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In-Service Teacher Education: US Deborah Hughes Hallett Department 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)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 textCurriculum 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 impactDuring 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 adaptProviding 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)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 parentsExhibiting support for policies that teachers want to persuade administrators or parents Answers to “landscape questions”: How topics fit together, where mathematics is appliedAnswers 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) orOrganizations may be private NGOs (Teachers as Scholars) or –Paid for by school districts or grants Based at a university (CRR and IM&E)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 staffStates 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:During school year: One or two day sessions, orOne or two day sessions, or 2 hours a night once a week for several weeks2 hours a night once a week for several weeks Often not paid but usually give PDPs (professional development points, )Often not paid but usually give PDPs (professional development points, ) During summers:During summers: One day to one month, more hours per dayOne day to one month, more hours per day Often give college credit or count toward master’s degreeOften 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: Disagree12345Agree 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 Des David 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: Consumer Price Index (CPI) Data is as follows:

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Economists’ View of the CPI Data Converting 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

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Biologists’ Use of Algebraic Structure V 0 is initial velocity of chemical reactionV 0 is initial velocity of chemical reaction [S] 0 is initial concentration of substrate[S] 0 is initial concentration of substrate V max, K M are constantsV max, K M are constants 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/ V 0 and 1/[S] 0

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

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