International Journal of Mathematical Education in Science and Technology
ISSN: 0020-739X (Print) 1464-5211 (Online) Journal homepage: http://www.tandfonline.com/loi/tmes20
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 To link to this article: http://dx.doi.org/10.1080/0020739X.2016.1192232
Published online: 15 Jun 2016.
Submit your article to this journal
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tmes20 Download by: [Umeå University Library]
Date: 15 June 2016, At: 04:03
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. Kulaksiza and Johan Lithnerb,c
Downloaded by [Umeå University Library] at 04:03 15 June 2016
a Department of Psychology, Umeå University, Umeå, Sweden; b Department of Science and Mathematics Education, Umeå University, Umeå, Sweden; c Umeå Mathematics Education Research Centre, Umeå University, Umeå, Sweden
ABSTRACT
ARTICLE HISTORY
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 transferappropriate 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.
Received February KEYWORDS
Creative mathematical founded reasoning; algorithmic reasoning; effortful struggle; transfer-appropriate processing
1. Introduction 1.1. Learning mathematics A main goal in mathematics is to help students understand, judge, perform, and use mathematics in a variety of mathematical situations. However, much of the time in classrooms is devoted to solving tasks, using pre-specified procedures, otherwise known as algorithms (e.g. [1,2]). Algorithms are quick, reliable, time savers, and they prevent miscalculations given that they include a finite sequence of executable instructions that take care of the difficulties in the task. However, the reason that it is quick and reliable is that it is designed to
CONTACT Bert Jonsson
[email protected]
© Informa UK Limited, trading as Taylor & Francis Group
Downloaded by [Umeå University Library] at 04:03 15 June 2016
2
B. JONSSON ET AL.
avoid meaning; therefore, it does not give rise to any deeper understanding of the principles of mathematics.[3] The use of pre-defined algorithms can reduce the working memory (WM) load of complicated calculations [4] and can subsequently free up cognitive resources for more advanced problem-solving.[5] However, if all or almost all learning is accomplished using pre-defined procedures without considering the underlying intrinsic mathematical properties, there is an apparent risk that students’ task-solving procedures are based on the superficial characteristics of the tasks, leading students to engage in the un-reflected 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 superficial versus a more deeper/conceptual understanding of mathematics has previously been investigated and described in different terms. Hiebert and Lefevre [7] argue that conceptual 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] In this study, we performed a follow-up of two previous studies, which provided evidence that ‘creative mathematical founded reasoning’ (CMR), was superior with regard to mathematical task performance when compared to ‘algorithmic reasoning’ (AR).[9,10] The main focus is whether the specifics associated with CMR and AR practice, respectively, elicit different processes known to support performance. In particular, it was of interest to pursue whether the differences in test performance after practicing with either AR and CMR tasks could be explained by a process denoted as the hypothesis of ‘effortful struggle’ (ES), as opposed to the hypothesis of ‘transfer-appropriate processing’ (TAP). It has in a recent study been shown that quizzes and unite exams that require more effort in terms of higher order thinking promote not only deeper conceptual understanding but also lead to superior performance on more low (thinking) level tasks. However, quizzes and exams that require less effort did not promote deeper conceptual understanding and did, therefore, not lead to better performances on lower level thinking tasks.[11] Below we elaborate on the constructs TAP, ES, and the significance of task design. We also elaborate on AR and CMR reasoning according 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-first theories arguing that procedures are acquired before concepts (e.g. [12]); Concepts-first theories arguing that concepts are acquired before procedures for the opposite (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 effectiveness 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 underlying processes of TAP and ES in relation to the specifics associated with AR and CMR 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 effortful struggle students allocate in their task-solving attempt?
INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY
3
Downloaded by [Umeå University Library] at 04:03 15 June 2016
1.2. Transfer-appropriate processing The TAP phenomenon states that if there is a close relationship between how information 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 recognition test. Franks, Bilbrey, Lien, and McNamara [16] further conducted a study to investigate the effects of priming in relation to the same or different tasks. They used different 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 different task conditions. In mathematics and mathematics education there are many studies that have been investigating effects 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 important for children with mathematical learning difficulties.[18] A specific approach taking into account TAP was Reed, Corbett, Hoffman, Wagner, and MacLaren [19] study using cognitive tutoring (CT) instructions. The findings were consistent with the TAP approach, students who used CT instructions were helped on subsequent task involving CT instructions. A result that is in line with Adams et al. [20] study on problem-solving in which an increased similarity between instructions and practice situations was found to enhance performance. With regard to phonological encoding, Mulligan and Picklesimer [21] found that phonological 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 greater reading fluency in the isolated word test after the isolated word training. The authors 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, practicing 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 imaging (fMRI) and electroencephalography (EEG). Ritchey, Wing, LaBar, and Cabeza [24] presented positive, negative, and neutral emotional scenes during encoding and during a subsequent recognition task. The results revealed that emotional arousal was associated with 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 Tulving [25] presented visual words paired with sounds during encoding. It was shown that
4
B. JONSSON ET AL.
remembering those visual words activated the same auditory brain regions that were active during encoding.
Downloaded by [Umeå University Library] at 04:03 15 June 2016
1.3. Effortful struggle With regard to mathematics, Niss [26] argued that ‘the students need to be engaged in activities where they have to “struggle” (in a productive sense of that word) with important mathematics’ (p.1304) in order to achieve desirable learning outcomes. In a mathematics education 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 specific activities that lead to both effortful and productive struggle. However, in memory research, there are supportive studies showing that more ‘effortful struggle’ in terms of effortful retrieval facilitate later performance (e.g. [27–29]), which has also been proven to be effective in teaching situations (e.g. [30–33]). The retrieval effort 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 beneficial strategy they denoted as ‘desirable difficulties.’ Hence, ‘any time that you, as a learner, look up an answer or have somebody tell or show you something 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 mathematics, 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 off 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 five different continents.[39] Textbooks also commonly provide templates in the form of pre-defined formulas/procedures illustrating how to solve the different 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 effort (struggle). However, in a typical classroom situation, a teacher or textbook provides a set of tasks accompanied by template algorithms that can be used to solve the task. This is then followed by massive repetition of the provided algorithm, leading to an un-reflected use of the same algorithm without nearly any conceptual understanding of the tasks.[6] And thus often with almost no requirement for effort. How the tasks are designed is, therefore, an important key for introducing aspects of ‘desirable difficulties’ and/or struggle (e.g. [26,37]). This study concerns a particular component of effortful struggle in mathematics learning: the struggle related to the construction of task solution methods. Solving tasks by using pre-defined templates does not include this particular component of struggle.
INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY
5
Downloaded by [Umeå University Library] at 04:03 15 June 2016
1.4. Task design Chapman [40] argued that appropriately designed mathematical tasks will (1) promote students’ 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] showed that participants practicing with ill-structured tasks (tasks that possessed unknown parameters, 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 practice session. However, at post-test, the pattern was reversed; participants practicing on illdefined tasks outperformed those who had practiced on well-defined tasks. It was argued that working with tasks that are higher in complexity facilitated students’ ability to develop structures that were beneficial 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 effort.[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 construct their target knowledge [3] but also the opportunity to struggle with the task.[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] suggested that a 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 reasoning (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 arguments based on the intrinsic mathematical properties of the task (see [2] for details). This CMR definition is similar to common definitions 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-defined procedures are commonly used. These types of reasoning are in this study viewed in relation to specific 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 neuroscientific support for the AR–CMR distinction Using Lithner’s [2] framework as points of departure, Jonsson et al.[9] conducted a study to examine the effects of CMR and AR practice on subsequent test performance. Participants
Downloaded by [Umeå University Library] at 04:03 15 June 2016
6
B. JONSSON ET AL.
practiced either the AR or CMR tasks and were tested on three types of tasks one week later. The first 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] used a subset of the same tasks; they evaluated test performance and its relation to brain activity. Both studies showed that the participants who practiced mathematical task-solving without any algorithmic support (CMR) outperformed those that had been provided with algorithmic support (AR) on memorybased 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 participants had an easier time accessing their memory of the solution method, as reflected 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 effort 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 effort, a higher level of ES, which was important for the subsequent test task performance. ES was discussed in relation to the retrieval effort hypothesis, which assumes that the more demanding or effortful retrieval is during practice, the better that the same material will be remembered later.[46] However, as pointed out in both Jonsson et al.[9] and Karlsson Wirebring et al.[10], there is an alternative 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] and Karlsson Wirebring et al.[10] discussed and argued against TAP, it was not further investigated. This study uses the task used in Jonsson et al.[9] and Karlsson Wirebring et al.[10] and investigated the effects of TAP and ES with regard to the specific characteristics of AR and CMR tasks, respectively. In the Materials section, the AR and CMR tasks specifics and their relationships to TAP and ES are described.
