Creative and algorithmic mathematical reasoning: effects of transfer-appropriate processing and effortful struggle

Abstract

Two separate studies, Jonsson et al. (J. Math Behav. 2014;36: 20–32) and Karlsson Wirebring et al. (Trends Neurosci Educ. 2015;4(1–2):6–14), showed that learning mathematics using creative mathematical reasoning and constructing their own solution methods can be more efficient than if students use algorithmic reasoning and are given the solution procedures. It was argued that effortful struggle was the key that explained this difference. It was also argued that the results could not be explained by the effects of transfer-appropriate processing, although this was not empirically investigated. This study evaluated the hypotheses of transfer-appropriate processing and effortful struggle in relation to the specific characteristics associated with algorithmic reasoning task and creative mathematical reasoning task. In a between-subjects design, upper-secondary students were matched according to their working memory capacity.The main finding was that the superior performance associated with practicing creative mathematical reasoning was mainly supported by effortful struggle, however, there was also an effect of transfer-appropriate processing. It is argued that students need to struggle with important mathematics that in turn facilitates the construction of knowledge. It is further argued that the way we construct mathematical tasks have consequences for how much effort students allocate to their task-solving attempt.

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Creative and algorithmic mathematical reasoning:
effects of transfer-appropriate processing and
effortful struggle
Bert Jonsson, Yagmur C. Kulaksiz & Johan Lithner
To cite this article: Bert Jonsson, Yagmur C. Kulaksiz & Johan Lithner (2016): Creative and
algorithmic mathematical reasoning: effects of transfer-appropriate processing and effortful
struggle, International Journal of Mathematical Education in Science and Technology
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY, 
http://dx.doi.org/./X..
RESEARCH ARTICLE
Creative and algorithmic mathematical reasoning: effects of
transfer-appropriate processing and effortful struggle
Bert Jonsson a, Yagmur C. Kulaksizaand Johan Lithnerb,c
aDepartment of Psychology, Umeå University, Umeå, Sweden; bDepartment of Science and Mathematics
Education, Umeå University, Umeå, Sweden; cUmeå Mathematics Education Research Centre, Umeå
University, Umeå, Sweden
ARTICLE HISTORY
Received  February 
KEYWORDS
Creative mathematical
founded reasoning;
algorithmic reasoning;
effortful struggle;
transfer-appropriate
processing
ABSTRACT
Two separate studies, Jonsson et al. (J. Math Behav. 2014;36: 20–32)
and Karlsson Wirebring et al. (Trends Neurosci Educ. 2015;4(1–2):6–14),
showed that learning mathematics using creative mathematical rea-
soning and constructing their own solution methods can be more
ecient than if students use algorithmic reasoning and are given
the solution procedures. It was argued that eortful struggle was the
key that explained this dierence. It was also argued that the results
could not be explained by the eects of transfer-appropriate pro-
cessing, although this was not empirically investigated. This study
evaluated the hypotheses of transfer-appropriate processing and
eortful struggle in relation to the specic characteristics associated
with algorithmic reasoning task and creative mathematical reasoning
task. In a between-subjects design, upper-secondary students were
matched according to their working memory capacity.
The main nding was that the superior performance associated with
practicing creative mathematical reasoning was mainly supported
by eortful struggle, however, there was also an eect of transfer-
appropriate processing. It is argued that students need to struggle
with important mathematics that in turn facilitates the construction
of knowledge. It is further argued that the way we construct mathe-
matical tasks have consequences for how much eort students allo-
cate to their task-solving attempt.
1. Introduction
1.1. Learning mathematics
Amaingoalinmathematicsistohelpstudentsunderstand,judge,perform,andusemath-
ematics in a variety of mathematical situations. However, much of the time in classrooms
is devoted to solving tasks, using pre-specied procedures, otherwise known as algorithms
(e.g. [1,2]). Algorithms are quick, reliable, time savers, and they prevent miscalculations
given that they include a nite sequence of executable instructions that take care of the dif-
cultiesinthetask.However,thereasonthatitisquickandreliableisthatitisdesignedto
CONTACT Bert Jonsson bert.jonsson@umu.se
©  Informa UK Limited, trading as Taylor& Francis Group
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2B. JONSSON ET AL.
