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This paper reports how the context in which a mathematics item is embedded impacts on students' performance. The performance of Year 10 students on four PISA items was compared with performance on variants with more familiar contexts. Performance was not better when they solved items with more familiar contexts but there was some evidence that items requiring the secondorder use of context were more influenced by an alteration of context than items of firstorder use. Recommendations for further study are included.Télécharger gratuit ERIC ED572560: Item Context Factors Affecting Students' Performance on Mathematics Items pdf
Item Context Factors Affecting Students’ Performance on
Mathematics Items
Felipe Almuna Salgado Kaye Stacey
The University of Melbourne The University of Melbourne
<felipea@ student .unimelb.edu.au> < k.stacey@unimelb.edu.au >
This paper reports how the context in which a mathematics item is embedded impacts on
students' performance. The performance of Year 10 students on four PISA items was
compared with performance on variants with more familiar contexts. Performance was not
better when they solved items with more familiar contexts but there was some evidence that
items requiring the secondorder use of context were more influenced by an alteration of
context than items of firstorder use. Recommendations for further study are included.
In the 1980s and 1990s advances in cognition and measurement theories led researchers
to consider how assessment could be developed to ensure that meaningful aspects of
students" learning and thinking are captured. In this manner, different curricula and
standard documents around the globe started to develop new forms of connectedness of
mathematical content by focusing on mathematical modelling, contextualisation and
problem solving (Stillman, 2000). The use of mathematical tasks in context was hoped to
foster motivation for the learning of mathematics, and demonstrate and teach students to
deal with the links and transfers between mathematics and the real world.
Mathematical tasks in context are particularly relevant in PISA, the Programme for
International Student Assessment conducted since 2000 by the Organisation for Economic
Cooperation and Development (OECD). Blum (2002) notes that since 2001 (the first time
PISA results were published), researchers have debated the role of item context in students’
performance; some researchers believe it makes items simpler but others consider that
including contextual elements makes items significantly more challenging for students (de
Lange, 2007). As a result, this paper investigates specifically how a systematic alteration of
the context in which a PISA item is embedded impacts students’ performance.
Key Terminology
Context of Mathematical Items and Tasks
There are several names in the mathematics education literature for what this paper
(along with many other researchers) calls context, such as cover history, thematic content,
and setting. In order to have a more refined understanding of the term context, Clarke and
Helme (1998, p. 132) define figurative context as “the scenario where the task is
encountered”. Busse and Kaiser (2003) further refine figurative context into the objective
figurative context and the subjective figurative context. According to these authors, the
objective figurative context is related to the “description of the scenario given in the task”
(p. 4) whereas the subjective figurative context refers to the “individual representation of
the objective figurative context” (p. 4). The objective figurative context is “often implicitly
meant by researchers when referring to the context” (Busse & Kaiser, 2003, p. 4). For this
reason, the term objective figurative context is referred to as (item) context in this study
except when otherwise stated. The context also needs to be distinguished from the
2014. In J. Anderson, M. Cavanagh & A. Prescott (Eds.). Curriculum in focus: Research guided practice
(Proceedings of the 37 th annual conference of the Mathematics Education Research Group of
Australasia) pp. 5562. Sydney: MERGA.
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stimulus— the actual material about the item context that is presented by the item to the
student (e.g., in words, pictures and graphs).
Mathematical Core
The phrase mathematical core is used to describe the underlying mathematical model
and solving challenges inherent in a problem. In this study, items are described as having
the same mathematical core if they have the same or very similar underlying mathematical
model, similar calculational demands, etc.
Levels of Context Use
The very first distinction among levels of context use can be traced back to de Lange
(1979). This was refined by OECD (2009) to distinguish three levels of context use. Two
of these levels are common in PISA items. These levels were borrowed for this research.
