LUMOS Learning in Undergraduate Mathematics

Report
Capturing Undergraduate
Learning
An Ako Aotearoa/TLRI Project
Bill Barton, Judy Paterson,
Greg Oates, Caroline Yoon
The idea behind the Project #1
The team consists of 31 people, mostly from this
department, and the funding body.
We share the assumption that things can be improved.
Indeed, we share an assumption of continual, rational
development.
How would you improve undergraduate
learning?
Think for a moment what you
would do to enhance students’
undergraduate mathematics
experience.
How would you know that an improvement
had taken place?
The idea behind the Project #2
In this project we are
turning our attention
to learning ata class
level.
We want to understand how the way we
deliver a course affects the student learning
that results.
The idea behind the Project #3
Yes, we do believe that we can significantly enhance
learning in an undergraduate mathematics degree.
Sources of this belief are:
(i) research knowledge1
(ii) the questioning of the dominance of traditional
lecturing2
(iii) our observations of the conservative response
to the changing educational environment
(technology3a, student body3b, economic
pressures3c).
1.
2.
3a.
3b.
3c.
D Holton (Ed.) (2002) The Teaching & Learning of Mathematics at University Level. The 11th ICMI Study. Kluwer Academic Pubs.
A Ryan (2012) Massive Black Mirror. Times Higher Education, 4 October, 2012.
This Department
E. Anderson (2003) Changing US Demographics and American Higher Education, New Directions for Higher Education, No. 121
The assumption that lecturing to larger audiences is the only way to increase student/staff ratios.
The idea behind the Project #4
But we do not believe in change for change’s sake.
So, we want to establish a rational basis for the
improvement of undergraduate teaching.
That is the main idea behind the project.
We aim to develop a way to know how a course
contributes to the desired learning outcomes for
undergraduate mathematics.
Only if we can do this do we have a reason for
changing the way we do things now.
The idea behind the Project #5
NOTE:
We do not expect that this research will result in
fundamental change to all courses, or, say, the demise
of lecturing.
Rather, we expect that we will obtain evidence that
different types of courses contribute to student
learning in different ways, and therefore the
department might decide to arrange things so that
students have a range of opportunities to meet the
learning expectations of lecturers and employers.
Components of the Project
Developing a Course Learning Profile 1. Identify the broad spectrum of desired learning outcomes
2. Find ways to observe these learning outcomes
3. Analyse and report learning outcomes for courses as a Course
Learning Profile (CLP)
Course Innovations
1. Team-based learning
2. Intensive Technology
3. Low lecture
Project Extensions
1. Canterbury & Victoria Mathematics & Statistics Departments
2. Law, English, Psychology and Dance at Auckland
Main Component of the Project
Developing a Course Learning Profile 1. Identify the broad spectrum of desired
learning outcomes
2. Find ways to observe these learning
outcomes
3. Analyse and report learning outcomes for
courses as a Course Learning Profile (CLP)
Developing a Course Learning Profile
Our first task is to identify, and categorise, ALL the learning outcomes
desired in an undergraduate mathematics course.
1.
2.
3.
4.
Those desired by the lecturers of the course
Those desired by lecturers of subsequent and graduate courses
Those desired by the university in Graduate Profiles
Those desired by employers of mathematics graduates
For Example
1.
2.
3.
4.
5.
6.
7.
8.
Mathematical content
Mathematical skills
Mathematical processes
Mathematical habits
Attitudes towards mathematics
Mathematical communication
General thinking and learning behaviours
……. ????
Observing & Reporting Learning Outcomes
1.
Mathematical content—examinations, assignments
2.
Mathematical skills—examinations, assignments
3.
Mathematical processes—observation in tutorials?
specially designed tasks?
4.
Mathematical habits—observations? self-report?
5.
Attitudes towards mathematics—surveys
6.
Mathematical communication—observations in tutorials?
assignments?
7.
General thinking and learning behaviours—self-report?
class participation observations?
8.
……. ???? ????
Other Components of the Project
Course Innovations
1. Team-based learning
2. Intensive Technology
3. Low lecture
A Team-Based Learning model of
delivery:
1. Shifts responsibility towards the students.
2. Provides feedback to students and lecturers.
3. Uses some lecture time for students to work in teams
on tasks that apply ideas and concepts.
4. Allocates students to teams as fairly as possible for
the duration of the course
This approach is used in Maths 326
See www.teambasedlearning.org/
The Readiness Assurance Process (RAP)
The students do a multiple choice test on the
pre-reading.
They then do the same test in their team.
How?
IF-AT scratch and win score cards
Immediate feedback for students
… the instant feedback makes sense - to learn
from our mistakes and adapt to our environment
this kind of testing is far more beneficial than a
number out of 10 you receive a week later.
Feedback to lecturers
Feedback is most powerful when it is
from the student to the teacher. When
teachers are open to feedback from
students, then teaching and learning can
be synchronized and powerful. Feedback
to teachers helps make learning visible.
(Hattie 2009)
Hattie J (2009) Visible Learning; a synthesis of over 800 meta-analyses relating to achievement London;
Routledge
• Business as usual for much of the course
• What to do with the lecture time we saved?
• Take a holiday?
• Students do tasks in teams – apply ideas
• What sort of tasks have proved successful?
Example of a Maths 326 task – the
team hands in one A4 sheet.
• A parliament with 99 seats has three parties,
A, B and C. Laws can be passed by a coalition
of parties with at least 50 seats.
• The largest party, C, splits into two factions.
Investigate what happens to the power of the
parties when party C splits into two smaller
parties.
• You might ask these or other questions about
the shifts in power when a party splits:
– Will the power of the two factions add to
the pre-split power of Party C?
– Can a dummy player gain power because of
the split?
– How much shifting in power can occur?
A good solution to this task will ask and
answer further questions about the effects of
the split of Party C.
What sorts of things do we want to
take a long, hard look at?
• What sort of questions do we hear students
ask as they work? Are they good questions?
• What evidence can we find of mathematical
behaviour that we value – “I wish my students
would……”
• I am sure they are behaving more like
mathematicians but how can I convince other
people that it is happening?
Other Components of the Project
Course Innovations
1. Team-based learning
2. Intensive Technology
3. Low lecture
Intensive Technology Innovation
• We are talking about “Effective” use of
technology, not indiscriminate use;
• Its here, and its use is growing rapidly – we
cannot ignore it if wish to prepare our
students for mathematics in a modern age;
• TSG 13: ICME-12: Teaching & Learning of
Calculus - 17 presentations (12 countries);
Technology was a major theme, even though
two large Technology-specific TSG’s
Ponce-Campuzano & Rivera-Figuero (Delta 2011):
Value of CAS (various technologies) to reveal particular
aspects, especially when considering domains of antiderivatives, where CAS may provide inconsistencies and
alternatives to by-hand solutions.
Compared the solutions provided by a variety of CASsoftware (Derive 6.0, Scientific Work Place 5.5,
Mathematica 8.0, Wolfram Alpha) when used to compute
antiderivatives of functions.
The many examples where different CAS-technologies and
by-hand computations yield different results are very
interesting.
Tasks & Assessment: Definitions
Technology Trivial: 1999 Maths 102
2x  5
If f ( x) 
, find f ( x)
3x  1
Can still ask such questions in the skills tests for skills deemed
necessary.
Technology Neutral: 2007 Maths 108
Technology Neutral……
Thomas & Klymchuk (Delta 2011)
Sketch the graph of a function f(x) such that it is
continuous on 0  x  3 and 3  x  5, and
f ( x)  0 for 0  x  5, x  3.
Does limx3 f ( x) exist for your function?
Technology Active (CAS-positive)
Lin & Thomas – Proceedings of Delta 2011.
Technology can help from an exploratory perspective, which will help
understanding, but it will not answer the question directly.
This is neither a technology neutral or technology trivial question.
Technology Active – more examples
Thomas & Klymchuk (Delta 2011)
1. If

