Using mathematical inquiry to engage student learning within the overall curriculum

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Using Mathematical Inquiry to Engage Student Learning
within the Overall Curriculum
Jill Fielding-Wells and Katie Makar
The University of Queensland
Mathematics is often perceived as a stand-alone subject in the school curriculum. When used as a tool to
examine cross-curricular content, mathematics can enable deeper understanding of the context under
investigation (Makar & Confrey, 2007). A study was designed to investigate opportunities and challenges
that emerged when students addressed authentic interdisciplinary problems using an inquiry-based
approach. This paper aims to identify aspects of students’ engagement across two cohorts of a year 5 (age 9 -
10) classroom in Australia. Using a framework developed by Kong, Wong & Lam (2003), the paper discusses
students’ affective, behavioural, and cognitive engagement with mathematics during four integrated
curriculum units over the course of a year. Results suggest that in both cohorts, students initially struggled
with the shift from teacher-directed to student-driven learning within an inquiry-based, interdisciplinary
environment. By the end of each year, however, the students had developed observable improvement in their
ability to engage on multiple dimensions within the framework with ill-structured mathematical problems
encountered across content areas. Implications for mathematics education research are addressed.
Introduction
Study within the disciplines has been a traditional model of studying knowledge. It allows for a high
degree of specialisation and provides for disciplinary knowledge to be taught by someone well-versed
in the knowledge, skills, and ways of knowing (epistemology) in that discipline and who has developed
their understandings within that community of practice. Integrated curriculum in schools often falls
short of the authenticity intended and instead lapses into thematic work that lacks the depth of
understanding possible within discipline areas (Beane, 1995; Vars, 1997). Mathematics is particularly
under-utilised or misunderstood in the context of integration. Integration of mathematics into other
curriculum areas often suffers from the ‘handmaiden syndrome’ (Wineburg & Grossman, 2000), where
mathematics plays a secondary role in the learning of content. That is, calculations are used within the
problem, but mathematical understanding is not necessarily advanced. What is the potential for
engaging students in solving problems which extend their learning both in mathematics and other
content areas?
This paper reports on insights that emerged from a class of fifth grade students (aged 9-10) from two
different cohorts (2006, 2007) in the first two years of a larger longitudinal study that examined how
teachers and students engaged in inquiry-based mathematics. Within the first few months of the study,
it became apparent that there was a potential to take advantage of opportunities for developing
students’ understandings in other curriculum areas using mathematical inquiry. In this paper we focus
on the way in which mathematics problems that crossed curricular boundaries supported both deeper
understandings and greater engagement.
Literature
Interdisciplinary learning and inquiry in mathematics
Integration of curriculum has been heralded as building student engagement and access to challenging
content (Apple & Beane, 1995). Interdisciplinary learning has been a popular idea in school reform for
several decades, yet its implementation in classrooms has yet to achieve the kinds of potential foreseen.
Researchers have identified several important reasons for integrating content across disciplines

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(Grossman, Wineburg, & Beers, 2000). The first is that the focus of integration is typically on the big
ideas in a discipline rather than narrow fact-based perspectives. Second, authentic problems are rarely
constrained to a single discipline and are often highly motivating. Most problems in life are ill-defined;
that is, their problem definition is ambiguous or has many open constraints (Reitman, 1965). Eraut
(1994) argues that unlike well-defined problems, ill-defined problems have no solution or multiple
solutions, and small changes in the problem definition often require large changes in the solution
process. This is a notable difference than the way context is treated in mathematics problems in
schools, where dissimilar contexts in an exercise set typically require the same mathematical
procedure. Most researchers consider inquiry to be the process of solving or addressing authentic, ill-
defined problems, although there is no precise definition of inquiry agreed on by researchers
(Anderson, 2002). “The presence of ambiguity and limitations and in its applications is thus a major
force for inquiry” (Borasi, 1992, p. 164). This attention to problem definition and context are a critical
distinction between inquiry and conventional problem-solving in mathematics.
Inquiry has been carried out in school science with positive outcomes, but little research has been done
on inquiry in mathematics. According to Borasi (1992):
Mathematical applications require not only good technical knowledge but also the ability to take into
account the context in which one is operating, the purpose of the activity, the possibility of alternative
solutions, and also personal values and opinions that can affect one’s decisions. Unfortunately, none of
these elements is usually recognised as relevant to mathematical activity by people who have gone
through traditional schooling. (p. 160)
Open-ended contexts provide interesting possibilities for integrating mathematics into other content
areas. By engaging students in open-ended problems, there is potential for them to develop not only
stronger content knowledge, but also resilience, flexibility, generative thinking, and persistence.
Research on student experiences with the shift from conventional mathematics and reform-based
mathematics (including inquiry) suggests that student perceptions of their experiences can vary widely.
The outcomes of a study by Star and his colleagues (Star, Smith, & Jansen, 2008) found that across
multiple kinds of experience with reform-based mathematics, the students described little in common.
Star et al. argue for the importance of engaging students in discussing differences between the two
contexts of learning mathematics (traditional and reform-based), giving them a way to recognise and
articulate impacts of their learning experiences and to develop a metalanguage to talk about learning.
Student engagement
Optimising student’s engagement in academic settings is essential as it has been thoroughly
documented as a predictor of academic success (Fredricks et al., 2004). Students who do not actively
participate in classroom activities are at increased risk of school failure. Student engagement has been
researched by educators and psychologists for many decades now, providing an extensive research base
(Fredricks, Blumenfeld, & Paris, 2004). Research on student engagement shows it to be a highly
complex and multi-faceted construct, with three commonly identified dimensions: affect, behaviour
and cognition. Social aspects have also been considered important in informing the overall
understanding of student engagement (Lutz, Guthrie, & David, 2006). It is important to note that
engagement is not a dichotomous state, but rather exists along a continuum and is observed
subjectively by the researcher.
Affective engagement has been variously described as including aspects such as anxiety, interest, and
boredom (Connell & Wellborn, 1991 cited in Kong, Wong, & Lam, 2003); as interest, achievement
orientation, anxiety and frustration (Kong et al., 2003); and identification with teaching staff or peers,

