Using a Children’s Thinking Using a Children’s Thinking Approach to Change Prospective Approach to Change Prospective

Elementary Teachers’ Efficacy and Elementary Teachers’ Efficacy and Beliefs of MathematicsBeliefs of Mathematics

AERA Paper SessionAERA Paper SessionSpring 2007, ChicagoSpring 2007, Chicago

Sarah Hough, Ph.D.

David Pratt, Ph.D.

David Feikes, Ph.D.

Keith Schwingendorf, Ph.D.

OutlineOutline

Defining the ProblemDefining the Problem Context of StudyContext of Study Samples from SupplementSamples from Supplement Literature ReviewLiterature Review ResultsResults DiscussionDiscussion Future Studies and New Future Studies and New

Possible GrantsPossible Grants

Breaking the Traditional Beliefs about Breaking the Traditional Beliefs about MathMath

Many preservice teachers Many preservice teachers come from traditional come from traditional mathematics classrooms mathematics classrooms where procedural knowledge where procedural knowledge is revered over conceptual is revered over conceptual understanding and understanding and investigation (Ma, 1999)investigation (Ma, 1999)

Math Teachers Need to KnowMath Teachers Need to Know

There appears to be a There appears to be a disconnect between math disconnect between math needed to know in needed to know in elementary school and that elementary school and that which is taught in math which is taught in math content courses in college.content courses in college.

Nathan & Koedinger (2000).Nathan & Koedinger (2000).

Beliefs and Knowledge about MathBeliefs and Knowledge about Math

Beliefs about mathematics determine future teaching Beliefs about mathematics determine future teaching beliefs about mathematics and are resistant to beliefs about mathematics and are resistant to changechange

Many elementary school teachers have limited Many elementary school teachers have limited understanding of the subject matter they are understanding of the subject matter they are responsible for teaching.responsible for teaching.

(Ball, 1990; Ma, 1999; Mewborn, 2000)(Ball, 1990; Ma, 1999; Mewborn, 2000)

A Special Type of Math Content Course A Special Type of Math Content Course is Neededis 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).National Research Council (2001).

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 TeachersMathematical Content Course Supplement for Elementary Teachers

Focus on How Children Learn MathematicsFocus on How Children Learn Mathematics

Context Supporting Change in BeliefsContext Supporting Change in Beliefs

The CMET supplement contains:The CMET supplement contains:

problems and data from the National Assessment of problems and data from the National Assessment of Educational Progress (NAEP)Educational Progress (NAEP)

our own data from problems given to elementary our own data from problems given to elementary school children school children

questions for discussionquestions for discussion

- Supplement to existing textbook- Supplement to existing textbook- Aligned to content typically taught- Aligned to content typically taught

Context Supporting Change in BeliefsContext Supporting Change in Beliefs

These descriptions are based on current research and include: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)

Qualitative Results from Phase 1 of StudyQualitative Results from Phase 1 of Study

What is the Focus of the CMET materials?What is the Focus of the CMET materials?

Understanding Mathematics ConceptuallyUnderstanding Mathematics Conceptually Multiple Solutions to ProblemsMultiple Solutions to Problems Less Rule OrientedLess Rule Oriented Focus on Process, not final answerFocus on Process, not final answer Children’s Thinking ImportantChildren’s Thinking Important Problem Solving EmphasizedProblem Solving Emphasized Discovering PatternsDiscovering Patterns

Math as a ProcessMath as a Process

“ “The math is the same in this course The math is the same in this course as in different courses because it is as in different courses because it is dealing with numbers. However the dealing with numbers. However the class is different because class is different because the final the final answer is not he most important partanswer is not he most important part of the problem. By far the greatest of the problem. By far the greatest thing to learn is the understanding of thing to learn is the understanding of how the solution was reached.” how the solution was reached.”

Children’s Thinking ImportantChildren’s Thinking Important

The math isn't high school or college level The math isn't high school or college level math. The math in this course requires math. The math in this course requires more thinking and attempting to more thinking and attempting to understand how understand how children understandchildren understand the the concepts.concepts.

Multiple SolutionsMultiple Solutions

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

Literature ReviewLiterature Review

Mathematical Knowledge Necessary for Mathematical Knowledge Necessary for TeachingTeaching

Children’s Thinking ApproachChildren’s Thinking Approach Self-Efficacy for Understanding and Teaching Self-Efficacy for Understanding and Teaching

MathMath

MKNTMKNT

- 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)

Children’s Thinking ApproachChildren’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).

EfficacyEfficacy

Bandura: “people’s beliefs about their Bandura: “people’s beliefs about their capabilities to produce designated levels capabilities to produce designated levels of performance and exercise control over of performance and exercise control over events that affect their lives”events that affect their lives”

Efficacy has been shown to lead to:Efficacy has been shown to lead to: Greater levels of planningGreater levels of planning Increased enthusiasmIncreased enthusiasm Being more committed to the professionBeing more committed to the profession

Efficacy and InstructionEfficacy and Instruction

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

Efficacy and Teaching Strategies in Efficacy and Teaching Strategies in Mathematics InstructionMathematics Instruction

Using manipulatives was strongly embraced by Using manipulatives was strongly embraced by the preservice teachers with the highest degree the preservice teachers with the highest degree of mathematics teacher efficacyof mathematics teacher efficacy

Similarly, teachers with the lowest efficacy for Similarly, teachers with the lowest efficacy for teaching expressed concern over using teaching expressed concern over using manipulatives as a teaching aid.manipulatives as a teaching aid.

