Christian Urff 2025 (updated text to a version of 2011)

Introduction

Mathematical tools are central teaching and learning instruments that allow for the hands-on exploration of fundamental mathematical concepts (Krauthausen, 2012, 2022). Unlike physical tools that learners can manipulate with their own hands (e.g., moving, turning, or rotating objects), the "real" possibilities for interaction with virtual tools are limited to digital interactions. With traditional desktop computers and laptops, interactions primarily occur via mouse clicks or keyboard input, while touchscreens in tablets enable direct haptic feedback through multi-touch gestures (Agostinho et al., 2015). AR glasses now also allow input using natural hand gestures (pinching, rotating, etc.).

With virtual tools, the actual actions are performed by the computer and visually rendered – the user initiates the action. This is clearly illustrated by the example of a virtual twenty-square grid: Counters can be virtually flipped, but not by actual physical turning. Instead, the software executes the flip operation with a click or touch gesture and displays an animation of the action. I refer to such computer-executed but user-initiated operations as computer-based or virtual actions (Urff, 2010). Computer-assisted actions are characterized by the fact that the action is initiated by the user, but a significant part of the execution is supported by the computer.

Relationship between virtual and physical actions

Restrictions and requirements

One could initially argue that the concrete possibilities for action, and thus the action experiences, are significantly limited in virtual actions. This is certainly true if only tactile-haptic experiences are considered (Moyer-Packenham & Bolyard, 2016). The child cannot physically feel a virtual flipper, cannot smell it, and the act of turning it over is not directly linked to a motor movement of the flipper itself.

This limitation is the decisive reason why virtual tools should be used whenever possible. Continuation and expansion Virtualized learning tools should be used to complement, not replace, the use of physical learning materials (Krauthausen, 2022; Bouck et al., 2020a). This is especially true for primary and preschool education, as primary experiences play a crucial role in developing understanding during this phase of development. A child will likely only be able to correctly interpret a virtualized flipping operation on the computer if they have already flipped a physical counter themselves or observed this action being performed with a physical tool.

Current research shows that combining concrete and virtual learning tools (so-called "blended manipulatives") can be particularly effective (Ahmad et al., 2024; Yakubova et al., 2024). The sequence and integration—first concrete/physical, then virtual—remains didactically significant. But why should virtual learning opportunities be offered in addition to concrete tools?

Didactic potential of computer-assisted actions

If the visual-auditory and interactive experiential possibilities of virtual actions are examined more closely, computer-based actions, as a continuation of real actions through technological possibilities, offer some didactic potential that can expand physical-concrete action experience:

1. Cognitive relief and shifting through automated action execution

By having the computer handle the execution of actions, more cognitive resources remain available for actual mathematical learning (Sweller, 2020; Paas & van Merriënboer, 2020). Decoupling the impulse to act from its execution can reduce the child's non-learning-relevant motor and cognitive load (Skulmowski & Xu, 2022). In terms of cognitive load theory, this corresponds to the principle of reducing the extraneous cognitive load, i.e., cognitive loads that are not necessarily relevant to learning (Castro-Alonso, 2020).

Example calculation field: While a physical twenty-square requires manually sliding tiles onto the grid, engaging the child in this motor activity, this process is automated on a computer. With a virtual twenty-square, a click or touch automatically animates the insertion of one or more tiles. The child doesn't need to worry about the correct execution of the tile movement—which, at least with sufficient primary experience, is irrelevant for mathematical concept formation—but can observe the insertion operation and its effects, focusing attention on the change at the symbolic level (the sum increases). This allows for experimentation and potentially a better understanding of the relationships, as more cognitive resources are available for observing the effect of the action. Well-designed virtual learning tools, therefore, in conjunction with appropriate tasks, have the potential to reduce cognitive load and facilitate the operational understanding of the learning situation through such partial automation, as experimenting with different action options, such as "What happens to... if...?", is significantly simplified.

2. Conceptual control of action execution

While actions on physical work equipment allow not only mathematically meaningful actions (e.g., placing a tile) but also actions that have no mathematical-conceptual equivalent (e.g., overlapping tiles, misplacements, mismatched representations), computer-assisted actions can be limited to conceptually consistent mathematical actions (Moyer et al., 2002; Bouck & Park, 2018).

