developing numerical ability in children with

DEVELOPING NUMERICAL ABILITY IN CHILDREN WITH MATHEMATICAL DIFFICULTIES USING ORIGAMI1,2 ÁGOTA KRISZTIÁN AND LÁSZLÓ BERNÁTH Institute of Psychology, Eötvös Loránd University, Budapest, Hungary

HAJNALKA GOMBOS

Metropolitan Educational Service, Budapest, Hungary

LAJOS VERECZKEI Department of Behavioural Sciences, University of Pécs, Hungary

This is a manuscript and may not reflect the final version. Citation details follow: Krisztián, Á., Bernáth, L., Gombos H., & Vereczkei L. (2015). Developing numerical ability in children with mathematical difficulties using origami. Perceptual and Motor Skills, 121 (1), 233-243. doi: 10.2466/24.10.PMS.121.c16x1

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Correspondence should be addressed to Ágota Krisztián, Department of Educational

Psychology, Institute of Psychology, Eötvös Loránd University, H-1075, Budapest, Kazinczy u. 23-27. Hungary (e-mail: [email protected], [email protected])

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Aknowledgement: The authors are grateful to Andrea Béki, Klára Csabai and Katalin Zsirka

Klein for their help in the organization of the experiments, as well as to Eszter Kovács and George Bernath for their comments regarding the text.

Developing Numerical Ability in Children with Mathematical Difficulties Through Origami

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DEVELOPING NUMERICAL ABILITY IN CHILDREN WITH MATHEMATICAL DIFFICULTIES USING ORIGAMI Keywords: mathematical difficulties, numerical ability, spatial abilities, origami Summary Certain aspects of numerical processing show a connection with spatial abilities (Dehaene, Spelke, Pinel, Stanescu & Tsivkin, 1999). Spatial abilities may be enhanced through the practice of origami (Taylor & Hutton, 2013, Cakmak, Isiksal & Koc, 2014). We therefore followed the assumption that the development of spatial abilities will lead to the development of numerical processing. We have examined whether spatial and numerical skills can be developed using origami and the folding of 3D shapes. During the course of the 10 weeks of the training program, consisting of weekly 60 minute sessions, the performance of children with mathematical difficulties showed considerable improvement in spatial and numerical tasks as opposed to the control group of children with mathematical difficulties.

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DEVELOPING NUMERICAL ABILITY IN CHILDREN WITH MATHEMATICAL DIFFICULTIES USING ORIGAMI

In recent years, a growing number of empirical studies have reported a significant link between mathematical and spatial ability based on Dehaene’s (1992) triple code model. The model (Deahaene, 1992) assumes that numerical processing is possible on a preverbal level using analogue magnitude representation. The analogue magnitude representation contains the mental number line which stores numbers in a non-discrete form, as a continuum, and which is – among other things – active in tasks involving estimations (Deahaene, 1992). Dehaene, Spelke, Pinel, Stanescu and Tsivkin (1999) found that the areas of the brain active during the process of estimation are associated with visuospatial functions, while during exact arithmetic tasks, activity appears in areas associated with verbal functions. Other results also suggest that numerical processes are connected with spatial representations in common circuits of the parietal cortex (Hubbard, Piazza, Pinel & Dehaene, 2005). Research conducted with 0-3 day old neonates has also shown that they can link longer lengths with auditory numerosity, which indicates that this link between space and gradation is not something acquired but inherent (de Hevia, Izard, Coubart, Spelke & Streri, 2014). Nevertheless, the fact that this inherent nature does not make the role of experience negligible is crucial in terms of development. The role of genetic factors and environmental effects were investigated by Tosto, Hanscombe, Haworth, Davis, Petrill, Dale et. al. (2014) who examined 4174 pairs of 12 year old monozygotic and dizygotic twins. Based on their results, it seems that the relationship between spatial and mathematical abilities is 60% genetically determined and 40% environmentally determined.

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While fMRI examinations (Dehaene et. al. 1999, Hubbard et. al. 2005), neonatal examinations (de Hevia et. al. 2014) and the role of genetic factors (Tosto et. al. 2014) proves that spatial and numerical skills are connected, on a behavioural level, early studies seemed to contradict this. Concerning behavioural data, more than 30 years ago Lean and Clements (1981) found that university students who used verbal methods for processing numerical information showed better mathematics performance than those who preferred visual processing. Moreover, according to multiple regression analysis, they found that spatial ability has little effect (R2 =10%) on mathematical performance. However, they equated visual and spatial processing, while the two are distinct. For example, Hegarty and Kozhevnikov (1999) showed that while schematic spatial representation has a positive correlation with mathematical problem solving (r= 0.48), pictorial representation has a negative marginally significant correlation (r= -0.34, p=0.056). More recent studies have also found a connection between numerical and spatial abilities: spatial skills at age 5 can significantly predict approximate symbolic calculation tasks (for example: “Sarah has 20 cookies and she gets 16 more. John has 45 cookies. Who has more?”) at age 8 years (Gunderson, Ramirez, Beilock & Levine, 2012). A 4-year longitudinal study with a sample of primary school children showed a significant connection between spatial ability and numerical ability (addition and subtraction, counting) (Lachance & Mazzocco, 2006). University students have shown a similar relationship in their performance in mental rotation tasks and numerical cognition (number magnitude comparison, number-line mapping;Thompson, Nuerk, Moeller & Kadosh, 2013). Spatial abilities can be developed by practice. For example, more than half a century ago, in his pioneer research, Brinkmann (1966) studied eighth grade students who completed

