With enormous thanks to Anita Campbell for taking these notes.
Blogging from The Tenth Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics
Stuart Torr Centre for Research in Engineering and Science Education.
Sasol Inzalo Foundation
Stuart is a PhD student of Tracy Craig.
Mathematics is very abstract. Students sometimes complain about this. Developing abstract reasoning is key in mathematics.
Abstraction has 2 components
- A process involving decontextualizing and generalising
- Abstract objects and concepts, e.g. the economy, justice, equality, numbers, functions
An abstract object is the end result of an abstract process.
Two schools about dealing with abstraction:
1) Cognitive/empirical approach (Piaget)
- Empirical abstraction. Recognise similarity between objects
- Pseudo-empirical. End product like number 5
- Reflective abstraction. Performing operations on
2) Socio-cultural / dialectical approach (Davyov, Russian inspired by Vygotsky)
- Recontextualisation. Making new links with objects
- Non-linear / dialectic development. Back and forth between objects.
“Ascension to the concrete,” vague to elaborate. Developing an understanding of a mathematical object by refining …
Children are capable of abstraction, e.g. stick drawing of a dog.
Maths Education thinking on processes and objects:
- Sfard – reification. Think of mathematical objects in operational or structural ways.
- Dubinsky – APOS: Action, Process, Object, Structure
- Gray and Tall – Procepts
Shifting between levels is difficult. Focusing attention is important.
It’s difficult to investigate abstraction empirically or determine what caused the shift. Like Father Christmas – you can see the results but it’s hard to catch in the act. (Simon)
Learning through activity (Simon)
AIC (Hershkowit)
Abstraction and Construal level
High level = abstract
- Coherent
- Superordinate
- Simpler
- Goal dependent
Concrete property can be high-level. “Getting it right” is not a consideration.
Abstract representations aren’t good on their own – you have to get the right one.
Construal level theory (CLT)
Distance and abstract thinking go together. To get students to think in a more abstract way, get them to think about something occurring in a far-away country or the moon and they will be primed to think more abstractly.
Far mode and Near mode
Star Wars advert, “In a galaxy far, far away …” primes you to think of the characters differently to if the setting was nearer to you.
Intuitions focus influence.
E.g. perception drawings that you can recognise if you are familiar with the object drawn abstractly.
Implications for maths education
General purpose tool for focusing attention. You’re more likely to see that 
Performance on word problems (Schley & Fujita).
An extremely large number is far mode thinking. In early algebra, they progress to 100 and then generalise to n.
Most of the constructs we use are bipolar and we impose meaning. (Kelly, psychological theory). You can make meaning by using a triple and finding similarities between two and seeing how they differ from the third.
Placing undergraduate mathematics assessment on the national Higher Education agenda
Deborah King and Cristina Varsavsky (Cristina presented)
Context
- Changing higher education landscape
- Development of threshold learning outcomes (TLOs) in Science and Mathematics
How do we demonstrate that graduates have the outcomes we desire? That is the reason for assessment.
There was not much in literature (or last Delta conference) on assessment. This motivated their work.
Assessment in undergraduate mathematics
- High failure rates
- Students focus on answers, not on the logic followed to obtain the answers
- Shopping for marks is common (viewing scripts for marks rather than learning)
However
- Assessment practices have not changed for decades.
- 70% is closed book exams
- Little variety in other 30%
- Ticks, crosses, normalising marks is common.
Why are we not using criteria? (Like in humanities, etc.)
It’s too hard:
- Workload for lecturers
- Even more students will fail
- Fear of scrutiny
Start a national conversation around assessment to solve the problem of applying assessment criteria. Conversations covered much more. 20 – 30 participants in workshops, held around Australia.
Challenges and Insights
- Unaware of standards agenda – for ‘central’ staff to deal with
- Not used to talking about assessment in departments
- Individual activity, no communication between tutors, lecturers, coordinators
- The same piece of work received very different marks by maths educators.
- It’s difficult to articulate how to allocate marks.
- Communication of expectations is mostly verbal, through role modelling or model solutions
- Opposite practices regarding strictness / leniency as term progresses.
- Tutors left to interpret how to allocate part marks.
- Unusual to have marking meetings with tutors.
Use of rubrics
Trialing different versions in workshops in different cities.
Looked in literature, couldn’t find much regarding higher education. Much in high schools but they were too broad to adapt for higher education.
Clear learning outcomes need to be established before developing assessment tasks and rubrics.
Workload: an initial investment but bears fruit down the track.
Generic assessment criteria don’t work. Taylor criteria around specific task.
Better effort from students if they are clear on assessment criteria.
Resource: Maths Assess book, contains basic principles of assessment and exemplars of typical assessment tasks.
Mathsassess.org
2.5 year project.
Can download book from website.
Ask ‘what are you giving marks for?’ his is the first step. Students tend to get or loose marks for the same thing several times in an assessment.
Are students interested in reading assessment criteria? Some not. Less arguing for marks because they know how they are being assessed.
Moodle can pop up a rubric as work is submitted.
Opening Real Science
Leigh Wood
Working with pre-service teachers.
$2.3 million grant (R23 million)! A large project.
Aim to link ‘real’ scientists with educators to improve the outcomes in STEM for school students.
Challenge: Values of maths and science department were about the discipline; the values of education department were about the students.
Australia has a new maths curriculum, including statistics and financial maths.
Online modules, each 4 weeks work.
- To infinity and beyond
- Statistical Literacy
- Smart budgeting
- Investing and Protecting, includes superannuation, credit, protecting through insurance
Scale is a theme used across all modules.
Design principles
Inquiry-based … (see slide)
Design requirements
On-line delivery, currently as a moodle site hosted by Macquarie University. Available for others to use.
Google Opening Real Science and look for link to Moodle site.
Sarah.rosen@mq.edu.au for guest access.
Investigating how past experiences in mathematics have influenced pre-service primary teachers
Dilshara Hill, Macquarie University
Pre-service Primary Teachers
Characteristics observed:
Maths anxiety, more so than other maths students.
Aim to see if and how past experiences have influenced them, also their attitudes and past level of mathematics.
Literature
Cobb(1986) beliefs are related to social experiences, such as in classrooms.
(others)
Data collected
- Personal info
- Attitudes
- Past experiences
87% females, 71% 18 – 25 yrs old.
Maths level:
- No yr 12 maths (22%)
- Non-calculus based course (51%)
- Calculus-based course (17%)
(Typical of previous years too.)
Circle the words from 6 positive, 6 negative works, could add own
More negative (40%) than positive words (30%). 28% had both positive and negative experiences.
Asked about positive or negative experiences
48% only positive
25% only negative
9% neutral / no response
(rest had both positive and negative experiences)
8 positive and 10 negative types of experiences
Top positive experiences
- Good marks
- Good teacher (90% said teacher contributed to negative experience)
Top negative experiences
- Teacher (94% said teacher contributed to negative experience)
- Not understanding work
Found a relationship between past experiences and attitudes
People with
Found a significant relationship between past experiences and attitudes
82% who had positive experiences did maths to year 12
95% who had positive experiences did maths to year 12
Discussion
- Teachers have a huge influence on their students.
- Implications for educators:
- Are educators sufficiently aware of how much impact they could have on their students?
- Do they set reasonable goals to give students opportunities for success?
- Addressing negative attitudes of pre-service primary teachers
- Mathematics in today’s society
Is there an assumption that mathematics is not valued or desires in primary education? Good students are not expected to go into primary education.
- Prerequisite mathematical knowledge. Should year 12 Maths be a pre-requisite for primary school teachers? Yes, says Dilshara.
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