Transcript Reconceptualizing Mathematics: Courses for Prospective and Practicing Teachers Susan D. Nickerson Michael Maxon San Diego State University.

Reconceptualizing Mathematics:
Courses for Prospective and
Practicing Teachers
Susan D. Nickerson
Michael Maxon
San Diego State University
Teachers matter.
From a carefully chosen
sample of 2525 adults,
representing a crosssection of
U. S. adults, an
overwhelming majority
agreed that improving the
quality of teaching was the
most important way to
improve public education.

Every Child Mathematically Proficient from the
Learning First Alliance, 1998

Before It’s Too Late: A Report to the Nation from
the National Commission on Mathematics and Science Teaching
for the 21st Century, 2000

What Matters Most: Teaching for America’s
Future and No Dream Denied from the National
Commission on Teaching and American’s Future, 1996 & 2003

The Mathematical Education of Teachers from
the Conference Board of the Mathematical Sciences, 2001

Mathematical Proficiency for All Students from
the RAND Mathematics Study Panel, 2003


Educating Teachers of Science, Mathematics,
and Technology from the National Research Council, 2001
Undergraduate Programs and Courses in the
Mathematical Sciences from MAA
Consider the following:
A teacher wants to pose a question that will show him
whether his students understand how to put a series of
decimals in order from smallest to largest. Which of the
following sets of decimal numbers will help him assess
whether his pupils understand how to order decimals?
Choose each that you think will be useful for his purpose
and explain why.
.123 1.5
.60 2.53
.6
4.25
2
.56
3.14 .45
.565 2.5
Or would each of these work equally well for this purpose?
There is a growing understanding
that the mathematics of the
elementary and middle school is not
trivial, and that teachers need more
preparation and different preparation
than has been common.
Looking ahead
Recommendations
Our focus with examples of content
and instructional possibilities
Guiding Questions
Examples of our PD structures
and delivery
In 1998, the National Research
Council (NRC) appointed a committee
of mathematicians, scientists,
mathematics and science educators
including K-12 teachers, and a
business representative to investigate
ways of improving the preparation of
mathematics and science teachers.
The Committee’s 2001 report,
Educating Teachers of Mathematics,
Science, and Technology, includes
recommendations about the
characteristics teacher education
programs in mathematics and science
should exhibit.
Programs should have the following
features:
• Be collaborative endeavors developed
and conducted by mathematicians,
education faculty, and practicing K-12
teachers;
• Help prospective teachers to know well,
understand deeply, and use effectively the
fundamental content and concepts of the
disciplines that they teach;
• Teach content through the perspectives
and methods of inquiry and problemsolving
Programs should have the following
features:
• Be collaborative endeavors developed
and conducted by mathematicians,
education faculty, and practicing K-12
teachers;
• Help prospective teachers to know well,
understand deeply, and use effectively
the fundamental content and concepts of
the disciplines that they teach;
• Teach content through the perspectives
and methods of inquiry and problemsolving
Collaborative endeavors are a part of our:
• Pre-service courses
• Professional development:
summer courses, usually at the
university (3 days to 2 weeks)
partnership with several districts,
urban & rural
classes through the university
on-line hybrid courses
Integrating Learning
Mathematics with Practice
Director of Mathematics
Site Mathematics
Administrators
Mathematics
Instructors
(SDSU)
Teachers in
program
All Teachers at site
Math Resource Teacher
Math Teacher Ed
Instructors
(SDSU)
Programs should have the following
features:
• Be collaborative endeavors developed
and conducted by mathematicians,
education faculty, and practicing K-12
teachers;
• Help prospective teachers to know well,
understand deeply, and use effectively
the fundamental content and concepts of
the disciplines that they teach;
• Teach content through the perspectives
and methods of inquiry and problemsolving
What content preparation do teachers
need to have to teach middle school
mathematics well?
The 2001 document from CBMS, The
Mathematical Education of Teachers
recommends that for teaching middle
school mathematics: At least 21 semester
hours of mathematics, some of which
should include a study of elementary
mathematics, some of which should focus
on the middle grades, and some of which
include the study of mathematics proper.
Content recommendations in MET
are made in four areas:
1. Number and Operations
2. Algebra and Functions
3. Measurement and
Geometry
4. Probability and Statistics
1. Number and Operations
• Understand and be able to explain
the mathematics that underlies the
procedures used for operating on
whole numbers and rational
numbers.
Example 1: Multiplication of
Fractions
“Juanita had mowed 4/5 of the lawn,
and her brother Jaime had raked 2/3
of the mowed part. What part of the
lawn had been mowed and raked?”
2 4