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a 3-week training course of programmed instruction about lines, angles and plane figures. The training group performed better on spatial visualization than the control group. Playing video games has also been associated with improved spatial visualization test scores (Dorval & Pepin, 1986), which is likely due to improvement of spatial skills. Moreau (2013) reported that a 3-week video game training program improved performance in mental rotation tests. After 3-weeks of spatial visualization training, samples of fifth to eighth grade students, all showed improvement in spatial ability (Ben-Chaim, Lappan & Houang, 1988). A concise account of the improvability of spatial abilities was given by Uttal et. al. (2013) based on the meta-analysis of 217 training studies. As spatial abilities do not have a generally accepted definition, they propose a 2-dimensional classification system based on Newcomb and Shipley (2015). One dimension is the intrinsic/extrinsic, the other the static/dynamic dimension. The 4 spatial skills: intrinsic and static (perceiving objects), intrinsic and dynamic (producing complex units from smaller ones, 2D to 3D transformations, mental rotation), extrinsic and static (understanding abstract spatial concepts, e.g. a plumbline) and extrinsic and dynamic (visualizing a complex of objects from another viewpoint, e.g. Piaget’s Three Mountains Task). They placed the training programs into 3 categories. First and second was indirect training: video games and semester-long course, the third was practice of spatial tasks directly (Uttal et. al., 2013). Based on the results of the meta-analysis, spatial skills seem to be malleable, training courses effective and transferable. The direct development of intrinsic dynamic spatial abilities and its effects on arithmetic performance were examined by Cheng and Mix (2014). They studied children of 68 years of age and found an improvement in mental rotation tests after 40-minute mental rotation exercises, while a control group without the practice showed no improvement. The authors also noted that children who had completed the mental rotation exercises performed 6


better in certain mathematical tasks (missing term problems, e.g. 3+_=8), but not in numberfact problems (4 + 5 = _) and two- and three-digit calculation problems. This suggests that practice of mental rotation tasks may have additional effects on certain arithmetic skills. Origami is a combination of mental rotation, mental folding and motor activity and has an enhancing effect on spatial ability (Shumakov & Shumakov, 2000). The effects of indirect development on spatial abilities were examined by Taylor and Hutton (2013) in a 6 week course with one 90 minute session per week. They practiced origami, single-sheet pop-up paper engineering, and applied paper engineering exercises with fourth-grade students. The experimental group performed better in two tasks than the control group: in the mental folding task, the difference was marginal (p=0.054), while in the Make a Die test, the difference was significant. The mental rotation test showed no difference between the groups. Cakmak, Isiksal & Koc, (2014), also using indirect development methods throughout a longer period of time – origami exercises over a 10 week course saw that 4-6 grade children’s performance in the Spatial Ability Test improved significantly. Besides improving spatial skills, origami has been shown to affect the level of geometric reasoning and geometry achievement, even after a mere 4 weeks in tenth grade students (Arici & Aslan-Tutak, 2013). Based on these results, we can posit that origami improves spatial abilities, while improved spatial abilities lead to improved mathematical skills among various aged students with normal mathematical abilities. The assumption follows that practicing spatial task like origami might also be applied to students who have difficulties in mathematics to improve their arithmetic skills. To test this, we devised an indirect spatial training course for fifth and sixth grade students with mathematical difficulties. We examined whether intrinsic and dynamic spatial skills and numerical skills show improvement after the training program. 7


