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

Introduction

Mathematical tools are central teaching and learning tools that allow students to explore fundamental mathematical concepts through hands-on interaction (Krauthausen, 2012, 2022). In contrast to physical tools, which learners can manipulate with their own hands (e.g., moving, tilting, or turning objects), the "real" action options with virtual tools are limited to digital interactions. On traditional desktop computers and notebooks, interactions primarily occur via mouse clicks or keyboard inputs, while touchscreens in tablets enable direct haptic feedback through multi-touch gestures (Agostinho et al., 2015). AR glasses now also enable input using natural hand gestures (finger placement, rotation, etc.).

The actual actions in virtual tools are performed by the computer and visually rendered – the user merely provides the targeted impulse to execute the action. This becomes clear with the example of a virtual twenty-square field: Turning pieces can be virtually turned over, however, not by actually turning them over physically; rather, the turning operation is performed by the software via a click or touch gesture, and an animation of the turning operation is displayed. Such operations performed by the computer but initiated by the user are referred to as computer-assisted 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 tangible tools

Restrictions and requirements

First, one could argue that the concretely perceivable action options, and thus also the action experiences, are severely limited in virtual actions. If only the tactile-haptic experience options are considered, this is certainly true (Moyer-Packenham & Bolyard, 2016). The child cannot physically feel a virtual turning piece, and the act of turning it over is not directly linked to a motor movement on the piece itself.

This limitation is one of the key reasons why virtual tools are considered Continuation and expansion They should be used as a complement to working with physical tools and not as a substitute (Krauthausen, 2022; Bouck et al., 2020a). This is especially true for primary and preschool learning, because during this developmental phase, primary experiences play a central role in building understanding. A child will probably only be able to correctly interpret the virtualized turning operation on the computer if they have already turned a turning piece themselves or at least observed this action on the physical tool.

Current research shows that combining physical and virtual resources (so-called "blended manipulatives") can be particularly effective and leads to improved learning outcomes (Ahmad et al., 2024; Yakubova et al., 2024). The sequence and interconnection—first physical, then virtual—remains didactically important. But why should virtual exploration opportunities be offered in addition to physical resources?

Didactic potential of computer-assisted actions

If the visual-auditory and interactive experience possibilities of virtual actions are examined more closely, computer-supported actions, as a continuation of real actions, offer new didactic potential through the technological possibilities:

1. Cognitive relief and shifting through automated action execution

By allowing the computer to take over the execution of the action, more cognitive resources remain available for the actual mathematical learning (Sweller, 2020; Paas & van Merriënboer, 2020). The decoupling of action impulse and execution can reduce the child's non-learning-relevant motor and cognitive load (Skulmowski & Xu, 2022). This corresponds to the principle of reducing extraneous cognitive load from the Cognitive Load Theory (Castro-Alonso, 2020).

Example calculation field: While in a physical twenty-field game, tiles must be pushed onto the field by hand, and the child is occupied with this motor movement, this happens automatically on the computer. In the virtual twenty-field game, a click or touch automatically animates the insertion movement of one or more tiles. The child does not have to worry about correctly executing the tile movement – which is irrelevant for mathematical concept formation – but can observe the insertion operation and its effects and focus their attention on the change on the symbolic level (the sum increases), thereby better understanding the connections through experimentation. Well-designed virtual manipulatives can actually reduce cognitive load through such partial automation and at the same time support learning, especially since experimenting with different options in the sense of "What happens to ... if ...?" is made much easier.

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 that are essential for the learner to develop and discover the didactically significant structural properties. All other operations that would otherwise be possible with analogue tools due to their physical properties are limited. For example, the virtual twenty-field ensures 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 disabilities (Park et al., 2022; Yakubova et al., 2024). Especially when a profound understanding is not yet present, incorrect actions can lead to fundamental misunderstandings. For example, if a placement error on the concrete twenty-tile square causes the iconic and symbolic representation to mismatch (e.g., 14 tiles are on the square, but the number 15 is next to it), children often attempt to subsequently rationalize this discrepancy and construct faulty mental models. Virtual tools avoid such errors by only allowing conceptually correct operations and automatically ensuring consistency between 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 representations as a central element for 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.

Through this supported execution and dynamic visualization, the action experience is linked across multiple representational levels. This creates learning opportunities that can be significant for developing a deeper understanding of basic mathematical operations. This is particularly true for children who are not yet able to transfer the information completely or who are making errors. They can validate, modify, and correct their mental model through the actions. However, this networked, synchronous representation must be well-balanced in terms of the scope and design of the visual elements, because the networking across multiple representational levels can also result in additional cognitive load. Less is often more here: a reduction to a few interconnected representations at a time is recommended.

