Abstract
It is an increasingly common phenomenon that elementary school students are using mobile applications (apps) in their mathematics classrooms. Classroom teachers, who are using apps, require a tool, or a set of tools, to help them determine whether or not apps are appropriate and how enhanced educational outcomes can be achieved via their use. In this article we investigate whether Artifact Centric Activity Theory (ACAT) can be used to create a useful tool for evaluating apps, present a review guide based on the theory and test it using a randomly selected geometry app [Pattern Shapes] built upon different (if any at all) design principles. In doing so we broaden the scope of ACAT by investigating a geometry app that has additional requirements in terms of accuracy of external representations, and depictions of mathematical properties (e.g. reflections and rotations), than is the case for place value concepts in [Place Value Chart] which was created using ACAT principles and has been the primary app evaluated using ACAT. We further expand the use of ACAT via an independent assessment of a second app [Click the Cube] by a novice, using the ACAT review guide. Based on our latest research, we argue that ACAT is highly useful for evaluating any mathematics app and this is a critical contribution if the evaluation of apps is to move beyond academic circles and start to impact student learning and teacher pedagogy in mathematics.
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Notes
 1.
It is also available at: http://dlgs.unipotsdam.de/oer
 2.
The Math Learning Center (https://www.mathlearningcenter.org/resources/apps/patternshapes)
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Appendices
APPENDIX 1: ACAT Review Guide: An activity theory approach to reviewing math apps
Heiko Etzold & Ulrich Kortenkamp, University of Potsdam
Silke Ladel, University of Education Schwäbisch Gmünd
Kevin Larkin, Griffith University
Introduction
Mathematics education as a Design Science (Wittmann 1995) has the responsibility to help judge and review material for teaching and learning mathematics, including digital ones like computer software and apps for mobile devices. This review guide has been developed to fulfill that responsibility. The usual approach to evaluating the suitability of apps for teaching and learning are catalogues and categorisations, resulting in rankings (cf. Highfield and Goodwin 2013). However, these are not particularly useful to give hints about the suitability of the app for a certain subject matter or particular classroom situation. We propose another approach, a theoryguided approach to evaluating apps, without ranking them or even labeling them as ‘good’ or ‘bad’, but as a guideline to find ways to judge the deployment of specific apps in the classroom.
The theoretical basis for our review guide is activity theory, more specifically the ACAT model (ArtifactCentric Activity Theory, see Ladel and Kortenkamp 2014b). This model describes network of a subject (usually a student), an object (the mathematical subject matter), the mediating artifact (in our case, an app used by the student to work with the mathematical content), as well as rules (describing how the app should behave based on the mathematical object) and the group (the whole classroom situation).
It is not necessary to know the details of the psychological and pedagogical theory in order to understand and use the following guide. A helpful introduction for those who want to know more is the text of Kaptelinin (2014). For every step of the guide, we will give background information concerning the theoretical foundation.
The Review Guide
This review guide is organised into five steps that can be associated to five foci in the ACAT model. It is mandatory to follow these steps in the given order.
Starting from the mathematical content, we will analyse how students work with the app. From that, we will infer whether the app is able to help in supporting the acquisition of the desired content. Finally, concrete classroom situations can be discussed that are suitable for using the app. Based on this approach it can happen that the same app is perfectly suited for specific teaching situations, but not for others, even if the same topic is being handled. The stepbystep approach will guide the review to the essential and fundamental questions immediately. This can cause an early conclusion of the review after each step, if it is foreseeable that the app is not suitable for the planned instruction.
Thus, the review guide structures the decision process and makes it transparent to teachers. For every step, we list the possible data sources that can be used.
Step 1: What is the mathematical object of the app?
Identify the mathematical object, i.e. which concept, content or mathematics process is targeted by the app.
It is important to note that each app can address one or several mathematical objects. In the case of several objects, separate reviews for each object are necessary as each object will emphasise different learning facets and will therefore vary in efficacy across this range of facets.
Possible Sources

the app’s title and its official description at iTunes or Google Play;

additional material provided (e.g. downloadable worksheets);

external references (e.g. recommendations by peers who have used the app);

