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Edited by: Firat Soylu, University of Alabama, United States

Reviewed by: Kasia Muldner, Carleton University, Canada; Jennifer M. Zosh, Pennsylvania State University, United States

This article was submitted to Educational Psychology, a section of the journal Frontiers in Education

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Multiple kinds of manipulatives, such as traditional, virtual, or technology-enhanced tangible objects, can be used in primary education to support the acquisition of mathematical concepts. They enable playful experiences and help children understand abstract concepts, but their connection with cognitive development is not totally clear. It is also not clear how virtual and physical materials influence the development of different strategies for solving instructional tasks. To shed light on these issues, we conducted a 13-day intervention with 64 children from first grade, divided into three groups: Virtual Interaction (VI), Tangible Interaction (TI), and Control Group (CO). The VI group played a fully digital version of a mathematics video game and the manipulation of the blocks took place on the tablet screen. The TI group played the same video game with digitally augmented tangible manipulatives. Finally, the CO group continued with their classroom curricular activities while we conducted the training, and only participated in the Pre and Post-Test evaluations. Our results highlighted that the use of tangible manipulatives led to a positive impact in children's mathematical abilities. Of most interest, we recorded children's actions during all the training activities, which allowed us to achieve a refined analysis of participants' operations while solving a number composition task. We explored the differences between the use of virtual and tangible manipulatives and the strategies employed. We observed that the TI group opted for a greater number of blocks in the number composition task, whereas the VI group favored solutions requiring fewer blocks. Interestingly, those children whose improvement in mathematics were greater were the ones employing a greater number of blocks. Our results suggest that tangible interactive material increases action possibilities and may also contribute to a deeper understanding of core mathematical concepts.

Learning mathematics at an early age is fundamental to ensuring academic success in STEM (science, technology, engineering, and mathematics) disciplines and maximizing future integration into professional life (Wang and Goldschmidt,

In the current study, we used the activity of composing and decomposing sets of manipulatives representing numbers, an exercise that has been traditionally practiced with concrete material in order to foster an understanding of numerosity (Geary et al.,

In mathematics curricula, teaching is frequently supported by tangible objects (three-dimensional models of geometrical shapes, etc.) that help young students to better understand abstract concepts, for instance in the acquisition of cardinality (Geary et al.,

Following this vein, the acquisition of the number concept—one of the building blocks of mathematical learning—would benefit from direct interaction with objects (Dienes,

The sensitivity to numerosity is improved gradually as the infant develops (Izard et al.,

Nevertheless, children are not able to explicitly identify simple quantities involving numbers from 1 to 4 until 4 years old, and up to 5 until 5 years old. To do so, different skills must be developed such as counting and conceptual subitizing; the combination of two “subitizable” numbers, for e.g., recognizing the presence of a 3 (***) and a 4 (****) and implicitly composing a set of 7 (*******) (Steffe and Cobb,

To foster the conceptualization of unit items children may rely on hand actions such as pointing or grasping (Steffe and Cobb,

Children's addition and subtraction strategies also evolve during childhood. For instance, in order to solve 9 + 8, 4 to 5-year-old children would count from 1 to 9 for the first addend and then from 9 to 17 for the total sum (“counting all strategy”; Fuson,

Interaction with objects may supports the development of different strategies by diminishing cognitive load and freeing up working memory, given that the perceived entities are cognitively available through the objects that represent them in space (Manches and O'Malley,

With the appearance of digital technologies, researchers have been exploring how the manipulation of digital (Yerushalmy,

In this study, we focus on the kinds of actions virtual and physical manipulatives offer and their impact on numerical learning. On one hand, interaction with virtual manipulatives is limited to dragging objects on the screen, but it still allows children to displace, join and isolate objects as traditional manipulatives allow (Moyer-Packenham and Westenskow,

Technology-enhanced tangible manipulatives offer several advantages when compared with traditional or virtual manipulatives (Moyer-Packenham and Westenskow,

We recruited participants from one state school in Montevideo (Uruguay) with a medium-high sociocultural status consisting of 64 children (three classrooms) from first grade. All children had an informed consent form signed by their parents or legal guardians. A research protocol was approved by the Local Research Ethics Committee of the Faculty of Psychology, and is in accordance with the 2008 Helsinki Declaration. We employed a quasi-experimental design and each classroom became one of the following experimental groups: Control (CO), Virtual Interaction (VI), and Tangible Interaction (TI).