This ensures that only those operations are offered as actions that are important for the learner to develop and discover the didactically significant structural properties and thus for understanding. Other actions, which are nevertheless possible with analog tools due to their physical properties, are limited. For example, with the virtual twenty-frame, it is ensured that:

• never more than 20 tiles can be placed on the field,
• red tiles are always structured in a coherent manner and are therefore easier to understand (if automatic structuring is switched on),
• the 5 and 10 structuring is automatically taken into account and
• the iconic and symbolic representation always match each other.

This conceptual structuring Conceptual scaffolding using digital tools supports learning, especially for children with learning difficulties (Park et al., 2022; Yakubova et al., 2024). Particularly when a deep understanding is lacking, incorrect actions can lead to fundamental misunderstandings. For example, if a placement error on a physical twenty-square causes the iconic and symbolic representations to misalign (e.g., 14 tiles are on the square, but the number 15 is next to them), children often try to rationalize this discrepancy afterward, sometimes constructing flawed mental models in the process. Virtual tools can better prevent such misunderstandings by allowing only conceptually correct operations and automatically ensuring consistency across all representational levels.

3. Synchronized representation across multiple representation levels

The computer-assisted execution of the action makes it possible to synchronize actions across multiple representational levels (Krauthausen, 2012, 2022). This enables the dynamic networking Different forms of representation are a central element for a deep mathematical understanding (Moyer-Packenham & Bolyard, 2016).

Example: If a summand on the calculation field is reduced by one, then:

• the corresponding number is reduced (symbolic level),
• the sum is reduced by one,
• At the same time, on the iconic level, a tile is removed from the twenty-field and
• the total visible quantity is reduced by one element.

This supported execution and dynamic visualization link the action experience across multiple representational levels. This creates learning opportunities that can be significant for developing a comprehensive understanding of fundamental mathematical operations. This is especially true for children who, while capable of performing this transfer themselves, are still prone to errors. They can expand, validate, modify, and correct their mental model of the relationships between the representations through these actions – provided they are given appropriate tasks (for example: "What happens to the twenty-frame if I add five red tiles? Think about it first, then check."). However, this interconnected, synchronous representation must be well-balanced in terms of the scope and design of the visual elements, as linking across multiple representational levels simultaneously can also create additional cognitive load. Less is often more here: a reduction to a few interconnected representations at a time is therefore recommended.

4. Realization of didactically valuable actions that are not objectively possible

Of particular interest is the visualization of action processes that illustrate core concepts in mathematics education, but which cannot be realized at all or only with disproportionately high effort using physical teaching aids.

Example “Power of Five”: While it is not easy to simultaneously place quantities of five onto a physical twenty-square grid, this is quite possible virtually (Krauthausen, 1995, 2012). On a virtual twenty-square grid, portions of five and one can be inserted. This allows the child to experience firsthand that it is more efficient (and requires fewer "clicks") to assemble six counters from a five-square and a single counter, rather than inserting six single counters.

Example of a hundred square and place value system: Here, for example, quantities can be inserted into bundles of ten or as units, making place value notation clearly tangible across different levels of representation. Some apps (e.g., www.lernsoftware-mathematik.de/zahlen) can even dynamically visualize the bundling and unbundling process, for example, by automatically grouping ten units into one ten.

While these actions are theoretically possible with physical materials, they would require significant cognitive resources. Grouping ones into tens requires first identifying ten ones, grouping them together, and exchanging them with a ten-rod. By the time this action is completed, a child has likely lost sight of the concept of place value.

