Computers in mathematics education – Training the mental number line

Computers in Human Behavior 48 (2015) 597–607

Contents lists available at ScienceDirect

Computers in Human Behavior journal homepage: www.elsevier.com/locate/comphumbeh

Literature Review

Computers in mathematics education – Training the mental number line Korbinian Moeller a,⇑,1, Ursula Fischer a,1, Hans-Christoph Nuerk b,a, Ulrike Cress a a b

Knowledge Media Research Center, Tuebingen, Germany Eberhard-Karls University, Tuebingen, Germany

a r t i c l e

i n f o

Article history: Available online 4 March 2015 Keywords: Mental number line Computer-supported learning Numerical development

a b s t r a c t Number magnitude is often described to be represented along a mental number line. In children, the accuracy of this mental number line seems to be associated with other basic numerical/arithmetic competencies. As a consequence, specific number line trainings have been developed. Evaluations indicated improvements not only in number line accuracy but also other numerical (e.g., magnitude comparison) and arithmetic tasks (e.g., mental addition) not trained directly. Importantly, this was observed for both board games as well as first computer-supported number line trainings. However, computer technology progresses rapidly. In this literature review, we specifically focus on the issues of multi-player learning environments and embodied interactions as new opportunities for training the mental number line. Regarding multi-player environments we discuss the adaptivity of learning environments needed to ensure balanced success rates in such trainings. As regards embodied interaction, we elaborate on new trainings allowing for bodily experiences of numerical concepts with new motion sensitive input devices and tangible user interfaces combining benefits of physical manipulatives with digitally provided symbolic information. We conclude that the latest developments in computer technology open up new directions for the training of the mental number line in particular and numerical/arithmetical competencies in general. Ó 2015 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

4.

5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear numerical representations in mathematics education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Training the mental number line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Computer technology to train numerical competencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Computer-supported mental number line trainings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Socially interactive number line trainings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Embodied number line trainings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Bodily movements in number space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Empirical evaluation of the effectiveness of number line trainings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Tangibles in mathematics intervention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Computer-supported trainings in formal mathematics education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⇑ Corresponding author at: Knowledge Media Research Center, Schleichstrasse 6, 72076 Tuebingen, Germany. 1

E-mail address: [email protected] (K. Moeller). Both authors contributed equally to the article and should be considered as shared first authors.

http://dx.doi.org/10.1016/j.chb.2015.01.048 0747-5632/Ó 2015 Elsevier Ltd. All rights reserved.

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1. Introduction

2. Linear numerical representations in mathematics education

It is widely agreed that the spatial representation of numbers, often described by the metaphor of a mental number line, is one of the basic representations underlying successful number processing and arithmetic (Booth & Siegler, 2006; Dehaene, Piazza, Pinel, & Cohen, 2003). Upon this mental number line, numbers are represented spatially in ascending order corresponding to their magnitude. First postulated back in 1970 by Restle, evidence for such a spatial representation of number magnitude has not only been observed in adults (e.g., Dehaene, Bossini, & Giraux, 1993). Rather, there is now accumulating evidence for systematic spatial–numerical associations in first-graders (van Galen & Reitsma, 2008), pre-counting children (e.g., Patro & Haman, 2012), and even pre-verbal infants (e.g., de Hevia & Spelke, 2010). Interestingly, the mental number line relates number magnitude and space with a specific spatial directionality. In western cultures with left-to-right reading and writing, smaller magnitudes are systematically associated with the left side and larger magnitudes with the right side of space (see Wood, Nuerk, Willmes, & Fischer, 2008 for a meta-analysis). Additionally, it was observed that even just looking at a number can induce spatial shifts of attention (Fischer, Warlop, Hill, & Fias, 2004) while the spatial direction of eye or head movements predicts number magnitudes produced in random number generation tasks (e.g., Loetscher, Schwarz, Schubiger, & Brugger, 2008). Thus, the association of numbers and space is automatic, bidirectional and present from early age on. Importantly, the mental number line representation is associated with children’s numerical competencies (e.g., Booth & Siegler, 2006; Geary, Hoard, Nugent, & Bailey, 2012). Therefore, the mental number line is a promising representation to be trained in numerical education and intervention. In children, the mental number line representation is usually assessed by the number line estimation task, in which participants have to estimate the location of a given target number (e.g., 17) on a presented number line (ranging, for example, from 0 to 100; Siegler & Opfer, 2003). While the spatial–numerical demands of this task are undisputed, the underlying processes are discussed more controversially. On the one hand, Moeller, Pixner, Kaufmann, Nuerk (2009); see also Moeller & Nuerk, 2011) argue that number line estimation is influenced by children’s understanding of the base-10 place-value structure of the Arabic number system (i.e., their understanding of the composition of the Arabic number system in units, tens, etc.). On the other hand, Barth and Paladino (2011) suggest systematic influences of proportion judgment strategies on estimation performance. Thus, when training number line estimation, it is not only the spatial nature of the number magnitude representation which is practiced, but also other representations (e.g., place-value representation) or strategies (e.g., proportion judgment) which may be recruited to map numbers onto space. Interestingly, however, while there is a reliable association between number line estimation performance and arithmetic abilities, the correlation between the above described directional aspect of number-space associations and arithmetic capabilities has been controversially discussed (Cipora & Nuerk, 2013; Dehaene et al., 1993; Fischer & Rottmann, 2005). For the development and implementation of successful mental number line trainings aimed at fostering numerical development, training the continuous spatial character of the mental number line seems more promising than focusing on the directional aspect of the mental number line (see Patro, Nuerk, Cress, & Haman, 2014 for a discussion). Interestingly, this is also reflected in popular instructional materials for early mathematics education.

