1 Introduction

Researchers and educators agree that promoting children’s mathematical competencies prior to first grade is important (e.g., Claessens & Engel, 2013). Towards this end, several studies have investigated preschool teachers’ knowledge for engaging children with mathematical activities, including the way they design mathematical activities to implement in their classrooms (e.g., Tirosh et al., 2011). However, young children often spend a great deal of time in informal environments such as the home, after-school programs, and libraries (Hofferth & Sandberg, 2001) in the care of parents, grandparents, neighbors, and other responsible adults (Gambaro et al., 2025; Pilarz, 2018). In some countries (e.g., Spain, China, United States) and for various reasons (Covid-19 virus; single-parent families), adults other than parents (e.g., grandparents) play a significant role in children’s lives (Alonso et al., 2024; Guan et al., 2022; Lehti et al., 2019) where learning opportunities arise, even during leisure activities (e.g., games, art activities) (Alonso et al., 2024). In general, various adults contribute to children’s educational development and may thus contribute to children’s mathematical development. Yet, adults are not always aware of what mathematics children could or should learn or how to go about helping them learn mathematics (Cannon & Ginsburg, 2008).

Considering that nearly all adults are potential, if not actual, caregivers, we began investigating adults’ knowledge and beliefs regarding promoting young children’s mathematical learning. In our previous study of 90 adults (Barkai et al., 2022), we requested participants to describe situations where numerical ideas were raised by children on their own, and then to describe situations where numerical ideas were raised with adult involvement. Most adults were parents or grandparents of children aged 3–6 years, none were early childhood educators. Findings indicated that two-thirds of the participants were able to recall situations in both cases. However, in the cases of adult involvement, only 20% of participants described situations where the adult was the initiator of the numerical discussion. This last result indicated a need to raise adults’ awareness of how they could be more proactive and involved in promoting children’s numerical competencies.

To follow up, we (Levenson et al., 2022) requested the same participants to suggest activities they could carry out with children with the aim of promoting children’s verbal counting, as well as object counting skills. In that study, even when explicitly requested to suggest a verbal counting activity, over half of the participants suggested an object counting activity, indicating lack of awareness that verbal counting is an important skill, separate from object counting. Furthermore, many of the suggested activities were general, without indications of such details as to how many objects they would set out to be counted or how those objects would be laid out to count. Most suggested activities did not refer to any of the counting principles mentioned by Gelman and Gallistel (1978).

We then asked ourselves, how might adults plan for activities with young children if they had some knowledge of what it means for young children to learn about numbers and counting and if they were specifically requested to include details of the planned activity? Acknowledging that those with positive attitudes are more likely to engage children with mathematical activities (Blevins-Knabe et al., 2000), we decided to offer an elective graduate course, entitled Early childhood numerical thinking: Theory and research to practicing and prospective mathematics teachers working towards their master’s degree in mathematics education. The aim of the course was to raise participants’ awareness of number competencies developed prior to first grade, to increase their knowledge of children’s development of those competencies, as well as the tasks that might promote early number knowledge and competencies. Within this context we requested participants to plan verbal and object counting activities for young children. Our aim was to investigate what we call adults’ knowledge of content and playful learning, that is, adults’ knowledge of activities that can promote numerical thinking within a playful context.

2 Background

2.1 Verbal and object counting

This section serves as the background for analyzing the verbal and object counting competencies participants chose to promote in their planned activities. Both verbal and object counting skills are foundational to early mathematics learning (Clements & Sarama, 2014). Verbal counting entails reciting numbers in the conventional order and knowing the principles and patterns in the number system as coded in one’s natural language (Baroody, 2006). Fuson (1988) found that learning to count verbally occurs in two phases. At first, children learn to recite the conventional number words in order and consistently. During the second phase, called the elaboration phase, children come to realize that the chain of numbers can be broken up and that parts of the chain may be produced starting from a number other than one.

