Transcript Indezine Template

Advanced Topics
in High School
Mathematics:
Enrichment or
Advancement?
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
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!
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
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
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
Academic Program:
One-year intensive Masters Degree in Secondary Mathematics
Education
Why Math?
Because it’s…
• The language of the sciences.
• The way in which we describe nature.
• The art of problem solving.
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.
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!
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.
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.
Content Progression of Math Major
• High school: Algebra I, Geometry, Algebra II,
Trigonometry, maybe some Calculus
• Lower division: More Calculus, multi-var Calc,
Differential Equations, Linear Algebra, etc.
• Upper division: Abstract Algebra, Real
Analysis, Topology, Advanced Probability, etc.
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.
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.
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.
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: x0  1,b 2a is the vertex, why is e
special, why does long multiplication of decimals
work, product of negatives is positive, where
quadratic formula comes from
Class structure
• Warm-up questions: Moment of reflection
0
(why is x  1 )
• 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
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?
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?
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.
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?
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?
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
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)
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)
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?
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
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)
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?
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
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
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