The zeroorcler use of context is when the item generally involves just mathematical
terms, shapes, data and the translation of textually packaged mathematical problems (e.g.,
about triangles). Alternatively, some realworld terms might be included just to camouflage
or add a little interest to the mathematical operations required. An early years’ example is
‘four more than three apples’. PISA has used almost no zeroorder context items. First
orcler use of context takes place when the context is “needed for solving the problem and
judging the answer” (OECD, 2009, p. 31). Finally, the secondorder use of context takes
place when “students need to move backwards and forwards between the mathematical
problem and its context in order to solve the problem or to reflect on the answer within the
context to judge the correctness of the answer” (OECD, 2009, p. 31). Information drawn
from knowledge of the context (not just the problem statement or known mathematical
facts) is required to solve the problem. A similar classification of the levels of context use
is provided by Stillman (1998). According to her, context can function as a border (zero
order/firstorder), a wrapper (firstorder) or tapestry (secondorder and more) for a
problem. As discussed below, these definitions of levels need clarification in further work.
Research into Context
In school mathematics, when formulating mathematical representations of tasks
embedded in contexts, factors such as (a) activation of realworld knowledge, (b) students’
engagement, and (c) students’ familiarity with the task context are found to be important
elements affecting the task accessibility to students and performance on these tasks,
although the effects and their exact causes are not yet well known. Studies related to the
factors above will be now briefly discussed.
Context and Activation of RealWorld Knowledge
Two seminal studies in this area are by Carraher, Carraher and Schliemann (1985,
1987). In Carraher at al. (1985), the performance of five young Brazilian street sellers
(aged 9 to 15 years old) on mathematical tasks presented in realworld contexts (involving
selling, giving change) was better than on schooltype mathematical tasks and on context
free symbolic tasks involving identical numbers and operations. The researchers explained
the better performance by differences in the way students approached the realworld
versions because in the simulated store contexts students mentally worked with money (a
concrete realworld construct), which changed the arithmetic demand of the tasks. Carraher
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et al. (1987) conducted a followup study with 16 Brazilian third graders. These authors
found that students showed significant differences in performance when they solved
simulated store contexts (outside school contexts, usually presented in verbal form), than
problems inside school contexts presented in written form, and symbolic computation
exercises; Carraher et al. (1987) found that realworld knowledge facilitated the task’s
accessibility, hence it could lead students to a greater performance.
Baranes, Perry, and Stigler (1989) intended to replicate Carraher et al. (1987)’s findings
with Year 3 American students (who do not work as street sellers and almost certainly had
more consistent schooling) but found no contextual effects in either performance or
strategy use for success with the American sample (/z=18); that is to say, the students did
not generally activate their realworld knowledge and representation of it in the solution of
the tasks. Participating students sometimes activated their realworld knowledge when the
numbers used in the word problems induced students to use “a culturally constituted
system of quantification, such as money” (Baranes et al., 1989, p. 316).
Context and Engagement
In the examination of the positive effect of context on students’ performance, better
performance by students can be associated with more involvement with the task context.
Stillman (2002) investigated exhaustively how fortythree Year 11 and 12 students used the
task context during modelling tasks in selecting the mathematics they needed initially, in
keeping on track throughout the particular task they were attempting (and generally in their
task solving), and as a final check at the end of a task. Her study found that mathematical
modelling tasks resulted in more understanding and thus greater involvement with the
context than standard application tasks; and that better performance was correlated with
medium to high involvement with the context. Nevertheless, students who performed well
in mathematics were found to be an exception: they did well with low involvement.
Context Familiarity
A wellknown postulate is that a familiar context “provides a less abstract and more
directly experienced grounding for the new domain, thus enhancing the use of particular
strategies that allow for more efficient processing” (Huang, 2004, p. 279). From a
theoretical stance it follows that familiar contexts might enhance students’ performance.
This has been found in studies such as that of Chipman, Marshall, and Scott (1991).