avoidmeaning;therefore,itdoesnotgiverisetoanydeeperunderstandingoftheprinci-
ples of mathematics.[3]Theuseofpre-denedalgorithmscanreducetheworkingmem-
ory (WM) load of complicated calculations [4] and can subsequently free up cognitive
resources for more advanced problem-solving.[5]However,ifalloralmostalllearningis
accomplished using pre-dened procedures without considering the underlying intrinsic
mathematical properties, there is an apparent risk that students’ task-solving procedures
arebasedonthesupercialcharacteristicsofthetasks,leadingstudentstoengageinthe
un-reected use of those algorithms.[2,6] As such, being able to recall the algorithms in
their original form, without any conceptual understanding of them, does not promote the
type of learning that most teachers are aiming for. The distinction between attaining a more
supercial versus a more deeper/conceptual understanding of mathematics has previously
been investigated and described in dierent terms. Hiebert and Lefevre [7]arguethatcon-
ceptual knowledge is characterized as knowledge that is rich in relationships, and that the
interconnection within the networks is as important as the discrete pieces of information
themselves. Conversely, procedural knowledge is based on one’s familiarity with the rules
and procedures needed to solve the tasks (see [7] for an overview). However, there are
also studies showing that this relationship can be bidirectional.[8]Inthisstudy,weper-
formed a follow-up of two previous studies, which provided evidence that ‘creative mathe-
matical founded reasoning’ (CMR), was superior with regard to mathematical task perfor-
mance when compared to ‘algorithmic reasoning’ (AR).[9,10] The main focus is whether
the specics associated with CMR and AR practice, respectively, elicit dierent processes
known to support performance. In particular, it was of interest to pursue whether the dier-
ences in test performance after practicing with either AR and CMR tasks could be explained
by a process denoted as the hypothesis of ‘eortful struggle’ (ES), as opposed to the hypoth-
esis of ‘transfer-appropriate processing’ (TAP). It has in a recent study been shown that
quizzes and unite exams that require more eort in terms of higher order thinking pro-
mote not only deeper conceptual understanding but also lead to superior performance on
more low (thinking) level tasks. However, quizzes and exams that require less eort did not
promote deeper conceptual understanding and did, therefore, not lead to better perfor-
mances on lower level thinking tasks.[11] Below we elaborate on the constructs TAP, ES,
and the signicance of task design. We also elaborate on AR and CMR reasoning accord-
ing to Lithner [2] model of mathematical reasoning. We do recognize that there are other
models and perspectives of mathematical reasoning touching on the distinction between
conceptual and more procedural knowledge; such as the procedures-rst theories arguing
that procedures are acquired before concepts (e.g. [12]); Concepts-rst theories arguing that
conceptsareacquiredbeforeproceduresfortheopposite(e.g.[13]) and an iterative model
arguing that conceptual and procedural knowledge are mutually independent.[8,14]
However, the main focus in this paper is not to evaluate the eectiveness of Lithner [2]
model in relation to other models of mathematical reasoning (though we provide empirical
data replicating previous studies and supporting the model per se). But to investigate the
underlyingprocessesofTAPandESinrelationtothespecicsassociatedwithARandCMR
practice and test tasks. This addresses a broad and fundamental but not easily answered
question; does the way we construct tasks has consequences for how much eortful struggle
students allocate in their task-solving attempt?
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 3
1.2. Transfer-appropriate processing
The TAP phenomenon states that if there is a close relationship between how informa-
tion is encoded and subsequently retrieved, then performance is facilitated because of this
close relation. TAP has been examined in numerous studies. Morris, Bransford, and Franks
[15] investigated whether performance on recognition tests was related to the similarity
between the previous acquisition tasks and subsequent recognition tests. It was found that
semantic acquisition was superior to rhyme acquisition on a standard recognition test,
whereas rhyme acquisition was superior to semantic acquisition on a rhyming recogni-
tion test. Franks, Bilbrey, Lien, and McNamara [16] further conducted a study to investi-
gate the eects of priming in relation to the same or dierent tasks. They used dierent
word groups involved in a lexical decision task, animacy judgment task, liking judgment
task, hardness judgment task, and bigness judgment task. The results supported TAP, as it
was found that the same task conditions led to greater priming than did the dierent task
conditions.
In mathematics and mathematics education there are many studies that have been inves-
tigating eects of TAP in terms of creating so-called ‘realistic teaching situations’, hence the
in-class tasks should be applicable in the real-world context. The general idea is that math
learning should be as close to the real-world application as possible, after which it gradual
becomes more formal and general (e.g. [17]). An approach that seems even more impor-
tant for children with mathematical learning diculties.[18] A specic approach taking
intoaccountTAPwasReed,Corbett,Homan,Wagner,andMacLaren[19]studyusing
cognitivetutoring(CT)instructions.ThendingswereconsistentwiththeTAPapproach,
students who used CT instructions were helped on subsequent task involving CT instruc-
tions.AresultthatisinlinewithAdamsetal.[20]studyonproblem-solvinginwhich
an increased similarity between instructions and practice situations was found to enhance
performance.
With regard to phonological encoding, Mulligan and Picklesimer [21]foundthatphono-
logical led to better recollection than did semantic encoding on a rhyme recognition test.
Martin-Chang and Levy [22] found similar results in their experiment, in which words
in isolation and in context were presented to elementary school students. Students showed
greaterreadinguencyintheisolatedwordtestaftertheisolatedwordtraining.Theauthors
also discovered that the children were able to identify the words that were presented in
isolation more accurately and quickly. In addition, Nungester and Duchastel [23]further
showed that practicing multiple choice items yielded better performance on a multiple
choice retention test when compared to practicing on short answer items; likewise, prac-
ticing on short answer items produced better performance on a short answer retention
test.
Additional support for the TAP phenomenon comes from brain imaging studies –
mainly those using non-invasive techniques, such as functional magnetic resonance imag-
ing (fMRI) and electroencephalography (EEG). Ritchey, Wing, LaBar, and Cabeza [24]pre-
sented positive, negative, and neutral emotional scenes during encoding and during a sub-
sequentrecognitiontask.Theresultsrevealedthatemotionalarousalwasassociatedwith
the level of similarity between the encoding and recognition tasks, which could be observed
as increased hippocampal activation. In an EEG study, Nyberg, Habib, McIntosh, and Tul-
ving [25] presented visual words paired with sounds during encoding. It was shown that
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4B. JONSSON ET AL.
remembering those visual words activated the same auditory brain regions that were active
during encoding.
1.3. Eortful struggle
With regard to mathematics, Niss [26] argued that ‘the students need to be engaged in activ-
itieswheretheyhaveto“struggle”(inaproductivesenseofthatword)withimportantmath-
ematics’ (p.1304) in order to achieve desirable learning outcomes. In a mathematics educa-
tion review, Hiebert and Grouws [1] suggested that ‘struggle’ is necessary for the students to
develop their conceptual understanding. From a mathematics education perspective, little
is known regarding how to translate this understanding into specic activities that lead to
both eortful and productive struggle. However, in memory research, there are supportive
studies showing that more ‘eortful struggle’ in terms of eortful retrieval facilitate later
performance (e.g. [27–29]),whichhasalsobeenproventobeeectiveinteachingsitua-
tions (e.g. [30–33]). The retrieval eort hypothesis has also been found to facilitate transfer,
which means that learning generalizes to other formats or contexts beyond the one in which
it was initially learned; this is often viewed as an important aim for learning (e.g. [34,35]).