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and sense of belonging (Horn-Hasley), 2007. In the schooling context, affective engagement can be
considered the beliefs, attitudes and emotions as experienced by students.
Behavioural engagement is identified by Fredricks et al. (2004) in three ways. The first is as positive
conduct, the following of rules and the maintaining of compliant behaviour. The second is through
measures of effort, persistence, concentration, attention, questioning and communicating. Finally,
engagement can be seen as school commitment, identifiable through such measures as school
representation in extracurricular activities.
Connell and Wellborn (1991; cited in Kong et al., 2003) identify cognitive engagement as a measure of
psychological investment in learning, a desire to go beyond basic requirements and the desire for
challenge. This includes flexibility in problem solving, industry and resilience. There is a distinction in
cognitive engagement between students adopting surface strategies, for example, memorisation as
distinct from deeper strategies such as integration and justification. In school, cognitive engagement
can be considered to include the way in which students attend to information, store information in
memory, access knowledge and use that knowledge to think and solve problems. Kong and his
colleagues (2003) identified significant markers of cognitive engagement as the use of surface
strategies (memorisation, practicing, handling tests), use of deep strategies (understanding the question,
summarising learning, connecting new knowledge with old), and reliance (on parents and teachers).
The complexity and inter-relatedness of the construct itself makes the study of the components of
engagement difficult to study individually (Fredricks et al., 2004; Kong et al., 2003). For example, a
student who works diligently to complete a highly interesting, yet complex problem may be
behaviourally engaged and also highly affectively and cognitively engaged. However, a student
working on multiple simple, mundane algorithms may be behaviourally engaged and yet bored,
frustrated and mentally unchallenged. Therefore it is not always possible to identify engagement
through observation alone as, in both of these instances, the student would appear to be engaged.
Teaching approaches which have been identified as improving engagement are encouraging curiosity,
creating cognitive dissonance, teacher enthusiasm (Dolezal, Wesh, Pressley, & Vincent, 2003),
relevance and authenticity (Sarker & Frazier, 2008).
Theoretical Framework
A framework for conceptualising and measuring engagement in mathematics was developed by Kong,
Wong, and Lam (2003) through research and validation, resulting in the identification of significant
markers of engagement. These markers were used in our study as a framework for investigating,
categorising, and interpreting student engagement, and are as follows:
Affective engagement
 interest
 achievement orientation
 anxiety
 frustration
Behavioural engagement
 attentiveness
 diligence
 time spent on task
 non-assigned time spent
on task
Cognitive engagement
 surface strategies (memorisation,
practising, test taking strategies)
 deep strategies (understanding,
summarising, connections, justifying)
 reliance (on teaching staff/parents)
The study by Kong et al. (2003) found that most of the students in their study were desirous of
achieving well in mathematics, likely a reflection of societal expectations. These students reported
feeling particularly anxious during mathematics lessons and more specifically in test situations.
Students also reported fear of mathematics and expressed that their anxiety reflected in poor

