Advanced Topics in High School Mathematics: Enrichment or Advancement ?

1. Advanced Topics in High School Mathematics: Enrichment or Advancement?

2. Who & Why of Math for America DC? • Sarah (education), John, Michael, Chris (math) • Investing in the Human Infrastructure (ARRA) • Mathematics as the core to STEM developmental knowledge. • Teaching as one of the most important profession • DC as the most important city …. Well, maybe

3. Mathematicians and Educators • Collaboration between Mathematicians and Educators • Challenging Fellows Development as Mathematicians • Developing Teachers for Urban Classrooms. • Can Schools of Education and Departments of Mathematics and Statistics work together? • ….. Maybe so!

4. Recruitment Model • Nationally recruitment model to bring talented mathematicians to DC Secondary Mathematics Classrooms • Washington, DC as well as • Other MfA programs in New York, Berkeley, Boston, Los Angeles, San Diego & Utah

5. How the Fellowship Works Year 1 • Prepare to become a secondary school math teacher • Masters level teacher preparation program with tuition and fees paid by MƒA DC • Extensive student teaching experience • $22,000 MƒA DC stipend • Pre-service professional development

6. How the Fellowship Works Years 2-5 • Teach in a Washington DC public or public charter school • Receive a regular teacher’s salary and MƒA stipends of $12,500 per year • Mentoring, coaching and other support services • Participate in professional development activities

7. Academic Program: One-year intensive Masters Degree in Secondary Mathematics Education

8. Why Math? Because it’s… • The language of the sciences. • The way in which we describe nature. • The art of problem solving.

9. Why Teaching? Because of the… • AHA! moments. • Challenges and rewards. • Nationally, only 40% of teachers have degrees in mathematics. • The opportunity to instill a love of math in someone with unlimited potential.

10. Why DC? Only 36% of eighth graders scored “proficient” or above on their standards-testing in 2008-2009. Many schools in DC are serious about high mathematics performance, and there is much work to be done!

11. A new kind of capstone course for Math for America fellows Goal: High school teachers should have the habit of mind to explore why the high school mathematics they teach is true and use this knowledge to relate it to the bigger mathematical picture.

12. My postulates (the problems) • Although knowledge of certain facts and technical fluency is important, reasoning is often missing in instruction. • Teachers learn to teach from the way they were taught. • Knowledge and habits of mind can be compartmentalized by topic.

13. Content Progression of Math Major • High school: Algebra I, Geometry, Algebra II, Trigonometry, maybe some Calculus • Lower division: More Calculus, multi-varCalc, Differential Equations, Linear Algebra, etc. • Upper division: Abstract Algebra, Real Analysis, Topology, Advanced Probability, etc.

14. Where’s the “Why?” • HS: Almost none – taught as rules. Often many teachers don’t know reasons why things work. NCLB may exacerbate. • LD: Professors occasionally show the why in lecture, but don’t test or ask on homework. Students ignore it. • UD: Cornerstone of the lecture and frequent in homework and/or exams. Take home idea: “Why” is important to the adv. math, but has no relevance to high school.

15. When do you ask “Why” about High School Math? Answer many “why” questions in UD classes, but about the subject matter itself (e.g. why must a subgroup be normal for cosets to have a group structure?). Prove Fundamental Theorems (Arithmetic, Algebra, Calculus) in UD. These are technical nor necessarily the questions they would have.

16. Idea of the course • Ask and encourage the “why” questions about parts of the high school curriculum. Should not be our whys, but their whys. • Develop the skill/habit of mind to answer these questions. Math should primarily stay at the high school level • Encourage this mindset in their own teaching.