Bjork and colleagues (e.g. [36,37]) noted that the initial cost during practice facilitates
later performance, a benecial strategy they denoted as ‘desirable diculties.’ Hence, ‘any
timethatyou,asalearner,lookupananswerorhavesomebodytellorshowyousome-
thing that you could, drawing on current cues and your past knowledge, generate instead,
you rob yourself of a powerful learning opportunity’.[37, p.61] In relation to mathemat-
ics, Rohrer and Taylor [38] showed that spaced practice was superior to mass practice for
later tests. It was also shown that a mixture of problem types during practice, which boosts
later test performance, actually impeded practice performance. This cross-over interaction
from practice to test shows that a ‘struggle’ with tasks during practice pays o on a later
test, but this is probably experienced as a non-constructive strategy given that it impedes
practice performance. However, to simply tell the students, ‘you have to struggle with the
tasks to learn’ is not very constructive, especially when mathematical textbooks arrange
tasks in an orderly fashion and use mass practice strategies that follow an introductory
example, as found in a comparison of common textbooks from 12 nations in ve dierent
continents.[39]
Textbooksalsocommonlyprovidetemplatesintheformofpre-denedformu-
las/procedures illustrating how to solve the dierent tasks. This ‘help’ often removes the
opportunity for students to extract the conceptual meaning of the tasks. An ‘opportunity’
that often is and has to be associated with certain amount of eort (struggle). However, in
atypicalclassroomsituation,ateacherortextbookprovidesasetoftasksaccompaniedby
templatealgorithmsthatcanbeusedtosolvethetask.Thisisthenfollowedbymassive
repetition of the provided algorithm, leading to an un-reected use of the same algorithm
without nearly any conceptual understanding of the tasks.[6]Andthusoftenwithalmostno
requirement for eort. How the tasks are designed is, therefore, an important key for intro-
ducing aspects of ‘desirable diculties’ and/or struggle (e.g. [26,37]). This study concerns
a particular component of eortful struggle in mathematics learning: the struggle related
to the construction of task solution methods. Solving tasks by using pre-dened templates
does not include this particular component of struggle.
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 5
1.4. Task design
Chapman [40] argued that appropriately designed mathematical tasks will (1) promote stu-
dents’ conceptual understanding of mathematics, (2) retain their interest in the task, and
(3) optimize their learning. In the context of productive failure, Kapur [41,42]showedthat
participants practicing with ill-structured tasks (tasks that possessed unknown parame-
ters, multiple solutions) perform worse than participants who were able to practice with
well-structured tasks (tasks that possessed few parameters that could vary) during the prac-
tice session. However, at post-test, the pattern was reversed; participants practicing on ill-
dened tasks outperformed those who had practiced on well-dened tasks. It was argued
that working with tasks that are higher in complexity facilitated students’ ability to develop
structures that were benecial for problem-solving. These results indicate that how the tasks
are designed is important for enhancing reasoning, task-solving, and conceptual learning.
Unfortunately, students do not acknowledge how they learn and do not appreciate learning
approaches that requires cognitive eort.[43] This adds to the above argument and is in line
with the focus of this study; how tasks are designed is important in order to invite/‘force’
students to allocate (appropriate) resources to the tasks at hand.
1.5. Mathematical reasoning and didactical situations
In addition to how tasks are designed and distributed, and mixed with other tasks, it is
important to create situations where students are not only given the responsibility to con-
struct their target knowledge [3]butalsotheopportunitytostrugglewiththetask.[26,37]
In Brousseau’s [3] devolution of a problem approach, students have to take responsibility
for the task-solving process, or at least part of it. The teacher’s responsibility is to provide a
situation that allows the students to work and struggle with the task’s solutions; the teachers
are to refrain themselves from interfering in the process rather than communicating their
knowledge of how to solve the task.
In a model based on the theory of didactical situations,[3] Lithner [2]suggestedthata
key variable when learning mathematics through task-solving is the reasoning that students
activate. Lithner [2] focused on two types of reasoning: creative mathematical founded rea-
soning (CMR) and algorithmic reasoning (AR). CMR includes two main components: (1)
the solver constructs a solution that is new to her, and (2) the solution is supported by argu-
ments based on the intrinsic mathematical properties of the task (see [2] for details). This
CMR denition is similar to common denitions of mathematical problem-solving (e.g.
[44,45]). AR tasks are, on the other hand, designed to mimic a school context in which
examples and pre-dened procedures are commonly used. These types of reasoning are in
this study viewed in relation to specic task characteristics, denoted as AR and CMR tasks.
AR tasks can be solved as CMR tasks; however, it is also possible to solve these tasks with
no or little conceptual understanding by simply applying the provided algorithms. Hence,
‘AR tasks’ include given solution methods and they are expected to trigger AR, while ‘CMR
tasks’ include no given method and they are expected to trigger CMR.
1.6. Behaviour and neuroscientic support for the AR–CMR distinction
Using Lithner’s [2] framework as points of departure, Jonsson et al.[9]conductedastudyto
examine the eects of CMR and AR practice on subsequent test performance. Participants
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6B. JONSSON ET AL.
practiced either the AR or CMR tasks and were tested on three types of tasks one week later.
The rst test task required them, within 30 seconds, to read the task, retrieve the appropriate
formula from memory, and write down the formula. The second and third tasks required
them to reconstruct or construct the appropriate solution to the task.
