Getting the point: Tracing worked examples enhances learning

Learning and Instruction 35 (2015) 85e93

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Getting the point: Tracing worked examples enhances learning Fang-Tzu Hu, Paul Ginns*, Janette Bobis The University of Sydney, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 March 2014 Received in revised form 24 September 2014 Accepted 5 October 2014 Available online

Embodied cognition and evolutionary educational psychology perspectives suggest pointing and tracing gestures may enhance learning. Across two experiments, we examine whether explicit instructions to trace out elements of geometry worked examples with the index finger enhance learning processes and outcomes. In Experiment 1, the tracing group solved more test questions than the non-tracing group, solved them more quickly, made fewer errors, and reported lower levels of test difficulty. Experiment 2 replicated and extended the findings of Experiment 1, providing evidence for a performance gradient across conditions, such that students who traced on the paper outperformed those who traced above the paper, who in turn outperformed those who simply studied by reading. These results are consistent with the activation of an increasing number of working memory channels (visual, kinaesthetic and tactile) for learning-related processing. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Cognitive load theory Embodied cognition Worked examples Tracing

1. Introduction 1.1. Cognitive load theory Cognitive load theory (CLT; Sweller, Ayres, & Kalyuga, 2011) foregrounds the role of human cognitive architecture in predicting whether instructional designs will support learning. The theory holds effective problem-solving is made possible by a large, wellorganised network of schemata held in long-term memory; however, the construction and automation of schemata requires conscious processing in a working memory limited in capacity and duration when information is novel. CLT researchers have tested a range of instructional redesigns targeting different hypothesised sources of working memory load. Earlier investigations (e.g., Cooper & Sweller, 1987; Sweller & Cooper, 1985) focused on redesigns that reduced extraneous cognitive load, i.e., working memory processes unrelated to schema construction and/or automation. Subsequent investigations of intrinsic cognitive load (e.g., Pollock, Chandler, & Sweller, 2002) theorised this source of load as a function of the number of interacting elements a learner must consciously attend to while learning. Lastly, germane cognitive load has been positioned as working memory capacity dedicated to the construction and automation of schemas (Paas & Van Gog, 2006). Recent critiques, however, have argued germane cognitive load can

be defined as the working memory resources available to address the element interactivity associated with intrinsic cognitive load (Sweller, 2010). The current formulation of CLT draws on evolutionary theorizing by Geary (2008), in particular the distinction between biologically primary knowledge and biologically secondary knowledge. The former is held to develop as a natural consequence of human genetic heritage; examples include learning to listen to and speak in a “mother tongue”, or recognise faces. Such skills are held to be acquired without conscious effort. In contrast, biologically secondary knowledge represents the knowledge corpus required to function in contemporary society. Cultural institutions such as schools and universities have emerged to support the slow, conscious and deliberate processes of learning to use historically recent artifacts such as writing systems and mathematics. Paas and Sweller (2012) argue that such evolutionary perspectives on educational psychology may provide the basis for novel cognitive load theory effects, with the potential for biologically primary knowledge to support teaching and learning of biologically secondary knowledge without imposing a substantial additional working memory load on learners. Embodied cognition, including the role of gestures in cognition, is discussed by Paas and Sweller as a promising source of evolutionarily informed scholarship for cognitive load theory. 1.2. Embodied cognition perspectives and the potential of gesturing

* Corresponding author. Faculty of Education and Social Work, University of Sydney, NSW, 2006, Australia. Tel.: þ61 2 9351 2611; fax: þ61 2 9351 5027. E-mail address: [email protected] (P. Ginns). http://dx.doi.org/10.1016/j.learninstruc.2014.10.002 0959-4752/© 2014 Elsevier Ltd. All rights reserved.

Reviewing the increasing emphasis on embodied cognition in cognitive science, Glenberg, Witt, and Metcalfe (2013) identified

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two general themes in embodiment scholarship. First, thinking is best understood as a function of the brain and the body interacting with the environment; thus, “thinking is grounded in the sensorimotor system” (Glenberg et al., 2013, p.576), rather than consisting of abstract symbol manipulation. Second, the need for the cognitive system to control action, i.e. interact with the environment, acts as a source of evolutionary pressure. One of the main ways in which we interact with the environment is with our hands. A rapidly expanding body of research has demonstrated that hand movement and position can substantially affect cognitive processing. In particular, pointing gestures, accompanied or not by touch, are of particular interest in the current study for their potential to affect information processing and subsequent learning. For the purpose of drawing attention, a pointing gesture apparently could serve as a primitive but effective attention-guiding cue, as people start using pointing to manage joint attention and interest as young as 12 months of age (Liszkowski, Brown, Callaghan, Takada, & de Vos, 2012). Studies of the interaction between visual attention and hand position also provide strong support for using pointing as an attentional cue. Positioning the hands near an object alters people's visual attention and perception towards that object, so the object will stand out from its surroundings (Cosman & Vecera, 2010), and will be scrutinised longer and deeper (Reed, Grubb, & Steele, 2006). In addition to pointing, hands support direct interaction with the environment through touch, often while simultaneously looking at or listening to stimuli. Similar to research reviewed above, this body of research has found synergistic effects on attentional processes when visual, auditory, and/or tactile inputs are synchronised (for a review, see Talsma, Senkowski, Soto-Faraco, & Woldorff, 2010). For example, Van der Burg, Olivers, Bronkhorst, and Theeuwes (2009) found when participants searched for line segments in a complex display including distractor line segments of various orientations and dynamically changing colour, search time and search slopes were substantially reduced when a tactile signal accompanied the target colour change. Based on studies of spatial cognition, pointing-based cueing may be particularly suitable for instruction with a high spatial content such as geometry, as pointing at an object leads attention to perceive that object in a more spatially oriented way (Fischer & Hoellen, 2004). Dodd and Shumborski (2009) found that encoding spatial arrays with pointing movements towards the visual display led to better memory performance, but not when participants pointed to all objects in an array. While their results indicated enhanced perceptual and motor traces for items selected for action (i.e., through pointing), they also found relatively impaired memory for items that had not been pointed at. Underpinning the various types of conscious cognitive activity discussed above is a working memory architecture consisting of channels for each of the sensory modes. Empirical research on the haptic working memory processor lags substantially behind research on the visual and auditory channels (for a review, see Kaas, Stoekel, & Goebel, 2008). Nonetheless, there is increasing recognition of the intersensory facilitation of visual processing by movement, such that Baddeley's (2012) most recent model of working memory speculates haptic sensory information, including kinaesthetic and tactile input, affects processing in the visuo-spatial sketchpad. Considering gesture more generally, Alibali (2005) identified a range of ways in which self-generated gestures might affect spatial cognition, including activating both lexical and spatial representations from long-term memory, increasing focus on spatial information, and helping to “package” spatial information with speech (cf. Alibali, Kita, & Young, 2000). This last possibility is particularly germane to the present study, with its focus on cognitive load. Ping and Goldin-Meadow (2010) argued gestures “can provide an overarching framework that serves to organise ideas conveyed in

