Mathematics Research In
High School?
TCM 2013
Dan Teague
NC School of Science and Mathematics
Goals of this Presentation
• Your kids can do it.
• You can do it.
• It is a uniquely valuable experience for
both of you.
Post-Calculus Opportunities for
Interested Students
• More Standard Courses (Differential Equations,
Multivariable Calculus, Linear Algebra, Proofs course)
• Mathematics Contests
• Math Club Activities
What is mathematics?
• How do you want your students to answer this
question?
• Based on their everyday classroom
experience, what answer do you think they
would give?
The Rules of the Game: Teaching Mathematics
The rules of that game are simple: we, the teachers show them
what to do and how to do it; we let them practice at it for a
while, and then we give them a test to see how closely they
can match what we did.
What we contribute to this game is called "teaching," what they
contribute is called "learning," and the game is won or lost for
both of us on test day.
Ironically, thinking is not only absent from this process, but in a
curious way actually counterproductive to the goals of the
game.
• Thinking takes time. Thinking comes into play
precisely when you cannot do something "without
thinking." You can do something without thinking if
you really know how to do it well.
• If your students can do something really well, then
they have been very well prepared. Therefore, if both
you and your students have done your jobs perfectly,
they will proceed through your test without thinking.
• If you want your students to think on your test, then
you will have to give them a question for which they
have not been fully prepared.
• If they succeed, fine; in the more likely event that
they do not, then they will rightfully complain about
not being fully prepared.
• You and the student will have both failed to uphold
your respective ends of the contract that your test
was designed to validate, because thinking will have
gotten in the way of the game.
• Considering how we mathematicians value thinking,
it is a wonder that we got ourselves into this mess at
all.
Dan Kennedy
Former Chair
AP Calculus Test Development Committee
What is mathematics?
• Mariane Moore:
Poetry is an imaginary garden
with real toads.
A research project will allow your
students to explore that imaginary
garden.
MOTIVATION
• Sheila Tobias: They Are Not Dumb, They Are
Different (Stalking the Second Tier)
• 1st tier These students will be mathematicians
regardless of what we do.
• 2nd tier These students can do whatever they
want, they choose not to do math and science.
“In humanities courses, you get
to use your own mind”
In mathematical research, students must
use their own minds.
112 REU’s this Summer
Chomp the Graph
Given a finite undirected graph G, players
alternate turns and remove either a single edge
or a vertex with all incident edges.
Whoever removes the last vertex, leaving their
opponent with the empty graph, wins.
A Wins
Characteristics of Good
Research Problems
1)
a low threshold of background knowledge.
2)
some “low hanging fruit”
3) with perseverance and creativity, will yield to a
variety of approaches.
4) wide array of extensions and directions in
which the work can progress .
Prime Derivatives
Given a positive integer x,
x' = 1 if x is a prime.
x' = a∙b' + a'∙b if x = a∙b (x is composite)
x' = 0 if x = 1.
Example: 6' = 3∙2' + 3'∙2 = 3∙1 + 1∙2 = 5
What numbers are
quadratic?
Is it true that for each positive integer x, there
exists an integer y such that x' = 2y?
How does this question relate to Goldbach's
conjecture?
Additional Questions
Since all composites are products of primes, we can
find the derivatives of all orders for all integers?
What patterns are observed?
Can you define an anti-derivative or solve
differential equations like x' = a∙x?
Develop integration by parts? Define the Prime
Derivative of rational numbers? Sequences of
rational numbers? Dervatives (mod p)?
Research? YES!
Some Guidelines for PostCalculus Reaearch
Start at the Beginning
• Don’t have students read other papers on the
problem.
• Create their own definitions, notations, and
approach. Let them make the problem their own.
• If they can’t recapitulate known results, they will be
unlikely to generate anything new.
• By recreating known results, they will develop
insight, intuition, and technique.
Keys to Mathematical Research
• You are not trying to solve a specific problem, although you
pose specific problems to solve.
• You are trying to develop a theory of the problem, to
understand all of its variations by understanding its
fundamental structure.
• Always start small. Understand completely the simplest case.
Don’t try to get too complicated too quickly.
• Create definitions and notation that is clear, simple, and
unambiguous.
Keys to Mathematical Research
• Once you understand a simple case, modify the problem:
a) Focus on the invariants.
b) What changes? What is it about the modification
that created the change?
c) What role does symmetry play? What role does
parity play?
• Make your conjectures as specific as possible. Test your
conjectures with new configurations, looking for counterexamples. Why must your conjecture be true? What
forces it to happen? Consider what would happen if they
were not true.
When can you share papers?
• Allow student groups to develop their
approach and generate some nice results
before sharing with others.
• If a group sees others doing something “more
sophisticated” they will likely alter their
approach and end up doing the same things in
the same way and getting stuck at exactly the
same place.
