Tutorial teaching to enable undergraduate students to make meaning with mathematics

Abstract

The poster reports from a study of teaching in one small group tutorial with first year mathematics students in a UK university. The aim was to characterise teaching that was designed to support students’ meaning making in mathematics. The research was developmental in that it contributed to the development of the teaching, as well as the tutor’s knowledge in teaching. Two researchers, one who was also the tutor, collected and analysed data in the form of recordings of each tutorial and reflections of the tutor. Analysis was grounded in the data and a theoretical construct, the Teaching Triad, was used to support analysis. Findings showed the tutor taking a questioning approach, seeking to probe students’ meanings, but needing to prompt frequently due to students very short and tentative responses. The study points to the difficulties in encouraging students to articulate their mathematical understandings.

Full text

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CERME9 (2015) – TWG14
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M. Gozde Didis1 and Barbara Jaworski2
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2 Loughborough University, Loughborough, United Kingdom
The poster reports from a study of teaching in one small
group tutorial with rst year mathematics students in
a UK university. The aim was to characterise teaching
that was designed to support students’ meaning making
in mathematics. The research was developmental in
that it contributed to the development of the teaching, as
well as the tutor’s knowledge in teaching. Two research-
ers, one who was also the tutor, collected and analysed
data in the form of recordings of each tutorial and reec-
tions of the tutor. Analysis was grounded in the data and
a theoretical construct, the Teaching Triad, was used to
support analysis. Findings showed the tutor taking a
questioning approach, seeking to probe students’ mean-
ings, but needing to prompt frequently due to students
very short and tentative responses. The study points to
the diculties in encouraging students to articulate
their mathematical understandings.
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In this paper, we focus on teaching in a small group
tutorial in the first year mathematics programme at
a UK university. We studied 10 tutorials in all and
focus here on one tutorial which addressed questions
in linear algebra. We are interested in studying re-
lationships between the planning for teaching and
teaching approaches, and the responses of students
in so far as they gave access to the students’ meaning
making in mathematics (Jaworski & Didis, 2014). Our
three basic research questions are:
1) What is the nature of the teaching manifested in
the tutorials?
2)
What student meanings can we discern and in
what ways?
3) In what ways can we link (1) and (2) and what is-
sues does this raise?
We took a sociocultural approach toward the research,
recognising the many factors that underpinned ac-
tivity in a tutorial, contributed to the interactions
between students and tutor, and to mathematical
meaning-making by the students. We see that, making
connections to the worlds of mathematics and beyond,
and processes of socialisation into culture and val-
ues are all central to how mathematics is taught and,
associated with this, how students make meaning in
mathematics.
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The tutor (the second author of this paper) planned
the tutorials and worked with the students. Her co-re-
searcher (the first author) observed and audio-record-
ed tutorials and transcribed the recordings. Our con-
versations we sat together aer tutorials were also
audio-recorded; capturing the tutor’s teaching reflec-
tions and our analytical discussion as two research-
ers. In working on our data we combined a grounded
approach with the application of the ‘Teaching Triad’,
a theoretical model which has been used extensively
for analyses of teaching (e.g., Potari & Jaworski, 2002).
We read and reread the data, developing a coding
scheme and seeking to make sense of the data in re-
lation to our research questions. In addition we ex-
plored the nature of the teaching using the Teaching
Triad. These forms of analysis were inter-woven
to provide a rich characterisation of teaching prac-
tice. Our approach was two-fold: (a) reflections on
the tutor’s discerning of meaning-making in the

Tutorial teaching to enable undergraduate students to make meaning with mathematics (M. Gozde Didis and Barbara Jaworski)
2278
tutorial in order to guide the teaching approach; (b)
the discerning of meaning making through analysis
of tutorial dialogue in order to link teaching with
learning.
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Analysis revealed many aspects of teaching and ways
in which the teaching approaches “in practice” related
to the approaches planned. A finer grain of analysis
showed that the codes relating to “tutor questioning”
were the most prevalent. Through questions, the tutor
not only invited and encouraged students’ participa-
tion, but also tried to control the mathematical focus
of the tutorial. She asked prompting questions to invite
students to respond and articulate students’ meaning,
trying to promote both individual meaning and collab-
orative meaning of students. The following dialogue
illustrates an example of the tutor’s approach.
Tutor: Now, Julia, what is the standard basis in
R3? [Prompting-Q]
S: (Julia) A matrix [SR-short/hesitant]
Tutor: Is a basis a matrix? [Prompting &
Probing-Q] [Mathematical challenge]
S: (Julia) No [SR]
Tutor: Ok, what is a difference? [Probing-Q ]
[Mathematical challenge]
S: (Julia) Vectors (student smiles) [SR-short/tenta-
tive]
Tutor: If I asked to write down the standard ba-
sis in R
3
, what would you actually write?
[Prompting-Q]
Students’ responses were mostly short, tentative
and hesitant. This led us to deduce that discerning
meaning making was difficult and time consuming.
University cultures and practices, in which students
are rarely expected to speak their mathematical
thoughts or engage in discussion, result in such en-
gagement being uncommon. Therefore, in relating
the teaching approach to students’ meaning making
with mathematics, we need to address further: (i) what
students expect from these tutorials and what their
readiness is to deal with the planned teaching; (ii)
what the tutor expects from the tutorial and wants
to see from students, and how this can be achieved;
and, (iii) what time factors influence the depth of stu-
dents’ meaning making.
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Jaworski, B., & Didis, M. G. (2014). Relating student mean-
ing-making in mathematics to the aims for and design of
teaching in small group tutorials at university level. In S.
Oesterle, P. Liljedahl, C. Nicol, & D. Allan (Eds.), Proceedings
of the 38th Conference of the International Group for
the Psychology of Mathematics Education and the
36th Conference of the North American Chapter of the
Psychology of Mathematics Education, Vol. 3 (pp. 377–
384).Vancouver, Canada: PME.
Potari, D., & Jaworski, B. (2002). Tackling complexity in mathe-
matics teaching development: Using the teaching triad as
DWRROIRUUHǍHFWLRQDQGDQDO\VLVJournal of Mathematics
Teacher Education, 5(4), 351–380.