Transcript
Page 1: A Capstone Course for Future Teachers of Secondary Mathematics

A Capstone Course for A Capstone Course for Future Teachers of Future Teachers of

Secondary MathematicsSecondary Mathematics

Development and Implementation

Lisa Rezac and Melissa Shepard LoeUniversity of St. Thomas

Ninth Annual AMTE Conference January 2005

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Presentation OutlinePresentation Outline Motivation and Justification (MET, PMET)Motivation and Justification (MET, PMET) Implementation: Implementation:

• Course catalog descriptionCourse catalog description• Syllabus and Course TopicsSyllabus and Course Topics

Bridges Through Number TheoryBridges Through Number Theory Assessment TopicsAssessment Topics Exposition Project: Historical researchExposition Project: Historical research Exploration Project: Contemporary Exploration Project: Contemporary

researchresearch Discrete MathematicsDiscrete Mathematics Topics in “Higher Mathematics”Topics in “Higher Mathematics”

• Triumphs, Disappointments, DifficultiesTriumphs, Disappointments, Difficulties

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Motivation and Justification:Motivation and Justification: CBMSCBMS: Conference Board of the : Conference Board of the

Mathematical SciencesMathematical Sciences METMET: Publication from CBMS: The : Publication from CBMS: The

Mathematical Education of Teachers Mathematical Education of Teachers (Parts I and II) published by the MAA(Parts I and II) published by the MAA

PREP/PMETPREP/PMET: Professional : Professional Enhancement Programs of the MAA Enhancement Programs of the MAA and Preparing Mathematicians to and Preparing Mathematicians to Educate TeachersEducate Teachers

MNBoTMNBoT: MN Board of Teaching: MN Board of Teaching

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MET Document ObservationMET Document Observation

“ “There is evidence of a vicious cycle in There is evidence of a vicious cycle in which too many prospective teachers which too many prospective teachers enter college with insufficient enter college with insufficient understanding of school mathematics, understanding of school mathematics, have little college instruction focused on have little college instruction focused on the mathematics they will teach, and then the mathematics they will teach, and then enter their classrooms inadequately enter their classrooms inadequately prepared to teach mathematics to the prepared to teach mathematics to the following generations of students.”following generations of students.”

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MET Document MET Document RecommendationRecommendation

“… “… mathematics departments can support mathematics departments can support the design, development, and offering of a the design, development, and offering of a capstone course in which conceptual capstone course in which conceptual difficulties, fundamental ideas and techniques difficulties, fundamental ideas and techniques of high school mathematics are examined of high school mathematics are examined from an advanced standpoint.”…”Such a from an advanced standpoint.”…”Such a capstone sequence would be most effectively capstone sequence would be most effectively taught through a collaboration of faculty with taught through a collaboration of faculty with primary expertise in mathematics and faculty primary expertise in mathematics and faculty with primary expertise in mathematics with primary expertise in mathematics education and experience in high school education and experience in high school teaching.”teaching.”

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Our Philosophy: Our Philosophy: Capstone courses should…Capstone courses should…

Be taken after most math content courses Be taken after most math content courses are completed, so students have a solid are completed, so students have a solid background of traditional mathematical background of traditional mathematical knowledge.knowledge.

Build bridges between the content in the Build bridges between the content in the required college courses and fill in gaps from required college courses and fill in gaps from within the courses.within the courses.

Maintain a high level of mathematical Maintain a high level of mathematical content and rigor and also address related content and rigor and also address related pedagogical issues. pedagogical issues.

Most importantly, connect college level math Most importantly, connect college level math courses and the high school curriculum.courses and the high school curriculum.

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Description from the Course Catalog:

This course gives students a sense of the history, applicability and currency of one or more mathematical ideas and serves as a capstone mathematics course for students seeking to teach secondary mathematics. In the course, students make substantial oral and written presentations on topics carefully selected to have a strong relationship to secondary school mathematics. They use publications, e.g. The American Mathematical Monthly, Mathematics Magazine, Mathematical Intelligencer and Scientific American, as well as standard texts as sources for their work.

ImplementationImplementation Advanced Mathematics: Exploration and ExpositionAdvanced Mathematics: Exploration and Exposition

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Partial SyllabusPartial Syllabus

Title:Title: Math 450 Advanced Mathematics: Exploration and Math 450 Advanced Mathematics: Exploration and ExpositionExposition

CreditsCredits: 4 semester credits (one regular course): 4 semester credits (one regular course)RequiredRequired: Of Math majors with a Secondary Ed. Co-major. : Of Math majors with a Secondary Ed. Co-major.

