Strengthening the Problem-Solving Abilities of Our Students

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Strengthening the Problem-Solving Abilities of
Our Students

• Emma Carberry, School of Mathematics and
  Statistics.

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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.

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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.

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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.

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

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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.

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• 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.

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• 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.

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How best to help?

• Students need to learn independence.
• But that doesnt mean we should leave them to
  figure it all out for themselves.

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• 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.

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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.

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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.

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Helps students learn material, but didnt 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.

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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.

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• 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.

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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.

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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 anothers work
• I circulate and help each group.

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• 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.

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• 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

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• 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.

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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
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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
2011
4.28
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Tutorial classes and/or laboratory classes were
worthwhile.
Mean
4.27
2006
2009
4.58
2011
3.94
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Overall I was satisfied with the quality of this
unit of study.
Mean
4.45
2006
4.41
2009
2011
4.24
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• 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.

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• 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.