Philipp pm slides

The Not-So-Elementary Task of Preparing
Elementary School Teachers to
Understand Elementary Mathematics
Yopp Distinguished Speaker Series
University of North Carolina, Greensboro
September 8, 2014
Dr. Randy Philipp
San Diego State University
RPhilipp@mail.sdsu.edu

Abstract
!If elementary school mathematics is so elementary, then
why do so many people struggle to understand? Using
video of children's mathematical thinking, we will
consider some of the complexities of seemingly simple
topics of whole numbers and fractions, highlighting
some of the issues related to mathematical
understanding, and I will share how a focus on
children’s mathematical thinking motivates prospective
elementary school teachers to more deeply engage with
mathematics. !

Thank You!
UNCG Mathematics Education Group in the
School of Education
The Graduate Students
James D. and Johanna F. Yopp

Presentation Plan
• What does it mean to do mathematics?
• Consider issues related to mathematics
content.
• There is a problem with how PSTs are
learning mathematics, but we found one
approach.
• I’ll share 4 principles of mathematics
teaching and learning addressed by a focus
on children’s mathematical thinking.

But first: A secondary question to keep in
mind today
2014 AMTE Session organized with Rob Wieman
Is Video the New Manipulative in Education?
-Maarten Dolk, Freudenthal Institute

Seeing Fractional Relationships
!
!

Manipulatives are Tools
• With manipulatives, do we assume that the
mathematics is embedded in the representation, for
all to see?
• What are students seeing?
• With video, do we assume the pedagogical
principle is explicit for all to see?
• What are students, teachers, or colleagues seeing?
Ball, D. L. (1992). Magical Hopes: Manipulatives and the Reform of Math Education. American Educator(Summer), 14-18,
46-47.
Thompson, P. W. (1995). Notation, convention, and quantity in elementary mathematics. In J. T. Sowder & B. P. Schappelle
(Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 199-221). New York: State University of
New York Press.
Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in
Mathematics, 47, 175-197.

Define Mathematical Proficiency
• Concepts !
• Procedures !
• Problem Solving!
• Reasoning and Justifying!
• Positive Outlook!

Define Mathematical Proficiency
• Concepts (Conceptual Understanding)!
• Procedures (Procedural Fluency)!
• Problem Solving (Strategic Competence)!
• Reasoning and Justifying (Adaptive Reasoning)!
• Positive Outlook (Productive Disposition)!
National Research Council. (2001). Adding it Up: Helping
Children Learn Mathematics. Washington, D.C.: National
Academy Press.!

The Strands of
Mathematical Proficiency
National Research Council. (2001). Adding it Up: Helping Children
Learn Mathematics. Washington, D.C.: National Academy Press.!

The Strands of
The tendency to see sense in
mathematics, to perceive it as both
useful and worthwhile, to believe that
steady effort in learning mathematics
pays off, and to see oneself as an
effective learner and doer of
mathematics.!
Integrated and functional
grasp of mathematical
ideas!
The capacity to think logically about the
relationships among concepts and
situations, including the ability to justify
one’s reasoning both formally and
informally.!
Knowledge of procedures, knowledge of
when and how to use them appropriately,
and skill in performing them flexibly,
accurately, and efficiently.!
The ability to formulate
mathematical problems, represent
them, and solve them.!

The CCSS Process Standards
1
Make
sense
of
problems
and
persevere
in
solving
them.
2
Reason
abstractly
and
quan=ta=vely.
3
Construct
viable
arguments
and
cri=que
the
reasoning
of
others.
4
Model
with
mathema=cs.
5
Use
appropriate
tools
strategically.
6
AFend
to
precision
(in
language
and
math)
7
Look
for
and
make
use
of
structure.
8
Look
for
and
express
regularity
in
repeated
reasoning.

1
Make
sense
of
problems
and
persevere
in
solving
them.
2
Reason
abstractly
and
quan=ta=vely.
3
Construct
viable
arguments
and
cri=que
the
reasoning
of
others.
4
Model
with
mathema=cs.
5
Use
appropriate
tools
strategically.
6
AFend
to
precision
(in
language
and
math)
7
Look
for
and
make
use
of
structure.
8
Look
for
and
express
regularity
in
repeated
reasoning.

Students Learning Mathematics
Two 5th-Graders
(*This is video clip #9 on IMAP CD.)!
Circle the larger or write =.
4.7 4.70

Which of these strands do your students develop?