Others welcome, will count towards a general math major.Others welcome, will count towards a general math major.GoalsGoals: : Provide a bridge between upper level math coursesProvide a bridge between upper level math courses Explore connections to the high school curriculumExplore connections to the high school curriculum More – related to NCTM standards and MN state standardsMore – related to NCTM standards and MN state standardsAssessment:Assessment: Two individual projects, homework, final exam, Two individual projects, homework, final exam,

class participation, occasional quizzes, class discussionsclass participation, occasional quizzes, class discussionsTopics/Units:Topics/Units:I.I. Bridges Through Number TheoryBridges Through Number TheoryII.II. Assessment TopicsAssessment TopicsIII.III. Exposition: Historical researchExposition: Historical researchIV.IV. Exploration: Contemporary researchExploration: Contemporary researchV.V. Discrete MathematicsDiscrete MathematicsVI.VI. Topics in “Higher Mathematics”Topics in “Higher Mathematics”

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I. Bridges through Number TheoryI. Bridges through Number Theory

Using Simon Singh’s book, Using Simon Singh’s book, Fermat’s EnigmaFermat’s Enigma, we , we study the history of a mathematical theorem, study the history of a mathematical theorem, practice reading mathematics, and look at the practice reading mathematics, and look at the connections between abstract algebra, number connections between abstract algebra, number theory and analysis. theory and analysis.

Other possible choices could includeOther possible choices could include• The Crest of the Peacock by George The Crest of the Peacock by George

Gheverghese JosephGheverghese Joseph• Four Colors Suffice by Robin WilsonFour Colors Suffice by Robin Wilson• The Code Book by Simon SinghThe Code Book by Simon Singh• The Music of the Primes by Marcus Du Sautoy The Music of the Primes by Marcus Du Sautoy

For more possibilities, For more possibilities, see handout which lists see handout which lists sources and reviews.sources and reviews.

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II. Assessment TopicsII. Assessment Topics

Assessing student work:Assessing student work: • Solomon Friedberg, Solomon Friedberg, The Boston College The Boston College

Mathematics Case Studies ProjectMathematics Case Studies Project• Bonnie Gold et al, Bonnie Gold et al, Assessment Practices in Assessment Practices in

Undergraduate MathematicsUndergraduate Mathematics. . • Katherine MersethKatherine Merseth, Windows on Teaching Math, Windows on Teaching Math

Assessing Texts:Assessing Texts: reviewed the idea of reviewed the idea of mathematical definitions by specifically mathematical definitions by specifically looking at the idea of “limit” in many looking at the idea of “limit” in many different Calculus texts. Traced the MVT, different Calculus texts. Traced the MVT, Rolle’s Theorem, and the Fundamental Rolle’s Theorem, and the Fundamental Theorem of Calculus through the same set Theorem of Calculus through the same set of texts.of texts.

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III. Exposition Project:III. Exposition Project: Historical Research Historical Research

Individual presentation and paper on Individual presentation and paper on a historical search of a mathematical a historical search of a mathematical idea, person or theorem through idea, person or theorem through history. Must include a discussion of history. Must include a discussion of how the topic could be used in a high how the topic could be used in a high school classroom with reference to school classroom with reference to MN State Standards for curriculum. MN State Standards for curriculum. Their model is the “Bridges” study of Their model is the “Bridges” study of Fermat’s Last EnigmaFermat’s Last Enigma..

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Exposition Project TitlesExposition Project Titles A Wonder Tool Lost Forever (slide rule)A Wonder Tool Lost Forever (slide rule) Leonhard Euler: His Life and Contributions to MathematicsLeonhard Euler: His Life and Contributions to Mathematics African American Women in MathAfrican American Women in Math Georg Cantor and the infiniteGeorg Cantor and the infinite Fractal Geometry and the Mandelbrot SetFractal Geometry and the Mandelbrot Set Geometric Series, Descartes, and the fundamental theorem Geometric Series, Descartes, and the fundamental theorem

of calculusof calculus Hyperbolic GeometryHyperbolic Geometry Sophie Germain and her MathematicsSophie Germain and her Mathematics Game Theory and the Prisoner’s DilemmaGame Theory and the Prisoner’s Dilemma A Slice of Pi, Pi and eA Slice of Pi, Pi and e Map-Coloring and the Four Color TheoremMap-Coloring and the Four Color Theorem Fundamental Theorems of Algebra and CalculusFundamental Theorems of Algebra and Calculus Symbolic LogicSymbolic Logic TrigonometryTrigonometry Pigeonhole Principle and Ramsey TheoryPigeonhole Principle and Ramsey Theory

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Student Project RubricsStudent Project Rubrics The first time the course was run, students The first time the course was run, students

helped create the rubric(s).helped create the rubric(s). Rubric was handed out with the project Rubric was handed out with the project

assignment so they knew how the assignment so they knew how the instructor would be assessing them. instructor would be assessing them.