2. Aims and hypotheses In Jonsson et al.[9] and Karlsson Wirebring et al.[10] one group of upper secondary students 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 differences found in Jonsson et al.[9] and Karlsson Wirebring et al.[10] could be explained by ES and/or the TAP hypotheses. To maintain the same set-up as previously, the first analysis was targeting memory retrieval of the specific formula that was associated with the corresponding practice task. The analyses
INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY
7
Downloaded by [Umeå University Library] at 04:03 15 June 2016
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 subsequent 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 There were a total of 59 students (38 girls and 21 boys) from upper secondary schools with age ranging from 17–21 years old; the students’ mean age was 18 years and 9 months. Written 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 cognitive proficiency. WMC has been repeatedly identified as a significant 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: the phonological loop, the visuospatial sketch pad, and the episodic buffer.[49] Through these systems, WM feeds into and retrieves information from long-term memory. With regard to this study, the rationale was that mathematical task-solving includes the online manipulation of transient information, as well as the transfer of information to long-term storage – a process that relies substantially on WMC. WM is generally considered to have limited capacity and it is commonly measured using a dual task paradigm that combines a memory span task with a concurrent task (a complex WM task). 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 internal consistency.[51–53] The purpose for using operation span scores was to match participants in the CMR and AR practice conditions, ensuring that they were equal in terms of WMC.[9]
8
B. JONSSON ET AL.
4. Materials
Downloaded by [Umeå University Library] at 04:03 15 June 2016
4.1. AR and CMR tasks – an overall description The same layout structure was applied to both AR and CMR practice tasks. A basic calculator was presented on the screen to prevent simple mathematical calculations from causing any difficulties 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 different solution methods and were, as far as we know, novel to all participants. We have in pilot studies asked students whether these specific tasks were known to them or if they had encountered similar tasks. No student answered that they encountered these specific 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 confirmed that it was highly unlikely that the specific tasks have been used in the regular school work. 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 five 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 five) of those subtasks. The other four subtasks (not displayed) were identical, but they featured different numbers in the questions. When the participants had answered each subtask and clicked on the ‘next’ button, a correct answer was displayed on the screen. CMR Practice – Task Characteristics (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 example 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 first two subtasks. Figure 1(c) is an example of one (out of two) of those subtasks; the other subtask was identical, but it featured different 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 difference 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 and an example how to apply it.
4.3. Test tasks characteristics (Figure 1, right column) In Jonsson et al.[9] and Karlsson Wirebring et al.[10], the test tasks were all in CMR format, in this we added test tasks in AR format making it possible to evaluate effects of
9
Downloaded by [Umeå University Library] at 04:03 15 June 2016
INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY
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. In test task I (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
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 the participants had a time limit of 30 seconds to solve the task followed by a numerical 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.
Downloaded by [Umeå University Library] at 04:03 15 June 2016
4.4. Practice and test tasks and their relation to transfer-appropriate processing and effortful struggle Practicing with AR tasks I–V (Figure 1(a)) could facilitate AR test tasks II and III performance (Figure 1(b)) as the only difference was that different numbers were used, these similarities may therefore subserve TAP. Similarly, the only difference 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 different numbers were used; thus also subserving TAP. With regard to ES,[9] argued that CMR practice conducted without the support of formulas 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 subsequently facilitated, memory retrieval of the formula and/or (re)construction of the methods used to derive solutions. In addition, as shown in Dobson and Linderholm [52] generating responses – even relatively simple ones – requires more effort. 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 provided with the appropriate formula, and an example of how to apply it, reduces the demands 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 effort. 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 practicing with CMR tasks and subsequently being tested on memory retrieval (MR) of the 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 differed in terms of the amount of time available and had a correlation of .86. Test tasks II and III were, therefore, amalgamated into a composite 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 subsequently contrasted against other groupings as stated by the hypotheses.
INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY
11
4.6. Procedure
Downloaded by [Umeå University Library] at 04:03 15 June 2016
Both measures of cognitive proficiency and the practice- and test-sessions of AR and CMR tasks were conducted in groups in a classroom within the students’ own school using the participants’ own computers. Students were asked to log in to the same web page 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 performed either the AR or CMR test tasks, logging in on the same web page. The participants performed all the tasks individually and did not receive any assistance when completing the tasks and the webserver automatically stored the participants’ performance data. 4.7. Statistical analyses Analysis of variance (ANOVA) investigated potentially differences between groupings with regard to WMC, grades, and gender. An initial t-test evaluated hypothesis 1; that practicing 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 confirm.[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 dependent variable, was conducted. A significant main effect would indicate that the grouping of practice and test conditions differed with regard to test task performance. Subsequent planned contrasts evaluated whether such a main effect 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 specific contrasts of individual groupings. The illustration depicts the planned contrasts of the numerical test tasks. Planned contrast analyses use the whole data set, thereby increasing the power in the analyses.1 Cohen’s d was used for effect size measures. Therefore, effect sizes of 0.2, 0.5, and 0.8 are considered as small, medium, and large, respectively.[55] The average effect sizes across the different contrasts for TAP and ES, respectively, were calculated. Effect sizes in the direction of the a-priori stated hypothesis are displayed as positive effects and effect sizes in the opposite direction of the stated hypothesis as negative effect sizes. The average effect sizes for TAP and ES, respectively, were calculated with the effect sizes from each contrast as input.
5. Results An ANOVA confirmed that there was no difference between groupings regarding their WMC, F(3,52) = 0.37, p = .77. A chi-square analyses showed that there was no difference in gender distribution across groupings, χ 2 (3, N = 59) = 2.20, p = .53, χ 2 . One-way ANOVA’s showed that there were no grouping differences 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.
Downloaded by [Umeå University Library] at 04:03 15 June 2016
12
B. JONSSON ET AL.
Figure . Analyses of the composite numerical test task tasks from the overall comparison down to contrasts of specific conditions. The left column depict contrasts with regard to transfer-appropriate processing 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 effect of conditions F(3,58) = 12.63, p < .0001, np 2 = 0.41. The planned contrasts analyses presented below evaluated whether this main effect can be explained by TAP and/or ES and follow the steps seen in Figure 2.
Table . Proportion of correct responses during the test set for AR and CMR (mean values). Test conditions Memory retrieval Practice conditions AR CMR
Numerical test task
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.
INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY
13
Downloaded by [Umeå University Library] at 04:03 15 June 2016
5.1. Transfer-appropriate processing Contrast 1(left column). In the overall contrast, it was expected that the groups that received the same task format during the practice and testing sessions should outperform those that received different practice and test tasks (AR–AR and CMR—CMR > AR–CMR and CMR– AR). The analysis was non-significant, t(55) = 0.01, p = .99, d = 0.03. Contrast 2 (left column). An overall contrast without the additive influence of ES in the practice condition (AR–AR and CMR—CMR > AR–CMR), thus removing CMR–AR, revealed significant effects in favour of TAP, t(55) = 3.35, p = .01, d = 1.04. Contrast 3 (left column). For more specific 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 significant difference 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 analysis was significant in the opposite direction, t(55) = 2.22, p = .031, d = −1.03, possibly reflecting an effect of ES in the CMR practice condition. Two out of four contrasts were significant in the direction of the hypothesis and one in the opposite direction. The average effect size (Cohen’s d) was found to be 0.27, which is considered as a small effect size. 5.2. Effortful 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 significant 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 influences of TAP (CMR– AR > AR–CMR and AR–AR), thus removing CMR–CMR, revealed significant effects in favour of ES t(55) = 4.76, p < .001, d = 1.27. Contrast 3 (right column). For the more specific 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 significant effect 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 analyses revealed significant effects in favour of ES hypothesis, t(55) = 2.22, p = .03 d = 0.70. All the four contrasts were significant in the direction of the hypotheses. The average effect size (Cohen’s d) was found to be 1.34, which is clearly above 0.8 – the margin for a large Cohen’s d effect size.
