Teachers’ Specialized Content
Knowledge: Preparing Future
Elementary School Teachers to
Teach Mathematics
Elementary Pre-Service Teachers
Mathematics Project
Suzanne Chapin
schapin@bu.edu
Boston University
2
Why Focus on Teachers’ Knowledge?
• Pre-service and in-service elementary teachers do not
have a deep understanding of the mathematics they
teach (van Es & Conroy, 2009; Philipp et al., 2007; NRC,
2001; Ball, 1990)
• There is widespread agreement that teachers’
mathematical knowledge has a profound effect on
instruction and student achievement (NMAP, 2008;
NCTQ, 2008; Hill et al., 2005)
▫ To be effective, mathematics teachers must know the content they
will teach in a deep and connected way (Ball, Thames, & Phelps,
2008; Ma, 1999)
3
What Knowledge Should Teachers
Have?
• Profound Understanding of Fundamental
Mathematics (Ma, 1999):
▫ Knowledge that is deep, broad, and thorough
▫ Consists of knowledge packages
• Mathematical Knowledge for Teaching (Ball, Thames, &
Phelps, 2008):
▫ Knowledge of mathematics that is specific to the task
of teaching
▫ Includes several types of content knowledge (common,
specialized, and pedagogical)
Mathematical Knowledge for Teaching
(MKT)
Domains of Mathematical Knowledge for Teaching (Ball, Thames, & Phelps, 2007, p. 42).
Teachers with mathematical knowledge
for teaching can …
 Explain terms and concepts to students
 Interpret students’ statements and solutions
 Judge and correct textbook treatments of
particular topics
 Use representations accurately
 Determine sequences of instruction
 Ask questions that assist students in organizing
knowledge around important ideas
 Provide examples of mathematical concepts,
algorithms, or proofs
What else do we know about
mathematical knowledge for teaching?
• Linked to student achievement
• Can be improved through
experiences and instruction
• More than simply
▫ General mathematical knowledge
▫ General intelligence
▫ General pedagogical knowledge
What are some of the challenges
mathematics teacher educators face?
•
Students’ prior experiences and
beliefs
•
Mathematical skills
•
Functional fixedness
•
Decompression (inability to
unpack steps in a procedure)
Additional Challenges
• MKT can be hard for instructors (including
mathematicians) to get a handle on and to teach
• Curriculum is not transparent
• Enacting tasks to focus on MKT can be challenging
• Staying focused on the mathematics and not on how to teach the
math
• Unpacking the mathematics sufficiently and convincingly to help preservice teachers see what there is to learn and do
• Making visible the connections to the kinds of mathematical thinking,
judgment, reasoning one has to do in teaching
• Keeping the problems focused on MKT and not just “M”
Elementary Teachers Pre-Service
Mathematics Project
 Suzanne Chapin
 Ziv Feldman
 Johanna Bunn
 Nancy Anderson
 Matthew Chedister
 Alejandra Salinas
 Dicky Ng
 Jessica Shumway

Diana Cheng
Elementary Pre-Service Teachers
Mathematics Project (EMP)
• Overarching Goals:
▫ Improve prospective teachers’ knowledge of the mathematics
needed for teaching.
▫ Develop course materials for prospective teachers and
instructors that especially focus on specialized content
knowledge (SCK) and the processes of generalization,
justification, and communication.
Elementary Pre-Service Teachers
Mathematics Project
Year 1
Year 2
• Created instructional materials for
▫ Number theory
▫ Fractions
▫ Ratio and proportion
▫ Geometry
▫ Geometric measurement
• Field testing at Boston University,
Middle Tennessee State
University, and Utah State
University
• Piloted drafts of 28 tasks at 5
institutions in MA
• Revising student materials
• Refining support materials for
instructors
• Researching the efficacy of the
tasks and the role of discourse in
explaining and justifying
Snapshot of EMP Materials
• For use in mathematics content courses for elementary
pre-service teachers
• Replacement units consisting of instructional tasks
• 60 to 80-minute sessions
• Specified order for the sessions
• Instructor’s Guide for each session
EMP Design Principles
Design Features of Tasks
▫ Subsets of problems grouped around key ideas
▫ Multiple representations used to support reasoning
▫ Scaffolded questions posed to help PSTs identify key ideas
▫ Whole class discussion questions focused on explaining and justifying claims
▫ Use of classroom artifacts to simulate the demands of teaching
Implementation Features of Tasks
▫ Recurring cycles — PSTs work on subsets of problems, discuss these in small
groups, then discuss overarching questions as a class
▫ Emphasis during whole class discussion is on making claims, providing evidence,
and justifying those claims
Implementation Cycles
Adapted from Simon, 1994
Divisibility Task
Overview
Exploration of why the divisibility rules for 2, 4, 5, and 10
work.