a
0
(2x  1) dx  2
find the possible values of a.
2. Find the derivative of the function: y  ln2 sin(3x)  4
3. Find the derivative of the function:  1 1 dx
1 x
Tobin & Weiss (Delta 2011): Use of CAS in differential
equations in course examinations, ways of posing
questions to be CAS-active.
Question 1
Solve dy  xy  2 y  2 x  4 for y as a function of x
dx
This question involves solving a first order separable differential
equation which can be trivially done with just one command using
CAS.
Question 1A
Show that the following DE is separable and hence or otherwise solve
for y as a function of x. dy
dx
 xy  2 y  2 x  4
This allows the testing of separability and still gives a chance for an
answer to be found or checked by CAS, using an appropriate marking
schedule.
“Activating” technology trivial questions
This can be quite time-consuming if you have a data-base of
existing questions, and can take a while adjusting. However, it
becomes easier & you gradually build up a new database.
Example: Standard definite integration problems:
Find

7
2
(3x 2  2 x)dx What might we do to transform such a
question?
Consider:
If

7
2
7
( fx)dx  20, find  ( fx  4)dx
How about:
Still trivial?
2

b
a
b
( fx)dx  c, find  ( fx  4)dx
a
Often, technology-positive questions are more conceptually difficult
than the standard skills-style questions we may have posed in the
past;
Other Components of the Project
Course Innovations
1. Team-based learning
2. Intensive Technology
3. Low lecture
Low Lecture Innovation
The three key ideas behind this innovation are that:
(i) lectures are not the best means of imparting
information or developing skills, although they are
useful for overviews, “colour”, and modelling;
(ii) the best learning takes place when students are
themselves engaged, both individually and
together;
(iii) responsibility for content and skills learning will be
handed back to students using print and web
resources, but with the means for self- and
lecturer-monitoring of progress.
Low Lecture Innovation
A voluntary stream (max 32 students) of
MATHS 108 will be established in Semester 2,
2013, taught by the research team as extra to
load.
One lecture per week. Five Engagement
Sessions of 2hrs plus pre- and post-work.
These substitute assignments. Tutorials, tests,
exam all the same.
Significant on-line resources and monitoring
systems.
Timelines
Nov/Dec
2012
Identifying
Learning
Outcomes
Observing
Learning
Outcomes
Analysing &
Reporting
Outcomes
Developing
Innovative
Courses
Teaching &
Observing
Courses
Summer
2013
Sem 1
2013
Sem 2
2013
Summer
2014
Sem 1
2014
Sem 2
2014
Summer
2015
ILIUM
Investigating Learning In Undergraduate Mathematics
LUMOS
Learning in Undergraduate Mathematics and Other Subjects
(or Operationalising the Spectrum)
PLUMP
Profiling Learning in Undergraduate Mathematics Project
CLUMSY
Capturing Learning in Undergraduate Mathematics Spectrum
MULLET
Mathematics Undergraduate Learning Lectures Engagement Technology
CALCIUM
CApturing Learning Concepts In Undergraduate Mathematics
Learning
Undergraduate
Mathematics
Capturing
Profile
Low Lectures
Technology
Team Based Learning
Project
Investigating
Research
Spectrum
Send entries to Caroline ([email protected]) by midday 29th November
Thank you for your attention …
… and your future assistance.
Bill, Judy, Mike, Greg, Caroline,
Louise, Fiona, and Barbara
and 23 others.

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