These rubrics were also used by all the These rubrics were also used by all the students in the class to assess the students in the class to assess the presentations. Comments were typed and presentations. Comments were typed and used as anonymous feedback for the used as anonymous feedback for the presenters.presenters.

See actual rubrics in the handout.See actual rubrics in the handout.

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IV. Exploration Project: IV. Exploration Project: Contemporary ResearchContemporary Research

Individual exploration of a current Individual exploration of a current research article in research article in The College The College Mathematics JournalMathematics Journal or or Mathematics Mathematics MagazineMagazine and presentation in a and presentation in a seminar/colloquium style. Also must seminar/colloquium style. Also must include a discussion of how the topic include a discussion of how the topic might be used in a high school might be used in a high school classroom.classroom.

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Exploration ArticlesExploration Articles (more on handout)(more on handout) Pillow Chess, by Grant Cairns, Pillow Chess, by Grant Cairns,

Mathematics MagazineMathematics Magazine vol. 75, no. 3, vol. 75, no. 3, June 2002, pages 173-186.June 2002, pages 173-186.

A visit With Six, by Monte Zerger, A visit With Six, by Monte Zerger, College Mathematics JournalCollege Mathematics Journal, vol. 33, , vol. 33, no. 2, March 2002, pages 74 – 87.no. 2, March 2002, pages 74 – 87.

A Natural Generalization of the Win-A Natural Generalization of the Win-Loss Rating System, by Charles Loss Rating System, by Charles Redmond, Redmond, Mathematics Magazine, Mathematics Magazine, vol. 76, no. 2, April 2003, pages 119 – vol. 76, no. 2, April 2003, pages 119 – 126. 126.

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V. Discrete MathematicsV. Discrete Mathematics

Accessible to H.S. and college students, Accessible to H.S. and college students, NCTM Standard: Number and OperationNCTM Standard: Number and Operation

Other NCTM Standards: Problem Solving, Other NCTM Standards: Problem Solving, Reasoning and Proof, CommunicationReasoning and Proof, Communication

Univ. of St. Thomas and MN Board of Univ. of St. Thomas and MN Board of Teaching standards require discrete Teaching standards require discrete mathematicsmathematics

Rich source of problems for H.S., Rich source of problems for H.S., college, and “recreational mathematics”college, and “recreational mathematics”

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VI. Topics in “Higher Mathematics” VI. Topics in “Higher Mathematics” Connected to Problems in Connected to Problems in Secondary MathematicsSecondary Mathematics

Topics: induction, recursion, Topics: induction, recursion, divisibilitydivisibility

Text: Usiskin et al, Text: Usiskin et al, Mathematics for Mathematics for High School Teachers: An Advanced High School Teachers: An Advanced PerspectivePerspective

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Triumphs:Triumphs: Student project work and presentations showed Student project work and presentations showed

definite improvement from first to second projects. definite improvement from first to second projects. Some students who were average students in the Some students who were average students in the

typical classroom showed their true teaching typical classroom showed their true teaching potential in their projects.potential in their projects.

Communications with School of Education increased.Communications with School of Education increased. Awareness / appreciation of the work involved in Awareness / appreciation of the work involved in

doing doing mathematics. Students experienced working mathematics. Students experienced working on mathematics with classmates, and gave brief on mathematics with classmates, and gave brief presentations to explain solutions.presentations to explain solutions.

7 of the 8 students in Fall ‘04 volunteered at a 7 of the 8 students in Fall ‘04 volunteered at a regional NCTM conference in Mpls 2004, got exposed regional NCTM conference in Mpls 2004, got exposed to the “movers and shakers” in NCTM – better chance to the “movers and shakers” in NCTM – better chance of continued participation at such conferences.of continued participation at such conferences.

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Disappointments:Disappointments: Students unprepared for and unhappy with Students unprepared for and unhappy with

second project requirements. Although they were second project requirements. Although they were warned that this would be the hardest part of the warned that this would be the hardest part of the course, and that they needed to begin their course, and that they needed to begin their research during the first week of classes, this was research during the first week of classes, this was apparently not taken seriously. apparently not taken seriously.

Connections to the High School curriculum need Connections to the High School curriculum need to be more explicit.to be more explicit.

Heavy homework grading – big time demand.Heavy homework grading – big time demand. Procrastination - Some students tended to delay Procrastination - Some students tended to delay

starting work on the major projects until a week starting work on the major projects until a week or so before due date. Result – low quality work, or so before due date. Result – low quality work, stress, less attention to other course work for stress, less attention to other course work for 450!450!

