How do beginning preservice kindergarten teachers’ evaluations of mathematical apps provide insights into their understandings about bildung (danning) for young children?

Data Collection

The data were collected from 13 groups of KPTs (two to five people in each group) as part of a three-hour workshop on mathematics and play in a first-year course, “Language, Text and Mathematics”. This was part of the regular workshop provided to KPTs on playing as a mathematical activity (Bishop, 1988). The data can be considered as coming from a naturalistic setting and not part of an intervention. As such, the data provided baseline data about KPTs’ valuing of mathematics and how it could be learnt from apps and was appropriate for responding to the research question.

The KPTs engaged with apps, on tablets, for about one hour. Five digital games had been uploaded by the mathematics teacher educator, who was a colleague. The teacher educator chose apps of various qualities, in order for the KPTs to discuss what would be a high-quality app that could provide kindergarten children with learning opportunities through play. The teacher educator asked the KPTs to evaluate the apps according to whether their use in a kindergarten would be in line with the aims of the Framework Plan, with its emphasis on play (Ministry of Education and Research, 2017). They are first year students in kindergarten teacher education. They were familiar with the concept of danning from the pedagogy component of the “Language, Text and Mathematics” and as a part of danning and didactic practice course.

In this article, I focus on the KPTs’ evaluations of two apps: Bimi Boo and Tella. These apps were the ones that most groups evaluated (8 groups tested Bimi Boo, 10 groups tested Tella). Both apps presented a range of mathematical ideas. Bimi Boo is a commercially-designed app for children aged 2–6 for learning numbers from 1 to 20 (http://bimiboo.net/app/numbers/). Tella was designed by a set of government agencies, with the lead role being taken by staff at our institution (Western Norway University of Applied Science). The Tella website (http://tella123.org/#/om) describes the app as developing a sense of quantities and then numbers, before introducing children to simple addition and subtraction. All the tasks are done in a specific order. Originally aimed at children aged 5 to 8 years with learning needs in mathematics, it has also been promoted as being suitable for younger children.

With the consent of the KPTs, 13 groups were recorded using a screen recorder app, which simultaneously recorded the video and audio discussions via a tablet. The KPTs determined how long they recorded their interactions. The audio recordings were then transcribed. Eight groups discussed Bimi Boo from 2 minutes 4 seconds to 8 minutes 40 seconds. Ten groups discussed Tella from 2 minutes 42 seconds to 11 minutes 15 seconds.

Data Analysis

The transcripts were read carefully and then the comments made by the KPTs were class­ified as being related to one or more nodes of the ACAT model (Ladel & Kortenkamp, 2013). The original classification according to ACAT was done in collaboration with a colleague and presented as a conference paper (Vee & Meaney, 2022). Table 1 provides an overview of how I operationalised ACAT to the data by identifying key aspects connected to each of the nodes, in comments made by KPTs. The second column lists the key aspects of each node. These key aspects were based on Ladel and Kortenkamp’s (2013) descriptions of the nodes and checked against how the KPTs discussed the different apps. For example, when the KPTs talked about children, these were considered to be comments connected to the subject. When they talked about the mathematical focus of the app, this was classified as being related to the object. When they talked about how the apps engaged the children with mathematics, these comments were classified as being about the rules node. When the KPTs talked about how children could play with apps by themselves or with others, including adults, I classified these comments as being related to the group node. The third and fourth columns provide examples of comments made by the KPTs from the transcripts about to the two apps.

Once the data was classified according to the different nodes of the ACAT framework, then how the comments could be connected to KPTs’ views about children’s danning was identified. This was done in a similar way to that of the ACAT nodes. The descriptions of danning from earlier research in the earlier sections were used to identify key aspects. The descriptions of those key aspects were then checked against the comments made by KPTs. Table 2 provides an overview of how I operationalised aspects of danning in the KPTs’ discussions. The first column provides lists the key aspects of danning. The second column describes the key aspects and the last column provides examples from KPTs’ discussions about the key aspects of danning.

Object

The object is the mathematical content. When the KPTs discussed the mathematics in the apps, they focused on number comprehension, such as counting and quantitative reasoning. Although they did not discuss it explicitly, the KPTs could have considered that number comprehension as being important because of the goal in the Framework Plan about “numbers, quantities and counting” (Ministry of Education and Research, 2017, p. 53). The emphasis on number learning in the apps could reinforce the KPTs’ views about the importance of number comprehension for young children’s danning.

