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.

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 MathematicsFocus on How Children Learn Mathematics

Mathematical Content Courses for Mathematical Content Courses for Elementary TeachersElementary Teachers

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

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)

Problem cont.

Underlying beliefs about mathematics

Beliefs about mathematics determine future teaching beliefs about mathematics resistant to change

(Hill, Rowan, Ball, 2005)

Theoretical Perspective

Mathematical Knowledge Necessary for Teaching

Children’s Thinking Approach Self-Efficacy

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)

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).

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

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.

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)

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

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.

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.

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.

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.

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.

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.

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.

Phase II: Quantitative Results

Knowledge of Children’s Thinking Beliefs Efficacy

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

Knowledge of Children’s Thinking Results

Course 1: 4.1/5.0

Course 2: 3.8/5.0 n=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.

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)

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).

Beliefs Results

MANOVA test results indicates a significant move toward a more non-procedural view of math for both courses.

2.8 3.3 3.6Procedural Non-Procedural

Course 1-

pre

Course 1

post

Course 2-

post

Efficacy –Connection to the Classroom

Self-Reported Data collected at the end of two courses

Efficacy for two areas: Understanding Math and Teaching Math

Efficacy Results

T-test results indicate significance differences for both areas (p< .01)

Course 1: n=30 Course 2: n=24

Strongly disagree 3.2 4.03.1 4.2 Strongly agree

Understanding Math

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

3.72.6 4.0 Strongly agree

Teaching Math

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 

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

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

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

Learn About Children’s own AlgorithmsFiona 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

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.

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

Thank You!