speech, in effect chunking mental representations to reduce the load on working memory” (p.616). In cognitive load theory terms, mechanisms that act to chunk multiple elements of information into a single element are held to reduce intrinsic cognitive load and increase the opportunity for schema construction and/or automation. The present study extends such theorizing, testing if pointing and tracing gestures act to enhance learning of ideas conveyed in printed (textual and diagrammatic) instructional materials. 1.3. Pointing and tracing gestures in education There is a long history in educational practice of the use of pointing gestures to learn, as well as a gesture incorporating pointing, tracing a surface with the index finger. Learning to recognise letters of the alphabet by “Sandpaper Letters” is a method used extensively in Montessori schools for over a century. Students are encouraged to trace letters cut out of sandpaper with their fingers in the same sequence as writing the letter; while tracing, students simultaneously listen to the sound of the letter pronounced by their teacher (Montessori, 1912). This teaching technique works through a multisensory approach, involving simultaneous input from several modalities; students listen to the sound, look at its representation in the form of a letter, and feel the way it is written as they touch and trace the sandpaper letter. The learning benefits of tracing have been established across a number of recent experimental studies on letter learning and phoneme identification (e.g., Hulme, Monk, & Ives, 1987) as well as recognition of geometrical shapes in kindergarten children (Kalenine, Pinet, & Gentaz, 2011). Using a within-subjects design, Alibali and DiRusso (1999) tested preschoolers' accuracy in counting chips across a range of conditions (no gesture, puppet pointing, child pointing, puppet touching, and child touching), and found a clear positive gradient in counting accuracy across the above conditions (see Fig. 1, p.46). Alibali and DiRusso speculated the results could be explained by at least two processes: greater proximity of the finger to the chip when touching rather than pointing, and reduced working memory load by providing an external placeholder in the set of counted objects. Drawing on research discussed above, these results suggest the more sensory modalities are activated during the act of counting, the more accurate is performance; however, these results were generated during mathematical problem-solving, rather than instruction. Taken together, while the existing studies have demonstrated that finger pointing, touching and tracing can enhance task performance, it remains to be established whether such benefits extend to more complex instructions requiring higher levels of abstract thinking and problem-solving skills, and whether a similar gradient in performance is established when additional sensory modalities are recruited during instruction. Moreover, to the best of our knowledge, evidence for cognitive load explanations of pointing and/or tracing effects on learning outcomes e such as through subjective ratings of cognitive load e has not yet been provided. (In contrast, there is substantial evidence from dual-task studies for gesture's effects on cognitive load while processing information more generally; e.g., Ping & Goldin-Meadow, 2010). In an initial attempt to investigate pointing and tracing effects on cognitive load and learning, Macken and Ginns (2014) hypothesised that explicit instructions to point to related text and diagrammatic elements on heart anatomy and physiology, and trace out arrows indicating key blood flows across the heart's chambers, would enhance learning as measured on terminology and comprehension tests. Large statistically reliable effects of pointing and tracing were found on the above tests; however, there were no significant differences in post-instruction cognitive load ratings between conditions. Thus, a cognitive load

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explanation for the benefits of pointing and tracing therefore remains to be established. Alternative, potentially more sensitive methods of assessing subjective cognitive load are considered in the present study. For example, if tracing or pointing gestures act to chunk disparate elements of instructional text and diagrams into a single schema more effectively than visual study, then that schema should generate lower intrinsic cognitive load when activated into working memory for problem-solving during a subsequent test. 2. Hypotheses The literature reviewed indicates pointing and tracing gestures may be highly effective means of managing attention, acting as forms of biologically primary knowledge that could support learning of biologically secondary knowledge. Based on the theoretical frameworks of cognitive load theory and embodied cognition reviewed above, the series of experiments presented next tests the following hypotheses: Hypothesis 1: students who are instructed to trace out elements of geometry worked examples will outperform students who simply study the materials, as measured by error rates, time taken to solve problems, and test performance (Experiment 1). Hypothesis 2: students who are instructed to trace out elements of geometry worked examples will rate the difficulty of test items lower than students who simply study the materials, reflecting lower levels of intrinsic cognitive load as a result of enhanced schema construction (Experiment 1). Hypothesis 3: on the performance variates above, the effectiveness of index finger movements for learning will follow a gradient depending on how many working memory channels are activated. Thus, students who trace on the surface of instructional materials (i.e., generating activity in the visual, kinaesthetic and tactile sensory modes) should outperform those who trace in the air above the materials (i.e., generating activity in the visual and kinaesthetic sensory modes), who in turn should outperform those who simply study worked examples (i.e., visual sensory mode only) (Experiment 2). Hypothesis 4: students' reports of test difficulty, reflecting intrinsic cognitive load, will also follow a clear gradient, depending on how many working memory channels are activated. Thus, students who trace on the surface of instructional materials (i.e., generating activity in the visual, kinaesthetic and tactile sensory modes) should report lower levels of test difficulty than those who trace in the air above the materials (i.e., generating activity in the visual and kinaesthetic sensory modes), who in turn should report lower levels of test difficulty than those who simply study worked examples (i.e., visual sensory mode only) (Experiment 2).