Besides being able to count forward from one, several mathematics education researchers (Clements & Sarama, 2014; Ginsburg et al., 2016) as well as various curricula (Common Core State Standards Initiative in the United States (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010), and the Israel preschool curriculum (INMPC, 2010)), suggest additional competencies related to verbal counting, such as counting backward, skip counting, knowing the number that comes before or after some number, and counting forward and backward from some number other than one. These competencies are related to the ordinality aspect of number, i.e., “the capacity to place number words and numerals in sequence; for example, to know that 4 comes before 5 and after 3 in the sequence of natural numbers” (Verschaffel et al., 2017, p. 33). Studies have found that children may naturally apply these competencies when solving arithmetic problems, such as “counting down” (counting backwards) in a subtraction problem (Fuson, 1988). Counting backwards may be used by children in other capacities, as in time measurement, (e.g., counting back from nine to one and announcing that school is finished) (Papandreou & Tsiouli, 2022).

Object counting refers to counting objects for the purpose of saying how many. According to Gelman and Gallistel (1978), children need to master five principles of object counting: the one-to-one correspondence principle (i.e. assigning one count word to each object), the stable-order principle (i.e. reciting numbers consistently in the conventional order), the cardinality principle (i.e. knowing that the last number said when counting objects represents the quantity), the abstraction principle (i.e. knowing that any set of discrete objects can be counted), and the order-irrelevance principle (i.e. objects can be counted in any order). Successfully counting objects may depend on the number of objects to be counted, as well as the set-up of the objects (Briars & Siegler, 1984; Gelman & Gallistel, 1978). For example, kindergarten children may be able to count some number of objects placed in a row, but when the same objects are placed in a circle, children may err in their counting or not know where to begin their counting (Tsamir et al., 2018). Finally, engaging children with number activities, without talking to them, may not be enough. Ramani et al., (2015) found that caregivers’ talk about advanced concepts such as cardinality and ordinal relations, while engaging children in number-related activities, predicted children’s advanced number competencies that build on those concepts. Developing number competencies can take place during play activities. This is discussed next.

2.2 Playful mathematical activities

The aim of this section is to briefly describe different characterizations of play and playfulness, which in turn informed our characterization of playful activities. We begin, however, by first characterizing activities that are not considered playful. Fisher et al. (2013) described didactical instruction as a setting where the adult “acted as 'the explorer’ while the child passively watched and listened through each step of training” (p. 1874). Note that because the child is passive, we do not consider this type of interaction as playful. Similarly, Eason and Ramani’s (2017) design of a formal learning condition was characterized by decontextualized questions and limited opportunities for exploration. They further described this condition as one which limited children’s active role, for example, by gluing items together to restrict manipulation. The opposite of restricting manipulation, and in general, restricting movement, is active movement. In fact, when young children were asked what play means to them, many of them described physical activities, such as running around (Brockman et al., 2011). Running around, without having a specific aim such as training for a marathon, has also been called “exercise play,” defined as ‘‘gross locomotor movement in the context of play’’ (Pellegrini & Smith, 1998, p. 578). This type of play is frequently observed in preschool, and can include swinging, jumping, and running (Lindsey, 2014). Physical (gross motor) movement, such as walking a number line painted on the ground, has also been incorporated into what has been called mathematical guided play (Alvarez-Vargas et al., 2023). Thus, the first element we connect with playfulness is physical movement.

The terminology related to play within mathematics education varies and includes free play, play-based, guided play, game play, play context, and more. In the current study, participants were requested to design an activity that an adult could implement with a child, thus we do not consider it free play, which refers to children’s self-directed activities. Guided play involves adults who scaffold children’s learning (Fisher et al., 2013), providing meaningful contexts where children can actively explore mathematical ideas (Ramani & Scalise, 2020). Similarly, play-based pedagogy, play-based learning, and play-based curricula stem from a belief that “children’s learning should always be embedded in children’s imitative participation in meaningful cultural practices” (Van Oers, 2010, p. 29). Meaningful contexts are often thought of as those based on children’s own experiences and interests (Hirsh-Pasek et al., 2020) and can involve role play. For example, Van Oers (2010) described a situation where children were playing shoe-shop and trying on shoes. The teacher used this opportunity to engage the children with measuring and estimation. Although outsiders looking in might have deemed the activity as purely mathematical, it remained meaningful for the children because it was part of their shoe-shop play. Imaginary situations, where children can make-believe and take on different roles, elicits children’s personal values associated with the play situation, which in turn can promote meaningful mathematical learning (Li & Disney, 2023). Thus, a second component of play is the make-believe or imaginative context in which the learning takes place.