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performance. The expectation to achieve well in mathematics may be culturally biased as their study
took place in Confucian Heritage Culture (CHC) and students in their study experienced an
examination-driven curriculum and high-expectations (Kong et al., 2003). Despite the high regard held
for mathematics in CHC, Kong and his colleagues reported that a number of students identified that
they were tired of mathematics, having limited interest and finding exercises boring and onerous. They
related that some students in their study found mathematics interesting due to curiosity arousal and
subsequent satisfaction, and that achievement-oriented students focussed on achieving good results in
mathematics, but did not necessarily find mathematics interesting; the motivation for effort was to
achieve a good grade. Despite cultural differences, we found that the framework developed by Kong
and his colleague would be a useful foundation for our study.
Context, Design, and Methodology
The study presented here is embedded in a four year (2006-2009) longitudinal study designed to
understand the processes and practices of inquiry-based teaching and learning in primary school
mathematics. This paper focuses in particular on the affective, behavioural, and cognitive opportunities
and challenges that arose as students engaged with solving ill-structured authentic mathematics
problems that integrated with multiple subjects in the curriculum. Other results published elsewhere
(e.g., Makar, in press) report on the teachers’ evolving experiences with teaching mathematics through
inquiry. The research question being addressed in this paper is: What characteristics of students’
affective, behavioural, and cognitive engagement emerge when students address authentic mathematics
problems which have been integrated into multiple subject areas?
The research in the longitudinal study was designed using a Design Experiment (Cobb, Confrey,
diSessa, Lehrer, & Schauble, 2003) which allows the researchers to simultaneously study and work to
improve the teaching and learning opportunities within the study context. Results presented in this
paper come from two cohorts (2006, 2007) of a Year 5 (age 9-10) classroom in a suburban state
(government) school in Queensland, Australia. In each term (about 10 weeks, with 4 terms per year),
the class undertook an in-depth inquiry that engaged multiple content areas (Table 1).
Data were collected through research notes, researcher reflective journals, classroom video, and audio
recording of class discussions. Episodes were considered for analysis which would most likely
illustrate dimensions of students’ engagement. The Snack Box unit was chosen as a focus unit for its
potential illustrative properties as well as the opportunity that arose during this unit when the teacher
engaged students in discussion of their experiences with inquiry-based mathematics lessons in
comparison to their traditional mathematics lessons from the school-mandated curriculum. Portions of
this unit and the classroom discussions were transcribed as exemplars of student interactions. Research
notes and reflective journals were further used to compile examples of student engagement over the
two years reported here. The results presented in the next section describe examples of students’
engagement over the two years, including excerpts of student discussion, with particular focus on the
Snack Box unit (a detailed description of this unit follows Table 1).

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Table 1: Inquiry-based curriculum units implemented 2006-2007
Term
Dates
implemented
Unit Identifier
Inquiry problem
Content areas
Overview
1
Mar – Apr
2006
Commonwealth
Games
Are athletes getting faster?
Mathematics
Social Studies
Physical Education
Students sought longitudinal data from winning times in
events at the Commonwealth Games (teacher-designed)
2
Jun
2006
Kangaroos and
Resources
What patterns emerge between
availability of resources and
population levels of kangaroos?
Mathematics
Environmental
Science
Students investigated patterns in data observed from a role
play predator-prey game (source of game unknown)
3
Aug
2006
Typical Year 5
Student
Who is a typical year 5 student?
Mathematics
Social Studies
Health
Students design and implement a survey to find the ‘typical’
year 5 student, then search for students who match the
characteristics (Gideon, 1997)
4
Oct – Nov
2006
Flight
What is the best design for loopy
aircraft?
Mathematics
Physical Science
Technology/Design
Students investigate factors influencing flight (lift, drag, etc)
and design a scientific experiment to test the effect of
modifying various parts of a loopy aircraft (loop size and
placement)
1
Mar – Apr
2007
Citizenship
Survey
What are the community’s
attitudes towards citizenship?
Mathematics
Civics
Students survey members of the community about their
attitudes towards criteria for obtaining Australian citizenship
2
May – Jun
2007
Plants Growth
How do pollutants affect plant
growth?
Mathematics
Biology
Students design a scientific experiment to test the effect of
various household pollutants on the growth of bean plants
3
Sep
2007
Snack Box
What is the best design for a net
to package boxes of snack food?
Mathematics
English
Health
Students design packaging, nutrition information, and
advertising for a healthy snack
4
Oct – Nov
2007
Flight
Repeat (modified) of flight unit
from Term 4 2007

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Description of Snack Box Unit
A brief description of this unit of work is being provided to allow the reader to contextualise and situate
the students’ mathematical learning. The unit integrated Health and Physical Education (HPE), English,
Technology (Design) and Mathematics. Initial work in the unit surrounded the need for students to be
able to identify healthy food as part of the HPE Curriculum. Students were firstly exposed to the
nutrition panels of a variety of breakfast cereals and were requested to determine which was the
healthiest. They quickly discovered this was difficult as foods low in fat and sodium may be high in
sugar, or of little nutritional value, and therefore many factors needed to be considered. Students
learned how to read nutritional panels and make judgements about foods based on information given in
the panels. During this timeframe, they were also exposed to advertising as a media, deconstructing and
analysing the genre and its purpose and audience as part of the English curriculum. The students
designed their own healthy snack, with the requirement to make it conceptually appealing to children,
and then designed a package (creating the net), incorporating their knowledge of advertising in the
design. Their next task was to determine the optimal way to package four individual snacks together for
display on supermarket shelves. The students needed to determine what they felt were important
criteria in terms of design. They elected to consider ease of packaging the final product into cartons for
shipping, advertising display on supermarket shelves, ability to stack in a pantry, and minimal
packaging for environmental sustainability (necessitating a design which interlocked fairly closely or
tessellated). The students then considered options, designed and created both the packaging and its net,
and justified the design to the other groups in the class. Some of the design configurations chosen by
the students for trial are illustrated in Figure 1. Note that one of these design configurations (Figure 1d)
did not fit the criteria that the students had developed. The reasoning for considering this particular
arrangement of boxes will be discussed later.
Figure 1: Some of the configurations of boxes for which students chose to create nets
Results
The following section is divided into three parts. We first discuss students’ affective engagement in
working with the interdisciplinary, ill-structured inquiry problems posed in the units (Table 1).
Excerpts from classroom activities are provided to illustrate the use of Kong et al.’s (2003) framework
to gain insight into different aspects of students’ engagement. Following the section on affective
engagement, we consider behavioural elements of the students’ engagement, such as their diligence,
attentitiveness, and their time on task (both in class and outside of class). Finally, we consider cognitive
aspects of their engagement to focus on their strategies (surface and deep) and reliance during inquiry
(a)
(b)
(c)
(d)

