Developing university mathematics teaching
through collaborative inquiry: teacherresearchers working and learning
TSUKUBA -- SEAMEO -- 2016Collaborative teaching-research team
3 experienced teachers working with a mathematics module for first year
engineering students– 2 with long experience of teaching mathematics to engineering students:
roles – design, discuss, advise, analyse data
– 1 with long experience of mathematics teaching through inquiry approaches:
roles – design, teach, reflect, feed back, analyse data

1 educational researcher– With experience of working in university mathematics education:
roles – collect data, stimulate reflection and feedback, analyse data2

Innovation in teaching mathematics toengineering students (ESUM)• Seeking to promote engineering
students’ more conceptual
understandings in a first year
mathematics module

Teaching team designs tasks aroundmathematical topics to encourage
student engagement and inquiry inmathematics• Researcher observes and records all
lectures and tutorials, surveys students
and conducts post module focus groupinterviews

For example, a task relating to functions
with a focus on linear/quadratic
relationships

3Exploring functions and equations – an inquiry-based task
1a) Consider the function
f(x) = x2 + 2x (x is real)
Give an equation of a line thatintersects the graph of thisfunction:
Twice; Once; Never.
1b) If we have the function
f(x) = ax2 + bx + c.
What can you say about lines
which intersect this function
twice1c) Write down equations for threestraight lines and draw them in
GeoGebra
Find a (quadratic) function suchthat the graph of the function cuts
one of your lines twice, one of them
only once, and the third not at all
and show the result in GeoGebra.
Repeat for three different lines(what does it mean to be different?)

10MAA309 2 Polynomial Functions

4

The innovation and its goals (the micro)
4 new elements

Goals

•

Inquiry-based teaching/learning

To engage students in/with mathematics•

Geogebra environment for work on To encourage linking of different
representations
functions•

Small group activity in tutorials

To facilitate dialogue, interchange andsharing of ideas•

Small group project – assessed

To motivate activity

Jaworski & Matthews (2011)
Jaworski, Robinson, Matthews and Croft (2012)

Jaworski (2015)5

Macro factors• 2-semester module – 2 teachers – formal exam at end (innovation in firstsemester only -- 2 lectures & 1 tutorial per week for 13 weeks)• Given content specification (pre-calculus, calculus, vectors & matrices …)• Designated lectures and tutorials in traditional lecture theatres andseminar rooms• Students come from school learning/teaching experiences• Students’ wide range of mathematical experience (some don’t have A
levels in mathematics)• First year grades do not count to the degree
6The Sociocultural Scene• Considering the micro AND the macro– Micro: teaching/learning activity, interactions and insights– Macro: wider influences impacting on the micro• Sociocultural focus links teaching goals and classroom interactions withinstitutional, systemic and cultural influences:• e.g.,– considering the nature of the setting and how factors involved in the academic
infrastructure affect teaching/learning activity– looking at the cultural underpinnings of perspectives and actions

7

Developmental research: knowledge creation• Research which promotes development as well as charting or evaluating
it. Fundamentally an inquiry process• Teaching-research team – designing, teaching, analysing data, evaluating –insider researcher – knowledge in practice• Action cycle:
Plan  Act & Observe  Reflect & Analyse  Feedback• Researcher – observing, surveying, interviewing, analysing data – outsider
researcher – abstracted knowledge

8Inquiry communities -- three layers of inquiry
Inquiry in

Macro

Inquiry inInquiry in learning
mathematics

Microteaching mathematics
the research process9Outcomes from the ESUM Project• Centrality of questioning• Inquiry-based questions(tutorials)• Use of GeoGebra• Small group activity• Small group projects• Tests and Exam
• Student perspectives• Students responding in
lectures• Variety of groupresponses• Variety of responses• Variety of responses• Generally well done (Web PA)• Average score 10% higher
than previous• Evidence of conflict10

We feel being able to explore functions as agroup has helped our learning about
functions as we can discover together andask each other questions about how theywork and what they can be useful for, andwhere one student questions something,
the whole group benefits from theiranswer.
Understanding maths – that was the pointof GeoGebra wasn’t it? Just because I
understand maths better doesn’t mean I’ll
do better in the exam. I have done less past
paper practice.