3. Experiment 1 Experiment 1 was designed to explore whether a significant difference between instruction with tracing and without tracing could be obtained, using mathematics worked examples instructing angle relationship involving parallel lines. Research on the worked example effect (Cooper & Sweller, 1987; Sweller & Cooper, 1985) has demonstrated worked examples are an effective and efficient way to teach novice students mathematical methods and principles; worked-example-based instructional materials were used in Experiment 1 to determine whether instructions to trace might further enhance learning from worked examples.

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3.1. Method 3.1.1. Participants Participants were 42 Year 5 students aged between 10 and 11 years (M ¼ 10.50, SD ¼ .51) from an independent boys' school in Sydney, Australia. Students were novices with respect to the angle relationships in the instructional materials. They were randomly assigned to the tracing or the non-tracing condition. 3.1.2. Materials and procedure Students were tested individually, with each student being withdrawn from class for approximately 30 min. The experiment began with an initial instruction phase, identical for both groups. This phase was followed by an acquisition phase involving study, with or without tracing, of two worked examples; each worked example was paired by a similar practice problem. The experiment concluded with a test phase of six questions. Each test question was followed by a test difficulty rating. Initial instruction phase. Students had 5 min to study the three angle relationships involving parallel lines, including vertical angles are equal; corresponding angles are equal; and the sum of cointerior angles is 180 . For each angle relationship, instruction was provided including its definition, diagrams displaying the locations of the specific angles, and an example demonstrating how to use this angle relationship to solve a problem. Acquisition phase. All participants in the two conditions were then shown two worked examples applying the three angle relationships to find a missing angle. The first worked example from Experiment 1 with tracing instructions is given in Fig. 1. In the worked examples for the tracing group, every solution step was followed by instructions in brackets on tracing. Students were given 2 min to read and try to understand the solution steps, while using their index finger of their writing hand to trace out specified elements of the diagram following the instructions. Students in the non-tracing condition were instructed to read and try to understand the solution steps for 2 min, with their hands placed on their laps. Each worked example was paired with a similar practice problem, with a maximum of 2 min to solve the problem. Students who provided an incorrect answer were asked to try again within the 2-min time limit. Students who could not work out the correct answer when the time was used up were stopped. They were required to study the worked example again and then went back to solve the practice problem until the correct answer was attained. Among the 42 participants, one student in the tracing condition and two students in the non-tracing condition re-studied Worked Example 2. A Fisher exact test on number of participants restudying per condition was not statistically significant. Test phase. The test phase consisted of two basic questions, with similar diagrams and similar solution steps to the worked examples but with different numbers, and four advanced questions. Unlike the basic questions, which students could solve with two solution steps, all the advanced questions required a three-step solution, with varying permutations in the combinations of angle relationships across questions required to reach the correct solution. Surface features such as orientation and number of lines (parallel and non-parallel) presented were also varied compared to the worked examples and their practice problems. Students had up to 1 min to find a solution for each question. Test item difficulty self-reports. After each test question, students were immediately asked to rate the difficulty of the question they had just attempted as an indication of intrinsic cognitive load during the test phase; that is, students' experience of working memory load in recalling one or more schemas from long-term memory and keeping that knowledge active while solving test questions. Across the experiments reported in this article, a 5-point

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Fig. 1. Worked example with tracing condition instructions from Experiment 1.

illustrated subjective rating scale of test item difficulty ranging from 1 being “very easy” to 5 being “very difficult” was used. The design of the present measure of cognitive load diverges from Van Gog and Paas' (2008) recommendation to measure cognitive load during the test phase using mental effort ratings. Considering that it might be difficult for young students to comprehend the concept of “mental effort” in the commonly used mental effort rating scale €nboer, & Adam, 1994) or the metaphor of (Paas, Van Merrie “heaviness” used by Van Loon-Hillen, Van Gog, and Brand-Gruwel (2012) (illustrated using a cartoon figure holding increasing amounts of blocks), the notion of test question difficulty was used in the current rating scale. In the current experiment, a range of responses from 1 to 5 were possible, with two faces positioned above the 1 and 5 anchors to help students indicate how they felt while solving a test question. A smiling face was put on top of the number 1, indicating that a test question had been straightforwardly solved with little conscious effort, i.e. on the basis of suitable schema easily retrieved from long-term memory, generating little intrinsic load when comprehended in combination with other present elements of the test question. In contrast, a frowning face was put on top of the number 5, intending to capture an expression of considerable concentration during problem-solving based on the schema that might be incomplete and/or difficult to retrieve into working memory, as well as hold active in combination with other present elements of the test question. These two faces were drawn from the Faces Pain ScaledRevised (Hicks, von Baeyer, Spafford, van Korlaar, & Goodenough, 2001). 3.1.3. Data analysis Initial checks on the distributional properties of data under analysis used the ShapiroeWilks test of normality to evaluate distributional assumptions. Where non-normally distributed results (e.g., for error rates) necessitated the use of a non-parametric test (the ManneWhitney U using an exact p value; Mehta & Patel, 2012), a z-score associated with the ManneWhitney test was converted into the effect size r then converted to d using the typical transformation (see Chapter 7 of Borenstein, Hedges, Higgins, & Rothstein, 2009); 95% confidence intervals for d are also reported (Cumming, 2012). Where normality assumptions were not violated, the independent groups t-test assumption of equality of variances was assessed using Levene's test. Analyses of experimental data reported across the present article combined tests of significance, controlling the Type 1 error rate at 0.05, with estimates of the