3 5

2 4
35

8
15
4
5
refers to the entire lawn
8
15
refers to the entire lawn
2
3
refers to the part of the lawn
that has been mwed
o
Example 2: Why do we invert and
multiply when dividing fractions?
Write a story problem that can be represented and solved by
2 1/2 ÷ 1/3
First, divide by the unit fraction: 1 ÷ 1/3
1 ÷ 1/3 is 3, or 3/1
So 2 ÷ 1/3 is twice 3, or 6, or
in general k ÷ 1/n = k n
Continuing in this fashion, find 2 1/2 Ö1/3.
How many thirds are in 2 1/2?
1
2 3
4
5
6
7 1/2
2 1/2 Ö 1/3 = 2 1/2 x 3/1 = 7 1/2
Generalizing,
a  1  a  c  ac
b c b 1 b .
Next, consider 1 Ö2/3 ask:
How m any 2/3 are in 1
whole unit?
There are 1 1/2, or 3/2.
2 Ö 2/3 must have twice as many 2/3 as does 1,
So, 2 x 3/2 is 6/2 or 3
2. Algebra and Functions
• Understand and be able to work
with algebra as a symbolic
language, as a problem solving tool,
as generalized arithmetic, as
generalized quantitative reasoning,
as a study of functions, relations,
and variation, and as a way of
modeling physical situations.
2. Algebra and Functions
• Understand and be able to work
with algebra as a symbolic
language, as a problem solving tool,
as generalized arithmetic, as
generalized quantitative reasoning,
as a study of functions, relations,
and variation, and as a way of
modeling physical situations.
Coping strategies:
1. Just add
2. Guess at the operation to be used
Limited Strategies
3. Look at the number sizes and use those to
tell you which operation to use.
4. Try all operations and choose the most
reasonable answer.
5. Look for “key words.”
6. Decide whether the answer should be
larger or smaller than the given numbers,
then decide on the operation.
Desired strategy:
7. Choose the operations with the meaning
that fits the story. Perhaps draw a picture
to help understand the problem.
Consider this problem:
Dieter A: I lost 1/8 of my weight.
I lost 19 pounds.
Dieter B: I lost 1/6 of my weight, and
now you weigh 2 pounds more
than I do.
How much weight did Dieter B lose?
First, make a list of relevant quantities

Dieter A’s weight before the diet
Dieter A’s weight after the diet
Fraction of weight lost by A
Amount of weight lost by A
Difference in weight of A and B before diet
Difference in weight after diet

Etc.