What the KPTs could identify depended, not only on what the apps made available, but also on what the KPTs considered to be mathematics. Despite mathematics in the Framework Plan for Kindergartens (Ministry of Education and Research, 2017) being based on Bishop’s (1988) six universal mathematical activities, most KPTs restricted their discussions about mathematics to understandings about numbers in their evaluations. Previous research has also shown that kindergarten teachers tended to focus on numbers (Johansson, 2015) and so it could be a cultural expectation that young children should be taught to count in early childhood institutions. Although the KPTs were asked to reflect on the relationship between the apps and play, the KPTs did not mention problem-solving or playing, even though Bishop’s (1988) mathematical activity of playing has been linked to problem-solving (Fosse et al., 2020). Similarly, the KPTs could have made connections to locating in Bimi Boo, for example when they placed things inside the basket or items of clothing on the characters in the app. However, they did not mention these aspects as being about mathematics.

In their discussions, the KPTs considered that the Tella app provided young children with experiences to do with order, sorting, size, quantities, and counting, such as finding the biggest and the smallest items and ordering visual representations of items, according to size (see for example, Figure 2). One KPT in Group 17 stated, “It is about size and numbers. Measurement, volume and comparison.” Group 15 stated, “It is about largest and smallest, sorting, number and number words.”

With Bimi Boo, most of the groups identified the mathematical object as being about cardinal numbers, number representations and numeral recognition, including how to write numerals. Group 10 stated, “It is about number comprehension and writing numbers.” Similarly, Group 3 stated, “It is about numbers and quantity, cardinal numbers and number symbols.”

Such statements indicate that the KPTs focused on number comprehension, choosing to evaluate apps with this focus. This can be related to children’s danning, if the KPTs did so because they valued this as knowledge children needed to be a competent member of society. As Meaney and Christiansen (2020) described, potential interactions between children and these apps are likely to shape children’s interaction with mathematical ideas. Yet, most groups did not discuss the development of number comprehension beyond completing small tasks (discussed in the section on Rules) and focusing on individual ideas. This limited focus on number comprehension could restrict children’s possibilities for danning, by not highlighting other mathematical topics such as problem-solving or developing number comprehension beyond limited understandings. It may be that teacher educators need to challenge KPTs to recognise other mathematical activities (Bishop, 1988) that might be present when children engage with these apps and to show how number understandings are more than being able to count or write numerals.

Rules

Rules are about the design of apps and how they are expected to function. Moyer-Packenham et al.’s (2019) research indicates that the design features of an app influence the learning opportunities for young children. When the KPTs discussed how and in what way the app should present mathematics to children, they provided insights into what they considered would be appropriate for children’s danning in regard to how they should learn. Sometimes, the KPTs discussed how the rules of a game could be improved to provide better learning opportunities. For example, in discussing both apps, some groups identified that improved feedback could help children to complete the tasks correctly. This is linked to the subject node and the need for children to be successful when doing mathematics. The KPTs considered that the children should receive explicit information about what they had got wrong and how to fix it. Repetition and explanations were considered to be positive design features in both apps. These discussions provide insights into how the KPTs thought children should engage with mathematical ideas, which was through direct instruction and repetition of tasks. Direct instruction and repetition do not provide children with opportunities to make decisions for themselves and to learn through play as required by the Framework Plan (Ministry of Education and Research, 2017).

In the Bimi Boo app, a member of Group 3 said, “There is no feedback if you make a mistake, this should be explained.” In some groups, there were suggestions that Bimi Boo should be more flexible in how it responded to what a child might do. For example, in one task, the KPTs had to place a specific number of ribbons on a cat (see Figure 3). The KPTs found it frustrating when the ribbons had to be placed on the head and not on any other part of the cat. A member of Group 7 summarised the group’s feelings by stating, “It could be more flexible.”

In evaluating Tella, some groups also suggested that the rules for how to play the game could be more flexible. For example, when discussing how the children should “place things in the right order”, a member of Group 17 stated, “it could be from largest to smallest” (see Figure 4). For them, this inflexibility was exacerbated by the fact that there was no explicit information about what was incorrect. As one member of Group 17 said, “We do not get feedback on why we made a mistake.”