trials of the app.
Background
A central principle of activity theory is object orientation. As actions only exist (or at least can only be recognised) in relation to a specific object, activities are therefore directed towards the attainment of an object. Consequently, actions of students within an app can only be understood if the (mathematical) object of their actions is known. In this instance, the mathematical object is understanding how shapes can be manipulated to create new shapes.
Step 2: How do students interact with the mathematical object, mediated by the app?
Discuss what kind of interaction with the mathematical object the app offers to the students. For this, it is necessary to look at the separate interactions between subject and artifact, as well as between artifact and object. In examining these interactions we can ask: What actions does the app support? How does the app represent the mathematical object? How does the object influence the ‘behaviour’ of the app? What can students experience through the above?
Possible Sources

own systematic testing of the app,
Background
When designing an app, formulating the rules for user interaction would occur at this stage in the ACAT design process. As we are analysing an existing app, we immediately examine user interaction.
A core component of interest Activity Theory, and thus ACAT, is the process of Internalization and Externalization. External actions of the subject, such as pinchtozoom gestures to scale a city map represent internal actions – in this case, dilations that in turn are representations of student understanding. In a similar way, external actions can create internal representations. In order to understand this user interaction more fully, and to refer more concisely to the mathematical object, it is useful to subdivide this process between subject and object at their respective interactions with the artifact (Ladel and Kortenkamp 2014b).
The guiding questions thus follow back and forth between subject, artifact, object:

S → A: Which actions does the app support?

A → O: How does the app represent the mathematical object?

O → A: How does the object influence the ‘behaviour’ of the app?

A → S: What can students experience through the above?
Step 3: How does the interaction develop?
Structure the possible interactions by categorising them into activities, actions and operations (Leontiev 1972/1981):

activities are superordinate interactions directed by the subject’s motives – e.g. reading a city map to determine direction of travel;

actions are targeted, individual interactions – e.g. changing a map’s scale to get a more detail view of a certain section of a map;

operations are internalised interactions that do not need further thought and can be governed by instrumental constraints – e.g. the actual execution of pinchtozoom gesture to scale or of the drag gesture to move the map.
Elaborate upon how this categorisation changes while using the app, as actions can become operations during the learning process and allow for the creation of new actions in turn.
Possible Sources

discussion of hypothetical scenarios;

empirical tests.
Background
At this point in the review process, a specific view on the hierarchy of activities, actions and operations as another principle of activity theory is appropriate. At the same time, conclusions about possible developments of students’ learning can be drawn. A successful learning process is characterised by actions becoming operations in order to enable more complex actions.
Step 4: Is the app suitable for teaching and learning the mathematical object?
Compare the use of the app for the specific mathematical object, as uncovered in the guide, with knowledge from mathematics specific pedagogy, the discipline of mathematics, and psychology regarding the teaching and learning of the mathematics object in question. In other words, do the interactions identified and analyzed in steps 2 and 3 support the desired ideas, experiences, conceptions and competences, as required by quality mathematics education? In addition, is the technological design suitable for learning according to theories of high quality Human–Computer Interaction (HCI) design?
Possible sources

syntheses of the discussion above;

scientific background literature and references.
Background
The design of an app is guided by rules in the ACAT model, which in turn are guided by knowledge from mathematics education, HCI design, etc. This ensures that the app is indeed supporting student learning and that the targeted mathematical content can be taught or learned via its use.
Step 5: How can the app be used in classroom instruction?
Illustrate how the app might be used in the classroom. You can use the following questions as a guide:

can the app be used for individual, partner or small group work or is it limited to only one of these types of social interaction?

how can classroom discussions / collaborations / and interactions be fostered through activity with the app, either directly or indirectly?

what are possible provocations or tasks the teacher can provide students?

which kinds of differentiation, and which levels of difficulty, are possible?

is it an app for instructive (drill and practice) activities or is it discovery based and designed to introduce students to new content or to construct new ideas?