5. Adaptive support and differentiated learning paths

In addition to the pure experimental environment, virtual tools can also include support and differentiation offers, such as:

• Substantial tasksThis refers to tasks that encourage independent exploration and investigation of the connections. These tasks (for example, in the form of research questions) can be offered either within the app or externally. A good (virtual) learning tool is only as suitable for understanding-oriented learning as the accompanying tasks!
• Notes and feedback: Immediate feedback on actions taken during the exploration process can be offered – either on request or proactively by the digital learning system. This can be done visually or textually, for example, through visual cues (e.g., a flashing number of tiles, adding visual structuring features to the material) or a stimulating question ("Why don't you try this…", "Think about it…") at the appropriate point in the learning process. AI-supported learning guidance This can be a useful addition to relieve and supplement teachers during the exploration with the virtual learning tool.
• Decomposition of complex representations and operations: Complex operations can be broken down into individual steps, making them easier to understand and manipulate. It is also possible to decompose complex representations into interconnected sub-representations.
• Adjustable difficulty levels: The elements and functions of virtual learning tools can be adjusted manually depending on the learning level, or they can automatically adapt to the child's learning level and progress in an adaptive learning system.
• Documentation of learning paths through the recording and playback of actions. The fleeting representation of experimental actions can be transformed into a non-fleeting collection of documentation units (static – image or text) or action sequences (dynamic – video or description) for later analysis and discussion. This is possible even if the teaching tool itself does not offer these features (such as in the "Rechenfeld" app) thanks to permanently available tools like screen recording, screenshots, notes, or audio recordings. The collection of documentation units enables subsequent learning processes for the teacher, such as structuring the collection, presenting the results, exploring strategies, and providing diagnostic options (Wollring, 2008).

These options allow virtual tools to be personalized to meet learners’ learning needs.

Current developments and technologies for expanding virtual action experiences

Touch technology and embodied learning

Devices with multi-touch functionality allow for more direct haptic interactions than traditional mouse control. Therefore, devices with touch technology are now very popular in educational settings, especially in primary schools. Finger tracing and swiping gestures can be used as biologically primary knowledge, thus reducing cognitive load (Agostinho et al., 2015; Ginns et al., 2020). For example, the scale on a virtual number line can be quickly and dynamically changed with an intuitive swipe or zoom-in/out gesture using two fingers. Such interaction possibilities are not achievable with analog, static media.

Artificial intelligence and adaptive systems

Modern virtual learning tools can enable adaptive learning paths through AI-based systems that automatically adapt to the learning level and needs of individual students (Haryana et al., 2022). Such systems can dynamically regulate cognitive load and create optimal learning conditions. However, research in this area is still in its infancy. Many open questions remain, particularly regarding how generative AI should be meaningfully integrated to support learning rather than replace or hinder it.

Augmented and Virtual Reality

AR and VR technologies open up new possibilities for spatial-physical virtual interaction that integrate movement and learning by enabling three-dimensional manipulation of virtual objects and motion-based learning (Altmeyer et al., 2024). Such technologies offer particularly promising potential for spatial reasoning. Embodied learning approaches, in which physical movements promote mathematical understanding in an integrated way, for example by blending real-world experiences with virtual ones, are also made possible by AR and VR technologies.

Example AR Number Line: In the AR Number Line app, children can perform actions on a virtual number line. This number line can be projected directly onto a reference surface in their environment and potentially extended indefinitely. Through the tangible experience of moving forward and backward along the virtual AR number line, the relationship between the size of sets, their position on the number line, and the length ratios in the real environment (e.g., school hallway, schoolyard) can be established.

Critical reflection and limits

Despite the potential described, a critical look at virtual action possibilities remains necessary, especially with regard to their design and integration into substantial learning environments.

Not all virtual tools are automatically effective.

Research findings on the effectiveness of virtual learning tools are inconsistent. For example, a recent study shows that physical learning tools can lead to better learning outcomes in certain areas (such as fractions) than virtual tools (Al Mutawah et al., 2024). However, such results cannot be generalized to learning tools or the technology itself. Just as there are criteria for selecting and using tangible teaching aids, their use should be guided less by the technology itself and more by subject-specific pedagogical criteria. Design quality of the virtual tool is crucial (Moyer-Packenham & Bolyard, 2016), as is its integration into appropriately designed learning environments and questions.

Danger of cognitive overload

Paradoxically, poorly designed digital learning environments can also lead to increased cognitive load – for example, through unnecessary animations, distracting elements, or overly complex user interfaces (Skulmowski & Xu, 2022).

