InService Teacher Education US Deborah Hughes Hallett Department

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, Wiley 2011

Calculus: Derivatives From Calculus by Hughes Hallett, Gleason, Mc. Callum, 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 Mc. Callum, Connally, Hughes Hallett, et al

Common Core State Standards: Functions and Interpretation From Algebra by Mc. Callum, 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:

Economists’ View of the CPI Data Converting 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

Biologists’ Use of Algebraic Structure Michaelis-Menten Equation • • • V 0 is initial velocity of chemical reaction [S]0 is initial concentration of substrate Vmax, KM are constants How Do We Know If a Reaction Follows Michaelis-Menten?

Does a Chemical Reaction Follow Michaelis-Menten? Put in linear form, with variables 1/V 0 and 1/[S]0

Now Equation Has Linear Form: Variables are y = 1/V 0 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