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DifficultiesDifficulties Final Exam depends on student projects, needs Final Exam depends on student projects, needs

careful coordination.careful coordination. How can I call it a capstone course when I must How can I call it a capstone course when I must

cover new content?cover new content? How can I have a better connection with the How can I have a better connection with the

content of the high school curriculum?content of the high school curriculum? Motivating students to eliminate procrastination Motivating students to eliminate procrastination

and set (and meet!) intermediate goals for and set (and meet!) intermediate goals for projects.projects.

Helping students in managing multiple Helping students in managing multiple assignments / projects concurrently. Helping assignments / projects concurrently. Helping students to see the course as a tapestry of students to see the course as a tapestry of experiences and learning as opposed to a experiences and learning as opposed to a collection of independent disconnected collection of independent disconnected components.components.

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Student Descriptions of their own Student Descriptions of their own Persistent Mathematical DifficultiesPersistent Mathematical DifficultiesPossible opening activity for the course:Possible opening activity for the course:

In looking over your entire mathematical In looking over your entire mathematical career, identify three mathematical career, identify three mathematical concepts that seemed important, yet you concepts that seemed important, yet you had a hard time understanding them. had a hard time understanding them. Describe the specific difficulties that you Describe the specific difficulties that you had or have, and if you have overcome had or have, and if you have overcome them, how and when. You may also discuss them, how and when. You may also discuss a concept that you have not yet understood a concept that you have not yet understood or have given up trying to understand.or have given up trying to understand.

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If and only if If and only if proofsproofs

Symbols and Letter Symbols and Letter RepresentationRepresentation

Approximating f(x) based on Approximating f(x) based on a graph of f'(x)a graph of f'(x)

Origin of the trig Origin of the trig functionsfunctions SummationsSummations Three dimensional graphingThree dimensional graphing

Closure and Closure and ontoonto Polar coordinatesPolar coordinates Sequences and SeriesSequences and Series

Adding fractionsAdding fractions Finding PercentagesFinding Percentages Understanding trig functionsUnderstanding trig functions

Modular Modular ArithmeticArithmetic ProofsProofs FractionsFractions

VariablesVariables Imaginary NumbersImaginary NumbersGodel's Incompleteness Godel's Incompleteness

TheoremTheorem

u-substitutionu-substitution Proving G is a groupProving G is a group Mathematical InductionMathematical Induction

Fourier Fourier TransformTransform MechanicsMechanics

Metaphysics / Science and Metaphysics / Science and GodGod

ProofsProofsProbability and Probability and

StatisticsStatistics Real number lineReal number line

InductionInduction FunctionsFunctions Tangent linesTangent lines

Self-reported Student DifficultiesSelf-reported Student Difficulties

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Discussion of Student Discussion of Student DifficultiesDifficulties

Students still made mistakes when Students still made mistakes when describing concepts they thought describing concepts they thought they had overcome.they had overcome.

Students who could trace their Students who could trace their difficulties through several different difficulties through several different applications did a good job of applications did a good job of explaining concepts in an abstract explaining concepts in an abstract way.way.

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ConclusionConclusion We have discussed the motivation, We have discussed the motivation,

creation, implementation and results creation, implementation and results of a capstone course for future of a capstone course for future teachers of secondary education. teachers of secondary education.

We hope to continually work on We hope to continually work on improving this course and would like improving this course and would like to spend some time as a group on to spend some time as a group on any questions or comments you may any questions or comments you may have. We provide again our beliefs have. We provide again our beliefs on capstone courses as a set of on capstone courses as a set of possible discussion points:possible discussion points:

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Our Philosophy: Our Philosophy: Capstone courses should…Capstone courses should…

Be taken after most math content courses Be taken after most math content courses are completed, so students have a solid are completed, so students have a solid background of traditional mathematical background of traditional mathematical knowledge.knowledge.

Build bridges between the content in the Build bridges between the content in the required college courses and fill in gaps from required college courses and fill in gaps from within the courses.within the courses.

Maintain a high level of mathematical Maintain a high level of mathematical content and rigor and also address related content and rigor and also address related pedagogical issues. pedagogical issues.

Most importantly, connect college level math Most importantly, connect college level math courses and the high school curriculum.courses and the high school curriculum.

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CONTACT INFORMATIONCONTACT INFORMATION

Melissa Shepard Loe: Melissa Shepard Loe: [email protected]@stthomas.edu

Lisa Rezac:Lisa Rezac:[email protected]@stthomas.edu

University of St. ThomasUniversity of St. ThomasSt. Paul, MNSt. Paul, MN