Transcript Developing the Mathematical Knowledge Necessary for

Developing the Mathematical
Knowledge Necessary for Teaching
in Content Courses for Elementary
Teachers
David Feikes
Purdue University North Central
AMTE Conference
January 27, 2006
Copyright © 2005 Purdue University North Central
Connecting Mathematics for Elementary Teachers
(CMET)
NSF CCLI Grants: DUE-0341217 & DUE-0126882

Focus on How Children Learn Mathematics

Mathematical Content Courses for Elementary
Teachers
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Problem Solving
Sets
Whole Numbers
Number Theory
Integers
Rational Numbers – Fractions
Decimals, Percents, and Real Numbers
Geometry
More Geometry
Measurement
Statistics/Data Analysis
Probability
Algebraic Reasoning
CMET Supplement
CMET includes descriptions, written for prospective elementary teachers, on how
children think about, misunderstand, and come to understand mathematics.
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
For example, we discuss how linking cubes may embody the concept of ten in
understanding place value. At a more sophisticated level of mathematical thinking
Base Ten Blocks (Dienes Blocks) may be a better embodiment of the standard
algorithms for addition and subtraction.
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

CMET is not a methods textbook.

CMET is not a mathematics textbook.

CMET is a supplement for mathematical
content courses for elementary teachers.
Mathematical knowledge necessary for teaching is
fundamentally different, but very much related to:
1)
Mathematics Content Knowledge-- a textbook
understanding of mathematics.
2)
Pedagogical Content Knowledge-- how to teach
mathematics.
Mathematical knowledge necessary for teaching
can be developed through knowledge of children’s
mathematical thinking.
Survey Items:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Children learn mathematics through an understanding of
set theory.
The concept of 1-1 correspondence is an important
concept in the development of children’s ability to count.
Number is an easy concept to teach young children.
If children can count, they understand the concept of
number.
Initially addition is a counting activity for children.
The concept of ten is the basis for place value.
For children first learning multiplication, multiplication is
repeated addition.
Understanding multiplication will be easier for children
who have developed the concept of ten.
Children in second and even some first graders can
solve division problems by partitioning or using a
repeated operation.
In upper primary grades children often think of division
as the opposite of multiplication to divide.
11. Children initially do not see that addition is
commutative because addition is a counting activity
and they are counting differently depending on the
order of the numbers.
12. Initially, some children understand negative numbers
as a quantity representing a deficit.
13. Initially, some children understand negative numbers
as a location on a number line.
14. Sometimes children understand negatives in one
context and not the other.
15. Children without an understanding of negative
numbers frequently indicate that -7 > -3
16. The concept of repeated addition can be used to
explain the multiplication of a positive number times a
negative number.
17. Patterns are one way to illustrate a negative number
times a negative number e.g., -2 x 1 = -2, -2 x 0 = 0
therefore –2 x –1 = 2 and so on.
10.
Symbolic notation of fractions should be introduced
immediately with young children.
19. Teachers should always predivide shapes when
children are pictorially representing fractions.
20. In a fraction, the numerator and denominator are
related multiplicatively.
21. In a fraction the numerator and denominator are
related additively.
22. Some children believe 6/7 = 8/9.
23. Children are likely to cross multiply to solve ratio and
proportion problems.
24. Point, line, and plane are undefined terms to children.
25. Some children believe that perpendicular lines must be
horizontal and vertical.
26. Many children believe that parallel lines that are not
lined up are not parallel.
27. Mental imagery is essential to learning geometry.
28. Children use their concept images to determine if a
given shape is a triangle.
29. Learning area is about learning formulas.
18.
Conclusions:

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.

We believe preservice teachers will be more motivated
to learn mathematics as they see the context of
applicability (Bransford et al. 1999). 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.

Using knowledge of how children learn and think about
mathematics will also improve preservice teachers’
future teaching of mathematics to children.