Swars (2005)Swars (2005)

Results from Previous StudiesResults from Previous Studies

Demonstrated a possible connection Demonstrated a possible connection between efficacy and beliefs with pre-post between efficacy and beliefs with pre-post test responses. test responses.

After using the module, participants shifted After using the module, participants shifted to more positive efficacy beliefs and a to more positive efficacy beliefs and a more non-procedural view of math.more non-procedural view of math.

Pratt et. al (2006).Pratt et. al (2006).

Results from Previous StudyResults from Previous Study

SummarySummary

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.

MethodologyMethodology

Large midwest universityLarge midwest university Control and treatment groupsControl and treatment groups (n=138)(n=138) Beliefs about Math QuestionairreBeliefs about Math Questionairre Efficacy Beliefs Questions (Likert Scale)Efficacy Beliefs Questions (Likert Scale) IMAP surveyIMAP survey Correlations and ANOVA’sCorrelations and ANOVA’s

Math Beliefs InstrumentMath 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.

Results from StudyResults 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.

5.8

6

6.2

6.4

6.6

6.8

7

7.2

7.4

MATH Efficacy Teach Efficacy

Control

CMET

TEACHING Math BeliefsTEACHING Math Beliefs

Encouraging children to explore their own ways Encouraging children to explore their own ways of solving problems is as important as teaching of solving problems is as important as teaching them to follow procedures.them to follow procedures.

Rather than teaching mathematical ideas directly Rather than teaching mathematical ideas directly to students it is better to guide them in figuring to students it is better to guide them in figuring things out for themselves.   things out for themselves.   

Mathematical skills should be taught before Mathematical skills should be taught before concepts.   concepts.   

Beliefs about MathBeliefs about Math

Mathematics is a web of interrelated concepts Mathematics is a web of interrelated concepts and procedures. and procedures.

Mathematics is mainly about learning rules Mathematics is mainly about learning rules and formulas.   and formulas.   

Frequently when doing mathematics you are Frequently when doing mathematics you are discovering patterns and making discovering patterns and making generalizations. generalizations.

In mathematics there is one correct answer.In mathematics there is one correct answer.      

Beliefs about Children’s ThinkingBeliefs about Children’s Thinking

Children should master the basic facts before Children should master the basic facts before doing problem solving.doing problem solving.

Children solve and think about mathematics in Children solve and think about mathematics in a variety of ways and often different from a variety of ways and often different from adults. adults.

When children do not understand something in When children do not understand something in mathematics it is because they have not had mathematics it is because they have not had enough practice. enough practice.

Children learn mathematics better through Children learn mathematics better through extensive drill and practice rather than inquiry extensive drill and practice rather than inquiry and exploration.and exploration.

Difference in Beliefs ScoreDifference 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).

Correlations Between Teaching Correlations Between Teaching Efficacy and BeliefsEfficacy and Beliefs

Children solve and think about Children solve and think about mathematics in a variety of ways mathematics in a variety of ways and often different from adults.and often different from adults.

Children learn mathematics Children learn mathematics better through extensive drill and better through extensive drill and practice rather than inquiry and practice rather than inquiry and exploration.exploration.

Rather than teaching Rather than teaching mathematical ideas directly to mathematical ideas directly to students it is better to guide them students it is better to guide them in figuring things out for in figuring things out for themselves.themselves.

Correlations significant at p=.01Correlations significant at p=.01

Mathematical skills should be taught before concepts.Mathematical skills should be taught before concepts.

Rather than teaching mathematical Rather than teaching mathematical ideas directly to students it is better ideas directly to students it is better to guide them in figuring things out to guide them in figuring things out for themselves.for themselves.

MATH Efficacy TEACH Efficacy

DiscussionDiscussion 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 Using knowledge of how children learn and think about

mathematics may also improve preservice teachers’ future mathematics may also improve preservice teachers’ future teaching of mathematics to children.teaching of mathematics to children.

Current and Future ResearchCurrent and Future Research

Pre-Post Test of Control and Pre-Post Test of Control and Experimental groupsExperimental groups

Complete Analysis of IMAP online Complete Analysis of IMAP online surveys. surveys.

Secure funding for continued Secure funding for continued implementation of published materials implementation of published materials

Future Grant PossibilitiesFuture Grant Possibilities

How Children Learn Math (HCLM): How Children Learn Math (HCLM): Resource for TeachersResource for Teachers

How Children Learn Math (HCLM): How Children Learn Math (HCLM): Resource for ParentsResource for Parents

Both grants support teachers and caregivers Both grants support teachers and caregivers with resources that help them understand with resources that help them understand the way children think about math and how the way children think about math and how to teach math more conceptually.to teach math more conceptually.

Contact Us for More Info!Contact Us for More Info!

If you are interested in If you are interested in using the CMET materials using the CMET materials in your classroom or in your classroom or assisting us with potential assisting us with potential future grants, please future grants, please contact us!contact us!

dpratt@pnc.edudpratt@pnc.edudfeikes@pnc.edudfeikes@pnc.edu