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.
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NC School of Science and 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.
AP Calculus Test Development Committee
Poetry is an imaginary garden
with real toads.
A research project will allow your students to explore that imaginary garden.
In mathematical research, students must use their own minds.
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.
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 .
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
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?
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)?
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?
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.
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 opportunities.(time to work with students)
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.
Ralph Pantozzi, Kent Place School opportunities.
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 opportunities.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.