Ones Task
Below is the work of Terry, a second grader, who solved this addition problem and
this subtraction problem in May.
Problem A Problem B
1) Does the 1 in each of these problems represent the same amount?
Please explain your answer.
2) Explain why in addition (as in Problem A) the 1 is added to the 5,
but in subtraction (as in Problem B) 10 is added to the 2.

Example of Score of 4
Problem A Problem B
“No, in the addition problem it represents a ten because 9 + 8
= 17, but in the subt. problem it represents 100 or 10 tens.”
2) Explain why in addition (as in Problem A) the 1 is added to the 5,
but in subtraction (as in Problem B) 10 is added to the 2.
“In the addition problem it is a ten from the 17 so we add one
more ten to the 50 + 30. In the subtraction problem it’s not 1
more ten it’s 10 tens so we are taking 3 tens away from 12 tens.”

Example of Score of 0
Problem A Problem B
“In Problem a you adding one. In problem B you are
adding 10.”
2) Explain why in addition (as in Problem A) the 1 is added to the
5, but in subtraction (as in Problem B) 10 is added to the 2.
“Because in A you need to carry the 1 in 17 over. But in B
you are taking the 1 from the 4.”

Principal Investigators
Randy Philipp, PI
Vicki Jacobs, co-PI
Faculty Associates
Lisa Lamb, Jessica Pierson
Research Associate
Bonnie Schappelle
Project Coordinators
Candace Cabral
Graduate Students
John (Zig) Siegfried
Others
Chris Macias-Papierniak,
Courtney White
Funded by the National Science Foundation, ESI-0455785

Prospective
Teachers
Experienced
Teachers
Before
Professional
Development
Experienced
Teachers
w/2 years of
Professional
Development
Experienced
Teachers
w/4 or more
years of
Professional
Development
Ones Task
Group Means (Standard Deviations)
0–4 scale

Group Means (Standard Deviations)
Statistically Significant Differences
Prospective
Teachers
Ones Task
0–4 scale
Experienced
Teachers
Before
Professional
Development
Experienced
Teachers
w/2 years of
Professional
Development
Experienced
Teachers
w/4 or more
years of
Professional
Development
0.31 (0.80) 1.58 (1.26) 2.49 (1.22) 2.72 (1.05)
29 of 35 Prospective
Teachers Scored 0

The Strands of
?
The tendency to see sense in
mathematics, to perceive it as
both useful and worthwhile, to
believe that steady effort in
learning mathematics pays off,
and to see oneself as an
effective learner and doer of
mathematics.!

Example of
Mathematical Disposition
70
- 23
53
76
- 23
53
“Yes, math is like that sometimes.”

Mathematical Integrity
An education isn't how much you have
committed to memory, or even how much you
know. It's being able to differentiate between
what you do know and what you don't.!
!
(Anatole France, Nobel Prize-winning author)!

The Problem
Even when students attend a thoughtfully planned
mathematics course for prospective elementary school
teachers (PSTs), too many go through the course in a
perfunctory manner.
(Ball, 1990; Ma, 1999; Sowder, Philipp, Armstrong,
Schappelle, 1998)
Many PSTs do not seem to value the experience.

Why?
Many PSTs hold a complex system of beliefs
that supports their not valuing the learning of
more mathematics. For example, consider the
effect of believing the following:
“If I, a college student, do not know something,
then children would not be expected to learn it.
And if I do know something, then I certainly
don’t need to learn it again.”

In other words…
We, mathematics instructors of
PSTs, are answering questions that
PSTs have yet to ask.

Should we try to get PSTs to care more
about mathematics for mathematics sake?
This is one starting point.
We took a different starting
point, building on the work
of Nel Noddings (1984).

Why not start with what PSTs care about?
Children!
-Darling-Hammond Sclan, 1996

Circles of Caring
Children
Start
with
that
about
which
prospec=ve
teachers
care.

Circles of Caring
Children
Children's Mathematical
Thinking
When
PSTs
engage
children
in
mathema=cal
problem
solving,
their
circles
of
caring
expand
to
include
children’s
mathema=cal
thinking.

Circles of Caring
Children
Children's Mathematical
Thinking
Mathematics
And
when
PSTs
engage
with
children’s
mathema=cal
thinking,
they
realize
that,
to
be
prepared
to
support
children,
they
must
themselves
grapple
with
the
mathema=cs,
and
their
circles
of
caring
extend.