When taking a look at popular instructional material and visual conceptualizations used in formal mathematics education in primary school, one cannot fail to notice that these predominantly refer to some type of linear format and/or arrangement very similar to the metaphor of a mental number line. For example, common manipulatives such as base-10 blocks (consisting of cubes as single units and rods of 10 cubes) incorporate linear formations of tens (see Fuson & Briars, 1990, for a study on the effects of these base-10 blocks in teaching of first- and second-graders). Similarly, ten-frame tiles or twenty-frame tiles (in which single items can be placed) are also arranged in linear formations (of e.g., 5 or 10) and are widely used to convey basic numerical knowledge (see e.g., Losq, 2005, for a study on the benefits of ten-frame tiles). Some educators have also used bead strings up to hundred, structured in segments of ten by alternating colors, to promote basic understanding of number magnitude. A caveat all these materials and visual aids have in common is that it is possible for children to simply count the blocks, items or beads to produce a correct solution. While such counting strategies might be acceptable when children first enter school (e.g., Moeller, Martignon, Wessolowski, Engel & Nuerk, 2011a), they need to be overcome for children to transition to more effective calculation strategies. Moreover, the effectiveness of such manipulatives has been called into question, because it seems to depend heavily on teachers’ competencies in using them. Therefore, some educators have introduced empty number lines to improve children’s conceptualization of a certain number range and to illustrate the principles of arithmetic operations such as addition and subtraction. In this context, Kaufmann and Wessolowski (2014) argue that using such empty number lines can promote conceptual understanding and reduce the reliance on counting strategies. Empirically, Klein, Beishuizen, and Treffers (1998; see also Blöte, Van der Burg, & Klein, 2001) were able to promote second-graders competence in arithmetic operations with programs building on empty number lines. In sum, linear representations and number lines are now quite common in early mathematics education. We will briefly describe in the following how simple board games have advanced research on number line trainings before we review and discuss computersupported number line trainings.

3. Training the mental number line In recent years, several number line interventions have been developed. On a quite basic level, Opfer and Siegler (2007) gave second-graders feedback on their number line estimations and found performance improvements that were ‘‘strikingly abrupt, often occurring after a single feedback trial, and impressively broad, affecting estimates over the entire range of numbers from 0 to 1000.’’ (p. 169). Similarly, Siegler and Ramani (2008, see also Whyte & Bull, 2008) used linear board games, in which children had to move a playing piece along a linear and equidistant set of fields. The authors found that differences in number line estimations between low- and middle-income children were eliminated after just four 15 min sessions of training. In a series of follow-up studies, Ramani and Siegler (2008) observed positive transfer effects of such linear board games to children’s performance in magnitude comparison, counting, and numeral identification, which were not observed for an almost identical color board game. Importantly, these transfer effects were not attributable to the per se use of numbers in the board games. Playing circular (instead of linear) number board games did not improve preschoolers’

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K. Moeller et al. / Computers in Human Behavior 48 (2015) 597–607 Table 1 Number line trainings reviewed and discussed in the current paper. Program

Availability

Evaluation Evaluation study

Paper–pencil approaches and board games Quantities, Counting, Yes Yes Numbers Big Math for Little Kids Yes Yes

Number line estimation training Number board games

Krajewski, Nieding, and Schneider (2008) Greenes et al. (2004)

Content

Age group

Additional information

Basic quantity and number skills Numbers, shapes, measurement, operations, spatial relations Number line estimation

Kindergarteners

Only available in German

Pre-K to kindergarteners

Ó2015 Pearson Education

No

Yes

Opfer and Siegler (2007)

No

Yes

Ramani and Siegler (2008, 2011), Ramani et al. (2012), and Siegler and Ramani (2008, 2009)

Linear number board game Preschoolers

Yes Yes Yes

Clements & Sarama, 2008 Griffin (2004a, 2004b) Psycharis et al. (2009)

Preschoolers Pre-K to grade 8 12-year-olds

Yes

Wilson et al. (2009), and Räsänen, Salminen, Wilson, Aunio, and Dehaene (2009) Käser et al. (2013a)

Based on curriculum Based on curriculum Comparison and ordering of fractions Number sense and exact numerosities Basic numerical cognition, arithmetical abilities Basic calculation, number formats, base 10 principle

2nd to 5th graders

Basic function concepts Quantities, numbers, word problems, calculations Number line estimation, addition, subtraction

8th-graders 1st and 2nd graders

Computer-supported trainings Building Blocks Yes Number Worlds Yes Fractions Microworld Demo version (Part of ARI-LAB2) Number Race Free

Second-graders

Kindergarteners

Calcularis

Yes

Yes

Number Catcher

Free

No

No published studies available

Function Explorer Math games with Elfe and Mathis I Rescue Calcularis

No Yes

Yes Yes

Olson (1995) Lenhard et al. (2011)

See Calcularis

Yes

Kucian et al. (2011)

Socially interactive number line trainings Interactive Web Free test Platform version

No

No published studies available

Number line estimation, calculation

5th to 6th graders

Yes

Fischer et al. (2011)

Magnitude comparison

Kindergarteners

Yes

Link et al. (2014)

Second-graders

Yes

Link et al. (2013)

Number line estimation and place-value understanding Number line estimation

Embodied number line trainings Digital Dance mat No training Digital Dance mat No training Kinect training

