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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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Mathematics research in high school

Mathematics Research In High School?

TCM 2013

Dan Teague

NC School of Science and Mathematics


Goals of this presentation
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
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
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.



  • 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 mathematics1
What is mathematics? respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

  • Mariane Moore:

    Poetry is an imaginary garden

    with real toads.

    A research project will allow your students to explore that imaginary garden.


Motivation
MOTIVATION respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

  • 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
respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game. In humanities courses, you get to use your own mind”

In mathematical research, students must use their own minds.


112 reu s this summer
112 REU’s this Summer respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.


Chomp the graph
Chomp the Graph respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

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
A Wins respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.


Characteristics of good research problems
Characteristics of Good respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game. 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
Prime Derivatives respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

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
What numbers are quadratic? respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

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
Additional Questions respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

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
Research? respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game. YES!


Some guidelines for post calculus reaearch
Some Guidelines for Post-Calculus Reaearch respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.


Start at the beginning
Start at the Beginning respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

  • 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
Keys to Mathematical Research respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

  • 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 research1
Keys to Mathematical Research respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

  • 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 counter-examples. 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
When can you share papers? respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

  • 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 vs Research respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.


The role of rigor and proof
The Role of Rigor and Proof respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

  • Calculus

  • Number Theory

  • Combinatorics

  • Who makes the conjecture and how is it formed?


Making it work
Making It Work respective ends of the contract that your test was designed to validate, because thinking will have gotten in the way of the game.

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
Student characteristics? opportunities.

  • 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
Issues that can derail the project opportunities.

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)


Getting unstuck
Getting Unstuck opportunities.

  • 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
Additional Issues opportunities.

  • 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.


  • Support from school administration. opportunities.

  • 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
MAA PREP Workshop opportunities.


Prep workshop
PREP Workshop opportunities.

  • 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 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.



For more information on these summer programs, write to opportunities.Dan TeagueNC School of Science and [email protected]


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