This
idea
is
at
least
100
years
old.
John Dewey, 1902
Every
subject
has
two
aspects,
“one
for
the
scien=st
as
a
scien=st;
the
other
for
the
teacher
as
a
teacher”
(p.
351).
“[The
teacher]
is
concerned,
not
with
the
subject-‐maFer
as
such,
but
with
the
subject-‐maFer
as
a
related
factor
in
a
total
and
growing
experience
[of
the
child]”
(p.
352).
!
!
Dewey, J. (1902/1990). The child and the curriculum.
Chicago: The University of Chicago Press. !

Children
Mathematics
Looking at mathematics through the lens of
children’s mathematical thinking helps
PSTs come to care about mathematics, not
as mathematicians, but as teachers.

We tested this theory.
Major Result*
When compared to PSTs who did not learn about
children’s mathematical thinking, PSTs who
did learn about children’s mathematical
thinking
a) improved their mathematical content knowledge
and
b)
developed more sophisticated beliefs.
*Philipp, R. A., Ambrose, R., Lamb, L. C., Sowder, J. T., Schappelle, B. P., Sowder, L., et al. (2007). Effects of
early field experiences on the mathematical content knowledge and beliefs of prospective elementary school
teachers: An experimental study. Journal for Research in Mathematics Education, 38(5), 438 - 476.

Where Might We Infuse Children’s
Mathematical Thinking?
• Education Courses
• Mathematics Courses
• Hybrid Courses (CMTE)
– Link Content to Children
– Help PSTs Consider Future (e.g., Ali)

How did the focus on children’s
thinking help the PSTs?

A PST Speaks for Herself
Why study about children’s mathematical thinking while
learning mathematics?
CMTE Clip, 0:23 - 0:56

Four principles of mathematics and mathematics
teaching and learning addressed by focusing upon
children’s mathematical thinking.
Principle 1
The way most students are learning
mathematics in the United States is problematic.
Principle 2
Learning concepts is more powerful and
more generative than learning procedures.
Principle 3
Students’ reasoning is varied and
complex, and generally it is different from adults’
thinking.
Principle 4
Elementary mathematics is not
elementary.

Ally, End of Grade 5!
An average 5th-grade student at a high-performing
local school
Ally, VC #11, 1:35 - 2:30

Ally, End of Grade 5!
An average 5th-grade student at a high-performing
local school
Questions to Consider
How did Ally come to hold these conceptions?
What do teachers need to know to be able to hear
these conceptions in their students and support
them as they expand their reasoning to other
number domains?

NAEP ITEM: 13-year-olds
Estimate the answer to . You will
not have time to solve the problem using
paper and pencil.
€
12
13
+
7
8
a)1 b) 2 c)19 d) 21 e) I don’t know
14%
24%
28%
27%
7%
€
12
13
+
7
8
Is thought of as
12 7
13 8

Principle 1
Principle 2
Learning concepts is more powerful and
more generative than learning procedures.
Principle 3
Students’ reasoning is varied and complex,
and generally it is different from adults’ thinking.
Principle 4
Elementary mathematics is not elementary.

Felisha, End of Grade 2
Felisha learned fraction concepts using equal-sharing tasks in
a small group over 14 sessions (7 days). She had not been
taught any procedures for operating on fractions.
Felisha, VC #15, 1:39

Felisha, End of Grade 2
Questions to Consider
What do teachers need to know to help students
make sense of fractions as Felisha has done?
What do teachers need to know to be able to pose
a question that might help Felisha expand her
thinking?

Consider two problems
• A problem for secondary-school algebra students
• A problem for primary-grade students (K-2)

A Secondary-School Algebra Problem
19 children are taking a mini-bus to the zoo. They
will have to sit either 2 or 3 to a seat. The bus has 7
seats. How many children will have to sit three to a
seat, and how many can sit two to a seat?
x = # of seats with 2 children; y = # of seats with 3 children
x + y = 7 and
2x + 3y = 19
10
8
6
4
(2, 5)
2
- 2 2 4 6 8
- 2
2x+3y=19
x+y=7 2x + 3y = 19 and y = 7 - x
2x + 3(7 - x) = 19
2x + 21 - 3x = 19
-x = -2
x = 2 and y = 5

A Primary-Grade Problem
19 children are taking a mini-bus to the zoo. They
will have to sit either 2 or 3 to a seat. The bus has
7 seats. How many children will have to sit three
to a seat, and how many can sit two to a seat?
51% of kindergarten students in six classes (36/70)
correctly solved this problem in May.
How do you think they did that?
“Throughout the year children solved a variety of different problems. The
teachers generally presented the problems and provided the children with
counters. . . but the teachers typically did not show the children how to
solve a particular problem. Children regularly shared their strategies...
—Carpenter et al., 1993, p. 433. (JRME v.24, #5)

Principle 1
Principle 2
Learning concepts is more powerful and more
generative than learning procedures.
Principle 3
Students’ reasoning is varied and complex,
and generally it is different from adults’ thinking.
Principle 4
Elementary mathematics is not elementary.

