Content uploaded by James Russo
Author content
All content in this area was uploaded by James Russo on May 11, 2021
Content may be subject to copyright.
An instructional model to support
planning and teaching student centred
structured inquiry lessons
Maggie Feng
University of Sydney, NSW
<mfen5873@uni.sydney.edu.au>
Peter Sullivan
Monash University, Vic.
<peter.sullivan@monash.edu>
Janette Bobis
University of Sydney, NSW
<janette.bobis@sydney.edu.au>
Ann Downton
Monash University, Vic.
<ann.downton@monash.edu.au>
James Russo
Monash University, Vic.
<james.russo@monash.edu>
Melody McCormick
Monash University, Vic.
<melody.mccormick@monash.edu>
Sharyn Livy
Monash University, Vic.
<sharyn.livy@monash.edu>
Sally Hughes
Monash University, Vic.
<sally.hughes@monash.edu.au>
In this article, the authors from Monash University and the University of Sydney have
collaborated to present a research-informed model to support the planning and teaching of
mathematics, using a student centred structured inquiry approach.
9APMC 26(1) 2021
Introduction
ere is an old saying that, when translated into
English, states: “If you don’t know what harbour you
are going to, no wind is a good wind”. Of course,
if you have not chosen a destination, it makes no
dierence what direction the wind blows. e point
is, ideally lesson structures should match our goals for
student learning generally and mathematics learning
in particular.
On one hand, one of the goals of mathematics
teaching is to improve test scores of most
students, including performance on high stakes
assessments. e pedagogies associated with such
goals are familiar to teachers since they would have
experienced such teaching in much of their own
mathematics learning. is approach is described
below as Active Mathematics Teaching.
On the other hand, the goals of education
generally and mathematics education in particular
might also include developing student agency and
fostering inclusion. e Organisation for Economic
Co-operation and Development (OECD) (2019)
proposed aspirations for education globally in the
year 2030, two of which are relevant for the teaching
of mathematics. e rst is the need to maximise
opportunities of all learners, which emphasises the
importance of inclusive approaches to planning and
teaching. e second goal is preparation for uncertain
futures, which implies that capabilities for independent
thinking and creativity are not only important but
also can be fostered through education. Many systems
articulate similar aspirations using phrases such as
fostering creativity, developing curiosity, and building
resilience. Approaches to instruction that foster such
goals are described below as Student Centred Structured
Inquiry and are assumed to be less familiar to teachers.
10 APMC 26(1) 2021
Sullivan et al
Sullivan et al. (2020) describe the elements of such
teaching in detail. Caro, Lenkeit, and Kyriades
(2016) analysed results of PISA 2012 involving over
500,000 students and provide compelling evidence
of the eectiveness of this perspective, describing it as
cognitive activation.
e purpose of this article is to propose an
instructional model that can communicate to teachers,
various considerations and decisions associated with
student centred teaching. e model was developed
as part of the Exploring Mathematics Sequences of
Connected, Cumulative and Challenging tasks project1
(EMC3; https://www.emc3mathsteaching.com/
instructional-model).
Active Mathematics Teaching
e Active Mathematics Teaching approach
supports lessons in which teachers seek to develop
comprehension of specic mathematical concept or
to foster conceptual understanding of a procedure
or technique. Good, Grouws, and Ebmeier (1983)
explained such teaching includes addressing
prerequisite skills; lively presentations; assessment of
comprehension; controlled practice; seatwork; and
homework assignments; with the teacher having an
active role at each stage. is could be summarised as:
• Teacher poses some questions to check student
facility with pre-requisite skills.
• Teacher explains goals and solution methods,
including ways of setting out responses.
• Teacher poses further exercises and asks students
to work out the answers. Some students explain
what they have done.
• Further questions are posed in sets of similar
demand on students.
• Students’ responses to set exercises are corrected,
and some further examples are posed to check
both their accuracy and capacity to explain the
process they used.
