In-Service Teacher Education: US
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In-Service Teacher Education: US
Deborah Hughes HallettDepartment of Mathematics, University of Arizona
Harvard Kennedy School
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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
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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
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http://www.teachersasscholars.org/
http://crr.math.arizona.edu/
Some Organizations offering In-Service
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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
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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
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Types of Problems and Activities Teachers Have Found Useful,
and Why
First, what is the mindset of the students?
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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
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Calculus
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Rates and Concavity
From Precalculus bt Connally, et al, Wiley2011
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Calculus: Derivatives
From Calculus by Hughes Hallett, Gleason, McCallum, et al, Wiley 2013.
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Application to Medicine: Exponential Decay & Separable DesDavid Sloane MD (Harvard Medical School)
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Distance-Velocity Functions: Integrals-Derivatives
From TAS Seminar, Deb Hughes Hallett
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Scatter Plots(middle school)
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Famous Personalities
SCATTER PLOT•Estimated age
•Actual age
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Guess how old?Taylor Lautner
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How Old?• Miley Cyrus
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How old?• Barrack Obama
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How old?• Stefani Germanotta• a.k.a Lady Gaga
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Actual Ages…
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Graphs and Algebra(middle and high school)
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Basic: Functions and Graphs
From Functions Modeling Change by Connally, Hughes Hallett, et al
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Common Core State Standards:
Functions and Interpretation
From Algebra by McCallum, Connally, Hughes Hallett, et al
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Common Core State Standards:
Functions and Interpretation
From Algebra by McCallum, Connally, Hughes Hallett, et al
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Structure from the Viewpoint of Other Disciplines: Economics
and Biology
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Economists and Algebraic Structure:
y = 8.3132e0.0326x
R² = 0.9185
0
50
100
150
200
250
0 20 40 60 80 100
CPI
Years since 1913
CPI, with 1982-84 def ined to be 100
Consumer Price Index (CPI) Data is as follows:
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y = 0.0325t + 2.1214
0123456
0 50 100
y, ln
(CPI
)
t, years since 1913
Economists’ View of the CPI DataConverting to linear form enables an answer the question:“How fast has the CPI grown over last century?”
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Now Equation Has Linear Form:
Variables are y = ln(CPI) and x = Year
bmxy
CPI
1214.2Year0325.0)ln(
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Biologists’ Use of Algebraic Structure
• V0 is initial velocity of chemical reaction• [S]0 is initial concentration of substrate• Vmax, KM are constants
0
0max0 ][
][SKSV
Vm
Michaelis-Menten Equation
How Do We Know If a Reaction Follows Michaelis-Menten?
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Does a Chemical Reaction Follow Michaelis-Menten?
Put in linear form, with variables 1/V0 and 1/[S]0
max0max0
0max
0
0
1][
11
][][1
VSVK
V
SVSK
V
M
M
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Now Equation Has Linear Form:
bmxy
VSVK
VM
max0max0
1][
11
Variables are y = 1/V0 and x = 1/[S]0
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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
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Functions and Graphs: Interpretation
From Functions Modeling Change by Connally, Hughes Hallett, et al