No

numerical understanding and success in future arithmetic learning in a comparable way (Siegler & Ramani, 2009). Furthermore, Ramani and Siegler (2011) compared the effect of playing linear board games to other arithmetic activities and found that its specific benefits (i.e., improvement in magnitude comparison, numeral identification, and future arithmetic learning) were present in low-income children in particular. Finally, Ramani, Siegler, and Hitti (2012) brought these linear board games to the classroom and found that even in the classroom setting and instructed by briefly trained paraprofessionals, playing linear boards game led to more pronounced improvements than a color board game. Taken together, there is accumulating evidence that spatial–numerical games can successfully foster numerical and arithmetic skills in children. In this context, it has to be acknowledged that components training spatial–numerical associations have been employed in other multi-componential intervention or prevention programs such as Rightstart (e.g., Griffin, Case, & Siegler, 1994) or Big Math for Little Kids (Greenes, Ginsburg, & Balfanz, 2004). However, their specific effects were usually not evaluated because the evaluation of these intervention and/or prevention programs focused on the efficiency of the whole program and not specific components. Finally, it is important to note that all of these interventions were paper–pencil based (see Table 1 for an overview). However, we argue that computer-supported methods may considerably

Current version Ó2015

Ages 5–10

Only available in German, ÓDybuster 2014 Ó2011, INSERM-CEA cognitive neuroimaging unit Only available in German, published 2010

Ages 8–10

First-graders

increase both the motivational appeal but also training efficiency of spatial–numerical interventions in multiple ways: Computer technology may be particularly suited to allow for (i) implementing numerical trainings in general, (ii) ensuring adaptivity, accessibility and interactivity (i.e., collaborative and competitive gaming) of training approaches, and (iii) experiencing numerical concepts in an embodied fashion. Before we turn to these aspects of rather new approaches of computer-supported mental number line learning, we outline how computers may affect numerical learning in general. 3.1. Computer technology to train numerical competencies Over the last decades the use of computer technology to support learning within both formal as well as informal educational settings has become increasingly popular. The National Council of Teachers of Mathematics of the USA claimed already in 2000 that ‘‘technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning’’ (2000, p. 11). Recently, the actual benefits of computer technology in mathematics education were evaluated in a meta-analysis incorporating 46 primary studies with more than 36,000 learners. Li and Ma (2010) not only observed significant positive effects of computer technology on mathematics achievement but were also able to identify conditions particularly

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corroborating the beneficial effects of computer technology. Among others, they found that effects of computer technology on mathematics learning were more pronounced in elementary as compared to secondary school, and more beneficial for children with special needs in mathematics learning compared to general education students. This is of particular interest as it suggests that computer technology may be specifically suited to support teaching of basic numerical competencies in primary school. Additionally, it is very common that children with mathematics learning disabilities (MLD) exhibit particular problems with even the most basic numerical competencies (e.g., Dowker, 2005) with a subgroup of children with MLD exhibiting a specific impairment of their mental number line representation (e.g., Geary et al., 2012; Bartelet, Vaessen, Blomert, & Ansari, 2014). Given the beneficial effect of paper–pencil number line trainings on children’s numerical development and their mathematics achievement, the mental number line seems to be a reasonable candidate for computer-supported trainings. 3.2. Computer-supported mental number line trainings Several approaches already trained the mental number line representation using computer technology. In the following we review and discuss these approaches by first giving a brief overview of computer-supported trainings addressing the mental number line concept in different ways – from very circumscribed number line trainings to multi-componential training programs incorporating number line aspects among others. After this we discuss two specific aspects of computer technology that we find important for future applications supporting numerical/mathematical education: (i) The realization of interactive and adaptive training environments to increase and maintain motivation and (ii) the possibility of task-specific embodied experience of the concepts trained on the other hand. Current developments in educational technology make use of research on computer-based instruction (Kulik & Kulik, 1991) or more recently, ‘‘serious games’’, ‘‘game-based learning’’ or ‘‘digital educational games’’ (Amory, Naicker, Vincent, & Adams, 2008; Warren, Dondlinger, Steinand, & Barab, 2009). It has long been argued that (computer) games in general actively engage people, turn them into actors and discoverers, and enable intrinsic motivation and feelings of flow (Boyle, Connolly, & Hainey, 2011; Malone, 1980; see Sedig, 2007 for an operationalization of flow in mathematics learning software). Besides general motivational effects, they are also able to support learning on a cognitive level: By measuring a learner’s performance, computer-based games can give fast and individualized feedback, which additionally promotes learning (e.g., Cameron & Dwyer, 2005). They can adaptively provide different levels of task difficulty (e.g., Echeverría et al., 2012) and do so in rich multimedia environments that may include texts, audios, videos, or even simulations and 3D worlds (e.g., Miller & Hegelheimer, 2006). Computer-supported instruction was also found to induce or facilitate specific cognitive processes (see Connolly, Boyle, MacArthur, Hainey, & Boyle, 2012 for a systematic review of positive effects of game-based learning, see also Clark, 1994; Kozma, 1991 for early considerations). However, it is important to note that it does not seem to be the use of computer-supported learning environments per se which corroborates learning (see Livingstone, 2012 for a critical discussion). Instead, the benefits of computer-supported learning seem to depend on how it is actually implemented (e.g., Lei & Zhao, 2007; see Means, Toyama, Murphy, Bakia, & Jones, 2009 for a review on online learning). With regard to children’s numerical development, Samara and Clements (2006) highlight that to substantially contribute to early childhood mathematics education it is necessary that appropriate learning environments are employed