In an fMRI study, Karlsson Wirebring et al.[10]usedasubsetofthesametasks;theyeval-
uated test performance and its relation to brain activity. Both studies showed that the par-
ticipants who practiced mathematical task-solving without any algorithmic support (CMR)
outperformed those that had been provided with algorithmic support (AR) on memory-
based test tasks. This same group also outperformed the AR group on test tasks that allowed
for the reconstruction, or construction, of task solution methods one week later. The fMRI
analyses showed that CMR participants, to a lesser extent, taxed the brain region known to
be associated with mathematical processing: the angular gyrus. In addition, a second brain
region was found with relatively lower brain activity for the CMR when compared to the AR
participants: the left pre-central cortex/Brodmann area 6; this is an area involved in tasks
requiring working memory capacity (WMC). The fMRI study indicated that CMR partici-
pants had an easier time accessing their memory of the solution method, as reected in the
observed lower brain activity and given that they, to a lesser extent, needed to engage their
WM. It was argued that ES, in terms of cognitively more demanding tasks during practice,
is important for subsequent test task performance.
From a didactical situations perspective,[3] students must accept that the task is their
own problem, so as to solve it. They consequently need to put in more eort in generating,
justifying, and implementing their own solutions which, in turn, will facilitate subsequent
test task performance. In Jonsson et al.,[9] it was argued that practicing with CMR tasks in
contrast to practicing with AR tasks involved more eort, a higher level of ES, which was
important for the subsequent test task performance. ES was discussed in relation to the
retrieval eort hypothesis, which assumes thatthemoredemandingoreortfulretrievalis
during practice, the better that the same material will be remembered later.[46] However,
as pointed out in both Jonsson et al.[9]andKarlssonWirebringetal.[10], there is an alter-
native interpretation that might explain these results: TAP. TAP states that if there is a close
relationship between how information/tasks is initially encoded/learned and subsequently
retrieved/solved, performance is facilitated. Although Jonsson et al.[9]andKarlssonWire-
bring et al.[10] discussed and argued against TAP, it was not further investigated.
This study uses the task used in Jonsson et al.[9]andKarlssonWirebringetal.[10]and
investigated the eects of TAP and ES with regard to the specic characteristics of AR and
CMR tasks, respectively. In the Materials section, the AR and CMR tasks specics and their
relationshipstoTAPandESaredescribed.
2. Aims and hypotheses
In Jonsson et al.[9]andKarlssonWirebringetal.[10] one group of upper secondary stu-
dents practiced by AR tasks (i.e. including solution templates) and one group by CMR tasks
(i.e. not including solution templates), but both groups were tested on CMR tasks. The main
aim of this study was to investigate whether the dierences found in Jonsson et al.[9]and
Karlsson Wirebring et al.[10]couldbeexplainedbyESand/ortheTAPhypotheses.To
maintain the same set-up as previously, the rst analysis was targeting memory retrieval of
the specic formula that was associated with the corresponding practice task. The analyses
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 7
of the subsequent test tasks were used to evaluate the hypotheses of TAP and ES by targeting
the participant’s task-solving.
This study was therefore based on the following three hypotheses.
(1) It was expected that practicing with CMR tasks would better enhance performance
on the memory retrieval tasks than would practicing with AR tasks.
(2) According to the TAP hypothesis, it was expected that the practice format would
be related to the test format, in that practice with AR tasks would better enhance
test performance on AR test tasks than would practice with CMR tasks. It was also
expected that practicing with CMR tasks would better enhance test performance on
CMR test tasks than would practicing with AR tasks.
(3) According to the ES hypothesis, it was expected that practicing with CMR tasks
would better enhance performance than would practicing with AR tasks on subse-
quent test tasks, irrespectively of test task format. Thereby also providing support
for a cross-format transfer as a function of ES.
3. Methods
3.1. Participants
Therewereatotalof59students(38girlsand21boys)fromuppersecondaryschoolswith
age ranging from 17–21 years old; the students’ mean age was 18 years and 9 months. Writ-
ten informed consent was obtained in accordance with the Declaration of Helsinki, and the
study was approved by the Regional Ethical Review Board, Sweden.
3.2. Working memory capacity and matching procedure
When investigating school attainments in general, and the more cognitively demanding
subjects in particular, it is also important to investigate or control for variations in cogni-
tive prociency. WMC has been repeatedly identied as a signicant predictor of school
performance (e.g. [47,48]). WM is developed from the concept of short-term memory,
and the most common and validated model includes a central executive and three slave
systems:thephonologicalloop,thevisuospatialsketchpad,andtheepisodicbuer.[49]
Through these systems, WM feeds into and retrieves information from long-term mem-
or y. With re g ard to thi s s tudy, the ra t ionale wa s t h at mat h e matical ta s k -solving i ncludes
the online manipulation of transient information, as well as the transfer of informa-
tion to long-term storage – a process that relies substantially on WMC. WM is gener-
ally considered to have limited capacity and it is commonly measured using a dual task
paradigmthatcombinesamemoryspantaskwithaconcurrenttask(acomplexWMtask).
This study used a complex WM span task (operation span) developed by Unsworth and
Engle.[50] Operation span has been shown to have good test–retest reliability and inter-
nal consistency.[51–53] The purpose for using operation span scores was to match partic-
ipants in the CMR and AR practice conditions, ensuring that they were equal in terms of
WMC.[9]
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8B. JONSSON ET AL.
4. Materials
4.1. AR and CMR tasks – an overall description
The same layout structure was applied to both AR and CMR practice tasks. A basic calcula-
tor was presented on the screen to prevent simple mathematical calculations from causing
any diculties for the students when completing the tasks. The practice sessions consisted
of 14 corresponding AR or CMR task sets and were presented between subjects. The task
sets required dierent solution methods and were, as far as we know, novel to all partici-
pants. We have in pilot studies asked students whether these specic tasks were known to
them or if they had encountered similar tasks. No student answered that they encountered
these specic tasks, a few students claimed that they may have encountered similar tasks.