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activities. In each case, we contrast how these various aspects of engagement differed during the
interdisciplinary inquiry problems compared to their normal (workbook) mathematics problems. We
summarise the results by presenting the students’ categorization of the difference between ‘normal’ and
‘inquiry’ maths using Kong et al.’s framework.
Affective Engagement
Interest and Achievement orientation
The students’ interest in mathematics shifted over the course of the year, both in their enjoyment of the
subject and the locus of their motivation to achieve. At the beginning of the school year, the students
were asked to name their favourite and least favourite subjects, with the majority identifying
mathematics as a least favourite. Over time however, students began to separate ‘regular maths’ from
‘inquiry maths’ and became fairly articulate in identifying why they found inquiry maths more
interesting, despite challenges it posed. During the Snack Box unit in Term 3, the teacher questioned
the students during an activity:
Teacher: Are you actually enjoying this activity?
Students: (Together) Yeah!
Ben: … [because] I get to do something.
Gordon: … Making the box.
Isabella: That we get to discuss something we are doing rather than just writing.
Teacher: What are you finding the most challenging?
Jay: Like, actually making it [the net], …
Teacher: So do you think this would actually be harder than what you do in a [text]book?
Avril: Probably, [but this] is more interesting.
The locus of students’ motivation also shifted. In the first integrated inquiry unit with each cohort, they
often relied on the teacher as an arbiter. The students were competitive and keen to come up with the
‘best’ answer, with the teacher often being asked to determine which outcome was ‘best’ or whether
something was ‘right’ or being done ‘the right way’. They displayed confusion initially when the
teacher put the decision back on the class and pointed out that there were many possible methods and
many possible solutions.
Over the course of the year, the students developed an understanding that inquiry problems did not
result in a single correct answer; demonstrating a developing awareness of the importance of
identifying a best possible outcome based on constraints of resources, including time. The students’
interest and motivation appeared to stem from a changing perception of mathematics from a set of
problems that are right or wrong (where it was up to the teacher to tell them whether they were correct)
towards seeing mathematics as a result of negotiation and collaboration. Students’ ability to collaborate
noticeably increased across time as they began working together to achieve a class response rather than
an individual response. In the Snack Box unit for instance, students were able to identify that the ‘best’
box design at the completion of the work was dependent on a set of criteria that they themselves had
devised. They suggested that others might have different priorities and that there might be box designs
that they hadn’t even considered. The students relinquished ownership of individual designs and
constructively critiqued available models.
Interestingly, by this stage, the students no longer questioned the teacher as to whether they had found
‘the answer’. It appeared that over time the students shifted their focus from finding a correct,
individual answer, to providing responses which they could justify to the group. Accordingly, the

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motivation for the work had changed from obtaining a ‘good’ grade to a desire to discover a solution,
and it was not uncommon in either cohort to hear students request to skip breaks, specialist lessons and
other high interest activities in order to keep working (see Behavioural Engagement). Students
displayed significant interest in most aspects of all integrated, inquiry-based units. Greater engagement
was observed when in units (Table 1) that students were:
Interested in
- High interest: Plant Growth (2007)
- Low interest: Citizenship Survey (2007)
Could relate to
- Commonwealth Games (2006)
- Typical Year 5 Student (2006)
Perceived value
- Snack Box – tied to a real-world problem,
established in real context (2007)
Identified novelty
- Snack Box – one group developed an
impractical, but novel and highly challenging
box design (2007)
Experienced enjoyment
- Kangaroos and Resources (2006)
- Flight – making paper planes (2006, 2007)
Anxiety and Frustration
In the initial learning phases, the change in focus and control from teacher-led to student-led activities
was anxiety producing; students who had not been previously exposed to inquiry anticipated the
achievement of a correct answer and were concerned when this did not occur. However, as students
came to focus more on the process rather than on the ‘answer’, and learnt to assess their own
understandings and findings, they displayed less anxiety. From a teaching perspective it enabled the
teacher to identify student reasoning much more easily and therefore able to provide each student with
specific feedback. Anxiety also decreased as students came to realise the possibilities of alternate
approaches and were often able to work to their own strengths, for example kinaesthetic rather than
visual.
Overall, two primary sources of frustration were noted—task-oriented and socially-oriented. Task-
oriented frustration rose from the difficulties in achieving an optimal answer or determining ways to do
the task. For example:
Teacher: What does the group think they need to do next?
Thomas: I have no idea.
Teacher: If you get really stuck, I will help you, but I want you to go as far as you can before I do.
Amrita: We’ve gone as far as we can. Because we’ve tried a million times! … And we even ended up
scrunching up our last one and chucking [throwing] it in the bin.
Conversely, rather than decreasing engagement, allowing a period of frustration, during which the
students were able to reassess pathways, frequently induced positive engagement. Once students did
not panic or give up when they were having difficulties, the obstacles they encountered sometimes
created a new sense of empowerment in students to gain control over them. In this way, the students’