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

Particularly interesting is the visualization of action processes that illustrate core concepts of mathematics education, but cannot be realized with physical work materials at all or only with great effort.

Example “Power of Five”: While it's not easy to place quantities in five-piece portions simultaneously on the physical twenty-square board, this is easily possible in a virtual one (Krauthausen, 1995, 2012). In the virtual twenty-square board, portions of five and one can be inserted. This allows the child to vividly experience that it's more effective (and requires fewer clicks) to combine six reversible tiles consisting of a five-piece and a single tile, rather than inserting six individual tiles.

Example of a hundred square and place value system: Here, quantities can be entered in tens and ones, allowing place value notation to be clearly experienced across all display levels. Some apps can even dynamically visualize the bundling and unbundling process, for example, by automatically combining ten ones into one ten.

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 tasks: These are tasks that encourage independent exploration and research into the connections. These tasks (research questions) can be offered either within the app or outside of it.
• Notes and feedback: During the exploration process, immediate feedback on completed actions can be provided – either upon request or proactively through the digital learning system. This can be done visually or textually, for example, through visual cues (e.g., flashing the number of tiles, adding visual structuring features to the material) or a stimulating question ("Try this...", "Think about it...") at the appropriate time in the learning process.
• Gradual revelation: Complex operations can be broken down into individual steps. It is also possible to break down complex representations into interconnected subrepresentations.
• Adjustable difficulty levels: Various display options can be adjusted depending on the child's learning level or, in an adaptive learning system, can automatically adapt to the child's learning level and learning progress.
• Documentation of learning paths by recording and replaying. The fleeting representation of the experimental actions can be converted into a non-volatile collection of important documentation units (static - image or text) or action sequences (dynamic - video or description). Using permanently available tools such as screen recording, screenshots, and audio recordings, this is also possible if the tool itself does not offer this (as in the "Rechenfeld" app, for example). The collection of documentation units enables subsequent learning processes such as structuring the collection, presenting the results, strategy exploration, and diagnostic options for the teacher (Wollring, 2008).

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

Current developments and technologies

Touch technology and embodied learning

Devices with multi-touch functionality enable more direct haptic interactions than traditional mouse controls. Finger tracing and swipe 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 through an intuitive swipe or zoom-in/out gesture with two fingers. Such actions are not feasible with analog 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-geometric learning content by enabling three-dimensional manipulation of virtual objects (Altmeyer et al., 2024). Such technologies offer particularly promising potential for spatial imagination. Embodied learning approaches, in which physical movements promote mathematical understanding, for example, by blending real-life experiences with virtual ones, are also possible with 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 the environment and potentially extended indefinitely. This allows the relationship between the cardinality of quantities, their location on the number line, and the length ratios in the real environment (e.g., school hallway, schoolyard) to be established.

Critical reflection and limits

Despite the potential described, a critical look at virtual work tools remains necessary, especially with regard to their design and embedding in substantial learning environments.

Not all virtual manipulatives are equally effective

Research findings on the effectiveness of virtual teaching aids are not consistent. For example, a recent study shows that concrete manipulatives can lead to better learning outcomes in certain areas (such as fractions) than virtual ones (Al Mutawah et al., 2024). However, this is not applicable to all teaching aids. Just as there are criteria for the selection and use of physical visual aids, their use should be based less on the technology and more on subject-specific didactic 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). Reducing extraneous cognitive load requires careful instructional design.

Importance of the teacher

Virtual teaching materials do not replace professional subject-specific instruction provided by the teacher. Quite the opposite: Virtual teaching materials only unfold their full potential when their use is supported by appropriate tasks, preparation and follow-up, and discussion situations. Studies show that the way teachers use virtual manipulatives is crucial for learning success (Larkin, 2016). mathematics didactic competence the teacher remains central.

Conclusion and outlook

With appropriate design, virtual activities can be used to create opportunities for action and experience in mathematical learning that can offer real didactic added value compared to activities using non-virtual tools (Krauthausen, 2022; Bouck et al., 2020b). They are meaningful addition and continuation of real actions – not as a substitute. They often require analog primary experiences with physical work materials as a basis for interpretation and meaningful application. Key conditions for the profitable use of virtual tools and virtual actions are:

1. Subject-didactic foundation: Orientation towards proven mathematics didactic concepts and principles
2. Design quality: Careful design of virtual work materials based on findings from learning with multimedia and subject-didactic principles (Urff, 2014)
3. Integration of analog and digital: Virtual actions build on experiences with physical tools.
4. Teacher professionalization: Teachers need skills for the reflective use of digital media
5. Differentiation: Using technological possibilities for adaptive learning paths

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 should continue to investigate under which conditions which combinations of physical and virtual learning tools are optimal for which learning objectives and target groups.

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