The example provided is one of the tasks that were regarded as vaguely familiar to a few
students. We have also asked teachers and they conrmed that it was highly unlikely that
thespecictaskshavebeenusedintheregularschoolwork.
The test tasks were presented one week after practice. Figure 1(a–f) illustrates AR and
CMR practice and test tasks (practice tasks left column and test tasks in the right column)
from one of the 14 task sets that were used. These tasks are explained in detail below. See
[9] for more examples of tasks.
4.2. Practice tasks
AR Practice – Task Characteristics (left column, Figure 1). As in Jonsson et al.,[9]there
were 14 task sets; each task set had ve numerical subtasks, with a six-minute time limit
for each of the subtasks. Each subtask was presented together with a solution method in
a format familiar from mathematics textbooks, that is to say, a formula and an example of
how to apply it. Figure 1(a) shows an example of one (out of ve) of those subtasks. The
other four subtasks (not displayed) were identical, but they featured dierent numbers in
thequestions.Whentheparticipantshadansweredeachsubtaskandclickedonthe‘next’
button, a correct answer was displayed on the screen.
CMR Practice –Tas k C ha r a cter i s ti cs (left column Figure 1).As in Jonsson et al.,[9]there
were 14 CMR task sets. Each task set had three subtasks; however, no guidance (no exam-
ple and no formula) was provided on how to solve the tasks, and a 10-minute time limit
was given for each subtask. Numerical answers were required for the rst two subtasks.
Figure 1(c)isanexampleofone(outoftwo)ofthosesubtasks;theothersubtaskwasiden-
tical, but it featured dierent numbers. For the third subtask, the participants were required
to generate a mathematical formula in the format of a function (Figure 1(e)), which was
based on the previous subtasks. When the participants had answered a given subtask and
clicked on the ‘next’ button, a correct answer was displayed on the screen.
Note that the only dierence between AR practice tasks (I–V) and CMR practice tasks (I
and II) was that the AR task was provided with additional information in terms of a formula
andanexamplehowtoapplyit.
4.3. Test tasks characteristics (Figure 1, right column)
In Jonsson et al.[9]andKarlssonWirebringetal.[10], the test tasks were all in CMR for-
mat, in this we added test tasks in AR format making it possible to evaluate eects of
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 9
Figure . (a–f) Examples of the practice and test tasks: (a) AR practice tasks I–V; (b) AR test tasks II and III;
(c) CMR practice tasks I and II; (d) CMR test tasks II and III; (e) CMR practice task III; and (f) AR and CMR test
task I.
practicing with AR and CMR tasks on numerical test tasks presented in both AR and CMR
format. However irrespectively of practice condition the initial test task (memory retrieval)
was common for all participants. The ‘memory retrieval’ and the ‘numerical’ test tasks are
described below.
Memory retrieval.IntesttaskI(Figure 1(f)), the participants were given a 30-second
time interval; during that time, they were required to read the task, retrieve the formula
from memory, and write down a correct answer (a formula that could be used to solve the
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10 B. JONSSON ET AL.
task). The memory retrieval task was novel and identical for all participants irrespective of
whether they had practiced with the AR or CMR tasks. If it was solved correctly, then the
same formula could be used to solve test tasks II and III.
Numerical test tasks. The test tasks consisted of a numerical subtasks (test task II) where
theparticipantshadatimelimitof30secondstosolvethetaskfollowedbyanumerical
task (test task III), which was the same numerical task as in test task II; however, instead of
30 seconds, its time limit was 300 seconds. Figure 1(b) shows an example of AR test tasks
II and III, and Figure 1(d) provides an example of CMR test tasks II and III.
4.4. Practice and test tasks and their relation to transfer-appropriate processing
and eortful struggle
Practicing with AR tasks I–V (Figure 1(a)) could facilitate AR test tasks II and III perfor-
mance (Figure 1(b)) as the only dierence was that dierent numbers were used, these sim-
ilarities may therefore subserve TAP. Similarly, the only dierence between practicing with
CMR tasks I and II (Figure 1(c)) and being tested on CMR test tasks II and III (Figure 1(d))
was that dierent numbers were used; thus also subserving TAP.
With regard to ES,[9] argued that CMR practice conducted without the support of for-
mulas or any examples of how to apply them (Figure 1(b)) required more extensive ES
than did the AR practice. It was further argued that the more extensive ES associated with
CMR practice, did in turn support, the conceptual understanding during practice and sub-
sequently facilitated, memory retrieval of the formula and/or (re)construction of the meth-
ods used to derive solutions. In addition, as shown in Dobson and Linderholm [52] gener-
ating responses – even relatively simple ones – requires more eort. These arguments are
further supported by Karlsson Wirebring et al.,[10] who found that CMR practice reduced
the cognitive load and mathematical processing during a subsequent test, presumably as a
function of ES during practice. For AR practice tasks (Figure 1(a)), we argue that being pro-
videdwiththeappropriateformula,andanexampleofhowtoapplyit,reducesthedemands
for ES to a minimum. Hence, AR tasks all contain procedures that are necessary for solving
the task and this can ultimately be achieved with minimal eort.
4.5. Design
In a between-subjects design, using the matching procedure four independent groupings
of a practice and test condition were formed (i.e. (1) CMR–MR/CMR; (2) AR–MR/AR; (3)
AR–MR/CMR; and (4) CMR–MR/AR). A grouping such as CMR–MR/AR denotes prac-
ticingwithCMRtasksandsubsequentlybeingtestedonmemoryretrieval(MR)ofthe
formula (test task I). Immediately followed by being tested on AR test tasks (test tasks II
and III). The notation >and <denotes the hypothesized direction. A contrast such as
CMR–AR >AR–AR denotes the expectation that practicing with CMR tasks and being
tested on AR task is superior to practicing with AR tasks and being tested on AR tasks.