Four children (two from the VI group and another two from the TI group) failed to correctly answer 25% of the trials in our training game. Therefore, we performed subsequent analyses with the remaining 60 children (33 girls and 27 boys). Group descriptive information is shown in

Mean and standard deviations at pre- and post-tests by groups.

Passive Group (PA) | 20 | 6.6 (0.3) | 13 | 25.6 (5.7) | 28.8 (4.6) |

Virtual Interaction Group (VI) | 20 | 6.8 (0.5) | 11 | 31.8 (9.6) | 35.1 (9.3) |

Tangible Interaction Group (TI) | 20 | 6.8 (0.6) | 11 | 30.2 (10.3) | 34.4 (10.5) |

To evaluate the impact of both game modalities in the acquisition of mathematical abilities, we planned an intervention with three phases. A first and last phase of evaluations (Pre- and Post-Test), and a training of 13 days in between.

To evaluate children's mathematical abilities before and after training we used the third edition of the standardized Test of Early Mathematics Ability (TEMA-3, Bliss,

The three classes selected to participate in the study continued with their regular formal learning activities as part of the school curriculum. Apart from the fact that each class had a different teacher, teachers followed the same program and protocol, and were committed to giving the same math curricula information for the three classes. Both the TI and VI group played over 13 days (3 weeks). Sessions had a duration of 20 min each, from Monday to Friday. Two researchers were present in every session to help with any technical problems that may have arisen. In the first session, we introduced the game dynamics and made explicit the relation between size and value of each tangible and virtual block to facilitate effective use of manipulatives. The CO group continued with their regular curricular activities while the other two groups had 20 min per day of training. The CO group only participated in the Pre- and Post-Tests assessments.

The same evaluators assessed the groups again with TEMA-3 and the scores were analyzed in the same manner as in the Pre-Test evaluation.

The video game BrUNO was developed to give the learning activity a more attractive and playful format. We took gamification theory into consideration in order to incorporate some gamification elements in BrUNO, such as: microworlds, a main-character, a tutorial, several types of prizes, and funny sounds. During the development of BrUNO, we carried out two informal user tests to inform the game design (Marichal et al.,

BrUNO is a video game designed to work on additive composition. Children played BrUNO by using five types of blocks whose length and color were associated with their value (see

Block values, dimensions, and color.

To facilitate visual recognition of the location of the number required to build, a horizontal or vertical number line (depending on the scenario) is shown on the screen (see

Fully virtual version of BrUNO. Prize placed in number three (as indicated by the orange color). The player has already introduced 1 block of value 2. To reach the prize, he must add one block of value 1. In this example, a horizontal number line is present to help children locating numbers and to help in adding and subtracting operations.

We developed two conditions for the evaluation of manipulatives: the Tangible Interaction Group (TI) and the Virtual Interaction Group (VI). In both cases, children played BrUNO, but the interaction with the blocks differed. In the first case, children manipulated technology-enhanced tangible blocks, and in the second case, virtual blocks.

We designed a low cost tangible interaction device named CETA (Marichal et al.,

Tangible setting for BrUNO. Figure reproduced with author's permission (Marichal et al.,

We used the webcam of the tablet and a mirror to capture the image of the surface in front of the tablet holder in real-time. This image is constantly analyzed to detect blocks in the detection zone (for more details see Marichal et al.,

We designed a set of 25 blocks for 3D printing. The handling capabilities of the children at target age, the dimensions of the detection zone of the computer vision system, and the numeric quantities required by the different game challenges determined the dimensions of the blocks. All blocks contain magnets at their extremities, providing an affordance that increases the probability of joining blocks imitating the number line representation. Every block has a positive and a negative extremity. The concave and convex block's terminations constrain the way it can be joined. On the top face of each block we placed a set of colored markers (TopCodes; Horn,

The virtual version allows to play BrUNO without CETA device. The blocks are virtual and the child has to place them in the detection zone to submit its answer to the system (

We recorded the children's actions to trace the quantity and the type of blocks employed in children's solutions over time. This allowed us to analyze the game strategies developed by each group and follow the performance of every single participant. After each response our system recorded the following data: (1) the number required to form, (2) the number actually formed, and (3) the blocks used to form the number.

