In-Service Teacher Education: US

Deborah Hughes HallettDepartment 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/

http://crr.math.arizona.edu/

Some Organizations offering In-Service

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 DesDavid 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:

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:

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?”

Now Equation Has Linear Form:

Variables are y = ln(CPI) and x = Year

bmxy

CPI

1214.2Year0325.0)ln(

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?

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

Now Equation Has Linear Form:

bmxy

VSVK

VM

max0max0

1][

11

Variables are y = 1/V0 and x = 1/[S]0

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