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thinking grew increasingly flexible over time as they became more inclined to reconsider strategies and
reapproach them from alternative angles.
The second source of frustration was socially embedded and came from the students adjusting to
working collaboratively with their peers, needing to work together, listening to others’ ideas,
compromise, and accepting critique to achieve consensus. Comments such as the following were quite
typical.
Jason: Me and [student] can’t agree with each other, she just disagrees with everything I, we say.
Thomas: We can’t agree on anything, that’s the problem.
Angela: But I told them….
Frustration also resulted from students’ initial reluctance to let go of their own ideas or to have them
critically evaluated. Across the year however, students began to work more productively together and
to also develop improved understanding of group dynamics.
Jay: We’re having lots of arguments, not the constructive sorts of ones.
Avril: Well at first we were a group, and then we [split up], then everybody had problems so
we came back as a group and we finally found it out.
Teacher: So you found that once you worked together as a group –
Avril: Yeah. People cooperated.
Teacher: What did you learn from this?
Daniel: Teamwork
Angela: We learnt how to argue.
Thomas: … how to criticise constructively.
In the second half of the year, the teacher engaged the class in discussion about inquiry and traditional
mathematics and the students reported that inquiry enabled them to “cooperate”, “share answers”,
“share ideas”, “bounce ideas off people” and “listen to [others’] ideas”. Traditional maths was
described as “quiet” and “sit and listen”. The great majority of students argued that they much
preferred the inquiry way of interacting in the classroom. The students for whom this appeared the least
successful were those with less-developed social skills.
Behavioural Engagement
Diligence
Early in the year, there were observable tendencies in the students to want to surrender the inquiry
when anything became difficult. This was particularly noticeable in the Plant Growth unit as some
plants broke when they grew beyond 50 cm. The students were largely unwilling to modify or continue
their research until they were told quite firmly that it was not an option.
Teacher: A big problem … quite a few of [the plants] got snapped. … I’m wondering what we’re going
to do. Can I have some suggestions? …
Andrew: I think we should start over …
Jane: [But] you don’t know if more of them are going to snap [if we replant them] …
Sally: I think that we should just start over!

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Conversely, a great deal of resiliency and determination was noticed in the units conducted in the
second semester. One group in particular in the Snack Box unit chose an extremely difficult design to
create a net for (Figure 1d). They acknowledged that it was an unlikely design but requested permission
to go ahead out of curiosity. Most of the students persevered against a myriad of failed designs and one
who did express a desire to give up grinned when told it wasn’t an option and set back to the task.
Attentiveness and Time Spent on Task
The Kong et al. (2003) study had noted that the CHC students spent the majority of their time during
lessons listening and doing exercises, with variances noted in attentiveness and concentration span
throughout the lesson. In the current study, the teacher related increased attentiveness overall during
inquiry-based lessons as compared to conventional lessons, particularly during discussions. Although
only about one third of students were frequent participants in the more heated discussions, the
remaining students remained attentive throughout and contributed occasionally. Overall, the extent to
which the class paid attention and listened to fellow students was quite high in contrast to traditional
non-integrated mathematics lessons. As students became increasingly focused on addressing the
inquiry problem, they became more attentive to learning the skills and methods they needed in order to
progress on the question driving their inquiry. This attentiveness notably decreased in lessons that they
felt had little connection for them, sometimes to the extent of questioning the teacher as to the purpose
of their learning the content.
A conventional approach to comparing ‘time on task’ is by recording the number of students observed
to be on task over regular intervals (e.g., every 3-5 minutes). An independent observer was requested to
observe a heterogeneous group of students and, using a stopwatch, record the number of off-task
behaviours at five-minute intervals during inquiry-based learning, and again during traditional teaching
methods. This was rapidly noted as being problematic as a measure of student on-task behaviour. The
observer reported difficulty in categorising behaviour accurately as students in the inquiry lessons
could become quite noisy during [often highly engaged and animated] discussion regarding the focus of
the work and would often all be trying different exploratory procedures simultaneously. In contrast,
students during traditional work could appear engaged and yet be doodling rather than working when
approached closely.
On the other hand, time spent voluntarily on unassigned work could be considered a powerful identifier
in the Australian culture as relatively little extra work is expected of primary aged children outside of
school hours and this time is frequently spent on pursuit of hobbies or sporting interests. During several
of the inquiry units, students requested additional time in class to work on projects, especially with the
Snack Box unit, and also requested to be allowed to return to class in breaks and before school, or to
skip certain lessons in order to extend and complete work. Students also spent time outside of class
working on aspects of their inquiry unit without direction to do so (i.e., not assigned work). In many
cases this was small self-focussed activities, such as making additional paper planes (Flight unit),
which students requested to be tested under controlled conditions, and identifying alternate household
pollutants for testing (Plant Growth unit). However, in a few instances, the students took on quite large
projects; for example, one built a hot air balloon during the Flight unit and another independently
researched shelf height and advertising layouts of supermarket shelves in the Snack Box unit.

