## Elephant Delta Day 3 – Dr Alice Hui from The University of KwaZulu-Natal on Projective geometry as an undergraduate course: a tour of the three worlds of mathematics

Projective geometry as an undergraduate course: A tour of the three worlds of mathematics.

Attempt to persuade the audience to have projective geometry as an undegraduate course.

What is Projective Geometry?

Projective geometry describes objects as they appear, rather than as they are.

Two sides of a train track are parallel but look interest in the far end.

Taken from https://plus.maths.org/content/sites/plus.maths.org/files/features/projective/tracks1_small.jpg

• An axiomatic approach to study geometry
• A projective plane satisfies three axioms.
• Can allow students to study geometry using an axiomatic approach.
• Comparison to other axiomatic formulations that may be taught in undergraduate courses:
• We use axioms to study natural numbers, real numbers, group, ring, field, vector space, set theory. They are not geometric.
• Euclid’s Element is not rigorous in modern sense.

## Elephant Delta Day 3 – Professor Stephan Joubert from Tshwane University of Technology on Numerical solutions for the vibrating bar eigenvalue problem: a student exercise

Abstract of the problem:

In 1890 G.H. Bryan observed that when a vibrating structure is rotated with respect to inertial space, the vibrating pattern rotates at a rate proportional to the inertial rate of rotation. This effect, called “Bryan’s effect”, as well as the proportionality constant, called “Bryan’s factor”, have numerous navigational applications. Using a computer algebra system, we present a numerically accurate method for determining fundamental eigenvalues (and some of the overtone eigenvalues) as well as the corresponding eigenfunctions for a linear ordinary differential equation (ODE) boundary value problem (BVP) associated with a slowly rotating vibrating disc. The method provides easy and accurate calculation of Bryan’s factor, which is used to calibrate the resonator gyroscopes used for navigation in deep space missions, stratojets and submarines.

## Elephant Delta Day 3 – Dr Maritz Snyders from Nelson Mandela Metropolitan University on Online video-based tutorials for a first-year mathematics service course

Dr Maritz Snyders

• A tutorial is a method of transferring knowledge and may be used as a part of the learning process.
• A period of instruction given, especially to one or two students

What is the purpose of a tutorial/practical session?

• Allow the students to apply knowledge outside of the normal situation
• Force them to do something practical

• Small groups, face to face (25-30 students) weekly meetings
• Students work through provided exercises
• Guidance from lecturer or assistant when stuck

Problems:

• Scheduling due to variety of programmes doing same course
• Large numbers
• Available venues
• Available staff

Structure:

• Replacing face-to-face contact with exercises posted online
• Student completes exercises and submit assignments in their own time

• Fewer staff
• No scheduling problems
• Easier for large groups

• No immediate support
• Guessing
• No guarantee students actually write out detailed solutions
• Only written feedback without explanations

Can we design an online system that has all the advantages but without the disadvantages?…

## Elephant Delta Day 3 – Claire Cornock from Sheffield Hallam University on Teaching group theory using Rubik’s cubes

See presentation here.

Backgrounds: Maths at Sheffield, Hallam University:

• Real, practical, development of employability skills – in a very applied degree program
• Students struggle with abstract concepts

Pure maths modules:

• Knot theory, linear and discrete maths, number and structure, abstract algebra.
• Abstract Algebra: roughly 75% of students taking the module took the questionnaire.

Teaching methods:

• Groups of 30 students
• 2 hour workshops
• Support groups
• Partially printed notes
• All given a Rubik’s cube

They don’t have to be able to solve a Rubik’s cube!! They can always swap theirs for a new one.

Name different moves: ie. move front face 90 degrees clockwise. See here:

• subgroups
• generators
• homomorphisms
• equivalence relations
• permutations

Let $G$ be a group and let $g, h\in G$.…

## Elephant Delta Day 3 – Prof Maxine Pfannkuch from The University of Auckland on Visualizing chance in introductory probability

The first Judy Paterson Speaker

Visualising chance in introductory probability

The problem

• Random events and chance phenomena permeate our lives and environments: volcanic eruptions, epidemics, crashes – need probabilistic reasoning to understand these.
• Current teaching approach in intro prob is mathematical – obscures nature and role of randomness in processes and the nature of chance phenomena
• Many probability misconceptions are prevalent in people’s thinking (Kahneman, 2011)
• Appear resilient to teaching – how do we get students to confront these misconceptions?
• After 30 years of research a re-think is needed