Now we consider the values of the
quantities







Dieter A’s weight before the diet: ?
Dieter A’s weight after the diet: ?
Fraction of weight lost by A: 1/8
Amount of weight lost by A: 19 pounds
Difference in weight of A and B before diets: ?
Difference in weights after diets: 2 pounds (A is
2 pounds less than B
Etc.
Dieter A’s original weight (before
diet)
Before diet
Shows weight loss of
1/8 of original diet.
Weight lost was 19 lb.
After diet
A’s weight before diet: 19 x 8 = 152
A’s weight after the diet 152–19=133
B’s weight after the diet: 133 + 2 = 135
135 is 5/6 of B’s weight before the diet
so B weighed 162 pounds
B lost 162 – 135 = 27 pounds
2. Algebra and Functions
• Recognize change patterns
associated with linear, quadratic,
and exponential functions.
Example: Growing Dots
3. Measurement and
Geometry
• Identify common two-and three
dimensional shapes and list their
basic characteristics and properties
Example here GSP quadrilaterals
venn
4. Data Analysis, Statistics,
and Probability
• Draw conclusions with
measurements of uncertainty by
applying basic concepts of
probability
Ex: three card poker
Programs should have the following
features:
• Be collaborative endeavors developed
and conducted by mathematicians,
education faculty, and practicing K-12
teachers;
• Help prospective teachers to know well,
understand deeply, and use effectively
the fundamental content and concepts of
the disciplines that they teach;
• Teach content through the perspectives
and methods of inquiry and problemsolving
Cynthia has 1 cup of sugar and each
recipe requires: a) 1/2, b) 1/3, c)2/3,
d) 3/4, e) 4/5 of a cup of sugar. How
many recipes can she make?
Model and draw each case. What
number sentence describes each?
Later generalize for more or less
than one cup of sugar.
In a report by the MAA Committee on the
Undergraduate Preparation in Mathematics
(2004):
“..does not mean the same thing as
preparation for the further study of college
mathematics. For example, while
prospective teachers need knowledge of
algebra..the traditional college algebra
course with primary emphasis in developing
algebra skills does not meet the needs of
elementary [and middle school] teachers.”
Looking ahead
Recommendations
Our focus with examples of content
and instructional possibilities
Guiding Questions
Examples of our PD structures
and delivery
• Pre-service courses
• Professional development:
summer courses, usually at the university
(3 days to 2 weeks)
partnership with districts
classes through the university
on-line hybrid courses
Decisions about what to include rely on
answers to guiding questions
Guiding Questions:
1) What is the content at their grade level?
With what have they had only superficial
exposure? What would be difficult to learn
from the curriculum?
2) How does this content need to be
extended?
3) How does the research literature
characterize the difficulties & student
misconceptions?
4) Does this group of teachers have
particular needs?
1) What is the content at their grade
level? With what have they had only
superficial exposure? What would be
difficult to learn from the curriculum?
Ratio and proportional reasoning
Minimal and sufficient definitions of
quadrilaterals
Procedures can sometimes be learned
from the curriculum. When we talk
about content we are talking about
conceptual understanding and
procedural fluency.
2) How does this content need to be
extended?
3) How does the research literature
characterize the difficulties & K-12
student misconceptions?
Specialized knowledge for teaching
Difficulties in algebra include
•
•
•
•
•
Misunderstanding the equals sign
Comprehending use of literal symbols as
generalized numbers or variables
Expressing relationships in a variety of ways
such as tables, graphs, and equations
Understanding the role of the unit
Ratio and Proportional reasoning
Misunderstanding the equals
sign
Students tend to misunderstand the
equal sign as a signal for “doing
something” rather than a relational
symbol of equivalence.
Comprehending the many uses
of literal symbols