along effective teaching strategies, which requires professional development of teachers. Against this background, it is important to evaluate the effectiveness of computer-supported number line trainings in more detail. To date, quite a large number of computer-supported number line trainings already exist, and those presented cannot be considered to represent an exhaustive list. Most of the existing number line trainings are part of multi-componential training programs for the prevention or remediation of mathematical learning difficulties. While some of these programs are available online for free (e.g., The Number Race: http://www.thenumberrace.com/nr/home. php) others are distributed commercially (e.g., Calcularis: http:// www.calcularis.ch/int; Number games with Elfe and Mathis: http://www.psychometrica.de/rechenspiele1.html; Number Worlds and Building Blocks: http://www.sranumberworlds.com/; http:// gse.buffalo.edu/org/buildingblocks/ which overlap in several characteristics). In line with the meta-analysis by Li and Ma (2010), all of these programs also address basic numerical competencies and are targeted at rather young children from pre-kindergarten through elementary school (see also Clements & Sarama, 2011; Räsänen, Salminen, Wilson, Aunio, & Dehaene, 2009 for reviews). Moreover, all of them claim to be particularly suited as prevention or intervention for children at risk for or with mathematical learning difficulties. They do however differ considerably in the specificity with which the mental number line is trained. On the one hand, some programs (e.g., Calcularis, Number Worlds, and Elfe and Mathis) use number lines in a quite explicit way. In these programs, number line estimation is one component of the intervention. In Calcularis, for instance, a falling cone has to be landed as closely as possible to the target number on a given number line with the target number being presented as symbolic (i.e., Arabic digits), non-symbolic quantity (i.e., dot pattern) or a spoken number word (see Kucian et al., 2011; Käser, Baschera, Kohn, et al., 2013a, 2013b). On the other hand, the primary focus in The Number Race is on magnitude comparisons, which are presented in a symbolic or non-symbolic format. Children play against the computer and whenever they win they are allowed to move their playing piece an according number of fields. Thus, a number line is addressed in more or less the same way as in the board games used by Siegler and collegues (e.g., Siegler & Ramani, 2009) through movements along left-to-right ordered fields. Most of the multi-componential intervention programs presented here were evaluated in scientific studies that indicate their positive effects on children’s numerical and mathematical abilities (Number Worlds: Griffin, 2004a, 2004b; Building Blocks: Clements & Sarama, 2007; Clements & Sarama, 2008; Number games with Elfe and Mathis: Lenhard, Lenhard, Schug, & Kowalski, 2011; Calcularis: Käser et al., 2013a, 2013b). However, comparable to the case of paper–pencil programs these evaluation studies did not differentially evaluate effects of individual components, so that it is hard to appraise the efficiency of the number line components in particular. Although the Number Race is not strictly a multi-componential program aiming at enhancing children’s number sense as well as ‘‘to emphasize the association between representations of number and space’’ (Wilson et al., 2006a, p. 2), it is difficult to say whether the linear arrangement of the playing fields influences the training effect, because the trained content does not explicitly address the mental number line. However, there are several empirical studies indicating positive intervention effects of the Number Race (Wilson, Dehaene, Dubois, & Fayol, 2009; Wilson et al., 2006a; Räsänen et al., 2009) that also generalized to untrained numerical competencies (Wilson, Revkin, Cohen, Cohen, & Dehaene, 2006b). So far, only one study employed and evaluated a specific computer-supported number line training (Kucian et al., 2011). The authors used a preliminary game from the Calcularis program that they called Rescue Calcularis, in which

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Fig. 1. Panel A depicts an exemplary item from the Rescue Calcularis program (used by Kucian et al., 2011 and now integrated into the more comprehensive Calcularis program, see also Käser et al., 2013a, 2013b). Panel B depicts the estimations of five players in a multi-player session of the interactive competitive number line estimation game.

children had to indicate the position of a given target number – or the result of a calculation – on a presented number line by maneuvering a rocket to the estimated position (see Fig. 1, Panel A for an example). What further differentiated this study from all previous studies was that the authors not only investigated behavioral training effects but also their neural correlates. Behaviorally, a significant improvement of children’s number line accuracy and an increase in their arithmetic performance was observed. This was accompanied by reduced recruitment of brain regions commonly associated with number processing (such as the intra-parietal sulcus) after the training, possibly due to automatization of the respective cognitive processes. Additionally, findings also suggested partial remediation of deficient brain activation in participants with dyscalculia. In summary, this study indicates that computer-supported number line training leads to significant training effects on both the behavioral as well as the neural level. Interestingly, the number line concept has also been used to train more advanced numerical and mathematical competencies. For example, The Number Catcher (http://www.thenumbercatcher. com/nc/home.php) expands the range of The Number Race to multi-digit numbers. Even more complex content is addressed in the ARI-LAB2 environment, where fraction knowledge is trained by systematic use of number line information based on Thales Theorem (e.g., Psycharis, Latsi, & Kynigos, 2009, see also Booth & Newton, 2012 for fraction knowledge associated with number lines). And finally, Olson (1995) developed a framework for systematically using number lines in the teaching of algebraic functions. Taken together, there already are numerous computer-supported trainings addressing the concept of the mental number line in one or another way (see Table 1 for an overview). However, in the remainder of this article the specific focus will be on novel aspects of computer-supported instruction for the development of future number line trainings. We will discuss first attempts to implement social interactivity before elaborating on the potential of embodied interactions – from full body movements to tangible user interfaces. 3.3. Socially interactive number line trainings In most number line trainings, learners only interact with the learning environment, but not with one another. However, players of computer games seem to favor the opportunity to challenge and compete with one another (e.g., Weibel, Wissmath, Habegger, Steiner, & Groner, 2008). Depending on the trained content, collaborative or competitive types of social interaction are more or less suitable. In mathematical tasks that need elaboration and deep understanding, a collaborative reward structure would be more