Student Perspectives
I found GeoGebra almost detrimental
because it is akin to getting the question
and then looking at the answer in the backof the book. I find I can understand thegraph better if I take some values for x andsome values for y, plot it, work it out then I
understand it … if you just type in somenumbers and get a graph then you don’t
really see where it came from.

11

Learning from outcomes of research• Feeling pleased with what went well
• Feeding back into ongoing teaching and future teaching fromobservations and issues arising• Asking questions about how to improve on what did not go
well• Recognising the big issues in differing perspectives between
teaching team and student cohort (micro and macro)• Seeking ways of addressing these differences in perspective
12

Activity Theory -- Leont’ev’s three levels

activity<->motive, actions<->goals, operations<->conditions
Level
Teaching Team
1
Activity is mathematics teaching-learning. For the teacher(s) it
is motivated by the desire for students to gain a deep
conceptual-relational understanding of mathematics. We
might in this case call it “teaching-for-learning”
2

3

Students
For students the activity is learning within the teachingenvironment and with respect to many external factors (youth
culture, school-based expectations of university etc.) and is(probably) motivated by the desire to get a degree in the most
student-effective way possible.
Here, actions are design of tasks and inquiry-based questions – For students, actions involve taking part in the module:with goals of student engagement, exploration and getting
attending lectures & tutorials; using the LEARN page; using the
beyond a superficial and/or instrumental view of mathematics. HELM books; etc. with goals related to student epistemology.
Actions include use of GeoGebra with the goal of providing an So goals might include attending lectures & tutorials because
alternative environment for representation of functions
this is where you are offered what you need to pass the module;
offering ways of visualizing functions and gaining insights
clear views on what ought to be on offer and what you expect
into function properties and relationships. Actions include
from your participation; wanting to know what to do and how toforming students into small groups and setting group tasks
do it; wanting to do the minimum amount of work to succeed;with the goals to provide opportunity for sharing of ideas,
wanting to understand; wanting to pass the year’s work.
learning from each other and articulating mathematical ideas
Here we see operations such as the kinds of interactions used
Operations include degrees of participation – listening in a
in lectures to get students to engage and respond, the ways in
lecture, talking with other students about mathematics, reading awhich questions are used, the operation of group work in
HELM book to understand some bit of mathematics, using thetutorials and interactions between teachers and students. The
LEARN page to access a lecture, Powerpoint etc. The
conditions include all the factors of the university environment conditions in which this takes place include timetable pressure,13
that condition and constrain what is possible – for example, if fitting in pieces of coursework from different modules aroundActivity Theory analysis – Level 2 – Actions  Goals
Teaching Team

Students

For teaching team, actions are design of tasks andinquiry-based questions – with goals of student
engagement, exploration and getting beyond asuperficial and/or instrumental view of mathematics.
Actions include use of GeoGebra with the goal ofproviding an alternative environment for
representation of functions offering ways ofvisualizing functions and gaining insights intofunction properties and relationships. Actions
include forming students into small groups andsetting group tasks with the goals to provide
opportunity for sharing of ideas, learning from eachother and articulating mathematical ideas

For students, actions involve taking part in themodule: attending lectures & tutorials; using the
LEARN page; using the HELM books; etc. withgoals related to student epistemology. So goals
might include attending lectures & tutorials because
this is where you are offered what you need to pass
the module; clear views on what ought to be on offer
and what you expect from your participation;
wanting to know what to do and how to do it;
wanting to do the minimum amount of work tosucceed; wanting to understand; wanting to pass theyear’s work.

14

Emerging challenges• Recognising the differences in goals, arising from differingorientations and cultures, how can we take into account the macro
issues as we plan at the micro level?
• One response is to bring students into the planning, so that theygain insights into teaching goals and can respond, as insiders, fromtheir own perspectives and cultures – requires explicit effort toovercome power differentials.
• This was achieved in the SYMBoL project – another story!

15

Thank You ☺
b.jaworski@lboro.ac.uk

16

17

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