6. Discussion The analyses of memory retrieval confirmed hypothesis 1; that practicing CMR was superior compared to practicing AR with regard to subsequent memory retrieval. The analysis confirmed the results found in Jonsson et al.,[9] however this analysis did not address
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] and Karlsson Wirebring et al.[10]. To address this question we contrasted numerical test task conditions. The initial ANOVA revealed a main effect, indicating that groupings of practice and test conditions (AR–CMR; AR–AR; CMR–CMR; CMR–AR) differed with regard to numerical test task performance. This analysis was followed by contrasts analyses evaluating the effects of TAP and ES on the numerical test tasks. The analyses revealed significant effects 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, separately, and their combined effects and educational relevance are also addressed.
Downloaded by [Umeå University Library] at 04:03 15 June 2016
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 practice with AR tasks would improve test task performance on AR test tasks more than practice with CMR tasks would do. For contrast 1 (AR–AR and CMR–CMR > AR–CMR and CMR–AR), this prediction was not confirmed. For the contrast 2 excluding the additive effects of ES (AR–AR and CMR–CMR > AR–CMR), the prediction was confirmed. The condition-specific contrast (contrast 3) CMR–CMR > AR–CMR was significant. However, the significant effects 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. Effortful 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] The hypothesis received a clear support. Contrast 1 (CMR–AR and CMR—CMR > AR–CMR and AR–AR), as well as the overall contrast excluding the additive effects of TAP (CMR—AR > AR–CMR and AR–AR), together with the condition-specific contrasts (CMR—AR > AR–CMR and CMR—AR > AR–AR), all pointed to effects of ES. These results are in line with those of Stenlund et al.,[35] where it was shown that when practice required more effortful retrieval, it was more likely that cross-format transfer 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] and McDaniel, 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] and Karlsson Wirebring et al.[10], (i.e. CMR–CMR > AR–CMR).
Downloaded by [Umeå University Library] at 04:03 15 June 2016
INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY
15
The arguments for this effect are as follows: (1) the number of contrasts (ranging from the overall contrasts down to the specific comparisons) were in favour of ES relative to TAP; (2) the average effects size of the ES hypothesis were almost five times that of TAP; (3) CMR practice generated cross-format transfer; (4) for the overall contrasts (contrast 2, TAP), the effect of TAP was only seen when the effects of ES were removed; and (5) CMR practice led to significantly 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 effect.[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 effect 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, significant effects 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 effect. If the practice condition is associated with more struggle, as in CMR practice, the students will adjust their expectations 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. Effort 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 between practicing AR and memory retrieval that could affect the performances on the subsequent numerical test tasks. More specific, 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 amplified the effects 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 information from long-term memory (tests task I). For AR practice, on the other hand, it could be argued that repeated exposure to the formula (practice tasks I–V; Figure 1(a)) facilitated 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 amplified the effects
Downloaded by [Umeå University Library] at 04:03 15 June 2016
16
B. JONSSON ET AL.
of TAP on the subsequent numerical test tasks. See Xue et al. [62] for a discussion on the effects of repeated study. It was with the present design impossible to disentangle whether the effect is uniquely driven by task specifics associated with Lithner’s [2] model of mathematical reasoning or by a more general experiences of greater difficulties during CMR practice, that induced more ES, unrelated to CMR task specifics. 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 difficulties. 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 pointing out that the results of AR and CMR practice on CMR tests replicate that of Jonsson et al.[9] and Karlsson Wirebring et al.[10]. We argue therefore that the replication provides a firm 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 affected the results. The educational implications is that CMR task-solving ‘forces’/invites students to struggle with important mathematics that in turn facilitates the construction of knowledge. The study does indicate that the way we construct mathematical tasks have consequences for how much effort 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 struggle 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] for a discussion). In this way, practicing AR tasks did not yield better performance, irrespective of the test format used. It is, however, important to observe that TAP and ES are cognitive processes that can facilitate performance and, if combined, there are potentially additive effects that might be of benefit. Since task-specific characteristics have been declared as essential for achieving learning (e.g. [38–40]) it is imperative that task construction evokes, or at least considers, the principles of effortful retrieval/ES and TAP. Constructing tasks and teaching situations without considering how the brain processes information is something that should be avoided in order to effectively facilitate learning.
Note 1. ψˆ =
w jY¯ j
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 conflict of interest was reported by the authors.
