In-Service Teacher Education: US
Deborah Hughes Hallett
Department of Mathematics, University of Arizona
Harvard Kennedy School
The US Context
• Teachers are (too) busy keeping up with
everyday requirements (often non-academic)
• Curriculum is often not the teachers’ own; they
often work largely from a prescribed text
• During the school year, in-service sessions that
are immediately useful to teachers have the
most impact
What Works Best?
• Providing copies of materials that teachers can easily
copy or adapt
• Illustrating pedagogical techniques that can be used
in a classroom right away (don’t need equipment,
further training, etc)
• Exhibiting support for policies that teachers want to
persuade administrators or parents
• Answers to “landscape questions”: How topics fit
together, where mathematics is applied
http://www.teachersasscholars.org/
Some Organizations
offering In-Service
http://crr.math.arizona.edu/
Organizations
• Organizations may be private NGOs (Teachers
as Scholars) or
– Paid for by school districts or grants
• Based at a university (CRR and IM&E)
– Offer in-service teacher education to schools that
want it or pay for it; not mandated
• States and school districts may also have their
own teacher education staff
– May offer mandatory education sessions conducted
either by their own staff or by outsides
Scheduling
• During school year:
• One or two day sessions, or
• 2 hours a night once a week for several weeks
• Often not paid but usually give PDPs
(professional development points, )
• During summers:
• One day to one month, more hours per day
• Often give college credit or count toward
master’s degree
Types of Problems and Activities
Teachers Have Found Useful,
and Why
First, what is the mindset of the
students?
Student Beliefs About Mathematics:
Harvard Undergraduates
(about 1990)
Answer choices for each question:
Disagree
1
2
3
4
5
Agree
• A well-written problem makes it clear what method to use
to solve it
– Calculus students: 4.1, precalculus: 4.6
• If you can’t do a homework problem, you should be able to
find a worked example in the text that will show you how
– Calculus students: 4.1, precalculus: 4.7
• Review problems should have the section of the text they
come from listed after them in parentheses
– Calculus students: 4.2, precalculus: 4.8
Calculus
Rates and
Concavity
From Precalculus bt
Connally, et al,
Wiley2011
Calculus:
Derivatives
From Calculus by Hughes Hallett, Gleason, McCallum, et al,
Wiley 2013.
Application to Medicine: Exponential Decay & Separable Des
David Sloane MD (Harvard Medical School)
Distance-Velocity Functions: Integrals-Derivatives
From TAS Seminar, Deb Hughes Hallett
Scatter Plots
(middle school)
Famous Personalities
SCATTER PLOT
•Estimated age
•Actual age
Guess how old?
Taylor Lautner
How Old?
• Miley Cyrus
How old?
• Barrack Obama
How old?
• Stefani Germanotta
• a.k.a Lady Gaga
Actual Ages…
Graphs and Algebra
(middle and high school)
Basic: Functions and Graphs
From Functions Modeling
Change by Connally,
Hughes Hallett, et al
Common Core
State Standards:
Functions and
Interpretation
From Algebra by
McCallum, Connally,
Hughes Hallett, et al
Common Core
State Standards:
Functions and
Interpretation
From Algebra by
McCallum, Connally,
Hughes Hallett, et al
Structure from the Viewpoint
of Other Disciplines:
Economics and Biology
Economists and Algebraic Structure:
Consumer Price Index (CPI) Data is as follows:
CPI, with 1982-84 def ined to be 100
250
y = 8.3132e0.0326x
R² = 0.9185
CPI
200
150
100
50
0
0
20
40
60
Years since 1913
80
100
Economists’ View of the CPI Data
y, ln(CPI)
Converting to linear form enables an answer the question:
“How fast has the CPI grown over last century?”
6
5
4
3
2
1
0
y = 0.0325t + 2.1214
0
50
t, years since 1913
100
Now Equation Has Linear Form:
Variables are y = ln(CPI) and x = Year
ln(CPI)  0.0325 Year  2.1214
y  mx  b
Biologists’ Use of Algebraic Structure
Michaelis-Menten Equation
• V0 is initial velocity of chemical reaction
• [S]0 is initial concentration of substrate
• Vmax, KM are constants
Vmax [ S ]0
V0 
K m  [ S ]0
How Do We Know If a Reaction Follows Michaelis-Menten?
Does a Chemical Reaction Follow
Michaelis-Menten?
Put in linear form, with variables 1/V0 and 1/[S]0
K M  [ S ]0
1

V0
Vmax [ S ]0
KM
1
1
1



V0 V max [ S ]0 V max
Now Equation Has Linear Form:
Variables are y = 1/V0 and x = 1/[S]0
KM
1
1
1



V0 V max [ S ]0 V max
y  mx  b
What Should Calculus Preparation Look Like?
• Algebra: Develop insight into the structure of
expressions; achieve fluency through reasoning
• Functions and Graphs: Qualitative behavior,
parameters, families
• Modeling: Create a model; interpret results in
context
• Stamina and Strategy: Make repeated attempts, try
multiple approaches. Choose the best tools, graphs,
algebra, technology
Functions and Graphs: Interpretation
From Functions
Modeling Change
by Connally,
Hughes Hallett,
et al