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Cognitive Engagement
Surface Strategies and Deep Strategies
The inquiry approach used in this study necessitated a much greater use of deeper cognitive strategies.
As each topic was anchored by one single guiding question, the students were required to analyse the
question and consider the context. Therefore there was a much greater necessity for students to think
deeply and to understand what they were doing; surface strategies were not of significant use. For
example, one student remarked on the difficulty she encountered as she worked on designing a net to
enclose all four snack boxes, where multiplying the dimensions of one box by four didn’t work as it did
not account for arrangement of the boxes:
Amrita: It’s where you have just one item [one snack box] and to have to estimate something like if you
have four of them, by just using one of the boxes [pause] and figuring out where each box
would go, so you couldn’t multiply it [each dimension] by the same number.
Students were initially inclined to take a question as they saw it and try to answer based on impression
without logical evidence, research or validation. Later, students became more analytical of ideas,
considering their efficacy and practicality. Over the course of their experiences, students became
increasingly systematic, deconstructing and defining questions prior to determining possible pathways
and selecting methods. They were progressively more able to justify choices and decisions made.
Across the year, students also developed a willingness to critique and accept feedback against their own
decisions and to identify limitations.
Jason: It’s actually wasting more paper than a rectangular box because you wouldn’t surround the
box, you would just do it like a rectangular shape … so you’re not wasting any paper with a
rectangular prism, and with that [design, Figure 1d], you would because there’s like open area
you don’t need.
Inquiries were linked to real contexts based on topics across the curriculum and anchored in real-life.
Therefore there were many instances of students making connections to existing knowledge and the
real-world. These students are conceptualising a net used for packaging and considering why:
Isabella: … if we think back to the Golden Circle factory [previous years’ field trip], they do that with
their juice boxes. They have big rolls of poppers [juice box nets] …
Amrita: I’ve also realised that the [name of local shopping centre] uses that kind of shape to make those
bricks.
As the students became familiar with this style of learning, they became more creative in their
responses and often requested to extend certain projects and to attempt more difficult versions of
projects given, even when they did not know if there was an answer. In reference to Figure 1d in the
boxes diagram,
Thomas: I think we should do it because it’s really creative and if we do manage to make a net for it
then we can always try and pull it together and see what happens.
Whilst the majority of students enjoyed the deeper learning strategy, one student with particularly good
results on traditional test-taking, when in discussion with the teacher, stated:
Jason: I like the book, [the] book’s easier, I don’t have to think … this [inquiry] is more challenging.

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Another example of students’ movement from a surface strategy to a deep strategy was observed
during in the Flight unit. Students sought a design for their loopy aeroplane that would fly farther
(2006) or more accurately (2007). Students first flew their own design and their initial reasoning for
differences in the distances of flights relied on more superficial elements that were difficult to measure
or control (e.g., changing air currents or humidity of the classroom on different days, variations in
amount of tape used to construct the aircraft). Because of these elements, they first concluded there was
no way to compare different design elements of the aircraft. There were no practiced or memorised
procedures they could draw on to resolve the problem. A need to justify which differences had the
greatest impact on the distance or accuracy flown, however, enabled the teacher to introduce the
elements of a ‘fair test’ in science that supported students to consider constructing more experimental
design based processes by controlling for a specific set of variables (e.g., three set lengths and widths
of wing in three specified positions) and limiting design parameters (allowable lengths of tape, shutting
off fans and closing windows, prohibiting additional design ‘features’).
Reliance (on teaching staff/parents)
Students initially relied more heavily on the teacher to provide information, direction and feedback as
well as identifying and selecting methods to use. Students also wanted to know whether assumptions,
choices and decisions were ‘the right ones’. In particular, when initially encouraged to define a
problem, they often felt the need to cover all possibilities, creating approaches that would have been
unwieldy and improbable given the resource and time constraints, for example researching times for
every event across the Commonwealth Games or every possible type of household chemical for testing
on plants. Alternatively, there was a desire to force the design into classroom constraints artificially, for
example setting the sample size to 26 plants so that there was one for each student regardless of how
many were needed for experimental design.
Over time, students became more willing to pose questions, make suggestions and critique their own
and others ideas. They became far less reliant on adults and made more reliance on peers and self to
determine responses. In a discussion with the students on the differences between learning with
inquiry-based mathematics and traditional mathematics, students remarked:
Donna: In the old one [traditional approach], we’re not allowed to talk, you explain to us how to do it
… [in inquiry] we’re doing it ourselves.
William: You [the teacher] do nothing.
Students did not rely increasingly on parents to provide information, however, there was an increased
interaction between students and parents as they surveyed and discussed information with them in order
to develop more data for research and further understanding of issues. The parents were increasingly
treated as peers (a source of ideas/information to be evaluated) rather than as bearers of knowledge.
Parents were pleased with their children’s engagement and were sought to assist in locating information
to assist with decision-making, for example, one student requested that her mother take her to the
supermarket so she could question staff regarding heights of shelves and placement constraints when
advertising, a parameter that had not been identified by the teacher.
Summary
Over the course of the year, in both cohorts, mathematical inquiry enabled students to more deeply
explore complex connections of mathematics to the overall the curriculum. Indications were that
inquiry within an integrated context had a notable effect on students’ engagement in mathematics. This