We assumed that if the child wanted to respond with two blocks but put the first block in the detection zone while looking for the other, then we should develop a strategy to avoid considering this incomplete answer as a child's final solution. Thus, to avoid recording partial solutions we implemented what we call “action submit,” which consists of two steps. The first step is to wait for a stable solution. By stable solutions, we mean invariant responses by children for 1.5 s meaning that the blocks placed in the detection zone were not moved for 1.5 s and no blocks were added or removed. If this condition was completed, then we move to the second step in which the game character prepares itself for 1 s to execute the movement. If, during this time the child changed his or her answer, the time counter resets and “action submit” starts over again. If the answer did not change, the game character moves and the system records the blocks that composed the child's solution. To avoid duplicate responses (e.g., the child leaves the blocks in the detection zone and goes to the bathroom) we only registered the solutions that differed from the last recorded solution.

To test the effect of playing our training game over 13 sessions, we assessed the children's mathematics performance using TEMA-3 before and after training or without training as in the case of the CO group.

While we had a quasi-experimental design in which the groups were non-randomized at baseline, there were no significant differences between groups on Pre-Test, _{(2, 54)} = 20.9, _{Mean}: 32.54, VI_{SD} = 0.77; TI_{Mean}: 33.27, TI_{SD} = 0.74 and CO_{Mean}: 30.93, CO_{SD} = 0.86). However, only Post-Tests scores significantly differed when comparing TI vs CO (

We focused on the possible problem-solving strategies employed by the children when resolving the number composition task, and how the type of interaction could have affected their actions. To do so, we carried out exploratory analysis using participants' log files. It allowed us to observe which blocks were used to compose each number by all the participants, at every successful trial.

Firstly, we analyzed whether the number of blocks used to build the correct solution was different across groups. For example, to build the number 3, it is possible to use three blocks of 1 (“1-1-1”), one block of 1 and one block of 2 (“1-2”), or directly use one block of 3 (“3”). To evaluate how close the child was to using the minimum number of blocks that were necessary to build a number (one block in the case of numbers from 1 to 5, two blocks in case of numbers from 6 to 10, or three blocks if the number is greater than 10), we developed a score called the “Minimum Blocks Coefficient” (MBC). MBC is a metric that allows us to observe the different solutions in composing numbers while training additive composition. We aim to explore how children compose numbers using different types of manipulatives. For each correct solution it takes the minimum number of blocks necessary to build the number requested, and divides it by the number of blocks actually used. For example, in the case of number 3 the variant “1-1-1” becomes the score 1/3 = 0.33, because just one block is necessary to build the number (block of 3), and in reality, three blocks were used. The combination “1-2,” becomes 1/2 = 0.5, and “3,” becomes the score of 1.0. To calculate the MBC for one particular number and one particular group (TI or VI), we take all the correct solutions of the number formed by the participants of the group and calculate the mean value. Error rates were not analyzed because we observed that the tangible system required more time for the physical manipulation and during that time some partial solutions were recorded as errors before the child's final answer. For example, if the child wanted to respond with two blocks, but he or she put the first block in the detection zone while looking for the other and no changes occur in the detection zone for 2.5 s, the system registered the child's uncompleted solution as a response (error in this case). The algorithm is explained with more detail in the section “2.3.3.” For the aforementioned reasons we decided to only analyze the correct answers, so we were confident that we analyzed explicitly correct answers rather than random solutions.

We applied a two-way ANOVA considering the MBC as the dependent variable and Group and Numbers as the independent variables. Numbers is the variable that represents the number the child is asked to build. We divided all the Numbers that appear in the game (1–13) into three ranges based on the theoretical MBC that could be used for those numbers. Specifically, the theoretical MBC for numbers ranged from 1 to 5 is one block (i.e., they have the possibility to respond with a minimum of one block); for the numbers ranged 6–10 is two (i.e., they have the possibility to respond with a minimum of two blocks) and for the numbers ranged from 11 to 13 is three blocks (i.e., they have the possibility to respond with a minimum of three blocks).