The structuring of the tasks in Tella was also considered problematic, with the first-level tasks having to be solved before more advanced tasks could be attempted. As one KPT said in Group 10, “It’s a pity we cannot advance.” Related to this, some groups also described the tasks in the Tella app as being too similar and boring. They suggested that the children should have a range of choices about sizes and colours at the same level and should not have to wait till they reached the next level. They also felt that creating tasks involving only one small concept was not helpful for children’s learning. Group 11 suggested that it would be better if related concepts were not separated, “It should be that you press the largest, press the smallest, press the pad colour, use slightly different terms, which is largest, which is smallest, in between.” The KPTs noted that the design of the app focused learning on completing many small tasks, which was similar to what Christiansen and Meaney (2020) had found in their research. The KPTs’ evaluations indicated that they mostly valued young children making decisions and taking control in relation to the rate of their learning and the ordering of the tasks in the app. This is related to danning, in that it is about children becoming aware of their own rights and choices.

The KPTs generally considered the Bimi Boo app to be well designed, particularly due to its variation in tasks and the explanations provided to the children. Group 11 stated, “One of the best apps—very different every time.” Group 9 highlighted the feedback, “Very nice that there is an explanation in the game.” Group 10 described it as “very good and educational” because it did not just present facts that they had to be learnt, “It’s not cramming—kids do it and learn it.”

On the other hand, Tella was considered to present more abstract mathematical ideas, with the app instructing the children. For example, Group 5 stated, “It is very focused on mathematics,” where mathematics, as discussed with the object node, was about comparisons and numbers. The KPTs considered that the Tella app provided limited possibilities for learning because the children had to work with mathematical ideas in a specific order. For example, Group 15 stated, “It’s getting harder and harder.” Similarly, Group 9 discussed how “It gets harder when more things appear.” Some KPTs thought that the development of ideas went at an appropriate speed, “It works gradually with number comprehension” (Group 15). This suggested that the progression of tasks, as part of the design of the app, was seen by the KPTs to be connected with what children would be able to do on their own.

In their responses, the KPTs showed that they recognised how the rules of the different digital games could affect the children’s opportunities for engaging with mathematical ideas. These discussions provided insights into how the KPTs thought that mathematics should be learned, through apps. Focussing on the structuring of the tasks indicates that the KPTs considered that children’s possibilities for danning had to provide them with feedback on how to be successful so they could to be active in their learning. Providing small, manageable tasks was important to ensure children were successful, but if those tasks were too repetitive or easy, they could become bored. It may be that teacher educators need to encourage KPTs to understand how children’s engagement with open, problem solving apps also give them opportunities to be successful but also engage in play as required by the Framework Plan.

Subject

The KPTs’ discussions indicated that they saw the subject as being the child/ren, although at times they tended to merge their own needs into their understandings about children. For example, the KPTs frequently described both apps as being interesting for children. This seemed to be based on their understanding of what children could do or be interested in and how important it was for children to be successful in the tasks, in these two apps. This suggested that the KPTs considered that children needed to be motivated and active in their own learning as part of their development.

The KPTs described the Bimi Boo app as being most appropriate for younger children, but potentially boring for older children, as it was an easy game to play, with tasks being repeated at every level, with few new challenges. Group 19 described Bimi Boo as an app “For 2-year-old children, it’s fine but for 3-year-olds it’s boring,” while one KPT in Group 3 stated that it was “boring after a while because there is nothing new, maybe that’s why it’s easier for the little ones.”

The KPTs also identified that the children may not be able to trace the number symbols because they did not have the necessary fine motor skills. For example, a member of Group 3 stated, it is “intuitively difficult to draw the number inside the lines.” In connection to danning, the KPTs seemed to consider that it was important for children to be successful in completing the tasks so they would be motivated to learn and engage with mathematics. For example, they talked about why they did not get any feedback when they made mistakes in both apps. In Group 3, one KPT said, “We must get feedback if we make a mistake.” Similarly, in Tella, when the KPTs tried to place the items in the right order, “We do not get any explanation as to why we made a mistake” (Group 17). This suggests that the KPTs wanted to complete the tasks successfully and expected it also to be important for children to be successful, thereby motivating them to do more and to learn more.