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
Downloaded by [Umeå University Library] at 04:03 15 June 2016
References [1] Hiebert J, Grouws DA. The effects of classroom mathematics teaching on students’ learning. Second Handbook Res Math Teach Learn. 2007;1:371–404. [2] Lithner J. A research framework for creative and imitative reasoning. Educ Studies Math. 2008;67(3):255–276. doi: 10.1007/s10649-007-9104-2 [3] Brousseau G. Theory of didactical situations in mathematics. Dordrecht: Kluwer; 1997. [4] Raghubar KP, Barnes MA, Hecht SA. Working memory and mathematics: a review of developmental, individual difference, and cognitive approaches. Learn Individ Diff. 2010;20(2):110– 122. [5] Merriënboer JG, Sweller J. Cognitive load theory and complex learning: recent developments and future directions. Educa Psychol Rev. 2005;17(2):147–177. doi: 10.1007/s10648-005-39510 [6] Boesen J, Helenius O, Bergqvist E, et al. Developing mathematical competence: from the intended to the enacted curriculum. J Math Behav. 2014;33:72–87. doi: 10.1016/j.jmathb.2013.10.001 [7] Hiebert J, Lefevre P. Conceptual and procedural knowledge in mathematics: an introductory analysis. In: Hiebert J, editor. Conceptual and procedural knowledge: the case of mathematics. Hillsdale (NJ): Lawrence Erlbaum Associates; 1986. p. 1–27. [8] Rittle-Johnson B, Alibali MW. Conceptual and procedural knowledge of mathematics: does one lead to the other? J Educ Psychol. 1999;91(1):175–189. doi: 10.1037/0022-0663.91.1.175 [9] Jonsson B, Norqvist M, Liljekvist Y, et al. Learning mathematics through algorithmic and creative reasoning. J Math Behav. 2014;36:20–32. doi: http://dx.doi.org/10.1016/j.jmathb.2014.08.003 [10] Karlsson Wirebring L, Lithner J, Jonsson B, et al. Learning mathematics without a suggested solution method: durable effects on performance and brain activity. Trends Neurosci Educ. 2015;4(1–2):6–14. doi: 10.1016/j.tine.2015.03.002 [11] Jensen J, McDaniel M, Woodard S, et al. Teaching to the test …or testing to teach: exams requiring higher order thinking skills encourage greater conceptual understanding. Educ Psychol Rev. 2014;26(2):307–329. [12] Halford GS. Children’s understanding: the development of mental models. Hillsdale (NJ): Erlbaum; 1993 [13] Siegler RS, Stern E. Conscious and unconscious strategy discoveries: a microgenetic analysis. J Exp Psychol Gen. 1998;127(4):377–397. [14] Schneider M, Rittle-Johnson B, Star JR. Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Dev Psychol. 2011;47(6): 1525–1538. [15] Morris CD, Bransford JD, Franks JJ. Levels of processing versus transfer appropriate processing. J Verbal Learn Verbal Behav. 1977;16(5):519–533. Available from: http://dx.doi.org/10.1016/S0022-5371(77)80016-9 [16] Franks J, Bilbrey C, Lien K, et al. Transfer-appropriate processing (TAP). Memory Cognition. 2000;28(7):1140–1151. doi: 10.3758/BF03211815