effective than a competitive one (Johnson & Johnson, 1989). In contrast, a competitive reward structure seems adequate for simple and speed-related tasks that train automatic processes (Clifford, 1972), such as mental number line trainings. However, a competitive reward structure requires a quite homogeneous group of learners. Otherwise, low-achieving learners would continuously be outperformed by most of their competitors, leading to demotivation (Ames, Ames, & Felker, 1977). However, a homogeneous group of learners cannot be guaranteed and is often hard to realize. Therefore, there are specific demands on interactivity and adaptivity of the learning environment. In this vein, we recently developed a web platform with interactive browser games training both numerical as well as orthographical competencies (http://lernplattform.iwm-kmrc.de/). After creating an account, players enter the game zone and can choose a game. Among other arithmetic games (e.g., addition, Multiplication, etc.), we implemented a number line game in which players have to indicate the correct position of a given target number within a pre-defined number range. Players can choose from two number ranges (i.e., up to 100 or up to 1000) and two difficulty levels. In the easy condition, the number line ranges precisely from 0 to either 100 or 1000 and the start and endpoint are marked accordingly. This bounded version of the number line estimation task may be solved by strategies other than number line estimation such as proportion judgment (i.e., using halves and quarters of the range as additional reference points, e.g., Barth & Paladino, 2011). Therefore, the difficult condition involves variable start and endpoints that should discourage proportional strategies (e.g., positioning 37 on a number line with startpoint 12 and endpoint 64). The number line game can be played in either single- or multiplayer mode with up to five players. In the single player mode, the player competes with computer-controlled opponents; whereas the multi-player mode pitches human players against one another. The game is competitive because it is impossible for a player to place his/her estimate on the same position as another player. Additionally, each player’s estimate covers about 5% of the entire number line (see Fig. 1, Panel B for an example). This way, players are encouraged to make spontaneous estimations because a fast and accurate guess of one player makes it impossible for other players to win. It is simply not possible to place ones marker at the same or a very close position anymore. Also, this competitive gameplay should discourage alternative, more time consuming strategies (such as proportion judgments in the easy condition or calculating the difference between start and end point in the difficult condition). Thereby, the mapping between number and space is trained instead of alternative solution strategies. After all players have set their marker, the player with the estimation closest to the correct position of the target number (indicated by

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Fig. 2. Schematic illustration of the embodied training used by Fischer et al. (2011). Initially, the child stood on the central field of the dance mat. Then the item was presented and the child had to compare the target number in the circle to the standard number on the number line. In this case, 7 is larger than 4, so the correct response would be a jump to the right.

a green2 bar) earns a point. The player with the most points out of a predefined number of rounds wins the game. When implementing the number line game in a first practical test in the field, we found that success rates were very unbalanced. It became obvious that students’ numerical skill levels were heterogeneous and playing against students with a higher skill level was frustrating for students with low skill levels. This is why players’ skill levels need to be matched so that they apply more effort and play for longer periods of time (Liu, Li, & Santhanam, 2013). Therefore, adaptivity of the training environment was a crucial aspect that could make or break the whole idea of an interactive number line training (see also Wilson et al., 2006a, for adaptivity in The Number Race; Käser et al., 2012, 2013a, 2013b for adaptivity in Calcularis). In the further development of the number line game and the web platform, we will therefore implement inter-individual as well as intra-individual adaptivity. In multi-player games, inter-individual adaptivity is one possible solution to the problem of motivation and shall be achieved in two ways. First, each registered player will be assigned an automatically generated proficiency level that is updated with every completed game. It is then possible to selectively pair players by matching them according to their proficiency levels. Second, we will introduce individual handicaps such as varying speed and difficulty for players with a higher proficiency level (e.g., time delays), so that both players have a balanced chance of winning. On the other hand, in single-player games with computer-controlled opponents, intra-individual adaptivity is necessary. Based on the players’ previous performance, the activity of the computer-controlled player will be adapted to allow for an individual success rate of at least 50%. To realize such intra-individual adaptivity, it is necessary to update and adapt the parameters controlling the computer player on a trial to trial basis. After these modifications, it should be possible to balance success rates and keep up the motivation of all players regardless of their skill level. The ultimate goal will be to design interventions tailored to every individual’s needs. 3.4. Embodied number line trainings 3.4.1. Bodily movements in number space In recent years, digital media started to provide an additional feature that may also support learning in a specific way: Media nowadays enable ‘‘embodied interaction’’ (Dourish, 2001; Romero et al., 2007). Initially, dancing games on digital dance mats required subjects to move their whole body by stepping on different fields of the mat in a specific succession. Games featuring dance mats became widespread as part of the Sony PlaystationÒ 2 (PS2™) or MicrosoftÒ Xbox™. Nowadays, low-price USB versions of dance mats for PCs are still available. The next generation of devices allowing for embodied interactions was much more sophisticated but also more complex. Devices such as the Wii™ gaming console came with a motion sensitive controller, the Wii Remote™, that registered a player’s hand movements. The 2 For interpretation of color in Fig. 1 and 2, the reader is referred to the web version of this article.