Cycle 1
Explain rules for divisibility by 2, 5 and 10.
Cycle 2
Explain rule for divisibility by 4.
Cycle 3
Generalize and extend to other numbers (e.g., 8, 25) and
number theory concepts.
Summarize and Connect
Divisibility by 5 (Cycle 1)
a)
What is the rule for divisibility by 5?
b)
Consider the number 3,267, which can be written as 3,000 + 200 + 60 + 7. Is
3,000 divisible by 5? Is 200 divisible by 5? Is 60 divisible by 5? Is 7 divisible by 5?
c)
Fred has no idea if 3,000 is divisible by 5, but he knows that 3,000 = 3 x 1000. As
a teacher, how might you use this information to help him?
d)
Maria says she has another way to see if a number is divisible by 5. Here are some
examples of her strategy:
•
75 7 and 5. Seven is not divisible by 5 even though 5 is. So, 75 is not divisible by
5.
120  12 and 0. 12 is not divisible by 5 even though 0 is. So, 120 is not divisible
by 5.
255  25 and 5. 25 is divisible by 5 and so is 5. So, 255 is divisible by 5.
•
•
What is wrong with Maria’s reasoning? How can her strategy be amended so that
it would work?
Cycle 1 Continued ….
• Carol stated: “I can use what I know about
divisibility by 5 to explain why the rule for
divisibility by 10 works. I considered each of the
place values separately: thousands, hundreds,
tens, and ones.” Johan said, “I think about
remainders when dividing by 10.” What does
each student mean? Explain in your own words
why you only have to examine the ones digit
when deciding if a number is divisible by 10.
Group Discussion Question (Cycle 1)
Explain why looking at the digits in the ones place works
to check for divisibility by 2, 5 and 10 but not for
divisibility by other numbers such as 3.
PST’s ideas
▫ Ten is divisible by 2, 5 and 10.
▫ Any whole number can be thought of as a multiple of ten
plus some amount of units.
▫ Any multiple of 10 is divisible by factors of 10.
▫ Ten is not divisible by 3 and powers of 10 are not divisible
by 3, so we can’t simply examine the ones digit to see if it is
divisible by 3.
Divisibility by 4 (Cycle 2)
Group Discussion Questions (Cycle 2)
• Why might a teacher want students to construct numbers using
base-10 blocks when learning about divisibility by 4?
• How is the explanation for the divisibility rule for 4 similar to and
different from the explanations for the divisibility rules for 2, 5, and
10?
Generalizing and Extending (Cycle 3)
So far you've looked at divisibility rules in which the
divisors are factors of 10 (e.g., 2, 5, and 10) or factors of
100 (e.g., 2, 5, 10 and 4).
a)
Use what you have learned so far to explain why you examine
the last three digits of a number to see if it is divisible by 8 to
see if the original number is divisible by 8 (e.g., 23,816 is
divisible by 8).
b)
Make up a divisibility rule for 25. Explain why your rule
works.
Generalizing and Extending (Cycle 3)
• An important idea in mathematics is as follows: If n is a
factor of a number, a, and n is a factor of a number, b,
then n is a factor of their sum, (a + b). Explain the
connections you see between this idea and the divisibility
rules for 2, 4, 5, and 10.
Preliminary Testing of Efficacy of the Materials
• Does the use of the EMP instructional tasks result in
student achievement gains?
• Is there a difference in mathematics achievement
between the intervention group using the EMP tasks and
a comparison group of students?
Pre- and Post-Test Results – Geometric
Measurement
Sample: 54 PSTs at BU.
Mean number of points gained was 10.1.
Independent Samples t-Test conducted; df=53, p<0.001
Intervention vs Control at BU
Boston University Subjects
N=56
Table 1
Adjusted Means from Analyses of Covariance: The Elementary and Middle School
Number and Operations Content Knowledge Instrument Forms A and B
Adj. Posttest Mean
Obtained Post Test Mean
Comparison Treatment
F
Comparison Treatment
(n=19)
(n=37)
(n=19)
(n=37)
____________________________________________________________
Total Scores
***p<.001
12.44
15.40
14.08***
12.47
15.38
How EMP Tasks Support the Development of
PSTs’ Mathematical Knowledge
• Questions are scaffolded in each implementation cycle
• Implementation cycles build on prior cycles
• Cycles work together to help PSTs identify key mathematical ideas
that comprise robust justifications for the focus of the task (e.g.,
Why do these divisibility rules work?)
How EMP Tasks Connect to the
Mathematical Demands of Teaching
PSTs are asked to:
• Give or evaluate mathematical explanations and justifications
• Analyze students’ thinking and errors
• Choose and use representations
• Analyze textbook examples in terms of the mathematical demands
Ball, Thames, Phelps (2008)
Questions?