The Tella app was described as providing children with opportunities for exploring through movement, colours and the different sizes of items. For example, Group 7 stated, “Things move and it does not get boring,” while Group 11 said that “Movement and different sizes make it more exploratory.” This suggests that the KPTs considered exploration and curiosity as helping children to learn through their own actions, which is in line with their descriptions of danning (Grindheim, 2020).

For KPTs, it was important that children should be successful in their interactions with apps. Being successful would contribute to their motivation for learning, exploration and curiosity. This could increase opportunities for children to learn through play by making their own decisions, also aspects that are considered to contribute to children’s possibilities for danning connected to the Framework Plan (Ødegaard, 2019). Teacher educators may need to challenge KPTs in their pedagogical choices of apps that influence children’s opportunities for danning.

Groups

The group node in the ACAT model relates to how an app might be used in the classroom, or in this case in a kindergarten, and focuses on the expected kinds of interactions children could have from engaging with apps either individually or in groups. Co-operation as part of this node can be related to danning as it provides a different kind of cultural formation than when children are expected to learn on their own. Comments that were categorised within the group node ranged from whether the KPTs considered the control of inter­actions with the app lay with the child or whether there was a potential role for adults in controlling what happened in interactions with the app. This contributes to children’s danning because it is about how their social environment and to human relationships are connected to their learning.

The KPTs considered that with the Tella app, adults would need to provide explanations, as the app was too difficult for younger children to play on their own. This point was summarised in Group 9, “The adults can explain to the younger children about what is smallest and what is largest.” On the other hand, the KPTs considered that Bimi Boo was simple enough for children to play individually, “children can control the game themselves” (Group 3).

Underpinning these discussions, there seemed to be a view that the KPTs expected the children to use the app by themselves. If an adult was involved, it was to explain to the child what the app wanted them to do. In regard to danning, the KPTs seemed to expect children to become independent, responsible, and active on their own initiative, through playing digital apps. Teacher educators could broaden discussions about what the role of the adult could be when playing digital apps, as in the Framework Plan it is stated that staff shall be actively involved with the children when using digital tools (Ministry of Education and Research, 2017, p. 44).

Artifact

The artifact node focuses on the apps and how they acted as mediators between the object, the mathematical content and skills, and the subject, the children, with considerations of the rules and the group. For example, when the KPT identified how colour and movement in both apps make them interesting for children (see discussion about the Subject), they highlighted how the design of the app (the rules) contributed to children’s engagement in mathematical ideas. Thus, the KPTs implied that they expected that as part of their possibilities for danning, children would learn that materials needed to be interesting in order for them to engage with mathematical ideas.

The KPTs indicated that number comprehension tasks in the apps needed to be small and manageable so that children would be successful. It also seemed that the KPTs assumed the children would play the apps by themselves, with teachers or adults only needed if the apps were too difficult for the children to understand by themselves. As a result, parental control, repetition, and feedback built into the apps were considered good design features that would support children to work individually and be successful. The KPTs’ expectations about children working individually on the apps can be considered as providing opportunities for children’s danning.

The KPTs considered it problematic if the children could repeatedly fail when they engaged with the apps, as this would create a negative feeling towards learning mathematics. The KPTs wanted feedback and repetition in the apps because they considered that children needed to be successful when they were learning.

Identifying how the KPTs’ comments related to the different nodes in ACAT theory provided a way of understanding what they valued as possibilities for how mathematics education could contribute to young children’s danning. For example, how an app should teach children, what kind of ways children should interact with it and what kind of mathematics children should have experience with have the potential to contribute to children’s danning (see for example definitions of danning in Ødegaard, 2019).

The Norwegian Framework Plan for Kindergartens indicates that digital tools should encourage “children to play, be creative and to learn” (Ministry of Education and Research, 2017, p. 44). However, possibilities for children’s learning through play and being creative did not seem to feature in KPTs’ evaluations of the different apps, even though they had been asked to focus on play in their evaluations. Teacher educators might need to have discussions about the design features of apps which support creativity and co-operation with KPTs to broaden understandings about how these aspects could be seen, or not, in children’s engagement with apps. As it states in the current Framework Plan, for the mathematical goals, learning includes “discovering, exploring and creating structures that help children to understand relationships in nature, society and the universe” (Ministry of Education and Research, 2017, p. 53). Yet at the beginning of KPTs’ teacher education, they may have different understandings about the range of opportunities for danning that could be provided to young children, through their engagement with mathematical apps.