which requirements and competences are necessary to use the app?
Possible sources

additional teacher’s material;

trials with students;

results from experimental studies by other researchers;

imagination.
Background
Within both Activity Theory and ACAT, learning is never understandable as a pure individual activity of a student. It must always be seen in a social and corporate context where learning is common work. As noted by Giest and Lompscher (2004), in the classroom there is always a pädagogisches Gesamtsubjekt, such that the activities always occur in the contexts of interaction, communication or cooperation, including relationships between learners, teachers, and other participants.
APPENDIX 2 Sarah STEIN, Potsdam
ACAT Review for App Click the Cube
General Information
App: Click the Cube, version 1.1 (published on 2017/07/25)
Description in the App Store: The Click the Cube App provides students with opportunities to develop spatial ability using cube nets. Three different learning environments have been developed, each focusing on different aspect of spatial ability in the context of cube nets. In the first learning environment, a student has to decide if the illustrated net is a cube net or not by mentally folding the net. In the second learning environment, a net has to be supplemented with a sixth square, so that a net results in a cube net. In the third learning environment, the student has to dye the cube net according to a given coloured cube by rotating the cube. The environment includes several motivating elements for the students, such as collecting stars in order to receive a medal. Different differentiation elements are also provided, such as a slider, to help the student slowly build mental images of a cube.
Additional material: There is no additional material in the App Store description; however, from the description on the App Store a user can reach the website of the project Digitales Lernen Grundschule (Digital learning in primary school) of Potsdam University (dlgs.uni potsdam.de). There you can find this and other apps in Würfelwelten (World of Cubes). All these apps offer exercises for manipulating or visualising cubes.
What is the mathematical object of the app?
The mathematical objects of the app are cubes and their nets. The app focuses on the relation between 2D and 3D representations of cubes and should enable students to develop better spatial abilities.
How do students interact with the mathematical object, mediated by the app?
The app is organised into three parts. In the first part, students are asked to decide whether various nets are nets of a cube or not. In the second part, students are required to complete unfinished nets of cubes. In the third part, they colour a net to match a given cube. In all three parts the screen is divided into two areas. The left area contains the task and an overview of the medals and stars collected so far for correct answers. The right area shows the current cube or net for the task and it is possible to manipulate the object using various gestures on the screen.
In the first part, the user is presented with a slightly folded net of cube. The user can rotate and scale the net and use a slider to fold or unfold it gradually. The first operation, rotation, is done through pan gestures. It is not a requirement to start the gesture on the net as it is possible to start anywhere in the right area, allowing the student to see the full net in rotation. Through this gesture the user can see the object from all perspectives. With a pinchtozoom gesture, the user can change the scale of the net, to enlarge or shrink it on the screen. This enables the student to find the optimal position of the object for inspection. He or she can zoom into critical areas to obtain further information. The third operation is to use the slider, moving its indicator to the left or right with the finger. Moving the finger to the right will fold the net, that is, diminish the spatial angle between the faces of the net until it is at 90 degrees, moving the indicator to the left will open up the net until it is flat. Using the slider enables the user to extend their spatial impression of the net and encourages students to continue the folding or unfolding of the net mentally.
The net of the cube is coloured in white on one side and grey on the other. When folded, the grey side is on the inside and the white side is on the outside. Thus, a fully folded cube will appear white. The more the users move the slider to the right (thus folding the cube), the fewer number of points (stars) they receive for a correct answer. Not touching the slider will maximise the points if the correct answer is given when pressing the ‘Yes’ or ‘No’ button. Rotating or scaling the object does not influence the number of points. The reduction in points for using the slider is used consistently in all parts of the app. In order to receive as many points as possible, users are motivated not to use the visualized folding, but to find the answer with mental operations.
To start a new task, and to check or choose the answer, a user has to touch the corresponding blue words on the left side of the screen. Subsequently, an animation of the folding is shown on the right side. If two faces coincide this fact is highlighted in red. The user can still use the slider after this animation to further inspect the net and, if applicable, understand their mistake.
If enough stars have been awarded for an answer, the user will earn a medal. A total of four medals is possible.