The results showed that the type of manipulatives (TI or VI group) [_{(1, 126)} = 6.21, _{(2, 126)} = 10.8, _{Mean} = 0.65, TI_{SD} = 0.19, VI_{Mean} = 0.72, VI_{SD} = 0.15). These differences between TI and VI may be a result of the diverse composing strategies used when solving the number composition task.

The Minimum Blocks Coefficient (MBC) for each number the child was asked to build. We applied a linear model to data points with a 95% confidence level for each Experimental Group: Virtual Interaction (VI) and Tangible Interaction (TI).

Considering the variable Number, the number of blocks used were significantly fewer for the numbers ranging from 1 to 5 compared to the numbers ranging from 6 to 10 (

Participants reduced the number of blocks used during the 13 sessions that our intervention lasted (see

Minimum Blocks Coefficient (MBC) for each session and experimental group. We applied a linear model to data points with a 95% confidence level.

We explored the relationship between the number of blocks employed during the intervention (measured by MBC) and the amount of mathematical improvement (dScores: Post-Test scores − Pre-Test Scores) and found no correlation (

Further, we decided to analyze the differences in the number of blocks employed comparing the performance of the Better and Worse Improvers. Thus, we divided all participants by the median of the dScore comprising two groups. The Better Improvers were the children with a dScore above the median, while the Worse Improvers were the ones whose dScore was below the median (see

Minimum Blocks Coefficient by mathematics improvement for better and worse improvers. We applied a linear model to data points with a 95% confidence level.

We were also interested in the relationship between the Minimum Blocks Coefficient (MBC) and mathematical performance (Pre-Test scores). Analysis indicated that Pre-Test scores were positively correlated with the MBC (cor = 0.41,

Minimum Blocks Coefficient (MBC) by math performance (pre-test scores). We applied a linear model to data points with a 95% confidence level.

Our results indicate that the tangible manipulative group showed an advantage in mathematics scores after training compared to the control group. Our findings highlight the possibility of improving mathematical ability by practicing implicit number composition tasks assisted by tangible manipulatives.

We did not find significant differences either between the two types of manipulatives (virtual and tangible), or between virtual manipulatives and the control group when considering mathematical improvement tested by TEMA-3. It may be the case that virtual tangibles also have an impact in Post-Test scores, which was not observed due to the lack of statistical power of the present study.

We analyzed children's behavior during our intervention to look for possible differential profiles in their evolution during training. Our tablet-based intervention allowed us to record the children's responses every time they submitted a block to compose a number. Our results enabled us to reflect on the role of specific actions performed by children affecting the learning process, and how learning could be influenced by the interactive properties of the blocks rendered as a representational assistance (Manches and O'Malley,

It was observed that the TI and VI groups significantly differed in the numbers of blocks used to compose a number. VI employed significantly fewer blocks compared with TI, showing that the different type of manipulatives could have led to different problem solving strategies. TI children opted to compose numbers using more varied combination of blocks, i.e., they used more number composition strategies. This suggests that the affordances of physical objects do trigger more diverse solutions (Manches and O'Malley,

Our results are in accordance with Manches et al. (

When we analyzed strategies during training sessions we found that at the beginning of the training both groups employed more blocks to compose numbers with a tendency to diminish in the last sessions. This tendency to diminish may represent an approach to optimal performance (when the number is composed by the minimal quantity of possible blocks), probably reflecting learning toward increasing efficient and fastest strategies in number composition (Baroody and Dowker,

This is in line with the fact that composing and decomposing strategies becomes semiautomatic or automatic with effective and faster answers to basic number combinations. Children may automatize some combinations of a number through practice, resulting in an association with their counting knowledge. This association encourages efficiency, preventing children from repeatedly practicing all the possible combinations (Baroody,

Analyses showed that the mean of blocks used in the first three sessions was significantly smaller for the VI group, whereas both groups employed the same number of blocks in the last three sessions. This suggests that besides the tendency of both groups to optimize responses, they presented a different profile in their evolution during training. Children who used tangible manipulatives had the tendency to use more blocks and showed a more pronounced decrease in the number of blocks used during the intervention compared to children who used virtual manipulatives. This finding may be connected to the observed improvement in maths scores (measured by TEMA-3) for the TI group. The number of combinations used in the TI may have contributed to achieving mastery in mathematical knowledge, since mastery in basic number composition is enriched by experiencing more varied possibilities (Markman,

We did not find a correlation between the number of blocks employed by children and mathematical improvement in general (all children analyzed together). Nevertheless, when children were divided according to their improvement in mathematics (Post-Test − Pre-Test) after the intervention, it was observed that the greater improvement group showed a positive correlation between number of blocks employed and gain in mathematical knowledge, which was not found for the Worse Improvers.