However, test tasks II and III only diered in terms of the amount of time available and
had a correlation of .86. Test tasks II and III were, therefore, amalgamated into a compos-
ite score representing numerical test task-solving and are from here denoted as numerical
test tasks. The numerical test task performances associated with each grouping were subse-
quently contrasted against other groupings as stated by the hypotheses.
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 11
4.6. Procedure
Both measures of cognitive prociency and the practice- and test-sessions of AR and CMR
tasks were conducted in groups in a classroom within the students’ own school using
theparticipants’owncomputers.Studentswereaskedtologintothesamewebpage
and were directed to the cognitive test, which was used to match them into the separate
groups. Depending on the group they were assigned to, the participants were one week later
presented with either AR or CMR practice tasks, and after one additional week, they per-
formed either the AR or CMR test tasks, logging in on the same web page. The partici-
pants performed all the tasks individually and did not receive any assistance when com-
pleting the tasks and the webserver automatically stored the participants’ performance
data.
4.7. Statistical analyses
Analysis of variance (ANOVA) investigated potentially dierences between groupings with
regard to WMC, grades, and gender. An initial t-test evaluated hypothesis 1; that practic-
ing with CMR tasks would be superior with regard to memory retrieval (MR) one week
later compared to practicing with AR tasks. A result in line with the hypothesis would con-
rm.[9,10]
To pursue hypotheses 2 and 3 an ANOVA, with groupings of practice and test conditions
as a between subject factor and composite score of the numerical test tasks as the depen-
dent variable, was conducted. A signicant main eect would indicate that the grouping
of practice and test conditions diered with regard to test task performance. Subsequent
planned contrasts evaluated whether such a main eect could be explained by TAP and/or
ES. In line with the hypotheses these planed contrast analyses were a-priori stated, and
were thus always tested in the same direction (>). See Figure 2 for a visualization of the
analyses from the overall contrasts down to the specic contrasts of individual groupings.
The illustration depicts the planned contrasts of the numerical test tasks. Planned con-
trast analyses use the whole data set, thereby increasing the power in the analyses.1Cohen’s
dwas used for eect size measures. Therefore, eect sizes of 0.2, 0.5, and 0.8 are con-
sideredassmall,medium,andlarge,respectively.[55]Theaverageeectsizesacrossthe
dierent contrasts for TAP and ES, respectively, were calculated. Eect sizes in the direc-
tion of the a-priori stated hypothesis are displayed as positive eects and eect sizes in the
opposite direction of the stated hypothesis as negative eect sizes. The average eect sizes
for TAP and ES, respectively, were calculated with the eect sizes from each contrast as
input.
5. Results
An ANOVA conrmed that there was no dierence between groupings regarding their
WMC, F(3,52) =0.37, p=.77. A chi-square analyses showed that there was no dier-
ence in gender distribution across groupings, χ2(3, N=59) =2.20, p=.53, χ2.One-way
ANOVA’s showed that there were no grouping dierences with regard to age, F(3,58) =
0.30, p=.99 or mathematical grades F(3,58) =0.38, p=.76, respectively. These variables
were, therefore, excluded from the analyses.
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12 B. JONSSON ET AL.
Figure . Analyses of the composite numerical test task tasks from the overall comparison down to con-
trasts of specific conditions. The left column depict contrasts with regard to transfer-appropriate process-
ing and the right column with regard to effortful struggle. Significant contrasts in accordance with the
a-priori stated hypothesis are market with an asterisk (∗) and significant effect against the hypothesis
with a cross (×).
Table 1 displays the mean proportions of the test task scores, for the memory retrieved
tasks (test taskI; formula) and the numerical test tasks (composite of test task II and III).
The t-test evaluating hypothesis 1 (CMR–MR >AR–MR) revealed that practicing CMR
was superior to practicing AR when being tested on identical memory retrieval tasks one
week later t(57) =3.74, p<.0001, d=1.00.
To pursue hypotheses 2 and 3 an initial ANOVA followed by planned contrasts analyses
were conducted. The ANOVA revealed a main eect of conditions F(3,58) =12.63, p<
.0001, np
2=0.41. The planned contrasts analyses presented below evaluated whether this
main eect can be explained by TAP and/or ES and follow the steps seen in Figure 2.
Tab le . Proportion of correct responses during the test set for AR and CMR (mean values).
Test conditions
Memory retrieval Numerical test task
Practice conditions AR CMR AR CMR
AR . (.) . (.) . (.) . (.)
CMR . (.) . (.) . (.) . (.)
Note: The proportion of correct responses in each cell (e.g. AR–MR; AR–CMR) with standard deviation within
parentheses. The composite scores of numerical test task are based on an average of test tasks II and III.
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 13
5.1. Transfer-appropriate processing
Contrast 1(leftcolumn).Intheoverallcontrast,itwasexpectedthatthegroupsthatreceived
the same task format during the practice and testing sessions should outperform those that
received dierent practice and test tasks (AR–AR and CMR—CMR >AR–CMR and CMR–
AR). The analysis was non-signicant, t(55) =0.01, p=.99, d=0.03.
Contrast 2 (left column). An overall contrast without the additive inuence of ES in
the practice condition (AR–AR and CMR—CMR >AR–CMR), thus removing CMR–AR,
revealed signicant eects in favour of TAP, t(55) =3.35, p=.01, d=1.04.
Contrast 3 (left column). For more specic contrasts, it was predicted that practice CMR
would enhance test performance on CMR more than practice AR (CMR–CMR >AR–
CMR). The analysis revealed a signicant dierence in line with TAP, t(55) =2.16, p=
.035, d=1.03.