An increasing body of empirical research evidence determines, however, that
familiarity with a context can be associated with either negative or neutral impact on
students’ performance (Huang, 2004; van den HeuvelPanhuizen, 2005). Huang (2004)
investigated to what extent the performance and perception of task difficulty of 48 Taiwan
Year 4 students were influenced by familiar or unfamiliar contexts in four everyday
shopping mathematical tasks. The data did not support the hypothesis that familiar contexts
improve performance; instead students were statistically significantly better on tasks
embedded in unfamiliar contexts. Moreover, students spent a statistically significant longer
time in solving tasks with familiar contexts.
Van den HeuvelPanhuizen (2005) also acknowledged that familiar contexts are not
always helpful to students and may also generate difficulties in students’ problem solving.
Familiar contexts can hinder some students finding an answer, while others may focus on
contextual aspects and fail to engage with the necessary mathematics required to solve the
task. Boaler (1994) gave an example of this situation. She analysed performance of 50
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female students on two sets of questions intended to assess the same mathematical content
(equivalence of fractions) but set in different contexts (i.e. soccer season, planting plants,
cutting pieces of wood, and a fashion workshop). Results show that females underachieved
in task contexts with which they were probably more familiar (e.g. fashion rather than
soccer). They often took excessive account of contextual information in the tasks. Boaler
(1994) speculated that the relative underachievement on the fashion task was because the
attractive and familiar context distracted the students from the mathematical structure.
The above review has raised several issues concerning contextual influences on
students’ performance, which this study aims to examine. The results in the literature on
students’ performance with reference to the effects of context on mathematical tasks are
mixed; it can have a positive, a negative or a neutral impact for all the factors reviewed
above (i.e. familiarity, activation of realworld knowledge, etc.). It is also possible that
context will operate differently with different types of tasks (word problems, extended
modelling tasks, etc.). Hence, the effects of context on PISA mathematics items are
undetermined, due to the lack of studies addressing this issue.
The Study
This study is a pilot study to investigate the effects of (1) familiarity and (2) the level of
context use on students’ performance on PISA items. The item context was varied whilst
holding constant other features of the item that are known to affect performance
(performance is interpreted as average percent correct). There are three research questions:
1 . Is students’ performance better on items with more familiar contexts?
2. Is students’ performance on items that require secondorder context use more
affected by change of context than on items that require firstorder context use?
3. How does context affect students’ solutions of mathematics items and their
performance?
Selection, Characteristics and Creation of Test Items
Four PISA Mathematics items released after the 2000, 2003 and 2006 surveys were
selected (OECD, 2006). The items had approximately similar difficulty, and two were
judged by the researchers to require firstorder use of context, and two required second
order use of context (taking into account that the subjects are 15 year old students). Items
with low percentage correct were chosen because it was thought that the contexts used may
have made them more difficult, and also that any improved performance on items with
increased familiarity of the context could show up.
Table 1 gives the names and PISA codes of the selected items, along with the average
percentage of students in all OECD countries receiving full credit (OECD, 2009) and level
of context use, as judged by the researchers. Table 1 also names the four ‘PISAlike’ items
that were specially constructed for this study. Each PISA item had a PISAlike sibling item
where the context was changed to be more familiar to students, but with all other relevant
aspects of the problem held constant. To exemplify this, Pi A was about scheduling
internet chat between students in Germany and Australia accounting for time zone
differences. The context of its PISAlike sibling PLilA was to schedule TV broadcasts for
India of events at the Commonwealth Games that were being held in Australia at the time
of data collection. The timeliness of the Commonwealth Games, and frequent discussions
about when events are shown on TV, led the researchers to judge that this would be a more
familiar context for the sample population than scheduling internet chat. Held constant
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were the mathematical core of the problems, the level of context use, the required amount
of receptive and expressive communication, response type, etc. In this pilot study,
increased familiarity of the siblings was judged by the researchers. The PISA items and
siblings are fully described in Almuna Salgado (2010).