Contests vs Research
Contests
Research
Requires many tools/techniques/training
Understand background/problem
Finite undertaking/bite size problems
Skills/habits of mind related to problem solving
Comes with a guarantee of solvability
Open ended time frame
Success/failure is quantifiable/scored (performance recognized)
No guarantee of solutions possible
External source of problems
Evaluation of success more subjective
Master “tricks” of the trade
Internal source of problems
Competitive
Deeper understanding involved
Immediacy of reward
Delayed gratification
The Role of Rigor and Proof
• Calculus
• Number Theory
• Combinatorics
• Who makes the conjecture and how is it
formed?
Making It Work
1.
Have a variety of problems for students to choose from.
Be careful in letting student pick their own topics. They often
pick something too difficult or which requires significantly more
prior knowledge and techniques than they have at their disposal.
2.
Talk with Administration about your workload,
differences in grading, and having some process by which kids
can opt out.
3. Presentation of student work to math team, department,
headmaster, college counselors, and parents.
4.
Talk with Parents about grades, college
admissions opportunities.
5.
Let parents know that they aren’t supposed to
help the child.
6.
Talk with students about the difference in
studying and taking tests and doing a creative research
project.
7.
Include other faculty if possible.
Student characteristics?
• Students who want to go beyond the standard solution.
• Students who ask interesting questions .
• Students who are creative even if lacking some computational
capabilities.
• Students who have demonstrated independent learning and
persistence.
• Out of the box thinkers who can see things differently.
• Mathematical daydreamers.
• Mathematical artists who need time to work out their ideas.
• Students who can handle frustration well.
• Students who work well with others (or alone).
Issues that can derail the
project
1.
Failure to launch (we don’t know if what we are about to do will work, so
we don’t do anything)
2.
Only considers specific cases. Missing overlaps (paths->trees->bipartite->even
cycles)
3. Using techniques you don’t understand. Don’t recognize errors in their
work.
4.
Starting too hard. Being unwilling to do simple cases.
5.
Losing Interest. Doesn’t like the problem. Frustration with getting stuck.
6. Project gets lost in other school/extra-curricular work (priority of
“graded” work). Insufficient time in schedule to think about the problem.
Teacher
availability
(time to work
with students)
Getting Unstuck
1.
Suggest a specific special case. Suggest an overlapping problem
and compare.
2. Jump ahead. Work on a different part of the problem. Assume a
result and do a conditional argument. For a 2-regular graph that is a
cycle, ….
3. Take a break. Get away from the problem so you can come back
fresh.
4. Write down every idea, whenever it happens.
5. Keep a record of all your approaches and ideas, even those that
don’t seem to work.
Additional Issues
• Students will try to begin in the middle. Start too hard.
• Faculty frustration with students.
• Discovery is fun, writing is hard. Difficulty presenting their
work.
• Being helpful, but hands off.
• Student’s idea of a completed project differs from yours.
• All the variations of group work issues.
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Support from school administration.
Grading students and fitting into a school-wide grading system.
Balancing skills of research with research time.
Resource: Thinking Mathematically, John Mason, Prentice-Hall.
Fear of error, we don’t know if what we are planning to do will
work.
•
• Informing parents of the differences in research and “regular
classes
MAA PREP Workshop
PREP Workshop
• Have teachers experience first hand (if only for a short time) the
challenge and adventure of a research problem. To experience the
process they will lead their students through next year.
• Develop a group of collaborative teachers who can work together to
support their student’s research efforts (I’ll read yours if you read
mine).
• Backed up by research mathematicians if needed (I can’t tell if this
statement is false or just poorly written).
• Where should I suggest she go next? (What new problem does her
solution suggest?)
• New problems each year. Perhaps a national problem.
Ralph Pantozzi, Kent Place School
When I left the dormitory in Lincoln Nebraska, it was
actually the first time I noticed a sign that said "Welcome
Teacher Prep". What I experienced that week, however was
nothing like what one normally calls "teacher prep".
With two fellow educators who quickly became friends as
we worked together on a common problem, I had the
opportunity to "get messy" with some engaging mathematics.
While this "chip firing problem" was one that others have looked
at before, the workshop was set up so that we would experience
"the thrill of conjecture and the agony of counterexample" in an
environment where we were encouraged to "make our brains
hurt" - in the same good way we want our students to do in our
classes.
I came to the workshop because I hoped to experience the
kinds of creative work, both individual and cooperative, that I am
hoping to involve my students in in the coming years. I was not
disappointed. From the plane ride to Nebraska (where the number of
people looked at my scribbles of graphs over my shoulder and asked
me what I was doing with genuine curiosity) to the hours spent
working on the problem during the week, I felt that I was doing "real"
mathematical work as an individual and part of a community of
learners.
The work was "real" in that we needed to propose our own
ideas, test them out, gather information, and draw our own tentative
conclusions. In the daily classroom life of secondary school teacher,
this kind of work can fall by the wayside as we try to get to the
proverbial "end of the book". With my experiences in Nebraska, I have
been energized and motivated to make sure that the students that I
will be working with have the same opportunities to experience
mathematics in a form that will energize _them_ to seek to go past the
end of whatever book we're in.
Anja Greer Conference at
Exeter
For more information on these
summer programs, write to
Dan Teague
NC School of Science and
Mathematics
at
teague@ncssm.edu