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Quest for Engagement

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  1. Quest for Engagement John Hannah

  2. Quest University, Canada

  3. Summary • Quest University engages students • Objectives for an engaging mathematics course • Pedagogy, or hints for engaging students • Example(s) of me trying to do this • Conclusions

  4. Survey of Student Engagement • Level of Academic Challenge • Inclusion of Enriching Educational Experiences • Intensity of Student-Faculty Interaction • Use of Active and Collaborative Learning • Existence of a Supportive Campus Environment

  5. Level of academic challenge • How much has your coursework emphasised memorising, analysing, synthesising, making judgements or applying theories to new situations? • How much time do you spend preparing for class? • How many written assignments does each course require, and how long did they take to do? • How often have you worked harder than you thought you could to meet an instructor's standards or expectations?

  6. Level of Academic Challenge

  7. International comparison

  8. Maths Foundation at Quest • Goal: introduce students to the way mathematicians ask and answer questions about the world. • Instructor has lots of freedom regarding content and assessment, as long as they fit the aims of the course. • Students may never study mathsagain, so this could be their only contact with these ideas. • Need to engage the students and to spark their interest. • Contact time is 3 hours a day, and students should do out-of-classroom work for about 5 hours a day.

  9. Foundation Maths Courses • Maths: a historical tour of great civilizations • Mathematical puzzles • Modeling our world with mathematics • Money matters: mathematical ideas in finance • Spherical trigonometry • Visual mathematics • Doing mathematics

  10. Course objectives: what is maths? At the end of a mathematics foundation course, students should understand that mathematics is a process of abstraction of quantitative and spatial experience into a mental model capable of analysis.

  11. Course objectives: Rule of Three Students should recognize and work with the three most common modes of mathematical discovery and inquiry: symbolic (algebra), numeric (arithmetic), and visual (geometry).

  12. Course objectives: Problem solving Students should develop a proper attitude toward solving mathematical problems: • question formulation and clarification, • experimentation, • inference, • use of resources and technology, • refinement, attention to detail, and • final articulation.

  13. Course objectives: a bigger picture Students should place the mathematical conclusions in an appropriate context. This should include some of the following: • Appraise the significance of a solution for the empirical situation that the model represents, as well as its potential limitations. • Understand the role and value of deductive arguments (proofs) in mathematical thinking. • Identify and explore the implications of the conclusion for other mathematical questions and inquiries. • Evaluate how mathematical analysis has informed and is informed by larger cultural movements.

  14. Course objectives: another rule Students should communicate mathematics effectively to colleagues and client groups.

  15. Content: a vehicle for objectives • Numbers: integers, primes, rationals and reals; modular arithmetic; infinity. • Geometry: Euclidean geometry, Platonic solids, Euler characteristic, Eulerian paths, non-Euclidean geometry. • Probability: chance, randomness, independence; Bayes’ Theorem.

  16. Hints for engagement, I • This course should expose students to significant mathematical ideas. • Active learning is the key. Quest students will not be patient to observe and take notes on a mathematical process or theorem. They will wish to interact with it, to try it themselves, and learn from the experience. • Sometimes you may need to use a lecture format. In such cases, Quest students react best to interactive discussions. Ask questions, have them fill in the blanks, or anticipate the answer --- let them be involved.

  17. Hints for engagement, II • Students’ confidence varies. Math phobia can affect participation. If a student is quiet, it often indicates fear of the subject rather than natural shyness (the latter is rare at Quest). • Students should be made to feel that they can succeed. • Use small groups and the breakout rooms. Among friends, students are often less afraid to take risks and make mistakes. • Activities should allow for phobic students to achieve some success, while also challenging more experienced students.

  18. Hints for engagement, III • A culture of 100% attendance should be established early. The students who can least afford to miss classes are the ones who tend to do so. • Students are particularly good at oral presentations and often quite creative. Remind them to involve their audience. • Block teaching lends itself to depth rather than breadth. Focus on the core ideas and problems. Topics omitted might be covered in projects. • Feel free to link the course with other foundation classes. Students and colleagues will welcome breaking down the walls between classes.

  19. Example: what is a proof? Standard proof that √2 is irrational: Suppose √2=a/b where gcd(a,b)=1. Then a2=2b2 so a is even, say a=2c. Then 2c2=b2 so b is even too – contradiction. So √2 is irrational.

  20. Geometric proof that √2 is irrational

  21. Geometric proof that √2 is irrational

  22. Example: problem solving Fibonacci’s rabbits: • start with one pair, • pairs reach sexual maturity after one month, • mature pairs produce a new pair every month. How many pairs are there after 12 months?

  23. Rule of Three applied • Examining numerical data shows that the number an of pairs in the nth month satisfies an+1=an+an-1 • Graphing the data suggests an grows exponentially: an≈crn • Algebra finds the golden ratio: r=(1+√5)/2.

  24. Conclusions • Course Objectives • Think algebraically, geometrically and numerically. • Solve problems. • Step back and look at the big picture. • Communicate by talking and writing. • Engagement • Let the students explore, hypothesize, argue, explain.

  25. Quest for a new beginning