standardised mean difference effect size (d). Based on a major review of over 800 meta-analyses of educational research, Hattie (2009) suggested the following benchmarks for effect size magnitude: small d ¼ 0.20, medium d ¼ 0.40, and large d ¼ 0.60 and above. 3.2. Results The variables under analysis were number of errors, total time to solution of practice problems, and number of correct solutions to practice problems in the acquisition phase, and number of errors, total time to solution of test questions, number of correct solutions to test questions, and ratings of test item difficulty in the test phase. Means and standard deviations are provided in Table 1. Acquisition phase. Due to non-normal distributions of the data across both conditions as indicated by the ShapiroeWilks test, a ManneWhitney test was used to analyse number of errors. The mean rank of the tracing condition (Mean rank ¼ 19.93) was not statistically different to that of the non-tracing condition (Mean rank ¼ 23.07), U ¼ 187.50, p ¼ .346, d ¼ .36 [95% CI .97, .25]. The difference between the tracing condition's time to solution (M ¼ 100.76, SD ¼ 53.74) and that of the non-tracing condition (M ¼ 102.43, SD ¼ 49.50) was also not statistically reliable, t(40) ¼ .11, p ¼ .917, d ¼ .03 [95% CI .64, .58]. As for number of Table 1 Means and (standard deviations in parentheses) for acquisition phase errors, time to solution (Seconds), and number of correct solutions, and test phase numbers of correct answers and errors, total time for test (Seconds) and ratings of test difficulty. Variate Acquisition Phase Number of errors Time to solution Number of correct answers Test Phase A. Basic Questions Number of errors Time to solution (/120) Number of correct answers (/2) Test difficulty (/5) B. Advanced Questions Number of errors Time to solution (/240) Number of correct answers (/4) Test difficulty (/5)

Non-tracing

Tracing

.33 (.58) 102.43 (49.50) 1.90 (.30)

.14 (.36) 100.76 (53.74) 1.90 (.30)

.80 100.81 1.05 3.14

(1.03) (21.53) (.80) (.91)

.38 86.43 1.38 2.64

(.59) (26.27) (.86) (.82)

.52 215.33 1.71 3.13

(.68) (29.20) (.75) (.85)

.05 172.05 3.24 2.33

(.22) (41.36) (1.45) (.75)

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correct solutions across practice questions, the identical performance of the two groups prevented analysis of this variable. Test phase e basic questions. Due to non-normal distributions of the data across one or both conditions as indicated by the ShapiroeWilks test, a ManneWhitney test was used to analyse number of errors, time to solution, and number of correct answers. The difference in number of errors made by the tracing condition (Mean rank ¼ 19.07) and the non-tracing condition (Mean rank ¼ 23.93) was not statistically reliable, U ¼ 169.50, p ¼ .170, d ¼ .45 [95% CI 1.06, .17]. A marginally significant effect for time to solution was found: the mean rank of the tracing condition (Mean rank ¼ 18.24) was lower than that of the non-tracing condition (Mean rank ¼ 24.76), U ¼ 152.00, p ¼ .081, d ¼ .55 [95% CI 1.16, .07]. For number of test questions correctly answered, the mean rank of the tracing condition (Mean rank ¼ 23.98) was not reliably higher than that of the non-tracing condition (Mean rank ¼ 19.02), U ¼ 168.50, p ¼ .164, d ¼ .44 [95% CI .18, 1.05]. Lastly, on the overall ratings of test item difficulty, a marginally significant effect was also found: the tracing condition rated basic test questions as less difficult (M ¼ 2.64, SD ¼ 0.82) than the non-tracing condition (M ¼ 3.14, SD ¼ 0.91), t(40) ¼ 1.87, p ¼ .069, d ¼ .58 [95% CI 1.19, .04]. Test phase e advanced questions. Due to non-normal distributions of the data across one or both conditions as indicated by the ShapiroeWilks test, a ManneWhitney test was used to analyse number of errors, time to solution and number of correct answers. A significant effect for number of errors was found: the mean rank of the tracing condition (Mean rank ¼ 17.45) was lower than that of the non-tracing condition (Mean rank ¼ 25.55), U ¼ 135.50, p ¼ .007, d ¼ .98 [95% CI 1.61, .33]. A significant effect for time to solution was also found: the mean rank of the tracing condition (Mean rank ¼ 15.33) was lower than that of the non-tracing condition (Mean rank ¼ 27.67), U ¼ 91.00, p ¼ .001, d ¼ 1.15 [95% CI 1.80, .49]. For number of test questions correctly answered, the mean rank of the tracing condition (Mean rank ¼ 27.67) was reliably higher than that of the non-tracing condition (Mean rank ¼ 15.33), U ¼ 91.00, p ¼ .001, d ¼ 1.19 [95% CI .52, 1.84]. Lastly, on the overall ratings of test item difficulty, a marginally significant effect was also found: the tracing condition rated advanced test questions as less difficult (M ¼ 2.33, SD ¼ 0.75) than the non-tracing condition (M ¼ 3.13, SD ¼ 0.85), t(40) ¼ 3.21, p ¼ .003, d ¼ .99 [95% CI 1.63, .35]. 3.3. Discussion Experiment 1 was designed to test whether tracing out the graphical elements of geometry worked examples would enhance learning. Instead of studying worked examples in a conventional way, using only the eyes to read over and comprehend the materials, students in the experimental condition were instructed to use their index finger to trace out the corresponding elements of the diagrams after reading each solution step in the worked examples. It was hypothesised that students who traced elements of worked examples while studying during the acquisition phase would perform better on the subsequent test as measured by the number of correct answers, error rate, and time to solution across the acquisition and test phases. Moreover, it was predicted students in the tracing condition would consider the test items less difficult than students who simply studied the materials without any hand movement, as tracing was hypothesised to promote schema construction and/or automation. We predicted better constructed schemas would be more easily retrieved and applied at test; hence, students who traced should experience lower levels of intrinsic cognitive load, as indexed by difficulty ratings. Results supported hypotheses 1 and 2: the tracing condition significantly outperformed the non-tracing condition across a

