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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 • 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. • • • • • • • • • 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 [email protected]