Mathematical difficulties have not been expressly defined, most likely because it is less severe of a problem than dyscalculia and not as specific. Nevertheless, its occurrence is much more frequent than dyscalculia (Dowker, 2009). Karagiannakis, Baccaglini-Frank and Papadatos (2014) recommend a multidimensional system in which all too heterogeneous mathematical difficulties are differentiated within 4 subtypes: core number, memory retrieval and processing, reasoning, visual-spatial. This also seems to indicate that the improvement of spatial abilities decreases mathematical difficulties. In Hungary, children with weak mathematical performance are examined using Judit Dékány’s Dyscalculia Prevention Examination (Dékány, 1999; Dékány & Juhász, 2002), a series of tasks that examine without using exact grading the various sublevel abilities utilized in counting. Numerical and mathematical skills are examined by the following task types: assessing global quantity on one’s own body and on objects without counting, estimating and naming quantity relations, writing down and reading out numbers, determining quantity constancy, place-value notation, digit span, description and performance of basic mathematical operations. Special education teacher determine the diagnosis of the children based on their performance on these tests. Problems less severe than dyscalculia is diagnosed as mathematical difficulties. A new version of this examination process is currently being introduced which classifies children as having dyscalculia or mathematical difficulties based on a standardized score-system. We based our own examination on 5-6 grade students, as this is the age in which mathematical difficulties becomes apparent. The most likely reason for this is that Hungarian teachers focus on handling differences in individual performance in the first four school years 8


rather than on performance per se, and only at the end of the 4th grade do performance expectations gain significance (Nemzeti Alaptanterv{National Core Curriculum}, 2012). As a result, mathematical difficulties surfaces in 5th grade, thus, this is the age-group that sees an increase in the number of children sent to special needs specialists. These special education teachers instruct children diagnosed with mathematical difficulties through private sessions where tasks are fit to accommodate the child’s specific deficiencies in a playful manner, with more diverse methods than would be used in a normal mathematics class.

Method Participants The group of girls with mathematical difficulties was placed into an Experimental Group and into Control Group 1 using a random method. They had difficulty with the following mathematical problem-types: misjudging place-value, remembering numbers, applying appropriate calculation mechanisms in the case of 3-4 digit numbers, slowness, and uncertainty during calculation. All children continued their usual developmental sessions described above. The effects of these sessions were filtered out using Control Group 1 in which participants attended only their special needs sessions and not our origami classes. We also used another complementary control group (Control Group 2) whose members normal mathematical abilities, who did not attend either the traditional developmental classes or our training course. On the one hand, this served to control the effects of age-related development, and on the other hand enabled us to examine the development of the experimental group in comparison. The Experimental Group contained 13 fifth and sixth grade elementary students (M=12.1 years, SD=0.7 years). Control Group 1 had 12 children with mathematical 9


difficulties (M=11.9 years, SD=0.9 years). The Control Group 2 consisted of 12 children of corresponding ages with no mathematical difficulties (M=11.6 years, SD=0.5 years). Materials Origami: The children had to measure out a square on an A4 size sheet, which would serve as the basis for the origami work. During the course, we dealt with forms increasingly difficult tasks – in the onset we tackled a simpler tortoise, towards the end, a complex flower. The completed pieces of origami were then decorated individually by the children (see Fig. 1.).

------------------------- Insert Fig. 1. about here ------------------------- -

Folding spatial shapes: The pre-printed patterns were cut out by the children, who then proceeded to paste together the corresponding flaps to complete the shapes. Each successfully constructed shape was then individually decorated, for example, creating a mouse or a dog out of an octahedron (see Fig. 2.). There was no time pressure during the task: if someone was slower, the rest of the group could wait for them, or the group could start a new task, while the slower students could continue with their work in the following session.

------------------------ Insert Fig. 2. about here ------------------------ -

Measures 10


Spatial ability was measured with an adapted version of the Spatial Ability Test devised by Séra, Kárpáti & Gulyás (2002), which uses a series of tasks which incorporated counting two-dimensional shapes, mental rotation, mental knotting (see Fig. 3a.) and further tasks which involved creating previously observed spatial configurations by pasting prisms together (see Fig. 3b.). Each satisfactory solution merited 1 score-mark, the maximum score being 12.

----------------------------- Insert Fig. 3a. and Fig 3b. about here -----------------------------.-

Numerical ability was measured by 4 basic tasks: addition, subtraction, multiplication and division, 10 of each, with no time limit. As each satisfactory solution merited 1 scoremark, the score-range was 0-40. Addition and subtraction tasks involved four-digit numbers, while multiplication and division tasks involved one four-digit number to be multiplied or divided by a single digit number. Procedure The first author lead 3 separate experimental groups on different days of the week, with 4 and 5 participants. All three groups were tested individually, prior to the research completing a series of spatial ability and numerical ability tasks. One week after the completion of these tasks, all participating students of the experimental group participated in a 10 week long developmental training meeting once a week for 60 minutes. We recorded the spatial ability and numerical ability tasks with the experimental group and the two control groups one week after the completion of the training course. 11


Results

To determine whether the developmental sessions produced any improvement in spatial abilities, the data of spatial ability tasks was analysed using mixed ANOVA (3 group x 2 measurement) with Bonferroni multiple comparisons. The difference between the spatial abilities pretest and posttest (see fig. 4.) showed significant differences F(1,34)=12.28, p