996; Ross, 1992). Several studies have indicated a consistent relationship between teacher efficacy and classroom instructional strategies (Wertheim & Leyser, 2002) as well as willingness to embrace reform strategies (Hami, Czerniak, & Lumpe, 1996; Ross, 1992).

19. Efficacy and Teaching Strategies in Mathematics Instruction Using manipulatives was strongly embraced by the preservice teachers with the highest degree of mathematics teacher efficacy Using manipulatives was strongly embraced by the preservice teachers with the highest degree of mathematics teacher efficacy Similarly, teachers with the lowest efficacy for teaching expressed concern over using manipulatives as a teaching aid. Similarly, teachers with the lowest efficacy for teaching expressed concern over using manipulatives as a teaching aid. Swars (2005)

20. Results from Previous Studies Demonstrated a possible connection between efficacy and beliefs with pre-post test responses. Demonstrated a possible connection between efficacy and beliefs with pre-post test responses. After using the module, participants shifted to more positive efficacy beliefs and a more non-procedural view of math. After using the module, participants shifted to more positive efficacy beliefs and a more non-procedural view of math. Pratt et. al (2006).

21. Results from Previous Study

22. Summary A course focused on children’s thinking of mathematics can increase preservice teachers’ efficacy for understanding and teaching math as well as alter other beliefs regarding mathematics. This new knowledge and belief may ultimately change the way that these college students teach in the future.

23. Methodology Large midwest university Large midwest university Control and treatment groups (n=138) Control and treatment groups (n=138) Beliefs about Math Questionairre Beliefs about Math Questionairre Efficacy Beliefs Questions (Likert Scale) Efficacy Beliefs Questions (Likert Scale) IMAP survey IMAP survey Correlations and ANOVA’s Correlations and ANOVA’s

24. Math Beliefs Instrument A Math Beliefs Instrument, consisting of 12 forced choice items related to understanding of Math Concepts, Children’s Thinking, and Teaching Math was administered to preservice teachers. The Efficacy items on the questionnaire asked participants, on a scale of 1 to 10, how confident they felt in their (1) understanding of the mathematics topics covered in the course, and (2) their ability to teach these topics to children.

25. Results from Study Beliefs of efficacy were tested between control and CMET groups using two one way ANOVAs. Results indicated that significant differences between the two groups in Teaching Efficacy but not in Math Efficacy.

26. TEACHING Math Beliefs Encouraging children to explore their own ways of solving problems is as important as teaching them to follow procedures. Encouraging children to explore their own ways of solving problems is as important as teaching them to follow procedures. Rather than teaching mathematical ideas directly to students it is better to guide them in figuring things out for themselves. Rather than teaching mathematical ideas directly to students it is better to guide them in figuring things out for themselves. Mathematical skills should be taught before concepts. Mathematical skills should be taught before concepts.

27. Beliefs about Math Mathematics is a web of interrelated concepts and procedures. Mathematics is a web of interrelated concepts and procedures. Mathematics is mainly about learning rules and formulas. Mathematics is mainly about learning rules and formulas. Frequently when doing mathematics you are discovering patterns and making generalizations. Frequently when doing mathematics you are discovering patterns and making generalizations. In mathematics there is one correct answer. In mathematics there is one correct answer.

28. Beliefs about Children’s Thinking Children should master the basic facts before doing problem solving. Children should master the basic facts before doing problem solving. Children solve and think about mathematics in a variety of ways and often different from adults. Children solve and think about mathematics in a variety of ways and often different from adults. When children do not understand something in mathematics it is because they have not had enough practice. When children do not understand something in mathematics it is because they have not had enough practice. Children learn mathematics better through extensive drill and practice rather than inquiry and exploration. Children learn mathematics better through extensive drill and practice rather than inquiry and exploration.

29. Difference in Beliefs Score A one-way ANOVA was run on overall belief scores to test for differences. A significant result was found indicating a slightly higher score (about one half a standard deviation) for the CMET group. Multivariate Analysis of Variance (MANOVA) was run using these three variables as dependent variables and control versus treatment group as the independent factor. MATH and CT contributed most significantly to the difference between groups (MATH: F=10.9, p<.01; CT: F=6.3, p=.01).

30. Correlations Between Teaching Efficacy and Beliefs Children solve and think about mathematics in a variety of ways and often different from adults. Children learn mathematics better through extensive drill and practice rather than inquiry and exploration. Rather than teaching mathematical ideas directly to students it is better to guide them in figuring things out for themselves. Correlations significant at p=.01 Mathematical skills should be taught before concepts. Rather than teaching mathematical ideas directly to students it is better to guide them in figuring things out for themselves. MATH EfficacyTEACH Efficacy

31. Discussion 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 in content courses. 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 though the course isn’t focused on instruction, just students’ understandings) Using knowledge of how children learn and think about mathematics may also improve preservice teachers’ future teaching of mathematics to children. Using knowledge of how children learn and think about mathematics may also improve preservice teachers’ future teaching of mathematics to children.

32. Current and Future Research  Pre-Post Test of Control and Experimental groups  Complete Analysis of IMAP online surveys.  Secure funding for continued implementation of published materials

33. Future Grant Possibilities How Children Learn Math (HCLM): Resource for Teachers How Children Learn Math (HCLM): Resource for Teachers How Children Learn Math (HCLM): Resource for Parents How Children Learn Math (HCLM): Resource for Parents Both grants support teachers and caregivers with resources that help them understand the way children think about math and how to teach math more conceptually. Both grants support teachers and caregivers with resources that help them understand the way children think about math and how to teach math more conceptually.

34. Contact Us for More Info! If you are interested in using the CMET materials in your classroom or assisting us with potential future grants, please contact us! If you are interested in using the CMET materials in your classroom or assisting us with potential future grants, please contact us! dpratt@pnc.edu dfeikes@pnc.edu