Importance of the teacher

Virtual learning tools do not replace professional subject-specific pedagogical support from the teacher. On the contrary: Virtual learning tools unfold their full potential when their use is framed by appropriate tasks, preparation and follow-up, and learning environments. Studies show that the way teachers use virtual learning tools is crucial for learning success (Larkin, 2016). mathematics didactic competence the teacher remains central.

Conclusion and outlook

With appropriate design, virtual activities can create opportunities for action and experience in mathematical learning that offer significant didactic potential compared to activities using non-virtual tools (Krauthausen, 2022; Bouck et al., 2020b). They are considered meaningful addition and continuation Virtual actions should not be seen as a replacement for real-world actions. As a basis for interpretation and meaningful application, virtual actions often require primary, analogous experiences with tangible work materials. Key conditions for the successful use of virtual tools and virtual actions are:

1. Subject-didactic foundation: The design of virtual actions on teaching tools should be based on proven concepts and principles of mathematics education.
2. Design quality: Careful design of virtual learning tools based on insights from multimedia learning and subject-specific didactic principles is essential (Urff, 2014)
3. interplay of physical and digital (Duo of artefacts): Virtual actions build on experiences with tangible tools and can also be used effectively together as a „Duo of Artifacts“ (Bonow, 2020).
4. Teacher professionalization: Teachers need skills for the reflective use of digital media in order to utilize the specific possibilities of digital learning materials and actions and not simply use them as a replacement for analog media.

When designing virtual work tools and action offers, the following design principles are guiding from a subject-didactic perspective (cf. Urff, 2012):

• Design virtual tools so that the user is relieved of activities that are not relevant to mathematical concept formation when performing actions. This makes as many cognitive resources as possible available for mathematical exploration.
• Attempts to use virtual action offers and visualizations to depict the mathematical (thinking) operations to be promoted as best as possible.
• Design virtual resources so that the connection between different representations becomes understandable for children.
• Enable actions that are not or only very difficult to achieve with concrete work tools.

Digitalization and the resulting advance in digitalization in education offer opportunities, but should not lead to uncritical technological euphoria. The goal is not to replace traditional tools, but to expand the didactic repertoire. to expand wisely (Krauthausen, 2022). Future research and practice should therefore continue to critically examine under which conditions which combinations of concrete and virtual tools are optimal for which learning objectives and target groups.

---

literature

Agostinho, S., Tindall-Ford, S., Ginns, P., Howard, SJ, Leahy, W., & Paas, F. (2015). Giving learning a helping hand: Finger tracing of temperature graphs on an iPad. Educational Psychology Review, 27(3), 427-443. https://doi.org/10.1007/s10648-015-9315-5

Ahmad, K., Khalid, M., & Khan, S. (2024). The effect of concrete and virtual manipulative blended instruction on mathematical achievement for elementary school students. Canadian Journal of Science, Mathematics and Technology Education, 24(4), 515-532. https://doi.org/10.1007/s42330-024-00336-y

Al Mutawah, M., Thomas, R., & Khine, MS (2024). Impact of using virtual and concrete manipulatives on students' learning of fractions. Cogent Education, 11(1), Article 2379712. https://doi.org/10.1080/2331186X.2024.2379712

Altmeyer, K., Kapp, S., Thees, M., Malone, S., Kuhn, J., & Brünken, R. (2024). The impact of individual differences on learning with augmented reality. In F. Krieglstein et al. (ed.), Cognitive load theory: Emerging trends and innovations(pp. 112-128). MDPI.

Bonow, J. (2020). Analog and digital arithmetic triangles: The potential of combining teaching tools in inclusive settings. In S. Ladel, C. Schreiber, R. Rink, & D. Walter (Eds.), Research on and with digital media: Findings for primary school mathematics instruction (pp. 55–70). WTM – Verlag für wissenschaftliche Texte und Medien.

Bouck, E.C., & Park, J. (2018). A systematic review of the literature on mathematics manipulatives to support students with disabilities. Education and Treatment of Children, 41(1), 65-106.