13
engagement likely provided an impetus for them to sustain their focus on the problems over a
significant period of time, pushing them beyond their initial superficial response towards a more
significant and comprehensive solution.
The integration of mathematics with other curriculum areas likewise supported both understanding in
mathematics and within the content areas it connected to. Without the additional content areas, students
could not have authentically explored their mathematical understandings; without the mathematics,
they could not have investigated the problems at the level of rigour required to develop, evaluate and
justify their solutions. In addition, the meshing of mathematics with authentic cross-curricular problems
provided them with an appreciation of the utility of mathematics as a cohesive discipline to describe
and explore the world, rather than as a fragmented set of facts and procedures.
The students were able to recognise and articulate differences between their engagement with
mathematics within the inquiry-based contexts as distinct from traditional mathematics instruction.
After multiple inquiry units, the teacher engaged the 2007 cohort of students in a discussion to
ascertain their views on mathematics. The students overwhelmingly argued that the mathematics they
experienced through inquiry was far more interesting to them than the ‘old way’ of doing mathematics.
Below is a summary of their distinctions between ‘traditional’ and ‘inquiry’ maths, divided loosely into
dimensions of engagement.
Traditional Maths
Inquiry Maths
Affective
 Boring
 Do it because we have to
- get in trouble [if we don’t]
- curriculum says we have to
 Sit and look at pictures
[diagrams in the text] and
then answer only the
questions they ask
 Learn by mistakes
 Try different things, figure out what works best
Behavioural
 Just sit and listen
 Do a lesson, then just do
something else (advanced
student)
 Have to sit around, wait for
teacher
 Have to be quiet
 Cooperate, share answers/ideas
 Get to get in there and do it
 Talk about more stuff – talk about non-maths stuff
while you are ‘doing’ cutting and stuff – but then we
stop if we need to discuss something and then go
back to our conversation when we have figured it out
Cognitive
 Have to do it the way the
teacher teaches you
 Only one method – don’t
need to try things
 Don’t have the answer
 Have to work it out yourself
 Get to try different methods
 Learn more
 Think about ideas, get ideas from each other
 Get to bounce ideas off other people and listen to
their ideas
 Try more things
 Figure out what works best

14
Discussion and Conclusion
In this paper, we have looked at descriptions and shifts in students’ affective, behavioural and cognitive
engagement in mathematics that were observed when students took part in learning experiences which
embedded mathematics within the overall curriculum. The study suggested that by situating
mathematical problems within more authentic, interdisciplinary contexts, that these experiences could
generate positive outcomes in students’ engagement with mathematics.
Kong, Wong, and Lam (2003) Framework
The construct of engagement comprises dimensions of affect, behaviour, and cognition which are
overlapping and inter-related. For example, it is not clear-cut whether observations of students’
inclination to work on problems out of class would be categorised as interest (affect), time on task
(behaviour), or as negated reliance on the teacher (cognition). By drawing on Kong et al.’s (2003)
framework, a greater awareness of the diversity of students’ engagement patterns was observed by the
researchers. The framework therefore provided a multi-dimensional lens to gain insight into the
complexity of the concept of student engagement as a way to account for differences between
conventional and inquiry-based mathematics learning.
This complexity enabled the researchers to further understand several limitations of the framework for
categorising aspects of student engagement. Some aspects of the students’ engagement were linked to
their experiences in collaborating on problems together. However, it is unlikely that Kong and his
colleagues considered the social perspective in their study. In addition, while the framework allowed
for some identification of observed student behaviours, it did not explain the reasons for their
behaviours, for example, whether a students’ diligence in a problem was due to internal or external
motivation to persist. In fact in trying to categorise segments of classroom observations of student
work, it was clear that there was no way to look at only one domain of engagement alone. Rather, all
three needed to be considered concurrently.
Affective Engagement
Over the course of the two years documented here, it was likely that the context and interdisciplinary
nature of the problems under study had direct and indirect impacts on students’ affective engagement in
mathematics, especially in the dimension of interest. For example, in the Snack Box unit, the situated
nature of the problem within the study of Health (healthy foods) and English (advertisement) provided
a compelling context to study the mathematics. In short, students saw a purpose for the problem, even
in their choice to pursue the creation of a net for a box that was clearly not a logical choice within the
context of packaging (Figure 1d).
This context-driven nature of the problems under study also appeared to affect students’ achievement
orientation, although this was likely also related to the nature of inquiry process and the openness of the
problems under investigation. Rather than looking for an end to a question, the students showed a
desire to explore its multiple facets. Ironically, their achievement orientation appeared to decrease over
the course of the year as they shifted away from a necessity to obtain one correct answer.
The students’ increasing awareness that the problems they were engaging in did not have a single
correct solution may have diminished their anxiety levels. However, students experienced high levels
of anxiety initially as they developed their understanding of both the nature of the problems under
study and coping strategies for shifting from teacher-directed to student-directed inquiry. In contrast,
their levels of frustration tended to increase, although with noticeable shifts in their source, their
response to the frustration, and their locus of control. The frustration they experienced in dealing with