Often What We Think We are Teaching
is Not What Students are Learning
After students learn to apply the distributive
property to algebraic expressions, for example,
2(x + y) = 2x + 2y
…students often overgeneralize, for example,
(x + y)2 = x2 + y2

Often What We Think We are Teaching
is Not What Students are Learning

Often What We Find Students are Thinking
is Not What We Have Been Teaching!
At the time of this interview, Javier, in grade 5, had been in the
United States about one year, and he did not speak English
before coming to this country. Javier, VC #6, 0:00 - 1:10

Javier*, Grade 5
How many eggs are in six cartons (of one dozen)?
Javier thinks for 15 seconds, and says, “72.”
I: How did you get that?
Javier writes and says, “5 times 12 is 60, and 12
more is 72.”
I: “How did you know 12 x 5 is 60?”
J: “Because 12 times 10 equals 120. If I take
the [sic] half of 120, that would be 60.”
*At the time of this interview, Javier had been in the United States about one year, and
he did not speak English before coming to this country.

One Representation of Javier’s Thinking
6 x 12
= (5 x 12) + (1 x 12)
= [(1
2
x 10) x 12] + 12
= [1
2
x (10 x 12)] + 12
= [1
2
x (120)] + 12
= 60 + 12
= 72
(Distributive prop. of x over +)
(Substitution property)
(Associative property of x)
Place value
What would a teacher need to know to
understand Javier’s thinking, and where do
teachers learn this?

How do we, as teachers, respond when
faced with a child who has a more
mathematically creative or innovative
solution than we have?

Elliot, Grade 6: 1 ÷ 1/3?*
(*This is video clip #16 on IMAP CD.)
“One third goes into 1 three times because there
is three pieces in one whole.”

Elliot, Grade 6: 1 ÷ 1/3?*
(*This is video clip #16 on IMAP CD.)

Follow-up: 1 1/2 ÷ 1/3?
“There are three one-thirds in 1, and another 1/3 in 1/2, and
there is 1/6 left over.”
Elliot does not realize that the remaining 1/6 needs to
be seen as 1/2 of 1/3.

Elliot, Grade 6
Question for PSTs
Explain Elliot’s reasoning.
What does one need to understand to be in
the position to support Elliot?
Issue
Understanding is seldom black and white. Everybody
understands something about anything, and no one understands
everything about anything.

What knowledge do teachers need to
teach mathematics?

Common Content Knowledge
the mathematical knowledge teachers are responsible for developing
in students
Specialized Content Knowledge
mathematical knowledge that is used in teaching but not directly
taught to students
Pedagogical Content Knowledge
“the ways of representing and formulating the subject that make it
comprehensible to others” —(Shulman, 1986)

Common Content Knowledge
the mathematical knowledge teachers are responsible for developing
in students
Evaluate and understand the meaning of 12 ÷ 3.
Specialized Content Knowledge
mathematical knowledge that is used in teaching but not directly
taught to students
Write a real-life story problem that could be represented
by the expression 12 ÷ 3.
Pedagogical Content Knowledge
“the ways of representing and formulating the subject that make it
comprehensible to others” —(Shulman, 1986)
How might children think about the problem you wrote?

Our Approach
MCK
(CCK and SCK)
PCK
(KCS)

“Those who can, do. Those
who understand, teach.”
Shulman, L. S. (1986). Those who understand:
Knowledge growth in teaching. Educational
Researcher, 15(2), 4–14.

“Those who can, do. Those
who understand, teach.”
Shulman, L. S. (1986). Those who understand:
Knowledge growth in teaching. Educational
Researcher, 15(2), 4–14.
PCK
MCK

Is Video the New Manipulative
in Education?
• With manipulatives, do we assume that the
mathematics is embedded in the
representation, for all to see?
• What are students seeing?
• With video, do we assume the pedagogical
principle is explicit for all to see?
• What are students, teachers, or colleagues
seeing?
!

Philipp pm slides

More Related Content

What's hot

Similar to Philipp pm slides

More from kdtanker

Recently uploaded

Philipp pm slides