Bouck, E.C., Park, J., Sprick, J., Shurr, J., Bassette, L., & Whorley, A. (2017). Using the virtual-representational-abstract approach to support students with intellectual disabilities in mathematics. Focus on Autism and Other Developmental Disabilities, 33(4). https://doi.org/10.1177/1088357618755696

Bouck, E.C., Satsangi, R., & Park, J. (2020). The concrete-representational-abstract approach for students with learning disabilities: An evidence-based practice synthesis. Remedy and Special Education, 39(4), 211-228. https://doi.org/10.1177/07419325187926112

Bouck, E., Park, J., & Stenzel, K. (2020). Virtual manipulatives as assistive technology to support students with disabilities with mathematics. Preventing School Failure: Alternative Education for Children and Youth, 64, 1–9. https://doi.org/10.1080/1045988X.2020.1762157

Castro-Alonso, J.C. (2020). Latest trends to optimize computer-based learning: Guidelines from cognitive load theory. Computers in Human Behavior, 112, Article 106458. https://doi.org/10.1016/j.chb.2020.106458

Ginns, P., Hu, F.T., Byrne, E., & Bobis, J. (2015). Learning by tracing worked examples. Applied Cognitive Psychology, 30(2), 160-169. https://doi.org/10.1002/acp.3656

Haryana, MRA, Rambli, DRA, Sulaiman, S., & Sunar, MS (2022). Virtual reality learning media with innovative learning materials to enhance individual learning outcomes based on cognitive load theory. The International Journal of Management Education, 20(3), Article 100657. https://doi.org/10.1016/j.ijme.2022.100657

Krauthausen, G. (1995). The "Power of Five" and Thinking Calculations. In GN Müller & EC Wittmann (eds.), Counting on children (pp. 87-108). Primary School Working Group.

Krauthausen, G. (2012). Digital media in primary school mathematics lessons. Springer Spectrum.

Krauthausen, G. (2022). On the digitalization debate in elementary school mathematics education. In AS Steinweg (ed.), Mathematical education today and tomorrow: challenges and perspectives (pp. 25-40). University of Bamberg Press. https://doi.org/10.20378/irb-55799

Krauthausen, G., & Scherer, P. (2010). Dealing with heterogeneity: Natural differentiation in primary school mathematics teaching. IPN materials. http://www.sinus-an-grundschulen.de/fileadmin/uploads/Material_aus_SGS/Handreichung_Krauthausen-Scherer.pdf

Larkin, K. (2016). Mathematics education and manipulatives: Which, when, how? Australian Primary Mathematics Classroom, 21(1), 12-17.

Moyer, PS, Bolyard, JJ, & Spikell, MA (2002). What are virtual manipulatives? Teaching Children Mathematics, 8(6), 372-377.

Moyer-Packenham, P.S., & Bolyard, J.J. (2016). Revisiting the definition of a virtual manipulative. In PS Moyer-Packenham (Ed.), International perspectives on teaching and learning mathematics with virtual manipulatives (pp. 3-23). Springer. https://doi.org/10.1007/978-3-319-32718-1_1

Paas, F., & van Merriënboer, JJG (2020). Cognitive-load theory: Methods to manage working memory load in the learning of complex tasks. Current Directions in Psychological Science, 29(4), 394-398. https://doi.org/10.1177/0963721420922183

Park, J., Bryant, D.P., & Shin, M. (2022). Effects of interventions using virtual manipulatives for students with learning disabilities: A synthesis of single-case research. Journal of Learning Disabilities, 55(4), 325-337. https://doi.org/10.1177/00222194211006336

Skulmowski, A., & Xu, K.M. (2022). Understanding cognitive load in digital and online learning: A new perspective on extraneous cognitive load. Educational Psychology Review, 34(1), 171-196. https://doi.org/10.1007/s10648-021-09624-7

Sweller, J. (2020). Cognitive load theory and educational technology. Educational Technology Research and Development, 68(1), 1-16. https://doi.org/10.1007/s11423-019-09701-3

Urff, C. (2014). Digital learning media to promote basic mathematical skills.
Theoretical analyses, empirical case studies and practical implementation based on the development of virtual work tools. Man and Book

Urff, C. (2012). Virtual resources in primary school mathematics instruction. In S. Ladel & C. Schreiber (eds.): Learning, Teaching, and Research in Primary School (Volume 1) (pp. 59–82). Franzbecker