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range of variates. While there were no statistically reliable differences between conditions on acquisition phase variates, in the subsequent test, marginal effects for basic test questions were found on time to solution and ratings of test question difficulty, and large, statistically reliable effects for advanced test questions were found for number of errors, time to solution, test performance, and ratings of test question difficulty. The better test performance in combination with the lower level of test difficulty indicate that students in the tracing group constructed better problem-solving schemas from the instructional materials to handle the test questions with lower cognitive demand, compared to students in the non-tracing group. The significant advantages of the tracing condition over the non-tracing condition strengthen the argument that tracing out elements of worked examples facilitated schema construction over and above the typical benefits of learning from studying worked examples (Cooper & Sweller, 1987; Sweller & Cooper, 1985). 4. Experiment 2 The tracing instructions used in Experiment 1 aimed to activate haptic working memory resources during the learning process. However, since the haptic modality is a composite of the tactile and the kinaesthetic modalities (Kaas, Stoeckel, & Goebel, 2008), and active touch plays an essential role in haptic perception (Gibson, 1962; Klatzky & Lederman, 2003), it is at present unclear whether the differential activation of the tactile and kinaesthetic modes will affect learning outcomes. In a learning context, sometimes students will spontaneously hold a pointing finger making tracing movements in the air of new words or graphic shapes that they are trying to learn. In that case, students add haptic input coming from the kinaesthetic modality only into their learning processes. In contrast, when the participants in Experiment 1 were instructed to put their index finger on the piece of paper and trace out the worked examples on it, we hypothesised that sensory input from both the tactile and kinaesthetic modalities was received and incorporated with visual input into a representation initially held in working memory, then encoded into long-term memory. This raises the question of whether multiple non-visual sensory modalities are best used to maximise learning from worked examples (i.e., tracing with the finger on the paper), or whether similar results would be obtained if only one non-visual sensory modality was coupled with visual input (e.g., tracing with the finger in the air just above a worked example). This latter manner of embodied interaction with a worked example could be expected to enhance learning given the focussing of attention around perihand space (Cosman & Vecera, 2010; Reed et al., 2006). However, since haptic perception heavily relies on active touch, it is assumed that learning will be enhanced to a greater extent when both tactile and kinaesthetic representations of to-be-learned information are activated and integrated in working memory along with visual input. To address these questions, Experiment 2 tested hypothesised gradients across experimental conditions, predicting that students who traced on the surface of instructional materials (i.e., affecting visual, kinaesthetic and tactile working memory channels) would outperform those who traced in the air above the materials (i.e., affecting visual and kinaesthetic working memory channels), who in turn would outperform those who simply read worked examples (i.e., visual working memory channel only). A gradient in average test item difficulty ratings was also hypothesised, predicting that students who traced on the surface of instructional materials would report lower levels of test difficulty than those who traced in the air above the materials, who in turn would report lower levels of test difficulty than those who simply read worked examples.

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4.1. Method 4.1.1. Participants Participants consisted of 72 Year 5 students, including 56 boys and 16 girls, from 2 independent schools in Sydney, Australia. All participants participated voluntarily, and were aged between 9 and 11 years (M ¼ 9.94, SD ¼ .33). Participants were novices with respect to the three angle relationships in the instructional materials. They were randomly assigned to the tracing on the paper (tracing with touch) condition, the tracing above the paper (tracing without touch) condition, or the non-tracing condition. 4.1.2. Materials and procedure The same instructional materials and timings used in Experiment 1 were used in Experiment 2. The primary difference was in the acquisition phase instructions. Acquisition phase. In the worked examples for the “tracing on the paper” group, every solution step was followed by instructions in brackets on tracing. Students were given 2 min to read and try to understand the solution steps, while putting their index finger of their writing hand on the paper to trace out specified elements of the diagram following the instructions. Students in the “tracing above the paper” condition were instructed to keep their index finger about 5 cm above the paper and trace out specified elements of the diagram following the instructions without touching the paper. Lastly, students in the non-tracing condition were instructed to read and try to understand the solution steps, with their hands placed on their laps. Students who could not work out the correct answer when the time was used up were required to study the worked example again and then went back to solve the practice problem until the correct answer was attained. Across the 72 participants, one student in the tracing on the paper condition and one in the tracing above the paper condition re-studied both of the worked examples. One student in the non-tracing condition restudied Worked Example 1 only; four in the tracing above the paper condition and six in the non-tracing condition re-studied Worked Example 2 only. Using a Fisher exact test, there was a statistically significant ordinal association (Somer's d ¼ .16; Fisher exact test p ¼ .021) between condition and amount of additional learning opportunities. Importantly, however, the directionality of this association was in the opposite direction to results for the acquisition phase and test phase variates (see Section 4.2). Thus, any potential benefit of extra study during the acquisition phase did not translate to enhanced performance in the acquisition and test phases. Test phase. The test phase consisted of two basic questions, with similar diagrams and similar solution steps to the worked examples but with different numbers. Four of the advanced questions were identical to those used in Experiment 1. The two additional advanced questions required the same solution steps used in the similar questions but with an additional step of subtraction, as the target angle was divided into two adjacent angles by an extra line. Students had up to 1 min for each question. Test item difficulty self-reports. After each test question, students were immediately asked to rate the difficulty of the question they had just attempted as an indication of intrinsic cognitive load during problem-solving. This 5-point subjective rating scale was identical to the one used in Experiment 1. Data analysis. When testing hypotheses with sequence order (i.e., condition 1 > condition 2 > condition 3, or vice versa), using statistics that incorporate information about the hypothetical rank order will typically result in higher power compared to conventional analysis of variance procedures (McKean, Naranjo, & Huitema, 2001). Given the clear hypotheses described above regarding expected gradients on variates across conditions,

