Quest for Engagement John Hannah
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
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
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?
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.
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
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.
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).
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.
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.
Course objectives: another rule Students should communicate mathematics effectively to colleagues and client groups.
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.
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.
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.
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.
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.
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?
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.
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.