Creating sibling items was difficult, and reveals deep aspects about how context is
embedded in the problemsolving process. Finding an appropriate new context requires
imagination and understanding of students’ interests. Especially for the secondorder use
of context, although the basic underlying mathematical model may be the same, the new
assumptions and information required were often very different and changed the problems.
This foreshadows a need for further work to refine the notion of mathematical core. For
example, PiD Robberies gave a graph of the number of robberies in a town, and asked
students to evaluate a reporter’s statement that there had been a big increase. Truncated
axes made the graph potentially misleading. The mathematical core related to considering
both absolute and relative change. However, whether a change is truly large or small and
whether absolute or relative changes are most important varies markedly with context:
consider increased train fares, house prices, change in CO 2 concentration in the air or
average temperature of the earth. Context seems to be a dynamic feature of a task even
when the cognitive demand, competencies required and the model remain constant. In
summary, very careful consideration was needed to construct the siblings.
Table 1
PISA Items Selected for the Study and their Sibling Items
Item
PISA item
PISA item
Percentage
Item’s
Code of
Context of
code
name
code A
receiving
use of
sibling
sibling item
full credit A
context
item
PiA
Internet
Chat Relay
M402Q02
29%
First
PFilA
2010
Commonwealth
Games on TV
PiB
Rock
Concert
M552Q01
28% b
Second
PFilB
Car park
Support
Support for a
PiC c
for the
M702Q01
36%
First
PFilC
new shopping
President
centre
PiD
Robberies
M179Q01
30%
Second
PFilD
Train ticket
prices
A Source OECD (2009). B Field trial data (source OECD, 2013). C PiC was simplified for this study by
omitting one of the factors (survey time) given in M702Q01.
Participants and the Rotated Test Design
The target population in this study was students approximately 15 years old, because
PISA is administered at this age. The performance of thirty students from two Year 10
classes at one volunteer school in Melbourne, Australia, on four PISA items was compared
using paper andpencil tests. Participants were tested in a class period. Because of the
limited time for testing and potential interitem effects, a rotated design of three booklets
was used, so that students did not solve both an item and its sibling. This means that direct
comparison of performance at the level of the student was not possible. Each booklet
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contained PISA items and PISAlike items with first and secondorder use of context (see
Table 2). Booklets were distributed at random to 10 students each. Booklets also contained
some additional items not discussed here. To answer Research Question 3, a sample of
students was interviewed about their solution processes using stimulated recall. Space
limitations preclude a proper report here (see Almuna Salgado (2010)), although insights
have been included in the discussion section below.
Results
Table 2 presents the item names and booklets, sample size, distribution of students’
scores (assessed using the published PISA scheme), item means and standard deviations.
The more familiar context of the sibling items, compared to the PISA items, resulted in the
expected improved performance only for PiC and PLilC (Presidential poll compared to
shopping centre). Performance was not affected for the A sibling pair, and the more
familiar items for the B and D sibling pairs had lower performance. Overall more familiar
contexts did not improve performance. Instead, performance was worse with familiar
contexts, but not to a statistically significant extent ( F(l,78 ) = 1.179, p = 0.281).
Table 2
Distribution of Students ’ Scores (Items scored out of 2, sample size 10 students per item)
Item
Booklet
Full credit (2)
Partial credit (1)
No credit (0)
Mean
Item s.d.
PiA
A
50%

50%
1.00
1.05
PiB
C
50%

50%
1.00
1.05
PiC
C
60%
0%
40%
1.20
1.03
PiD
B
20%
40%
40%
0.80
0.78
Average performance on Pi
1.00
0.97
PLilA
C
50%

50%
1.00
1.05
PLilB
A
20%

80%
0.40
0.84
PLilC
B
50%
40%
10%
1.40
0.69
PLilD
C
0%
30%
70%
0.30
0.48
Average performance on PLil
0.77
0.76
Note: Partial credit not available on PiA, PiB and siblings based on the PISA marking scheme.