Downloaded by [Umeå University Library] at 04:03 15 June 2016
18
B. JONSSON ET AL.
[17] Van den Heuvel-Panhuizen M, Drijvers P. Realistic mathematics education. In: Lerman S, editor. Encyclopedia of mathematics education. Dordrecht, Netherlands: Springer; 2014. p. 521–525. [18] Goldman SR, Hasselbring TS. Achieving meaningful mathematics literacy for students with learning disabilities. J Learn Disabil. 1997;30(2): 198–208. [19] Reed S, Corbett A, Hoffman B, et al. Effect of worked examples and cognitive tutor training on constructing equations. Instruct Sci. 2013;41(1):1–24. doi: 10.1007/s11251-012-9205-x [20] Adams L, Kasserman J, Yearwood AA, et al. Memory access: the effects of factoriented versus problem-oriented acquisition. Memory Cognition. 1988;16(2):167–175. doi: 10.3758/BF03213486 [21] Mulligan NW, Picklesimer M. Levels of processing and the cue-dependent nature of recollection. J Memory Language. 2012;66(1):79–92. Available from: http://dx.doi.org/10.1016/j.jml.2011.10.001 [22] Martin-Chang S, Levy B. Word reading fluency: a transfer appropriate processing account of fluency transfer. Reading Writing. 2006;19(5):517–542. doi: 10.1007/s11145-006-9007-0 [23] Nungester RJ, Duchastel PC. Testing versus review: effects on retention. J Educ Psychol. 1982;74(1):18–22. doi: 10.1037/0022-0663.74.1.18 [24] Ritchey M, Wing EA, LaBar KS, et al. Neural similarity between encoding and retrieval is related to memory via hippocampal interactions. Cereb Cortex. 2012;23(12):2818–2828. doi: 10.1093/cercor/bhs258 [25] Nyberg L, Habib R, McIntosh AR, et al. Reactivation of encoding-related brain activity during memory retrieval. Proc Nat Acad Sci. 2000;97(20):11120–11124. doi: 10.1073/ pnas.97.20.11120 [26] Niss M. Reactions on the state and trends in research on mathematics teaching and learning: from here to utopia. In: Lester F, editor. Second handbook of research on mathematics teaching and learning. Charlotte (NC): Information Age Publishing; 2007. p. 1293–1312. [27] Carpenter S, Delosh E. Impoverished cue support enhances subsequent retention: support for the elaborative retrieval explanation of the testing effect. Memory Cognition. (2006). 34(2): 268–276. doi: 10.3758/BF03193405 [28] Glover JA. The “testing” phenomenon: not gone but nearly forgotten. J Educ Psychol. 1989;81(3):392–399. doi: 10.1037/0022-0663.81.3.392 [29] Pyc MA, Rawson KA. Testing the retrieval effort hypothesis: does greater difficulty correctly recalling information lead to higher levels of memory? J Memory Language. 2009;60(4):437– 447. Available from: http://dx.doi.org/10.1016/j.jml.2009.01.004 [30] Carpenter SK, Pashler H, Cepeda NJ. Using tests to enhance 8th grade students’ retention of U.S. history facts. Appl Cogn Psychol. 2009;23:760–771. doi: 10.1002/acp.1507 [31] Karpicke JD, Blunt JR. Retrieval practice produces more learning than elaborative studying with concept mapping. Science. 2011;331(6018):772–775. doi: 10.1126/science.119932 [32] Larsen DP, Butler AC, Roediger HL. Test-enhanced learning in medical education. Med Educ. 2008;42(10):959–966. doi: 10.1111/j.1365-2923.2008.03124.x [33] Wiklund-Hörnqvist C, Jonsson B, Nyberg L. Strengthening concept learning by repeated testing. Scand J Psychol. 2014;55(1):10–16. doi: 10.1111/sjop.12093 [34] Rohrer D, Taylor K, Sholar B. Tests enhance the transfer of learning. J Exp Psychol Learn, Memory, Cognition. 2010;36(1):233–239. [35] Stenlund T, Sundström A, Jonsson B. Effects of repeated testing on short- and longterm memory performance across different test formats. Educl Psychol. 2014:1–18. doi: 10.1080/01443410.2014.953037 [36] Bjork RA. Memory and metamemory considerations in the training of human beings. In: Shimamura J.M.A.P, editor. Metacognition: knowing about knowing. Cambridge (MA): The MIT Press; 1994. p. 185–205. [37] Bjork EL, Bjork RA. Making things hard on yourself, but in a good way: creating desirable difficulties to enhance learning. Psychol Real World. 2011:56–64. [38] Rohrer D, Taylor K. The shuffling of mathematics problems improves learning. Instruct Sci. 2007;35(6):481–498.