The second part of the app requires the student to complete a net of cube by adding missing squares. The new squares are glued automatically to the existing net if they are brought into the vicinity of an edge. At some edges it is not possible to connect a square. To check the current net for correctness the user presses the “Check” button. This will determine whether a correct net has been formed and at the same time the folding process will be demonstrated with an animation in the right area of the screen. Adding squares is only possible when the net is fully flat (the starting configuration in this and the next part of the app). Adding a missing square to the net is achieved by dragging the free square from its original position to the intended edge. If it is impossible to add the square there, the user is required to inspect the rules for nets of cubes more closely.
The third part of the app is concerned with colouring a given net of a cube. In the task area on the left a cube is shown, which can be rotated and scaled for inspection of all of its sides, using the same gestures as in part 1. Seven different colored circles, that can be used to colour the net, are shown at the top of the screen. In order to colour the net, a user has to drag the coulored circle from the top of the screen onto the square that should be colored. Again, it is possible to use rotation and scaling as well as the slider to control the amount of folding. The task for the user is to color the net such that its corresponding cube is colored in the same way as the “task cube” on the left. After the solution has been checked it is possible to rotate both cubes to find the difference, if there is one.
In general, the app provides the means to create a complete (mental) image of the cube or its net. The various interactions that are possible allow for an informed decision by the user about how much help is needed to solve each task. It is possible to use the app to interact with the cube or cube net for further inspection, but it is not necessary to solve the tasks (with the exception of rotating the task cube in part 3).
How does the interaction develop?
The user is operating mainly in the mind to act on the mathematical objects and has to fold the net mentally in order to determine whether it is the net of a cube, where to add the missing face, or how to rotate the cube mentally to assign the correct color. So the app is motivating ‘mental geometry’ in phases as described by Weigand (2013) i.e. (phase 1) presenting the task; (phase 2) imagination and mental operations, and (phase 3) presenting the solution. In phases 1 and 3, staged help may be available to simplify the activities. The second phase, on the other hand, should happen mentally without any aids. Preliminary stages of mental geometry or preparatory activities can include further aids for the second phase, for example gestures, geometric models for illustration or alternate solutions like figures.
In the app we consider, a figure of the net is shown in the task phase (1), which could be used for mental geometry immediately. The subject (i.e. the user) can rotate or scale the initial situation in order to be better prepared for phase 2. It might be the case that this already interferes with the second phase which should be mastered without a tool. In circumstances where the subject is not able to work on the task after it has been presented, the app offers interactive representations and acts as a tool for a preliminary stage of solving mental geometry tasks by visualising them. The rotation and scaling operations allow the subject to create visualisations from various perspectives that they can register mentally and might enable them to solve the task without any help the next time. Also, when touching the button for registering the solution, the animation of the folding is shown that can help the subject to check the correctness of their mental operations. As the app does not proceed to the next exercise automatically, but rather requires an active selection of the next task, it is still possible to repeat and confirm the actions and operations that lead to the solution. The app stimulates mental operations in various ways, so I consider it as a tool that helps students to acquire the competency of doing mental geometry.
Possible Improvements of the App:
When building a net, it is possible to place a square on, or rather in, the net. Sometimes this leads to flickering of overlapping faces. For a better understanding it would be helpful to prevent the possibility of overlapping faces.
Sometimes it is not possible to add a square at a certain edge, for example if this leads to a 2 by 2 square. While this is a reasonable restriction it can be irritating to students. A possible improvement could be an animation where a misplaced square is moving away from a forbidden edge after placement, in order to highlight that there is a problem with this placement. This could encourage the users to reconsider the conditions of cube nets in that situation.
In order to simplify the user interaction in the third part of the app the colouring of squares could be realized through a tap–tap gesture (first tapping the colour, then the square) instead of a drag gesture (dragging the color to the square).
It does not matter in the app whether a square is colored from the inside or the outside. However, it is difficult to see whether a coloured square will be on the inside or the outside in the folded cube. To avoid confusion the contrast between the inside and the outside should be increased. These aspects have been observed in tests with several people who had difficulty colouring the net.
Is the app suitable for teaching and learning the mathematical object?