analyses consisted of bootstrapped estimates of Spearman's rankorder correlation coefficient between the independent variable and median scores for each condition on dependent variables under analysis. This method is robust for analysing variances of experimental designs with an expected order of dependent variables (McKean et al., 2001), and has the benefit over alternative methods (e.g., Terpstra, 1952) of generating an effect size (Spearman's r) in addition to a p value. Values of Spearman's r presented below were accompanied by values of d, along with the 95% confidence interval for both r and d. Bootstrapped p values were onesided given the directional hypotheses used in the current experiment. Complementing the above analytic approach were multiple comparisons between conditions. Because data screening indicated non-normal distributions across most variates across all conditions, effect sizes and confidence intervals for each comparison were calculated using trimming and bootstrapping methods that are robust to variance heterogeneity and non-normality (Keselman, Algina, Lix, Wilcox, & Deering, 2008). Given that there were three levels of the grouping variable, Fisher's two-stage procedure was used to control the familywise error rate at 0.05; multiple comparison tests were assessed at a ¼ .05 only if an initial omnibus test was statistically significant (see Keselman, Cribbie, & Holland, 2004). 4.2. Results The variables under analysis were number of errors, total time to solution of practice problems, and number of correct solutions to practice problems in the acquisition phase, and number of errors, total time to solution of test questions, number of correct solutions to test questions, and ratings of test item difficulty in the test phase. Means and standard deviations are provided in Table 2. 4.2.1. Acquisition phase In order to test the hypothetical sequence order of students' performance (tracing on the paper > tracing above the paper > non-tracing for number of correct answers; tracing on the paper < tracing above the paper < non-tracing for number of errors and time to solution), Spearman's rank-order correlation coefficient between experimental condition and the three variates from the acquisition phase was estimated. A statistically reliable gradient was found for number of errors, r ¼ .24 [95% CI .45, .01], Table 2 Means and (standard deviations in parentheses) for acquisition phase errors, time to solution (Seconds), and number of correct solutions, and test phase numbers of correct answers and errors, total time for test (Seconds) and ratings of test difficulty. Variate Acquisition Phase Number of errors Time to solution (/240) Number of correct answers (/2) Test Phase A. Basic Questions Number of errors Time to solution (/120) Number of correct answers (/2) Test difficulty (/5) B. Advanced Questions Number of errors Time to solution (/360) Number of correct answers (/6) Test difficulty (/5)

Non-tracing

Tracing above the paper

Tracing on the paper

0.46 (0.93) 110.58 (54.07) 1.63 (.49)

0.08 (0.28) 126.00 (63.42) 1.63 (.65)

0.08 (0.28) 90.38 (47.12) 1.92 (.41)

.42 (.65) 82.88 (25.07) 1.25 (.74)

.50 (.93) 88.04 (30.31) 1.38 (.77)

.29 (.62) 71.79 (26.69) 1.67 (.70)

2.92 (.97)

3.06 (.88)

2.69 (.69)

2.21 (2.77) 320.54 (38.52) 2.25 (1.65)

1.63 (2.50) 303.67 (47.48) 3.00 (1.64)

1.04 (1.23) 292.46 (42.93) 3.92 (1.02)

3.58 (.74)

3.19 (.60)

3.12 (.62)