Contrast 4 (left column). It was also predicted that practice with AR would better
enhance test performance on AR than practice with CMR (AR–AR >CMR–AR). The anal-
ysis was signicant in the opposite direction, t(55) =2.22, p=.031, d=−1.03, possibly
reecting an eect of ES in the CMR practice condition.
Two out of four contrasts were signicant in the direction of the hypothesis and one in
theoppositedirection.Theaverageeectsize(Cohen’sd)wasfoundtobe0.27,whichis
considered as a small eect size.
5.2. Eortful struggle
Contrast 1 (right column). In contrast 1, CMR–AR and CMR–CMR >AR–CMR and AR–
AR, it was expected that practicing CMR would improve performance on subsequent tests,
more than practicing AR. The analysis was signicant and in favour of the ES hypothesis,
t(55) =3.10, p=.003, d=0.71.
Contrast 2 (right column).Contrast 2 without the additive inuences of TAP (CMR–
AR >AR–CMR and AR–AR), thus removing CMR–CMR, revealed signicant eects in
favour of ES t(55) =4.76, p<.001, d=1.27.
Contrast 3 (right column). For the more specic contrasts (CMR–AR >AR–CMR) it was
predicted that practice CMR would enhance test performance on AR more than practice
AR. The analysis revealed a signicant eect t(55) =5.99, p<.0001, d=2.69 in favour of
ES hypothesis.
Contrast 4 (right column). It was also predicted that practice with CMR would better
enhance test performance on AR than practice with AR (CMR–AR >AR–AR). The anal-
yses revealed signicant eects in favour of ES hypothesis, t(55) =2.22, p=.03 d=0.70.
All the four contrasts were signicant in the direction of the hypotheses. The average
eect size (Cohen’s d)wasfoundtobe1.34,whichisclearlyabove0.8–themarginfora
large Cohen’s deect size.
6. Discussion
The analyses of memory retrieval conrmed hypothesis 1; that practicing CMR was supe-
rior compared to practicing AR with regard to subsequent memory retrieval. The analy-
sisconrmedtheresultsfoundinJonssonetal.,[9] however this analysis did not address
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14 B. JONSSON ET AL.
whether the AR and CMR task characteristics associated with TAP and/or ES could explain
the results in this study or those presented in Jonsson et al.[9]andKarlssonWirebring
et al.[10]. To address this question we contrasted numerical test task conditions. The ini-
tial ANOVA revealed a main eect, indicating that groupings of practice and test condi-
tions (AR–CMR; AR–AR; CMR–CMR; CMR–AR) diered with regard to numerical test
task performance. This analysis was followed by contrasts analyses evaluating the eects
of TAP and ES on the numerical test tasks. The analyses revealed signicant eects of TAP
(hypothesis 2), however the evidence points to ES as a more likely explanation of the results
(hypothesis 3). Below, the results are discussed from the TAP and ES perspectives, sepa-
rately, and their combined eects and educational relevance are also addressed.
6.1. Transfer-appropriate processing
According to TAP, it was expected that practice with CMR tasks would enhance test task
performance on CMR test tasks, as compared to practice with AR tasks, and that prac-
tice with AR tasks would improve test task performance on AR test tasks more than prac-
tice with CMR tasks would do. For contrast 1 (AR–AR and CMR–CMR >AR–CMR and
CMR–AR), this prediction was not conrmed. For the contrast 2 excluding the additive
eects of ES (AR–AR and CMR–CMR >AR–CMR), the prediction was conrmed. The
condition-specic contrast (contrast 3) CMR–CMR >AR–CMR was signicant. However,
the signicant eects in the opposite direction, contrast 4 (AR–AR <CMR–AR) showed
that TAP-matched tests did not inevitably explain performance, thus pointing to ES as a
more substantial explanation for this result.
6.2. Eortful struggle and cross-format transfer
According to the ES hypothesis, it was predicted that CMR–CMR and CMR–AR would
outperform AR–AR and AR–CMR, as CMR practice requires more ES and thus results
in superior performance on subsequent tests.[9]Thehypothesisreceivedaclearsupport.
Contrast 1 (CMR–AR and CMR—CMR >AR–CMR and AR–AR), as well as the over-
all contrast excluding the additive eects of TAP (CMR—AR >AR–CMR and AR–AR),
together with the condition-specic contrasts (CMR—AR >AR–CMR and CMR—AR >
AR–AR),allpointedtoeectsofES.
These results are in line with those of Stenlund et al.,[35] where it was shown that
when practice required more eortful retrieval, it was more likely that cross-format trans-
fer occurred. In relation to Stenlund et al.,[35] the results of this study indicate that ES
facilitated subsequent task-solving and thus resulted in a cross-format transfer between the
CMR and AR tasks – a transfer that did not occur for the opposite practice test format (AR–
CMR) (see also Kang, McDermott, and Roediger [56]andMcDaniel,Anderson,Derbish,
andMorrisette [57]).
7. Conclusions
Although some support was found for TAP, our results indicate that ES is a more reasonable
explanation for the results presented in the Jonsson et al.[9]andKarlssonWirebringet
al.[10], (i.e. CMR–CMR >AR–CMR).