The app seems to be beneficial in developing the ability to do mental geometry and as such also to foster spatial abilities. In particular the visualisation of the connection between a net and a 3D object is helpful for the work with cube nets. The user can apply and check their knowledge about cube nets and extend their abilities in mental folding and in the mental imagination of this process. As a consequence, the app is suitable for practice phases. An introduction to the topic of (cube) nets should precede the use of the app. It is important to develop these abilities in class 3/4 (age 8–9), as it is a foundation for later teaching. For example, calculating surface areas of composite bodies (e.g. cuboids with an attached cylinder) is a topic where students show difficulties. It is a common mistake to just add the areas of the two nets without removing faces where the two bodies touch. As one cause for this can be a lack of spatial ability, the app is a helpful tool to prepare students to avoid such mistakes.
Using the app also opens new ways to work with cube nets and should be seen as a supplement to cutout nets and crafting cubes from these. The app cannot replace the haptic experience, but offers training that could not be done with paper or cardboard material or images alone. (See Bruner’s 1988 EIS approach.)
Furthermore, the app allows for differentiated learning. Again, it is possible to refer to the three phases in mental geometry. In the first phase, the task is presented together with a net. For higher achieving students (in this particular discipline) this can be solved solely through mental operations without further aids. Those students who need more assistance can use the onscreen operations for rotating and scaling the nets for a change of perspective. If this is not enough, they can also use the slider to fold the net slightly or completely, according to their needs. In this way the app can help to improve the mental geometry abilities of all users, starting at various levels of ability.
In conclusion the app seems to be well suited for the subject of cube nets and to help students to develop their spatial abilities while they work on objects in a way differentiated by their current abilities.
How can the app be used in classroom instruction?
In school education, there are several options to use the app. As the exercise tasks are delivered by the app, it can be used for individual work of students. For the teacher it is not necessary to provide examples so they can take care of certain students, i.e. lowachieving ones. These exercise sequences can be repeated several times during the school year. It is also possible to use this app for learning stations with the topic of cubes. The students would be able to talk about and compare their strategies. Possibly it is also usable for saving time and materials after the introduction of creating boxes etc.
There are, however, some difficulties when using the app.
For example, with a class size of 30 pupils, it will be difficult for the teacher to address the problems of each individual that result from using the app. On the basis of the medals it is not possible to say anything about the students’ skills. This is a critique of collecting stars and medals as points. If the user takes help when interacting with the app, they only get a part of a star. Because of filling the stars, it is not discernible whether the student solved 20 tasks with help or only 10 tasks without help as both options result in a medal. A better way would be a diagnosis tool where a teacher can see the student’s problems in order to respond adequately.
Additional notes:
Because there are only 11 different cube nets, it could be possible that users sometimes know all of them and they only tap the right answer without thinking about it. That should be prevented by a semifolded net and a random alignment at beginning, but it is not to be ruled out (see Huhmann 2013). One way to expand the app is to supplement it by cuboid nets. You can give similar tasks to be sure that students are not getting answers based on memorised learning. The mental folding of cuboid nets also offers new aspects, i.e. opposite edges with nonequal length. There are also 54 cuboid nets so the option to memorise them has been reduced.
In the second part the free square is difficult to move. This usability isn’t optimal and should be revised.
Another expansion could be an option for colorblinded people. For this purpose, all colours can be assigned by a unique structure
The rewarding of correct answers using medals isn’t enjoyable when using the app alone and once you have got all four medals there is no more progress. To solve this, there could be several stars (i.e. several colours), but that also depends on the target (diagnosis vs. collecting points).
One final point: Is it meaningful to reduce the score even when rotating the cube net? According to mental geometry (phase 2) it should not be allowed to rotate, but it contributes to understand the situation. One extension of the app (maybe for higher grades) could be to show the net and require the user to do any further operation mentally. With a correct answer you earn a yellow star and if you used the rotation option you only get a full blue star for a right answer.
Note
This is a translation of a German review by Sarah Stein, University of Potsdam. Translation by Heiko Etzold, Ulrich Kortenkamp and Kevin Larkin.
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Larkin, K., Kortenkamp, U., Ladel, S. et al. Using the ACAT Framework to Evaluate the Design of Two Geometry Apps: an Exploratory Study. Digit Exp Math Educ 5, 59–92 (2019). https://doi.org/10.1007/s4075101800454
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Keywords
 Mathematics education
 Mathematics apps
 Activity theory
 Artifactcentric activity theory
 Geometry
 App design
 App reviews