In other words, this approach can be described
succinctly and clearly, and presumably would be
familiar to all teachers. Active Mathematics Teaching
connects to what is described in the High Impact
Teaching Strategies (HITS) (Department of Education
and Training Melbourne, 2017) as Explicit Teaching
and Worked Examples.
While there are aspects of mathematics for which
such approaches are suitable, there are risks that
teaching in this way can make students dependent
on the teacher, as opposed to encouraging students
to think for themselves. is approach does not
specically make provision for the diversity of
achievement and motivation found in most classrooms
and, if this is the predominant method of instruction,
it results in alienating many students.
Student Centred Structured Inquiry
By comparison, Student Centred Structured Inquiry
approaches are more complex and less familiar to
teachers. To communicate the various associated
teacher actions, we developed an instructional model,
the schematic of which is presented in Figure 1. e
language of the instructional model draws heavily on
Smith and Stein (2011) which focuses on orchestrating
classroom discussions, an essential element of creating
opportunities for fostering student agency.
Figure 1. Student centred structured inquiry instructional model.
1. e authors are engaged in a project funded by the Australian Research Council, Catholic Education Diocese of Parramatta and Melbourne Archdiocese
Catholic Schools (LP 180100611). e views expressed are opinions of the authors who take full responsibility for the ethical conduct of the research and
preparation of the article.
e rst and arguably the most critical element of
the EMC3 instructional model is “Anticipate” which
is enacted prior to lessons as part of planning and is
summarised for teachers as:
• Identify specic learning goals including specic
questions and ways for prompting learning.
• Choose tasks based on mathematics learning
goals, the curriculum, and prior knowledge of
students.
11APMC 26(1) 2021
An instructional model to support planning and teaching student centred structured inquiry lessons
• Select resources, materials, and ways for students
to represent their thinking.
• Anticipate students’ solutions and strategies as
well as possible misconceptions.
• Plan enabling and extending prompts.
e rst phase in the lesson is “Launch” for which
our advice for teachers is as follows:
• Lead a preliminary activity, which can be a
uency exercise related to the content of the
lesson or a discussion to familiarise students with
the lesson context.
• Pose the main task without instructing students
on any solution path or method, with students
reading the question for themselves where possible.
• Invite questions to clarify language, materials and
representation.
We want to communicate to students (and their
teachers) that by transferring responsibility for learning
to students, they have to do the thinking when
choosing the type and format of their response. Such
choice is engaging and is part of fostering student
agency. Ideally the tasks posed are productively
challenging, build resilience and are mathematically
interesting so as to encourage curiosity.
e next phase is “Explore” in which teachers are
advised to:
• Allow individual think time after which students
might work collaboratively.
• Interact with students, observing and monitoring
how they are responding.
• Oer enabling prompts to students who are stuck
and extending prompts to students who have
nished.
• Select student work samples for subsequent
sharing.
• After around 10 minutes, if many students are
not progressing, encourage sharing of partial
solutions and/or discuss misconceptions that have
arisen.
ere are two key actions for teachers: one is
interacting with students to stimulate thinking and
persistence; the other is preparing for the subsequent
lesson phase.
e next phase, “Summarise/ Review” has two
interrelated aspects that occur contemporaneously. e
elements of “Summarise” are:
• Sequence the selected work samples.
• Support students in articulating solutions and
strategies by revoicing when necessary.
• Pose questions to stimulate student thinking,
connect mathematical ideas and build
understandings.
e teacher, at this phase, leads discussion of student
solutions, engages other students in thinking about
these solutions, and builds connections between the
solutions and the discussion that is prompted.
In “Review” we encourage teachers to:
• Synthesise, emphasise and record key mathematics
points building on student contributions.
is nal step involves a shift in the focus of
authority from students to the teacher.
Importantly, we see the launch-explore-summarise
process happening twice for each learning experience,
with the task for the second cyclebased on Variation
eory (Kullberg, Runesson, & Mårtensson, 2013).