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 15
Theargumentsforthiseectareasfollows:(1)thenumberofcontrasts(rangingfrom
theoverallcontrastsdowntothespeciccomparisons)wereinfavourofESrelativetoTAP;
(2)theaverageeectssizeoftheEShypothesiswerealmostvetimesthatofTAP;(3)CMR
practice generated cross-format transfer; (4) for the overall contrasts (contrast 2, TAP), the
eect of TAP was only seen when the eects of ES were removed; and (5) CMR practice
led to signicantly superior performance upon testing, even when the test tasks were in AR
format (CMR—AR >AR–AR), a cross-format transfer. This cross-format transfer could be
considered as a near-transfer eect.[58] In a study of WM training Dahlin, Neely, Larsson,
Backman, and Nyberg,[59] it was showed that training can be generalized to untrained
tasks as a function of training-induced changes that occur in brain activity. In Karlsson
Wirebring et al.[10], it was indeed, as pointed out in the introduction, discovered that CMR
practice, as compared to AR practice, resulted in decreased prefrontal brain activity at the
subsequent test. This was viewed as a reallocation of brain activity, which was interpreted
as a conceptually driven change in increased accessibility of the participants’ knowledge of
the task solutions, potentially freeing cognitive resources that could be otherwise allocated
to solving tasks on subsequent tests. See also Ischebeck, Zamarian, Schocke, and Delazer
[60] for similar arguments. The result that CMR practice was superior to AR practice on
subsequent test, even when the test tasks were in AR format (a cross-format transfer), is
particularly interesting because solving AR test tasks can be achieved simply by using the
algorithm and the examples provided – a procedure that does not require any mathematical
construction. It seems as CMR practice potentiated the eect of learning to an untrained
AR test task to greater extent that AR practice did, even though the practice and test tasks
both were in AR format. However there are, as pointed out, signicant eects of TAP. It is
possible that TAP contributes not only through the similarities between practice and test
conditions but also through a practice-test-expectancy eect. If the practice condition is
associated with more struggle, as in CMR practice, the students will adjust their expecta-
tions accordingly. Their expectations of the test session as a function of practice session
could therefore be associated with a feeling of ‘I need to struggle to achieve’ also in the
test condition. Eort is therefore transferred from practice to test, hence the ES becomes a
‘strategy’ and this strategy is therefore subject to TAP.
There are some limitations, with regard to this study. To maintain the similarity to the
previous studies we used the same practice and test tasks which means that the ‘retrieve
from memory test tasks’ (test task I) preceded the numerical test tasks (II and III). It could
be argued that there are similarities between practice CMR and memory retrieval and
betweenpracticingARandmemoryretrievalthatcouldaecttheperformancesonthesub-
sequent numerical test tasks. More specic, for CMR it could be argued that the processes
used to generate a formula (CMR practice task III; Figure 1(e)) and the subsequent attempt
to retrieve the formula (test task I; Figure 1(f)) tap into the same underlying processes. And
that this similarity amplied the eects of CMR practice on the subsequent numerical test
tasks as a function of TAP. But as pointed out by Karpicke and Zaromb,[61]self-generation
(CMR practice task III) does not involve the same process as when being asked to retrieve
informationfromlong-termmemory(teststaskI).ForARpractice,ontheotherhand,it
couldbearguedthatrepeatedexposuretotheformula(practicetasksI–V;Figure 1(a)) facil-
itated the memory retrieval of the same formula (test task I; Figure 1(f)). It can therefore
not be ruled out that the similarity between the repeated exposure of the formula during
AR practice and the subsequent memory retrieval of the same formula amplied the eects
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16 B. JONSSON ET AL.
of TAP on the subsequent numerical test tasks. See Xue et al. [62]foradiscussiononthe
eects of repeated study.
It was with the present design impossible to disentangle whether the eect is uniquely
driven by task specics associated with Lithner’s [2] model of mathematical reasoning or
by a more general experiences of greater diculties during CMR practice, that induced
more ES, unrelated to CMR task specics. However in Karlsson Wirebring et al.[10]the
brain activity associated with test task performances was found to be unrelated to how the
participants experienced test task diculties. Further studies will address this question also
for practice tasks.
Another potential limitation is the power in the analyses. In that respect it is worth point-
ing out that the results of AR and CMR practice on CMR tests replicate that of Jonsson et
al.[9]andKarlssonWirebringetal.[10]. We argue therefore that the replication provides a
rm basis for our attempt to disentangle the processes of TAP and ES, though it cannot be
ruled out that more power in the analyses in terms of more participants could have aected
the results.
The educational implications is that CMR task-solving ‘forces’/invites students to strug-
glewithimportantmathematicsthatinturnfacilitatestheconstructionofknowledge.The
studydoesindicatethatthewayweconstructmathematicaltaskshaveconsequencesfor
how much eort students allocate to their task-solving attempt. From a teacher education
perspective, it is therefore important that the help provided is appropriate, so that the strug-
gle becomes a facilitator of, and not an obstacle to, learning. The present study also argues
against the assumption that providing students with algorithms, which is common, leads to
superior performance (see Jonsson et al.[9] and Lithner 2008 [2]foradiscussion).Inthis
way, practicing AR tasks did not yield better performance, irrespective of the test format
used.
Itis,however,importanttoobservethatTAPandESarecognitiveprocessesthatcan
facilitate performance and, if combined, there are potentially additive eects that might be
of benet. Since task-specic characteristics have been declared as essential for achieving
learning (e.g. [38–40]) it is imperative that task construction evokes, or at least consid-
ers, the principles of eortful retrieval/ES and TAP. Constructing tasks and teaching situ-
ations without considering how the brain processes information is something that should
be avoided in order to eectively facilitate learning.
Note
1. ˆ
ψ=wj¯
Yj
Acknowledgments
We thank Tony Qwillbard for help with parts of the data collection and computer programming. We
also thank all the students for being part of this study and the teacher for helping out organizing the
data collection
Disclosure statement
No potential conict of interest was reported by the authors.
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 17
Funding
Kempe foundation to Johan Lithner; Umea University to Bert Jonsson
ORCID
Bert Jonsson http://orcid.org/0000-0002-5884-6469
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