Possible solutions

• Use a modelling approach to probability
• Construct a model (Konol and Kazak)
• Explore model behavior (Pratt, 2005)
• Use dynamic visualizations to give students an opportunity to:
• Experience random behavior through simulations
• Visualize chance through creating new representation infrastructure

Study: Part 1

Interviewed seven practitioners of probability (teachers,etc.) (at least 2 hours per interview)

• What probability concepts need to be promoted?

## Elephant Delta Day 2 – Prof Cristina Varsavsky from Monash University on Placing undergraduate mathematics assessment on the national Higher Education agenda

With enormous thanks to Anita Campbell for taking these notes.

Prof Cristina Varsavksy from Monash University.

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.

• 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.

## Elephant Delta day 2 – Prof Leigh Wood from Macquarie University on Opening Real Science

With enormous thanks to Anita Campbell for taking these notes.

Prof Leigh Wood from Macquarie University.

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.…

## Elephant Delta day 2 – Dilshara Hill from Macquarie University on Investigating how past experiences in mathematics have influenced pre-service primary teachers

With enormous thanks to Anita Campbell for taking these notes.

Dilshara Hill from Macquarie University.

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.…

## Elephant Delta Day 2 – Prof Leanne Rylands from Western Sydney University on Great Expectations: Expectations and attitudes of first-year mathematics students

Prof Leanne Rylands -image taken from here.

Live blogging: Note that these are notes I’ve taken live, but will edit this today into a more readable format. I want to put this up straight away though to see if I have any obvious misunderstanding. Equations will also be put into more readable format ASAP.

Context: 42,000 students at Western Sydney University – founded in 1989.

Background:

• A low level maths subject
• Poor mathematics background – getting worse
• Failure rate: 40-60% failure rate
• No maths prerequisites

Asked the students: Do you expect to fail? What grade do you expect to get?

All expected to pass – many expected to get very high grades

Students’ perspectives are different from the staff.

• What do students see at the start of the semester?
• Do their expectations change over the semester?

## Elephant Delta Day 2 – Karin Bothma from The University of Pretoria on Using clickers in the mathematics classroom: novelty or necessity

Karin Bothma – image taken from here.

Live blogging: Note that these are notes I’ve taken live, but will edit this today into a more readable format. I want to put this up straight away though to see if I have any obvious misunderstanding.

First time using clickers to teach mathematics.

Background: Large undegrad classes: 900 students for classes between 70 and 400.

High impact module (foundational)

Objective was successful learning.

Use clickers to promote active participation in class

Clickers are classroom communication system

Students used the clicker to respond to multiple choice questions on a screen

Class distribution of answers then displayed

To meaningfully influence student learning, clickers must be used skillfully…(Beatty, 2004)

Aim of the study:

How we do we design the questions appropriately.

Examine principles for effective use of clickers

CCS (classroom communication system) based pedagogy

Clicker questions: Designed with goals in mind:

1. Content goal-concept or concepts
2. Process goal – cognitive skill
3. Metacognitive goal-  beliefs about mathematics

Strategies:

1. Focus student attention (comparative questions)
2. Encourage cognitive processes (compare and contrast)
3. Encourage debate (set a question to highlight assumptions)
4. Provide opportunity for feedback

Questions that initiate discussion: shows a wide distribution in the answer histogram

Class discussion is essential for effective use in mathematics classroom (Kenwright, 2009)

Methodology

• TurningPoint: Voting within Powerpoint
• Clickers with multiple choice capability
• Each lecturer could set 3 questions per week
• Questionnaire on students’ experience distributed

Experience of lecturers:

• Setting multiple choice questions was not a challenge
• Intentionally designing and planning questions was difficult
• Time constraints or technical problems (sometimes questions not used)
• Practice of separate question and answer slides

Class discussion was based on how the students got the questions wrong

Misconceptions often illuminated

Metacognitive goals: Relational understanding

Experience of students: Students strongly agreed that clickers contribute to their learning, and that it improved their active participation

They found that clickers made the students aware of their own understanding.…