A Capstone Mathematics Course for Future Secondary Teachers

1. UTeach Institute – NMSI Conference May 20, 2014 A Capstone Mathematics Course for Future Secondary Teachers John Quintanilla Co-Director, Teach North Texas Professor of Mathematics and University Distinguished Teaching Professor Tress Kringen Alumna, Teach North Texas Mathematics Teacher, Jack E. Singley Academy (Irving ISD) Alyssa Mendez Student, Teach North Texas

2. Capstone Course “Why do I have to take Real Analysis?” “[T]he mathematics courses taken by prospective high school teachers [should] include at least a three-course calculus sequence, an introductory statistics course, an introductory linear algebra course, and 18 additional semester-hours of advanced mathematics, including 9 semester-hours explicitly focused on high school mathematics from an advanced standpoint. It is desirable to have a further 9 semester-hours of mathematics…” p. 55 History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

3. Courses for TNT Math Majors • MATH 2100: Functions and Modeling for Secondary Mathematics Instruction • TNTX 3100: Conceptual Algebra and Geometry • MATH 3850: Mathematical Modeling • MATH 4050: Advanced Study of the Secondary Mathematics Curriculum History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

4. Course Outline History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

5. Course Outline Prime Directive: Math majors should not be embarrassed about not knowing what they have not yet been taught. History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

6. Monday/Wednesday Lectures “Similar topics [to previous courses], but we covered wildly new nuances, and explained previous knowledge more in depth.” History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

7. Monday/Wednesday Lectures I think that the textbook was useful and appropriate for this course. I think the textbook will be a good resource for me to have five years from now. History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

8. Monday/Wednesday Lectures 1. Number Theory for Classroom Teachers 2. Polynomials and Rational Functions 3. Exponents and Logarithms History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

9. Monday/Wednesday Lectures • 1. Number Theory for Classroom Teachers • Prove the rule to check if a number is a multiple of 9. • Convert 857,536 into hexadecimal. • Convert into a fraction. • Find all with a delay of 1 digit and a block of 3 digits. • Find the decimal representation of D15C0 History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

10. Monday/Wednesday Lectures • 1. Number Theory for Classroom Teachers • Prove that • Prove that is irrational. • Prove that is both irrational and algebraic. History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

11. Monday/Wednesday Lectures • 2. Polynomials and Rational Functions • Prove the Rational Root Test. • Prove the Conjugate Root Theorem. • Explain why a sixth-degree polynomial has at most five “turning points.” History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

12. Monday/Wednesday Lectures • 3. Exponents and Logarithms • Find without a calculator. • Why are the Laws of Logarithms true? • Why does ? • Why does ? History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

13. Student Perceptions Before (After) taking this course, I understood the reasons why I had take theoretical courses like Real Analysis.

14. Student Perceptions Before taking this course, I thought I understood the concepts taught in secondary mathematics well enough to teach them. After taking this class, I now realize that I didn’t understand secondary mathematics as well as I had initially thought.

15. Certification Review • Long-range purpose: prepare students for the state mathematics certification exam • Discussed in office hours, not in class • Hopefully a relatively painless way to remind (or teach) students about the secondary mathematics curriculum • Hopefully not a huge time sink History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

16. Certification Review • Precalculus (8 weeks) • Functions • Systems of equalities and inequalities • Exponential and logarithmic functions • Trigonometry (3 weeks) • Polar coordinates and conic sections • Sequences and series • Calculus (5 weeks) • Limits and continuity • Differential calculus + applications (3 weeks) • Integral calculus • Combinatorics (1 week) • Basic probability and descriptive statistics (1 week) • ~15 problems, choose 5 • Pledge that all problems were attempted • Do NOT appear on exams or final History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

17. Student Perceptions Should the reviews be retained in future years? What percentage of problems were… History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

18. Friday Question-and-Answer Sessions • Inspired by example of Iowa’s Central College. • Students are given a list of difficult questions. • Students are called at random to the board to answer the next question on the list • Professor/MT plays the curious/obnoxious student • Grade obtained by peer evaluation • Reviews (or, in some cases, teaches) content without boring students to death • Addresses their own misconceptions in a low-key way • Complements the design/delivery of 5E lesson plans History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

19. Friday Question-and-Answer Sessions A student asks, “Is ?” History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

20. Friday Question-and-Answer Sessions A student asks, “My father was helping me with my homework last night and he said that the book is wrong. He said that and because and . He wants to know why we are using a book that has mistakes.” History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

21. Friday Question-and-Answer Sessions You ask a student to find 28% of 50. His solution: What would you tell the student? History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

22. Student Perceptions Should the Friday presentations be retained in future years? What percentage of the Friday problems were ___ for you to answer?

23. Student Perceptions Most questions were realistic that I should be prepared to answer. I feel better prepared to anticipate and respond to my future students’ questions.

24. Project: Finding Engagement Activities Basic premise: Authors of textbooks (and hence aspiring math teachers) seriously believe that the following problem actually piques student interest in mathematics. “Farmer Jones has to enclose a pen with 100 meters of fence. One side of the pen is already in place. What are the dimensions of the pen that has the maximum possible area?” History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

25. Project: Finding Engagement Activities • Answer 3 questions (150-250 words) from each of Algebra, Geometry, Precalculus, Prob/Stat • What interesting word problems can your students do now? • How can this topic be used in the future (extend from the past)? • How has this appeared in pop culture (high culture) (the news)? • Who contributed to the discovery (development) of this topic? • What are the contributions of various cultures to this topic? • How can technology be used to engage students with this topic? • Grading: Accuracy, Engagement and Understandability History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

26. Project: Finding Engagement Activities • All (ungraded) essays are amalgamated into a miniature reference book and returned to class. • The best essays (with students’ permission) are placed on my blog, http://www.meangreenmath.com. • One memorable example: order of operations. History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

27. Student Perceptions I feel like I have more ideas for capturing my future students’ interest in math. Should the engagement project be retained in future years?

28. Evidence of Course Effectiveness First Attempts on Texas 8-12 Mathematics Certification Exam P < 0.002 93% 83% 72% Number of students attempting exam

29. Perceptions of Capstone Course I hope future TNT students take Math 4050 in the same format. After taking this class, I feel much better prepared to be a math teacher.

30. Summary • Core components of our capstone course • Monday/Wednesday lectures • Certification reviews • Friday Q&A • Course project • Illustrates importance of theorem-proving skills • Low-key way to address student misconceptions • Low-key way to deepen knowledge of Precal/Calculus • Prepare students for certification exam and classroom • Complements the UTeach teacher-education sequence.

31. 4050 Webpage Friday Q&A, projects, certification reviews, and class outline MeanGreenMath.com Feel free to subscribe, promote on Twitter, etc. Conference Survey

32. Friday Question-and-Answer Sessions A student presents the following work: History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

33. Friday Question-and-Answer Sessions A student asks, “What is ?” History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

34. Friday Question-and-Answer Sessions A junior-high student is excited: “Miss Jones, I’ve found a new way to get the average of two numbers. Take 12 and 18, for example. Subtract 12 from 18; that’s 6. Divide by 2; that’s 3. Add 3 to 12, and you get the average, 15.” If you were Miss Jones, what would you tell the student? History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

35. Friday Question-and-Answer Sessions A student presents the following work: History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

36. Friday Question-and-Answer Sessions A student presents the following work: History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions

37. Friday Question-and-Answer Sessions A student asks: “I noticed the following pattern: Is there a reason why the last two digits are perfect squares? I know it usually doesn’t work out that way.” History Mon/Wed lectures Certification review Friday Q&A Projects Student perceptions