Therefore, children who showed a greater improvement tended to use more blocks. This outcome may suggest that an optimal performance in number composition (understood as fewer pieces used to form a number equals better performance) would not necessarily lead to a better learning experience. Another hypothesis would be that children who do not already have this mastery in number combinations, i.e., efficient, fast and accurate responses, would benefit more from employing manipulatives to solve additive composition and this might be the case for the “Better Improvers.” Children who improved at maths during training were the ones using more varied block combinations. This is connected to the fact that the use of a greater variety of strategies can result in a better learning outcome (Markman,

Interestingly, a negative correlation was found between mathematical scores at the Pre-Test (how good the children were at the beginning of the study) and the number of blocks employed. That is, being better at mathematics at Pre-Test implied the use of fewer manipulative blocks, probably due to a better knowledge of retrieval strategies while composing numbers (Rathmell,

It may seem contradictory that children who obtained the best scores at TEMA-3 (better at mathematics at baseline) used fewer blocks whereas the Better Improvers tended to employ more. However, according to Sarama and Clements (

These results also suggest that children who will benefit more from the use of manipulative blocks are the children who do not have already mastery in number combinations. The use of enhanced manipulatives may be more suitable for younger children who need to practice and automatize simple number combinations.

The present study has several limitations that should be considered when interpreting the results. It may lack statistical power since the number of participants in each group is small and for such reason, a larger confirmatory study is needed to strengthen the conclusions of the present study. The quasi-experimental design of the current study has more ecological validity (children were kept in their school groups), but it is susceptible to threats on internal validity compared to controlled experimental designs and for that reason we consider our results as exploratory and conclusions are drawn carefully.

Current findings indicate that the use of tangible manipulatives had a positive impact on mathematical learning. We were able to observe interesting relationships between the level of mathematics and the kind of manipulative strategies chosen by the children when solving number composition tasks. Our results suggest that tangible manipulatives increase action possibilities and may also contribute to a deeper understanding of core mathematical concepts. Playing the game BrUNO with tangible manipulatives promotes meaningful practice of more varied number combinations by encouraging children to focus on patterns and relationships in basic number combinations. In addition, we were able to observe how their responses pattern changed throughout the training leading to the use of less but efficient strategies in the last sessions which may reflect that they achieved mastery in doing such combinations. Thus, training in this basic combinations led to an improvement in mathematics and hopefully may lead children to effectively apply this knowledge in new and unfamiliar number combinations.

From an interaction design perspective (for more details regarding this research and perspective, see Marichal et al.,

All children that participated in this research had the informed consent form signed by their parents or legal guardians. The intervention current protocol was approved by the Local Research Ethical Committee of the Faculty of Psychology, and is in accordance with the 2008 Declaration of Helsinki.

AP: substantial contributions to the conception or design of the work, analysis and interpretation of data for the work, drafting and revising it critically for important intellectual content. FG: drafting the work or revising it critically for important intellectual content, interpretation of data for the work. EB: substantial contributions to the conception and design of the work and data acquisition. BF: drafting the work or revising it critically for important intellectual content, analysis, and interpretation of data for the work. GS: substantial contributions to the design of the work. SM: substantial contributions to the design of the work, drafting and revising it critically for important intellectual content.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to show our gratitude to the children and educators that participated in this work from Escuela Panamá, Montevideo, Uruguay. We also want to thank our colleagues Rita Soria, María Pascale, Mariana Rodriguez, Leonardo Secco, Matías Correa, Leandro Fernández, Dilva Devita, Mariana Borges, Gonzalo Tejera, Alvaro Cabana, Fulvio Capurso, and Rodrigo López who provided insight and expertise that greatly assisted the research and design of CETA; and Catarina Tome-Pires and reviewers for suggestions and comments that greatly improved the manuscript.