MicrosoftÒ Kinect™ sensor for the Xbox 360™ was one of the first camera-based systems that made controllers obsolete and registered as well as tracked players’ position and actions in 3D space. All these devices are constantly enhanced and refined, with both the Wii Remote™ as well as the Kinect™ now in their second generation. The Wii Remote™ was supplemented with the Wii Motion Plus™ expansion device to improve accuracy of motion control, and the new Wii U™ console comes with a new touchscreen controller, the Wii U GamePad™, with built in motion sensors. The Kinect™ 2.0 for Xbox One™, an upgraded version of its predecessor with improved depth perception, was released together with the new console in 2013. Kinect™ has also been released for WindowsÒ, and MicrosoftÒ has provided software development kits that make it possible to program specifically for Kinect™. All these systems measure user movements and/or gestures and also have a motivational appeal that can be beneficial for learning purposes. Yet, besides this general motivational appeal, bodily movements can also be used to support learning more specifically, given the task to be learned also has a spatial dimension. In case of numerical tasks, for instance, the movement of a learner may be congruent or incongruent to the direction of the internal number line representation. When any learning task activates the mental number line, movements that are congruent to this internal representation should corroborate task performance and even learning, whereas incongruent movements should hinder both. A first example of specific bodily number line training was implemented by Fischer, Moeller, Bientzle, Cress, and Nuerk (2011). The authors used a digital dance mat to train kindergarteners on the association of small numbers with the left and large numbers with the right side of space. In the dance mat training, children started standing on the central field of the dance mat. Performing a magnitude comparison task, children had to jump to the left when a presented target number was smaller than a presented standard number and jump to the right when the target number was larger than the standard (see Fig. 2). In a control condition, children only had to tick the larger number on a tablet PC with no spatial–numerical full-body movement. The embodied number line training was more effective than the control training. Children not only got more accurate in positioning numbers on a number line from 0 to 10 after the embodied training. The advantage of the embodied over the control training also generalized to children’s counting skills, which were not trained directly. Finally, the results of a mediation analysis argue against a general effect of the experimental condition (e.g., via motivational appeal) and hint at a specific effectiveness of the training. The authors found that children’s improvement in counting was mediated by their improvement in number line estimation and thus, by an increase in number line accuracy. These differential training effects in favor of the embodied over the control training were found in a categorical training of number line direction. However, because the mental number line is a continuous representation, it seems promising to train it in a more continuous fashion. This was pursued in another embodied training study with the dance mat and one using the Kinect™ sensor that will be described in turn in the following (see also Fischer, Link, Cress, Nuerk, & Moeller, 2015, for an overview).

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Link et al. (2014) aimed to not only address the mental number line representation in an embodied dance mat training. Instead, the place-value structure of the Arabic number system was also considered, because an early place-value understanding is fundamental for the development of later arithmetic abilities (e.g., Moeller, Pixner, Zuber, Kaufmann, & Nuerk, 2011b). Second-graders were trained in number line estimation. In particular, they had to grow (with a step to the right) or shrink a bar in length (with a step to the left) by either ±10 or ±1 by stepping on different fields of the dance mat until it reached the estimated position of a given target number. Because the ±10 fields reflected larger steps on the mental number line, we put the frontal fields of the dance mat on a step to increase the physical effort needed to do these larger steps – making the place-value structure physically tangible. Effectiveness of the embodied training was contrasted to that of two control trainings. In line with the results of Fischer et al. (2011), children’s improvements in number line estimation were more pronounced after the embodied than after both control conditions (controlling for the motivational appeal of the body movement and the trained numerical content). Closer inspection of the data revealed that children improved particularly in estimating target numbers for which the correct understanding of the place-value structure was specifically relevant (e.g., 29, where the confusion of tens and units would lead to a large estimation error on the number line). Using the Kinect™ sensor, Link, Moeller, Huber, Fischer, and Nuerk (2013) also pursued the idea of a continuous number line training. In their study, first-graders had to estimate the position of a target number within a given number range (i.e., 0–100) by walking along an up to 3 m long number line taped on the floor. Their movements were recorded and analyzed using the Kinect™ sensor for PC. Compared to a control condition training of the same task on a tablet PC, children again improved more in a paper–pencil version of the number line estimation task, indicating that the embodied condition was more effective in improving children’s number line accuracy. Moreover, reliable transfer effects to an addition task were observed. Children improved more after the embodied as compared to the control training in single-digit addition and even in single-digit additions requiring a carry operation. It has to be acknowledged that so far these embodied approaches to number line learning (see Table 1 for an overview) are restricted to single user environments. However, if success rates were balanced as described above, it would be possible to attach two dance mats to a PC and have two children perform the comparison task used by Fischer et al. (2011) against each other. For continuous movements, more refined camera-based tracking systems (such as Kinect™ 2.0) can now track more than one person at a time, so that a competitive version of the continuous number line training as used by Link et al. (2013) should also be possible. 3.4.2. Empirical evaluation of the effectiveness of number line trainings In almost all cases, evaluation studies were conducted by or with the involvement of the developers of the respective (computer-supported) number line trainings (see Table 1 for an overview). Importantly, these studies mostly support the effectiveness of number line trainings in general (e.g., Whyte & Bull, 2008; Siegler & Ramani, 2009) and computer-supported number line trainings in particular (e.g., Fischer et al., 2011; Kucian et al., 2011). This clearly indicates number line trainings to be an effective possibility of improving children’s numerical competencies. So far, the vast majority of these studies investigated either linear board games (e.g., Ramani & Siegler, 2011), simple computersupported number line trainings (e.g., Link et al., 2013), or number line trainings included in multi-componential programs such as Number Worlds (Griffin, 2004a, 2004b) or Calcularis (Käser et al., 2013a, 2013b). These multi-componential trainings usually build on a solid background of research in mathematics education.