15
the challenging level of the tasks provided new motivation to consider alternative ideas and probing
questions from peers or the teacher. In a traditional mathematics problem, students don’t ‘own’ the
task. Hence, frustration levels are connected to their difficulty with solving the problem set by the
teacher. They perceive these difficulties as their own shortcoming and internalise it as a personal
weakness rather than externalise it as the nature of the challenging problem.
Behavioural Engagement
In comparison to more conventional stand-alone mathematics lessons, students were observably more
attentive when working on the interdisciplinary, inquiry-based problems. It was possible that they knew
that discussions which emerged during learning activities could provide insight into the problems they
were working on. In contrast, it appeared that when students felt they could solve a conventional
problem in mathematics they ‘tuned out’ of the class discussion; or if they felt that the problems were
too tedious or too difficult to solve, they seemed to disengage from the lesson and doodle or chat with
their neighbour.
The nature of the problems they were studying potentially increased students’ diligence in working on
the problem. As illustrated earlier, many students were keen to continue working on problems even
during their lunch breaks or when the timetable required they move to another activity. In addition, the
social nature of the collaborative work may have allowed students to monitor their own engagement.
For less demanding periods of work, students could socialise as they worked without getting off task;
when the cognitive demand increased, student attention switched from social to cognitive energy.
Further, they complained bitterly when peers did not contribute to the task, with peer pressure often
triggering a group-imposed work ethic.
Cognitive Engagement
Finally, students’ cognitive engagement appeared to shift over the course of the year, when
encountering inquiry-based problems, from being satisfied with providing solutions to problems that
were based on common sense and superficial mathematics, towards deeper strategies that required
learning of new content. In engaging with their search for the best solution to a problem, students found
the need to seek more powerful forms of evidence to justify and defend their thinking. The negotiation
that ensued potentially drove them into more rigorous and powerful mathematical practices. This was
seen during the Flight unit when students moved from use of surface to deep strategies and began
controlling variables to isolate factors that influenced the distance or accuracy of the flights.
In addition, students were able to draw on the context of the problem to evaluate their solution and look
for ways to better serve the context under study, rather than the problem in isolation. For example, in
trying to determine whether people were getting faster over time by investigating the winning times in
the Commonwealth Games over several decades, the students debated whether observed improvements
in time were due to the athletes’ performance, the conditions of the race, or technological
advancements in sporting equipment. These debates drew them into a deeper analysis and compelled
them to seek out stronger evidence for their explanations (e.g., checking weather conditions during a
particular year’s marathon, or researching the decrease in drag attributed to full body suits in
swimming).
Reliance on the teacher was another area which included observable shifts during the year as students
worked on the inquiry-based integrated problems. In the initial unit of each year, students ‘waited’ for
the teacher to ‘tell’ them what to do at each stage of the investigation. While in the first year
(Commonwealth Games unit) this may have been due to the teachers’ difficulties in releasing control of
the pathway of the investigation (this was her first time teaching an inquiry-based unit), the same

16
patterns of students waiting for instruction appeared in the first unit of the second year (Citizenship
Survey) when the teacher had acquired significant experience in teaching inquiry. As the year
progressed (in both cohorts), students’ reliance on the teacher tended to shift from a dependence on her
for instructions towards more logistical/management needs (location of materials, moderating conflicts
in group dynamics, checking constraints for unconventional solution strategies). Further, the initiation
of questions surrounding the task shifted from the teacher posing questions to the class to guide their
thinking, towards the students posing questions to the teacher or their peers to probe ideas. During the
more conventional mathematics lessons during this time, the questioning patterns returned to being
teacher initiated and with reliance once again on the teacher for directions.
Using an inquiry approach to integrate mathematics into the overall curriculum
In this study, the use of mathematical inquiry situated within the overall curriculum potentially enabled
students to gain both a deeper understanding of mathematics and a better appreciation for its utility and
power as a tool to understand the world. Considering the dimensions of their engagement in this
process allowed the authors to begin to describe some of the changes which appear to occur when
students are involved in multiple episodes of cross-curricular, inquiry-based learning. Children need to
be engaged in rich educational activities in order to learn optimally. However, engagement is a
complex, multi-faceted construct with many inter-dependent and overlapping dimensions. Kong et al.’s
(2003) framework served as useful to consider each of these dimensions locally. Further, it needs to be
acknowledged that these exist along a continuum, with aspects such as surface strategies valued less
than deep strategies.
The consideration of a framework of student engagement as a tool for investigating has advantages,
however the authors felt that there were also significant limitations. For example, whilst behavioural
engagement is largely observable, affective and cognitive engagement are not so easily measured and
typically require input from the subjects themselves. Even behavioural engagement, often measured
using checklists, can be problematic in that students can be quite talented at appearing to work when in
fact they are not.
Research into student engagement within an integrated mathematics curriculum, and indeed in inquiry-
based learning in mathematics, has not been widely reported. Accordingly, this research is of an
exploratory nature, seeking to develop early understandings in this field. The observations made during
this study suggest its potential value in continuing to further develop understandings into the use of
authentic, contextually-based mathematical learning.
Acknowledgements
This study was funded by Queensland’s Department of Education, Training and the Arts; The
University of Queensland; Education Queensland; and the Australian Research Council. The authors
gratefully acknowledge the contributions of the students involved in this study, and of Lisa Lim for her
work in transcribing and annotating portions of classroom video.
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