Downloaded by [Umeå University Library] at 04:03 15 June 2016
INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY
19
[39] Jäder J, Lithner J, Sidenvall J. Reasoning requirements in school mathematics textbooks: an analysis of books from 12 countries. Manuscript in preparation; 2016. [40] Chapman O. Mathematical-task knowledge for teaching. J Math Teach Educ. 2013;16(1):1–6. doi: 10.1007/s10857-013-9234-7 [41] Kapur M. Productive failure. Cogn Instruct. 2008;26(3):379–424. doi: 10.1080/07370000802212669 [42] Kapur M. A further study of productive failure in mathematical problem solving: unpacking the design components. Instruct Sci. 2011;39(4):561–579. doi: 10.1007/s11251-010-9144-3 [43] Bjork RA, Dunlosky J, Kornell N. Self-Regulated learning: beliefs, techniques, and illusions. Annu Rev Psychol. 2013;64(1):417–444. [44] Schoenfeld A. Mathematical problem solving. Orlando (FL): Academic Press; 1985. [45] Niss M. Mathematical competencies and the learning of mathematics: the Danish KOM project. Paper presented at the Third Mediterranean Conference on Mathematics Education. 2003. p. 115–124. [46] McDaniel M, Roediger H, McDermott K. Generalizing test-enhanced learning from the laboratory to the classroom. Psychon Bulletin Rev. 2007;14(2):200–206. [47] Alloway TP. Working memory, but not IQ, predicts subsequent learning in children with learning difficulties. Euro J Psychol Assessment. 2009;25(2):92–98. doi: 10.1027/1015-5759.25.2.92 [48] Andersson U, Lyxell B. Working memory deficit in children with mathematical difficulties: a general or specific deficit? J Exp Child Psychol. 2007;96(3):197–228. doi: 10.1016/j.jecp.2006.10.001 [49] Baddeley A. The episodic buffer: a new component of working memory? Trends Cogn Sci. 2000;4(11):417–423. [50] Unsworth N, Engle RW. Working memory capacity and fluid abilities: examining the correlation between Operation Span and Raven. Intelligence. 2005;33(1):67–81. Available from: http://dx.doi.org/10.1016/j.intell.2004.08.003 [51] Conway ARA, Cowan N, Bunting MF, et al. A latent variable analysis of working memory capacity, short-term memory capacity, processing speed, and general fluid intelligence. Intelligence. 2002;30(2):163–183. Available from: http://dx.doi.org/10.1016/S0160-2896(01)00096-4 [52] Engle RW, Kane MJ, Tuholski SW. Individual differences in working memory capacity and what they tell us about controlled attention, general fluid intelligence, and functions of the prefrontal cortex. In: Miyake A, Shah P, editors. Models of working memory: mechanisms of active maintenance and executive control. Cambridge, UK: Cambridge University Press; 1999. p. 102–134. [53] Klein K, Fiss W. The reliability and stability of the turner and Engle working memory task. Behav Res Methods, Instrum Comput. 1999;31(3):429–432. doi: 10.3758/BF03200722 [54] Dobson JL, Linderholm T. The effect of selected “desirable difficulties” on the ability to recall anatomy information. Anat Sci Educ. 2014;8(5):395–403. doi: 10.1002/ase.1489 [55] Cohen J. A power primer. Psychol Bull. 1992;112(1):155–159. doi: 10.1037/00332909.112.1.155 [56] Kang SHK, McDermott KB, Roediger HL. Test format and corrective feedback modify the effect of testing on long-term retention. Eur J Cogn Psychol. 2007;19(4):528–558. doi: 10.1080/09541440601056620 [57] McDaniel MA, Anderson JL, Derbish MH, et al. Testing the testing effect in the classroom. Eur J Cogn Psychol. 2007;19(4):494–513. doi: 10.1080/09541440701326154 [58] Brehmer Y, Westerberg H, Bäckman L. Working-memory training in younger and older adults: training gains, transfer, and maintenance. Frontiers Human Neurosci. 2012;6:63. doi: 10.3389/fnhum.2012.00063 [59] Dahlin E, Neely AS, Larsson A, et al. Transfer of learning after updating training mediated by the striatum. Science. 2008;320(5882):1510–1512. doi: 10.1126/science.1155466 [60] Ischebeck A, Zamarian L, Schocke M, et al. Flexible transfer of knowledge in mental arithmetic — an fMRI study. Neuroimage. 2009;44(3):1103–1112. doi: 10.1016/j.neuroimage.2008.10.025
20
B. JONSSON ET AL.
Downloaded by [Umeå University Library] at 04:03 15 June 2016
[61] Karpicke JD, Zaromb FM. Retrieval mode distinguishes the testing effect from the generation effect. J Memory Language. 2010;62(3):227–239. Available from: http://dx.doi.org/10.1016/j.jml.2009.11.010 [62] Xue G, Dong Q, Chen C, et al. Greater neural pattern similarity across repetitions is associated with better memory. Science. 2010;330(6000):97–101. doi: 10.1126/science.1193125