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p ¼ .021, d ¼ .49 [95% CI 1.00, .01], but not time to solution, r ¼ .15 [95% CI .37, .08], p ¼ .108, d ¼ .30 [95% CI .79, .16]. The gradient for number of correct answers was also statistically reliable, r ¼ .30 [95% CI .10, .48], p ¼ .006, d ¼ .62 [95% CI .19, 1.08]. The omnibus test of an overall group effect was not statistically significant for time to solution, F(2, 28.19) ¼ 2.31, p ¼ .134. Omnibus tests using trimming and bootstrapping could not be calculated for number of errors and number of correct answers, possibly due to floor and ceiling effects respectively on these variates; however, omnibus tests based on ordinary least squares estimators were not statistically significant number of errors: F(2, 42.04) ¼ 1.83, p ¼ .173; number of correct answers: F(2, 44.49) ¼ 0.38, p ¼ .687. Test phase e basic questions. Spearman's rank-order correlation coefficient was also used to test hypothesised gradients between experimental condition and the four variates from the test phase (tracing on the paper > tracing above the paper > non-tracing for number of correct answers; tracing on the paper < tracing above the paper < non-tracing for number of errors, time to solution, and ratings of test item difficulty). The gradient for number of correct answers was statistically reliable, r ¼ .27 [95% CI .04, .48], p ¼ .012, d ¼ .55 [95% CI .08, 1.09]. The gradient for number of errors was not statistically reliable, r ¼ .10 [95% CI, .32, .13], p ¼ .195, d ¼ .20 [95% CI .67, .25], nor was the gradient for time to solution, r ¼ .15 [95% CI .37, .07], p ¼ .105, d ¼ .31 [95% CI .80, .13], or ratings of test item difficulty, r ¼ .07 [95% CI .31, .16], p ¼ .264, d ¼ .14 [95% CI .63, .32]. As in the acquisition phase, omnibus tests using trimming and bootstrapping could not be calculated for number of errors and number of correct answers, possibly due to floor and ceiling effects respectively on these variates; however, omnibus tests based on ordinary least squares estimators were not statistically significant (number of errors: F(2, 44.95) ¼ 0.47, p ¼ .628; number of correct answers: F(2, 45.93) ¼ 2.10, p ¼ .131). The omnibus test using trimming and bootstrapping was not statistically significant for time to solution, F(2, 29.26) ¼ 1.96, p ¼ .164, nor was the same test for ratings of test item difficulty, F(2, 28.29) ¼ 1.25, p ¼ .314. Test phase e advanced questions. Testing the same hypothesised gradients as for basic test questions, the gradient for number of correct answers was statistically reliable, r ¼ .41 [95% CI .22, .58], p < .001, d ¼ .89 [95% CI .45, 1.42], as was the gradient for time to solution, r ¼ .28 [95% CI .48, .06], p ¼ .010, d ¼ .57 [95% CI 1.07, .12], and ratings of test item difficulty, r ¼ .25 [95% CI .48, .02], p ¼ .015, d ¼ .53 [95% CI 1.07, .04]. The gradient for number of errors was not statistically reliable, r ¼ .16 [95% CI, .38, .07], p ¼ .093, d ¼ .32 [95% CI .82, .14]. The omnibus test using trimming and bootstrapping was statistically significant for number of correct answers, F(2, 27.23) ¼ 4.59, p ¼ .024. Multiple comparisons between conditions revealed a statistically reliable difference between the tracing on the paper and control conditions favouring the former condition, d ¼ .87 [95% CI .42, 1.65], and a difference between the tracing on the paper and tracing in the air conditions favouring the former condition that approached significance, d ¼ .43 [95% CI .04, 1.09]. The difference between the tracing in the air and control conditions was not statistically reliable, d ¼ .37 [95% CI .17, 1.10]. The omnibus test using trimming and bootstrapping was not statistically significant for time to solution, F(2, 29.80) ¼ 1.94, p ¼ .165, number of errors, F(2, 28.52) ¼ 0.89, p ¼ .426, or ratings of test item difficulty, F(2, 29.42) ¼ 2.41, p ¼ .124. 4.3. Discussion Considering the haptic sensory modality consists of the tactile and the kinaesthetic modalities, Experiment 2 was designed to further explore the tracing effect by examining whether tracing

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with or without the sense of touch would affect learning outcomes, compared to a control group relying on visual study only. It was hypothesised that, when students put their index finger on the paper and traced the worked examples on it, working memory resources from both the tactile and kinaesthetic modalities would become active to be used along with vision-based learning processes. This expansion of available working memory for learning was expected to be reflected in relatively better test performance, and relatively lower ratings of test difficulty, compared to the other conditions. In contrast, students who kept their index finger above the paper and made tracing movements were hypothesised to incorporate kinaesthetic input only with visual input in working memory, thus expanding working memory capacity available for learning but to a lesser extent. Students in the control condition who kept their hands on their laps without any movements were hypothesised to rely on visual working memory only to support learning. Based on previous research demonstrating that input from multiple modalities results in better performance on a mathematical task (Alibali & DiRusso, 1999), it was predicted that students in the tracing on the paper condition, who learned with three types of inputs from the tactile, the kinaesthetic and the visual modalities, would have the best performance at the post-test and report the lowest level of perceived test difficulty. Students in the non-tracing condition, learning with only visual input, would have the worst performance and report the highest level of perceived test difficulty. Students who traced above the paper were expected to perform mid-way between these extremes, reflecting the partial expansion of working memory capacity available for learning through the kinaesthetic channel, along with the focussing of visual attention on the instructional materials in perihand space. Results supported most of the hypotheses presented above, particularly with regard to problem-solving performance. First, during the acquisition phase, the hypothesised gradient was found for errors and number of questions correctly answered, but not time to solution. Second, during the test phase, the hypothesised gradient on basic test questions was found for test scores, but not errors, time to solution, or ratings of test question difficulty; for advanced questions, the hypothesised gradient was found for test scores, time to solution, and ratings of test question difficulty, but not number of errors made. These results stand in contrast, however, to those based on omnibus tests and follow-up multiple comparisons. Using such analyses, the only statistically reliable difference between conditions was found for advanced test question score, between the tracing on the paper condition and the control condition; the difference between the tracing in the air and tracing on the paper conditions may also have represented a real effect. In contrast, the difference between the tracing in the air and the control conditions was not statistically reliable. Taken together, we interpret the results of Experiment 2 as demonstrating that when instructional design of worked examples incorporates the kinaesthetic modality (i.e., instructions to move), the inclusion of the tactile modality (i.e., instructions to touch the materials) further enhances students' learning outcomes. 5. General discussion The present series of experiments explored whether explicit instructions to trace out elements of worked examples with the index finger would enhance novices' learning of geometry rules. CLT seeks to generate instructional designs based on wellestablished knowledge of human cognitive architecture. Historically, this has involved a consideration of working memory subsystems for processing visual and/or auditory information. The present study draws on notions of evolutionary educational psychology (Geary, 2008), embodied cognition perspectives (Glenberg