17. Our Whys versus Their Whys • Note: Students are so used to not thinking about high school math deeply, they don’t have any “why”s the first day. • Our whys: Fundamental Theorems, complete axioms of Euclidean Geometry, Construction of • Their whys: is the vertex, why is e special, why does long multiplication of decimals work, product of negatives is positive, where quadratic formula comes from

18. Class structure • Warm-up questions: Moment of reflection (why is ) • Class activities based in high school math that don’t require much outside of high school to solve them. • Final project: Ask something you don’t know about high school, answer it. • Lesson plan: Develop something for students that allows them to investigate

19. Class activities examples • Why does the sum of the digits work as a test of divisibility by 3? What other divisibility tests can you come up with? • We know the triangle congruence theorems – what are the quadrilateral congruence theorems? • What scores are possible in modified football with only 7 & 3 as possibilities? How would this work with general p & q?

20. Final projects • Can we have base 2.5? • The derivative of volume for a sphere is surface area, but this isn’t true for other solids. Why? • Why does Pascal’s triangle have so many cool features? • How are foci related in the different conic sections?

21. Positive results • For the most part, students produce high quality of work. • In beginning, all students can’t come up with topics they don’t know. By the end, they are much more cognizant of their math knowledge. • In past two years, at least one student who is not as engaged in the rest of their coursework shines in this environment. • Strong positive student responses.

22. Concerns & Further investigations • Must have people who are very fluent with HS math for this to work (HS math tests) • Does it have any impact in their teaching? Can it even have any impact in a NCLB environment? • Is one (summer) semester too short? Is it too late after they’ve already got a math major? • How even to assess?

23. Goals • What issues are present in a functioning classroom? • What are my assumptions about how people learn? • How to build an environment compatible with these assumptions? • What are the Common Core Standards? • How is some of the content learned in advanced courses related to high school mathematics?

24. Some Successes • Focus – talks in which the speaker had a specific goal in mind were successful • Examples – talks in which specific examples of phenomena were given were more successful • Research – positive feedback was received about talks which covered ongoing research • Interaction – most of the participants were student teaching, or were teachers with some years of experience; they were productive when allowed to be thoughtful about these experiences

25. Focus • Lessons learned teaching AP Statistics (Michael Costello) • Experiences at Washington Lab School (Rose Marie Russo) • Using exams as an assessment and learning tool (Lyn Stallings) • Writing about mathematics (Frances van Dyke)

26. Examples • Grading exams in such a way as to require that students engage with the material (Stallings) • Establishing expectations early in the year (Costello) • Using various “everyday” items to teach lessons about mathematics (Russo)

27. Research • Question : do students enrolled in Calculus I understand very much about graphs? • Answer: In general, no • Would they understand more if they were required to write short essays? • Is it possible to ask better questions?

28. Interaction • The M&M and Oreo experiments • Using (or modifying) games to be used as tools to explain material • Cataloging and explaining the nature of student mistakes

29. Assumptions • All the participants have a good knowledge up to the level of Calculus II • Most have a working knowledge of mathematics up to the level of linear algebra • Most are familiar with the sort of issues raised in courses like real analysis (for example: why are the real numbers uncountable)

30. Uncertainties • When are two functions equal? • What does the equation 5/6 = .833… mean? Why do we divide 6 into 5 to get this equation? • Whence equations like .999… = 1? • What is a real number? • Is the relationship between equivalent fractions similar to the relationship between equivalent real numbers?

31. Comments • Some of the talks were too general • Some of the talks presented ideas which might be unworkable in every possible setting • Perhaps the topic should have been introduced in advance of the talk, so that the participants would have the opportunity to think of questions to ask

32. Some Failures • It was a mistake to divide the course into two different parts; the course itself may have been too broad • The course evaluations suggested that the participants thought that there was a “silver bullet” where instruction is concerned; this seems to defeat the notion that it is acceptable to approach teaching in different ways • There should have been a greater opportunity to reflect on the nature of the talks

33. Some things to build on • Perhaps there should be two different courses: one would permits the speakers to treat the subject they introduced in a more thorough way (with more than one talk) • If the participants are student teaching at the time they are enrolled, they might keep a journal; this journal could include observations made about the sort of mistakes students make and theories about how to address these sorts of mistakes • The other course would explore thoroughly the issues raised in slide 8. The participants have a thorough enough knowledge of mathematics to contend with these topics (equations and equivalences), and with some thought could link them with the subject matter that they are liable to be teaching