Applying variation theory to task design involves
creating a new task from an existing task by keeping
most aspects the same, but deliberately varying some
aspects. e variant might be the context, with the
concept(s) staying the same. Alternatively, we might
vary the sophistication of the concept (or even the
concept itself), with the context staying the same.
e explicit intention is to consolidate the thinking
activated from the initial experience (Dooley, 2012).
is consolidation involves repeating the previous
three phases, noting that consolidation is often in a
subsequent lesson.
Advice to teachers for “reLaunch” is:
• Pose a further task that is a bit the same and a bit
dierent from the original to consolidate learning
from the rst task.
• Encourage students to apply and extend learning
from the rst task.
“reExplore” is described as:
• Use the same approach as for the exploration of
the initial task, including selecting work samples
(preferably from dierent students).
• Arm students who have applied new or more
sophisticated strategies from the earlier discussion.
“reSummarise/Review” is:
• Use the same approach as for review of the
initial task, emphasising overall learning and
anticipating how the new knowledge might be
used in the future.
ese textual descriptions are presented to teachers
along with the schematic in Figure 1.
12 APMC 26(1) 2021
Sullivan et al
e model explicitly addresses the aspects of the
HITS namely: Setting Goals, Structuring Lessons,
Collaborative Learning, Multiple Exposures,
Metacognitive Strategies and Dierentiated Teaching.
Interestingly, there is still Explicit Teaching in this
model although this happens after student experience
with the task rather than before, and there are also
Worked Examples although these come
predominantly from the students.
Conclusion
is article presented two models of instructional
processes in mathematics. Both are useful although
teachers are encouraged to be aware of the destination
goals associated with each model. e harbour for
the Student Centred Structure Inquiry instructional
model is student engagement and decision making,
along with the intention to include all students in
the learning.
Even though the EMC3 project oers examples of
tasks that are mainly open ended or open middled,
this instructional model works for any type of learning.
Examples of other types of learning experiences that
could follow this model are those based on
mathematical games and practical investigations.
References
Caro, D., Lenkeit, J., and Kyriades, L. (2016). Teaching strategies and
dierential eectiveness across learning contexts: Evidence from
PISA 2021. Studies in Education Evaluation, 49, 30–41
Department of Education and Training Melbourne. (2017). High
Impact Teaching Strategies. Downloaded in January 2021 from
https://www.education.vic.gov.au/documents/school/teachers/
support/highimpactteachstrat.pdf
Dooley, T. (2012). Constructing and consolidating mathematical
entities in the context of whole class discussion. In J. Dindyal,
L. P. Cheng, & S.F. Ng (Eds.). Mathematics education expanding
horizons: Proceedings of the 35th conference of the Mathematics
Education Group of Australasia (pp. 234–241). Singapore:
MERGA.
Good, T. L., Grouws, D. A., & Ebmeier, H. (1983). Active mathematics
teaching. New York: Longmans.
Kullberg, A., Runesson, U., & Mårtensson, P. (2013).e same task?
– Dierent learning possibilities. In C. Margolinas (Ed.),Task design
in mathematics education:Proceedings of the International Commission
on Mathematics Instruction Study 22 (pp. 609–616). Oxford, UK:
ICMI.
Organisation for Economic Co-operation and Development
(OECD) (2019). e future of education and skills Education 2030.
Retrieved from https://www.oecd.org/education/2030/E2030%20
Position%20Paper%20(05.04.2018).pdf
Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating
productive mathematical discussions. Reston VA: National Council of
Teacher of Mathematics.
Sullivan, P., Bobis, J., Downton, A., Hughes, S., Livy, S., McCormick,
M., & Russo, J. (2020). Ways that relentless consistency and task
variation contribute to teacher and student mathematics learning.
In A. Coles (Ed.)For the Learning of Mathematics Monograph
1: Proceedings of a symposium on learning in honour of Laurinda
Brown(pp 32–37). Canada: FLM Publishing Association.