A=L x W
40=5x
2a + 2b = 2(a + b)
Y = 3x + 5
Expressing relationships in a
tables, graphs, and equations
Three Burning Candles
Grou p 1
Grou p 2
Grou p 3
a. Which set of c andle s is represented by the graph?
b. What was the starting height of each candle ?
c. At what rate did each candle b urn? How do you know?
d. What is the slope of each line ?
e. Write an equation t o represent each candle’s burning.
f. Write an equation for a f ourth candle, Candle D,
whose graph is parallel to the graphs of Candle s A, B, a nd C.
Qu i c k T i m e ™ a n d a
T I F F (U n c o m p re s s e d ) d e c o m p re s s o r
a re n e e d e d to s e e th i s p i c t u re .
distance
(meters)
110
100
90
80
Rabbit: 8 m/s over (50 m) and 4 m/s back (50 m).
Average speed is 100 ÷ 18 3/4 = 5 1/3 m/s.
T
T urtle: 6 m/s for 100 m
70
R
60
50
Dashed line gives Rabbit's
average speed.
40
R
30
20
10
2
4
6
8
10
12
14
time (seconds)
16
18
20
Understanding the unit
a. Can you see 3/5 of something in this
picture? Where? Be explicit. (3/5 of what?)
b. Can you see 5/3 of something in this picture? Where?
How did you change the way you looked at the picture
in order to see 5/3?
c. Can you see 2/3 of something in this picture? Where?
d. Can you see 5/3 of 3/5 in this picture? How did you have to
change the way you looked at the picture in order to see
5/3 of 3/5
e. Can you see 1 ÷ 3/5?
Ratio and Proportional
Reasoning
Proportional reasoning has been called a
watershed concept, the “capstone of elementary
arithmetic and the cornerstone of all that is to
follow”
Situations that call for understanding the close
relationship between fractions and ratios, both of
which are represented by the same notation,
cause particular problems.
A donut machine produces 60 donuts every 5
minutes. How many donuts does it produce in
an hour?
a. Identify the rate in this problem. What unit rate is associated
with this rate?
b. Write a proportional statement that could be used to find the
number of donuts produced in an hour.
c. Made a graph of the donuts produced every 5 minutes and a
graph of the donuts produced every minute. Are the graphs
different? Why or why not?
d. What is the slope of each line?
Spanning the curriculum
Students must have an understanding of how
to write a useable mathematical definition
and establish equivalence among definitions.
Students develop a disposition for problem
solving.
Students must learn to use valid reasoning to
reach and justify conclusions.
4) Does this group of teachers have
particular needs?
Looking ahead
Recommendations
Our focus with examples of content
and instructional possibilities
Guiding Questions
Examples of our PD structures
and delivery
Pre-service courses for middle
school math teachers:
Min 32 units; 3 in each content area
5 courses for
elementary:
2 courses for
middle school:
Number & Op (3)
Math for Middle
School Teach (3)
Algebra (2 or 3)
Geo & Meas (3)
Prob & Stats (2 or 3)
Children’s
mathematical
thinking (1.5)
Transition to
Higher Math or
History of Math (3)
Other courses:
Calculus I
Transition to Higher
Math or History of
Math (3)
Statistics (3)
Number theory (3)
Math for Middle School Teachers
Pre-service course
Guided by NCTM Curriculum Focal Points (Gr 6-8)
• Division of Fractions
• Operations with Integers
• Ratio & Proportion
• Functions and relations
Children’s thinking in these same content areas
Professional development courses
• Immediacy. Teachers back in classes tomorrow,
bring questions about their own experiences and
students.
• Collaborative team encompasses school
administrators and educators with a focus on
scaffolding of ideas
• Usually more flexibility to design course particular
to group/team of teachers. Connections tight among
concepts.
Designing courses for professional
development
• Summer courses
• Partner with school district 1 to offer 4 courses to
qualify teachers for NCLB
• Partner with School District 2 to provide PD to 7th
through Alg 1 in pull out days (only 30 hours) each
of 2 years
Designing courses for professional
development
• Partner with School District 3 to provide PD to
middle school (either pull-out or after school)
• Partner with School District 4 to provide PD to 712 (summer and after school)
• On-line planned for middle school
• Teach content so they know well,
understand deeply, and use effectively the
fundamental content they teach
• Teach content through the perspectives
and methods of inquiry and problemsolving
It’s not just what we teach teachers,
but how we teach teachers.
Discussion
That students do not regard algebra as a
sense-making and useful subject is often
due to the way that algebra is often
taught and the way that students are
prepared for algebra.
Middle school teachers themselves must
make sense of algebra and its
underpinnings.
“Algebra has been experienced as an
unpleasant, even alienating event,
mostly about manipulating symbols that
do not stand for anything. (But) algebraic
reasoning in its many forms, and the use
of algebraic representations such as
graphs, tables, spreadsheets and
traditional formulas, are among the most
powerful intellectual tools that our
civilization has developed.” (Kaput 1999)