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However, the effectiveness of their number line components is hard to specify for those programs as they are usually evaluated as a whole rather than separately by their constituting components. Concerning the new approaches currently in development, so far only the embodied number line training approaches have been evaluated in intervention studies (Fischer et al., 2011; Link et al., 2013, 2014). Importantly, these studies provided converging evidence that an embodied component enhances the effectiveness of regular computer-supported number line trainings. Unfortunately, the socially interactive and adaptive training we discussed has not yet been evaluated regarding its effectiveness (but see Käser et al., 2013a, 2013b; Wilson et al., 2006a, 2006b for evaluations of adaptive mathematics learning environments). In sum, it can be said that (computer-supported) number line trainings seem to corroborate children’s numerical development and may thus be ready to be implemented into formal mathematics education. For the linear number board games Ramani et al. (2012) recently showed that these games are effective even in classroom settings. Thus, the perspective seems promising for other forms of number line trainings to also take the step into classrooms as a fruitful complement to above described instructional material already drawing on linear numerical representations (such as ten-frame tiles). However, the current state of research also suggests that the new promising possibilities offered by the progress in computer technology (i.e., embodied interaction and adaptivity of learning environments) still need further investigation to fully appraise their potential. However, when they build on recent successful implementation of computer-supported number line trainings, it seems promising that incorporation of more sophisticated technology will effectively enhance numerical learning. Nevertheless, it is important to note that a precondition for such an effective implementation is that developers take into account both empirical results but also educational standards to choose computer-supported learning environments appropriately for the respective content, consider effective teaching strategies, and ensure sufficient proficiency of teachers or trainers (cf. Sarama & Clements, 2006).

4. Future directions Computer-supported learning environments not only allow for embodied trainings of the mental number lines by means of bodily movements in number space but also an integration of physical manipulatives into computer-supported math interventions. Integrating manipulatives (such as blocks or sticks) into so-called Tangible User Interfaces (TUIs) creates a whole new possibility for the integration of tangible material with symbolic information. However, how the potentials of such new computer-supported learning environments (be it computer-supported or even future tangible number line trainings) can be integrated successfully into formal mathematics education remains to be investigated. In this outlook, we will discuss current developments and possibilities.

4.1. Tangibles in mathematics intervention A lot of intervention programs for children with mathematical difficulties in practice already make use of physical manipulatives that may at least partly draw on the number line concept (e.g., unit blocks that are assembled to tens rods, etc.). However, event digital interfaces can enhance the usefulness of these tangibles by combining them with the benefits of digital media, there is currently no specific number line training using TUIs. Nevertheless, there are applications of TUIs in mathematics education and intervention.

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From a theoretical point of view TUIs may be beneficial because children often are not explicitly taught on how abstract or symbolic learning content (such as numerical knowledge) is related to concrete physical manifestations. However, according to Pape and Tchoshanov (2001), the human brain is organized in a multiple representational patterning that consists of combinations of concrete, visual, and abstract representations (see also Bruner, 1966). Especially children may therefore benefit from the use of manipulatives when their association to abstract representations is emphasized (but see Uttal, Scudder, & DeLoache, 1997 for a more critical view). Considering this, Manches, O’Malley, and Benford (2009) elaborated on the advantages and limitations afforded by physical manipulatives by discussing the differences between virtual materials and physical learning materials. They argued that tangible materials may support learning by building on the advantages of physical manipulation while avoiding limiting exploration by constraining the range of actions and possible learning opportunities (as for example in textbooks). For example, an advantage of TUI is that they are able to provide an external record of previous states and actions. They also provide support for linking representations; meaning that they are interactive. This may actually be beneficial considering the many different ways in which numbers can be represented (Pape & Tchoshanov, 2001). Therefore, traditional manipulatives (such as unit blocks and tens rods) might not be well suited for children to connect intuitive physical understanding of numerical concepts with a formal understanding of mathematical expressions (Manches, O’Malley, & Benford, 2009). However, enriching traditional manipulatives with the opportunities of recent TUI developments may provide a promising perspective for future mathematics education. First examples for TUIs in mathematics education have already been implemented in various formats to convey different mathematical concepts. However, it seems that especially geometric concepts are suitable for TUIs. For example, Girouard et al. (2007) implemented Smart Blocks, a manipulative allowing for the exploration of volume and surface of 3D objects. The cube-shaped blocks are assembled via connectors, and allow children to experience hands-on manipulation, while RFID (= radio frequency identification) technology transmits information to a computer. This computer then provides information about surface or volume of the assembled object, which would otherwise not be directly accessible for children. The continuous feedback via the PC is then assumed to reinforce learning. Falcao, Meira, and Gomes (2007), on the other hand, suggested the development of a digital board that measures the surface of shapes that students can augment with sticks they place on the board. By placing sticks within a shape, they create divisions that the software attached to the board measures. Divisions with the same surface area would then automatically be colored in the same color. With the rising popularity of tablets, such a program could easily be implemented in real life. A program developed by Schneider, Blikstein, and Mackay (2012) named Combinatorix is a novel approach that combines tangible objects with an interactive tabletop. By rearranging physical tokens, students can explore, solve and understand probability problems. They not only see the effects their manipulations have on the problem space; a second screen additionally provides them with an abstract representation (e.g., a probability tree) of the associated changes. This integration of concrete and abstract material is supposed to improve students’ problem-solving skills. For an interesting overview over other approaches for the implementation of TUIs see Scarlatos (2006). Because TUIs can be used individually by each student while in the classroom, they provide real time feedback for each student and combine it with physical manipulatives to promote abstract learning (Girouard et al., 2007). This is one reason why TUIs are an interesting and promising approach to convey numerical concepts.

Although to our knowledge, so far no TUI was explicitly designed to allow for number line trainings, it would easily be possible to create such tangibles. For instance, by using tangible rods of different lengths that may be recognized by a touch sensitive surface, a program may output the actual numerical length of the rods, thereby creating a connection between a concrete manipulative and the symbolic representation of its length. Also, connecting multiple manipulatives in order to create an estimated length might be a promising approach to help children acquire number line estimation skills and maybe even the principles of addition and subtractions which have recently been associated with movements along the mental number line (e.g., McCrink, Dehaene, & Dehaene-Lambertz, 2007; Knops, Thirion, Hubbard, Michel, & Dehaene, 2009). In our opinion these and other possibilities of reflecting abstract numerical concepts and related procedure by TUIs may provide a huge potential for enhancing numerical learning and should thus be explored in future studies.