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et al., 2013) and seminal theorising and instructional design by educationalists such as Maria Montessori (1912), to expand the scope of considered working memory systems to those involving the hands. Specifically, we argue that pointing gestures are a form of biologically primary knowledge that may support the construction of biologically secondary knowledge, such as mathematical rules. We base this argument on developmental research showing the natural emergence, without explicit instruction, of pointing gestures (Liszkowski et al., 2012); basic laboratory studies demonstrating clear effects of hand position and pointing gestures on attentional processes (e.g., Cosman & Vecera, 2010; Reed et al., 2006); and educational research demonstrating the benefits for , & Sprengerlearning of tracing gestures (e.g., Bara, Gentaz, Cole Charolles, 2004; Hulme et al., 1987; Kalenine et al., 2011; Macken & Ginns, 2014). This theorising supported hypothesis generation across two experiments. Focussing on post-test results as the primary source of evidence for hypotheses, in Experiment 1, students instructed to trace elements of worked examples on parallel lines geometry solved basic test questions more quickly and rated them as easier, and solved more advanced test questions more quickly while rating them as easier and making fewer errors, than students in the nontracing condition. Students who traced also rated the test questions as lower in difficulty than students in the control condition, consistent with the argument that tracing reduced intrinsic load during the test phase. Results of Experiment 2 extended the above results, finding statistically reliable gradients in basic and advanced test scores, and advanced test question time to solution and ratings of test item difficulty. Taken together, these results are consistent with the hypothesis that the greater the number of working memory modalities (visual, tactile and kinaesthetic) activated during learning, the better the problem-solving schemas constructed, particularly given the strongest results were obtained with advanced test questions requiring transfer of learning from the worked examples studied (cf. Cooper & Sweller, 1987). These findings have a range of implications for educational practice. They indicate that learning of geometry principles by novices may be enhanced substantially by the simple addition of explicit instructions to trace elements of instructional materials. Such instructions were instantiated in paper-based instructional materials, representing a low cost for schools; however, some caution regarding cost effectiveness is warranted given the present results were generated under individualised instruction and testing conditions. We are cautiously confident such effects could be generalised to more realistic classroom settings, and to other subject areas, but further research is clearly required. Notwithstanding the limits of the present study, we note that the use of individualised instruction mimics the attention that students will receive from teachers, teachers' aides and parents during remedial work with students. Another aspect of the study that argues for caution relates to the differential findings in Experiment 2 depending on whether a “gradient” or omnibus test followed by multiple comparison” approach to analysis was taken. As discussed above, the omnibus approach has been argued to lack statistical power when there is a clear ordering in hypotheses (McKean et al., 2001). The only significant or near-significant multiple comparisons (following a significant omnibus test) were found on the number of advanced test questions solved correctly. These results aligned with the expectation that tracing on the paper would enhance learning, and the additional expectation that the greater the number of sensory modalities activated the more learning would be enhanced (i.e. comparing tracing on the paper with tracing in the air above the paper). Nonetheless, it is clear more research is needed to understand the mechanisms by which differing levels of sensory

modality activation may enhance learning. The research design and sample size used in Experiment 2 appeared sufficient for investigating gradient effects for some (but not all) variates, and larger studies may be required for more fine-grained investigations of tracing effects, whether on a surface or in the air. The present findings have some limitations that can inform subsequent investigations of tracing effects. First, hypotheses regarding intrinsic cognitive load during the test phase were tested using self-report scales. Future investigations should continue to gather process data to strengthen the case for a cognitive load interpretation of performance data. For example, future studies should collect students' ratings of cognitive load during both the acquisition and test phases, and supplement self-reports of cognitive load with eye-tracking data (Van Gog & Scheiter, 2010). Second, the present studies measured effects on learning immediately after instruction; future investigations with delayed post-tests will provide more robust tests of learning. Third, while the effects of tracing were strong for the inherently visuo-spatial-based mathematics materials used in the experiments described here, it remains to be seen how tracing instructions might be incorporated into topics that are less obviously visuo-spatial in nature. Fourth, the present series of experiments did not consider the role of any individual differences. The extent to which the effectiveness of tracing changes with prior knowledge would be worthy of investigation, since research on the expertise-reversal effect (Kalyuga, 2007) suggests that instructional designs that are effective for novicesdsuch as those recruited in the present experimentsdmay decline in effectiveness as prior knowledge increases. Another potential aptitude-treatment interaction might relate to spatial €ffler, 2010); if tracing effects operate substantially ability (Ho through their effects on spatial processing (cf. Alibali, 2005), then students higher in spatial ability might find instructions to trace relatively redundant. Lastly, and related to the previous point, research on effective sequencing of instructions involving tracing would be worthwhile. For example, in describing an instructional sequence for sandpaper letters, Montessori (1912) noted “the children, as soon as have become at all expert in this tracing of the letters, take great pleasure in repeating it with closed eyes, letting the sandpaper lead them in following the form which they do not see” (p. 276; italics in original). Such a sequence has considerable parallels with instructions used in generating the imagination effect (Ginns, Chandler, & Sweller, 2003), where students first construct a schema through studying materials, then partially or fully automate the schema by closing the eyes and imagining the instructions.

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