A third component of playful learning is being able to make choices. Although choices may be limited, they personalize the activity, increasing children’s interest, leading to increased engagement and involvement (Skene et al., 2022). The level of choice can be adaptable (King & Howard, 2016). Within a play context, adults may limit or increase the degree of choice based on the child’s learning and development. Notably, if children perceive that to some extent they have some choice, they feel that they are playing and are motivated to keep on playing (King & Howard, 2016).

Some researchers (Vogt et al., 2018) suggest that games, where there are a set of rules to follow (not necessarily games that have winners or losers) can still be considered play based. In the mathematics education literature, several studies have investigated children’s mathematical learning within the context of playing card games and board games. For example, Sonnenschein et al. (2016) investigated children’s number magnitude knowledge after playing a board game several times. Although children’s knowledge increased, the researchers suggested offering post games incentives, such as giving the children stickers when the game is completed to increase motivation and enthusiasm. This result hints that playing board games may be less engaging than playing make-believe. Ramani and Scalise (2020) noted the importance of adult scaffolding during card games, indicating that just playing the game, without talking about the numbers, may not be enough to increase children’s number knowledge. To sum up, characteristics of play mentioned in association with learning mathematics are physical engagement, imagination, games with rules, and giving the children choices.

In the current study, participants were asked to consider what concrete materials would be used in their activities. Of course, it is not the material that determines playfulness, but what one does with it. In Anderson’s (1997) study, parents were given multilink blocks, a story book, paper, and a workbook, items thought to be familiar to children. They were told to ‘play’ with their children while engaging them with mathematics. Although elements of play were not in focus, Anderson did notice if choice was given to the children. In some instances, the mother let the child choose which materials she wanted to play with (e.g., drawing on paper), but then directed the activity towards mathematics by asking the child how many pictures she drew. Anderson called this eliciting and redirecting the child’s desires. Other adults elicited and followed through on the child’s desires. In those cases, however, mathematics was not always part of the activity. Other studies noted that practically any object or material found in one’s environment may be used in a numerical activity, such as counting the crackers on one’s plate (Tudge & Doucet, 2004). Likewise, those same objects could be used during imaginative play. Thus, in the current study, we investigate how objects are used during a planned mathematics activity.

During play activities, adult input, especially in the form of questions, can impact children’s learning. This is discussed next.

2.3 Questioning as part of mathematical activities

Studies of interactions between adults (teachers, parents, child minders, and others) found that a major part of these interactions involve the adult posing questions. For example, in Anderson’s (1997) study, parents were given various materials and told to ‘play’ with their children while engaging them with mathematics. In her analysis, Anderson noticed that most parents took a questioning stance, focusing on eliciting children’s numerical knowledge (e.g., which tower has more blocks?). Bjorklund et al. (2004) termed this kind of behavior as giving a prompt (e.g., asking, how many?). Carlsen et al. (2010), in their study of kindergarten teachers’ questions, called such prompts suggesting action questions because this type of question leads the child to act before answering the question.

Questions that can be answered simply by retrieving information that is visually present (e.g., “How many?”, or “Can you point to the number two?”) are considered low-cognitive demand questions (Duong et al., 2021). On the other hand, questions that require thinking beyond what can be readily seen, such as, “How do you know?” questions, may be considered high-cognitive demand questions. Working with young children and mathematics, Sarama et al. (2012) found that such questioning practices not only promote mathematics, but also early language skills. When observing parent–child dyads, Duong et al. (2021) found that parents produced more low-cognitive demanding tasks than high-cognitive demanding tasks. Likewise, Anderson (1997) noted that some parents asked children to explain their answers (e.g., How did you know that?), but to a lesser extent. In general, “How do you know” questions are associated with a guided-play approach to learning, as opposed to a direct instruction approach (Weisberg et al., 2013).