4.2. Computer-supported trainings in formal mathematics education As already described above, it is not the increased use of information and communication technologies in formal (mathematics) education per se which leads to increased learning benefits (e.g., Livingstone, 2012). Rather, it seems to be the specific combination of the appropriate computer-supported learning environment for circumscribed topics, learning strategies, and trained educators (cf. Kreijns, Van Acker, Vermeulen, & Van Buuren, 2013 for motivating teachers to integrate computers in their teaching) which makes computer-supported learning effective (e.g., Attewell, Savill-Smith, & Douch, 2009, see Samara & Clements, 2006 for mathematics learning). Against this background, the question arises how far this is possible with computer-supported mathematics learning environments in general and computer-supported number line trainings in particular. Importantly, this question was increasingly pursued in recent years with promising results regarding the potential of computer-supported mathematics learning environments. For instance, Özyurt, Özyurt, Baki, and Güven (2013) provided evidence that an intelligent and individualized e-learning environment can be integrated successfully into formal mathematics education when teaching probabilities in secondary school. Moreover, Adams et al. (2014) observed reliable beneficial effects of a web-based tutoring system for learning about fractions. Interestingly, the tutoring system even involved a condition in which the spatial location of given decimals (e.g., 0.7) had to be placed on a number line (ranging from 0 to 1). These findings are promising when considering that the web-based learning platform described above reflects a quite similar approach of web-based mathematics learning. With these encouraging findings in mind, future studies might even be targeted at evaluating whether competitive (as in the described web platform) or rather interactive and collaborative environments such as shared displays and workspaces (e.g., Nussbaum, Alcoholado, & Büchi, 2015) lead to more beneficial effects in mathematics learning. Transferred to number line trainings, a collaborative setting might involve small groups of learners which have to come to a consensus about where to locate the respective target number on the number line. In such a case, cultural aspects (e.g., reversed number line orientation in right-to-left reading cultures, cf. Göbel, Shaki, & Fischer, 2011 for a review) but also more general aspects regarding the composition of the learner groups in computer-supported learning environments would have to be considered (e.g., Dascalu, Bodea, Lytras, Ordoñez de Pablos, & Burlacu, 2014; Zhang, Ordóñez de Pablos, & Xu, 2014).

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5. Summary and conclusions In the current article we described the idea of a mental number line reflecting the human representation of number magnitude, and evaluated in what way the development of children’s mental number line can be corroborated by computer-supported learning environments. Thereby, we attempted to add a new perspective to the integration of computer-supported trainings into mathematics education. It is important to note that there already are both physical but also computer-supported approaches to number line trainings for which scientific evaluation indicates significant training effects. Nevertheless, we argue that with the rapid progress in computer technology, increasingly sophisticated possibilities for training the mental number line are provided. We systematically elaborated on the issues of socially interactive learning environments as well as embodied interactions. With respect to socially interactive learning we suggest that adaptivity will be the crucial aspect of any learning environment aiming at allowing interactive and competitive number line trainings. It seems mandatory that mechanisms are implemented which guarantee balanced success rates in multi-learner trainings. Otherwise, children with difficulties in mathematics would lose motivation due to being constantly outperformed by more proficient players. Regarding the possibility of embodied interactions with the learning material offered by new digital media, we are confident that trainings allowing for bodily experiences of numerical concepts such as the mental number line should be particularly fruitful. New media devices such as the Kinect™ sensor allow for full-body movements along the mental number line. The created isomorphism between full-body movements in physical space and corresponding movements in number space should support children’s understanding of numerical quantities and their manipulation. Future studies should also look into the possibilities provided by tangible user interfaces for the purpose of embodied number line trainings as well as possibilities to successfully integrate computer-supported number line trainings into formal mathematics education. Against this background, exploiting the possibilities of computer technology seems specifically promising, as it opens up new directions for the successful computer-supported training of numerical representations. Acknowledgements Part of this review has been presented as a paper at the 21st International Conference on Computers in Education (ICCE) in Bali (2013) and is included in the conference proceedings [Wong, L.-H. et al., (Eds.) (2013). Proceedings of the 21st International Conference on Computers in Education. Indonesia: Asia-Pacific Society for Computers in Education]. Korbinian Moeller and Hans-Christoph Nuerk are members of the ‘‘Cooperative Research Training Group’’ of the University of Education, Ludwigsburg, and the University of Tübingen, which is supported by the Ministry of Science, Research and the Arts in Baden-Württemberg. Korbinian Moeller, Ulrike Cress, and Hans-Christoph Nuerk are members of the faculty of the LEAD graduate school funded by the Exzellenzinitiative of the German Science Foundation (DFG). References Adams, D. M., McLaren, B. M., Durkin, K., Mayer, R. E., Rittle-Johnson, B., Isotani, S., et al. (2014). Using erroneous examples to improve mathematics learning with a web-based tutoring system. Computers in Human Behavior, 36, 401–411. Ames, C., Ames, R., & Felker, W. (1977). Effects of competitive reward structure and valence of outcome on children’s achievement attributions. Journal of Educational Psychology, 69(1), 1–8. Amory, A., Naicker, K., Vincent, J., & Adams, C. (2008). The use of computer games as a tool: Identification of appropriate game types and game elements. British Journal of Educational Technology, 30, 311–321.

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