Research Question 2 hypothesised that changing context would change performance
more on items with secondorder than firstorder use of context. Since this hypothesis
predicted the magnitude of change, rather than whether one context of a pair made an item
easier or harder than another, the hypothesis was examined by analysing the absolute
values of the differences in performance. The data in Table 2 support this hypothesis.
Because there is no student level data on change, only the average changes for the items in
the sibling pairs can be compared. The absolute changes in mean scores for the two first
order sibling pairs are 0.00 and 0.20 respectively, and for the secondorder sibling pairs are
0.60 and 0.50 respectively. With only four sibling pairs, no statistical tests are appropriate,
but the data agree with the hypothesis.
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Discussion and Conclusions
The study aimed to examine differences in the performance of participants in two Year
10 classes when they solved PISA items with the same mathematical core whilst varying
the familiarity of contexts. Embedding items in a context more familiar to students did not
assist students’ performance. This might be explained by the fact that the greater familiarity
of items in this study was not empirically determined, but was only established from the
researchers’ opinions. It may also be explained if the new and more familiar sibling items
were not technically as well constructed as the multiplytrialled PISA items, but this is
unlikely because the PISA items were such a close model for their siblings.
Qualitative evidence from the interviews revealed that, in more familiar contexts,
students tended to bring personal information into arguments rather than using a
mathematical argument; a familiar context was sometimes interpreted and judged as
personal rather than from a mathematical point of view. Students’ comments suggested that
certain familiar contexts (e.g. train ticket prices) encouraged them to think and argue in
personal terms rather than mathematically. Students who considered that the absolute
proposed ticket price increase was a very large amount of money (which is part of their
subjective figurative context) may not have considered at all whether the proposed increase
was a small percentage, quite in line with normal inflation— indeed, they may not know
about the likely size of normal inflation. In addition, with this familiar context (i.e. train
ticket prices) some students were reality bound; they considered a familiar context as real
using Melbourne metropolitan ticket prices to judge their answers which produced a large
scenario for a narrow task. For example, some students took into consideration the
different kinds of tickets and prices in Melbourne; the subjective figurative context was
seen to guide the selection of arguments to communicate students’ answers. In this vein,
one difficulty with familiar contexts is that they tend to elicit responses in students that
may be based on integration of personal knowledge and values with mathematics in order
to build an intended solution. Familiar contexts also may be borderline cases where the
relatively stronger understanding of a task plays a role when students communicate an
answer; they may assume that it is not necessary to give a very detailed answer because
everyone already knows the arguments. For example, some students did not explain that
their answers were based on proportional reasoning (so they did not get full credit), perhaps
assuming that would be well understood.
On the other hand, the results in Table 2 provided some support for the hypothesis that
the items involving the secondorder use of context were more affected by change of
context than items involving the firstorder use of context. The experience of creating the
sibling pairs for the items with secondorder use of context also pointed in this direction.
The research hypothesis above seemed therefore to be supported, but the data set (using
two pairs of items for each level of context use) was too small to make a strong claim.
To conclude, the findings in this study are in line with the research literature, which
suggests that task context can impact on students’ performance in variable ways. This leads
us to infer that context needs more careful study with deeper analysis and experimental
control of the way in which context is involved. Plans for carrying out such further study
include (a) getting data from students on familiarity of the contexts, (b) extending the
operationalised definition of mathematical core beyond the notion of mathematical model
to include the mathematical competencies (OECD, 2009), mathematical procedure,
computation and strategies and solving challenges of every mathematical task, (c) using
more sensitive marking schemes than the PISA scheme, which is necessarily simple in
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order to be used by many teams for large sample sizes in many languages, and (d) refining
the definition and operationalisation of the levels of context use. Better information about
how context affects student performance might help teachers to instruct students how to
work more effectively with problems in context.
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