Strengthening the Problem-Solving Abilities of Our Students

1. Strengthening the Problem-Solving Abilities of Our Students Emma Carberry, School of Mathematics and Statistics.

2. Faculty of Science Graduate Attributes • Research and Inquiry: to be able to create new knowledge and understanding through the process of research and inquiry. • Personal and Intellectual Autonomy: to be able to work independently and sustainably, in a way that is informed by openness, curiosity and a desire to meet new challenges.

3. Faculty of Science Graduate Attributes • Communication: to recognise and value communication as a tool for negotiating and creating new understanding, interacting with others, and furthering their own learning. • Information Literacy: to be able to use information effectively in a range of contexts.

4. Faculty of Science Graduate Attributes • Ethical, Social and Professional Understanding: to hold personal values and beliefs consistent with their role as responsible members of local, national, international and professional communities.

5. We place a very high value on students learning to • develop new knowledge, • formulate and solve challenging problems, • think for themselves.

6. We recognise the importance of good communication, for example in • working effectively as part of a team • presenting solutions to a broader audience.

7. Aligning undergraduate teaching with these goals In Honours / Postgraduate studies, we focus on research and communication skills. The • one-on-one / small group interactions • higher level of preparedness of the students make this easier than in the undergraduate setting.

8. In undergraduate classes, we certainly give students problems to solve but there can be a disjunction between • the emphasis on research and communication skills stated in our graduate attributes, and • the emphasis in our classrooms on helping students acquire and strengthen these skills.

9. In undergraduate classes • time pressures often prompt us to focus on the acquisition of knowledge • class sizes make it more challenging to mentor students to think for themselves • students are less prepared.

10. How best to help? Students need to learn independence. But that doesn’t mean we should leave them to figure it all out for themselves.

11. We are like the leaders of an expedition. We need to provide guidance not only for acquiring technical knowledge but also for the more challenging goals: • becoming independent thinkers, • developing research and inquiry skills.

12. A practical case study In mathematics, tutorials are our golden opportunity: • smaller class sizes (≤30) • focus on problem solving • if the students come prepared, can engage with them at a time where they have struggled with a problem but it is not yet “done”.

13. Traditional mathematics tutorial • tutor summarises key aspects of theory and presents some problem solutions • tutor helps students individually • students work on problems they had not previously attempted/completed, both individually and in groups.

14. Helps students learn material, but didn’t align well with goals: • many students have not made a serious attempt at the problems before the tutorial • focus on individual learning rather than group interaction • few opportunities for “big picture” mentoring • few opportunities for improving students’ communication skills.

15. My goals First, create a situation where: • students are deeply engaged with a specific problem • the ground work is done but there is still more to discuss / learn about the problem • it is a collaborative environment • it is a supportive learning environment.

16. Then, moderate discussions to: • keep everyone thinking, in the moment and help students • analyse the structure of what they are trying to do • develop strategies for getting themselves “unstuck” • reflect upon what broader skills they have gained from solving a specific problem.

17. Differential Geometry Tutorials Motivation: • give strong, consistent message that tutorials are the core of the subject • explain what the goals are and how I expect them to be achieved • emphasise benefits • set clear expectations, make students accountable for preparing thoroughly.

18. Tutorial StructureFour types of questions: • Problems to write up and discuss • students bring written solutions • in small groups discuss and help one another • peer feedback on mathematics and exposition • mark one another’s work • I circulate and help each group.

19. Includes a challenging problem, I often ask someone with a good solution to present it. • Explanation complemented with strategic advice • how one could have thought of this. • what have we just learned, beyond the specifics? • If appropriate, I pose extension questions.

20. Problems for presentation and discussion • different groups assigned to present each week • questions encouraged, to build discussion • for each student there is both a “mathematics mark” and a “presentation mark” • detailed constructive feedback at end of class

21. Other required problems • problems I want the students to do each week, but which I am not expecting we will discuss in the tutorial. • Recommended problems • extra resources for the conscientious, builds repository of problems and solutions for exam study.

22. Some USE survey results The teaching in this unit of study helped me to learn effectively. Mean 4.08 2006 4.38 2009 2011 4.32

23. This unit of study helped me develop valuable graduate attributes [e.g. 1) Research and inquiry skills; 2) Communication skills; 3) Personal and intellectual autonomy; 4) Ethical, social and professional understandings; 5) Information literacy] Mean 3.78 2006 4.04 2009 4.28 2011

24. Tutorial classes and/or laboratory classes were worthwhile. Mean 4.27 2006 4.58 2009 2011 3.94

25. Overall I was satisfied with the quality of this unit of study. Mean 4.45 2006 4.41 2009 2011 4.24

26. “This is probably the first maths I've done in my (formal) education for which I had to think creatively beyond what had been taught.” “This is a hard unit, but the way it was taught made it enjoyable and relatively easy to follow but without making the unit easy. I love a challenge and that was this unit.” “The lecturer was very encouraging, and good at getting people involved.”

27. “The most impressive aspect of Dr. Carberry's teaching was the manner in which she ran tutorials. Students were allocated difficult problems, and each week a different student was asked to deliver a solution on the whiteboard to the rest of the class. While this was daunting for us at first, this approach not only developed our presentation skills -- the first and only undergraduate course I encountered that did so -- but it also created a fun and cooperative environment in which to learn.”