Several researchers used the terms open and closed questions when classifying adults’ questions during mathematical interactions. Carlsen et al. (2010) categorized questions as open if they inquired into how a child knew something, as in, “How did you decide that…?” Questions such as, “Why do you think that?” were categorized as argument questions. They found that the teacher in this study employed more open questions than suggesting action questions. Trawick-Smith et al. (2016) described open questions as those asking for an explanation, or asking a question that could potentially elicit different possible responses and that requires more than one word answers. Similarly, Bambha et al. (2024) coded questions as open if they included why and how questions. Questions were coded as closed if they included yes/no questions, or forced choice questions (e.g., either/or). In the current study, we appropriate Trawick-Smith et al.’s (2016) description of open questions, recognizing that such questions can encourage children to think more deeply about their actions and possibilities. We used the term closed question to include questions that require one-word answers and questions that may be seen as suggesting action or prompt questions.

The timing of specific interactions is also important. Follow-up questions, also called feedback questions, may be more important than initial questions (e.g., Hattie & Timperley, 2007). In a study of preschool teachers’ mathematical interactions, it was found that the quality, rather than the frequency of feedback based on children’s responses was critical (Hu et al., 2021). After a student responds to some initial prompt or question, a feedback question that is open-ended, such as “Why did you respond that way?” can push the student to reflect and engage in high-order thinking. On the other hand, merely giving prompts (Bjorklund et al., 2004) after a child errs, can leave the child unaware that he or she made a mistake, what the mistake was, or why it was a mistake. Studies have also found that some adults merely provide the correct answer (Vandermaas-Peeler et al., 2012), while others simply disaffirm or affirm the child’s answer (e.g., saying, “that’s not right,” or “that’s right!” or “well done”) (Bjorklund et al., 2004; Schnieders & Schuh, 2022).

To sum up, considering the importance of counting competencies, and that the playfulness of an activity can affect children’s attitude and motivation to learn mathematics (Ginsburg, 2006), and that engaging with children almost always involves questions, our research questions are as follows: Which numerical competencies do adults aim to promote through their planned activities? What elements of playfulness appear in those activities? What types of questions do adults plan to ask the children during the activity and when do adults plan to intervene in the activity?

3 Method

3.1 Participants and setting

The setting of this study was an elective graduate course, given three years in a row, entitled Early childhood numerical thinking: Theory and research. Participants were 61 students (labelled A1-A61) working towards their master’s degree in mathematics education, 59% of whom were experienced secondary mathematics teachers (with an average of 6 years’ experience) and the rest concurrently working towards their teaching degree. This context allowed us to work with adults who have a positive disposition towards mathematics and would be most likely interested (the course was an elective), if not enthusiastic, about promoting mathematical learning during early childhood. The aim of the course was to raise participants’ awareness of number competencies developed prior to first grade, to increase their knowledge of children’s development of those competencies, as well as the tasks that might promote early number knowledge and competencies. Most of the participants (77%) had some direct relationship with young children between the age of 3 and 6 years. Of those, 48% were parents, 32% were aunts/uncles, 10% were grandparents, and 10% had some other relationship (e.g., neighbors). IRB approval was received. The course instructor explained to the participants the aims of the research, that personal information would not be publicized, and that participation in the research was voluntary and not a requirement for the course. Those that opted in, signed an informed consent form.

The current study investigates results of a formative written assessment task given to participants after four 90-min lessons. During the first lesson, participants watched and then analyzed a video of a three-year old boy and his grandmother, engaging in various counting activities while baking cookies. Analysis of this video, as well as other video clips viewed during the course, focused first on the child’s ability to carry out the activity (e.g., Could the child count the cookies on the tray, and what were his difficulties?), as well as the adult’s role in the activity (e.g., What exactly did the grandmother ask her grandson to do? How were the cookies arranged on the tray?). During the second, third, and fourth lessons, participants read and discussed related research (e.g., Baroody, 2006; Clements et al., 2017; Gelman & Gallistel, 1978), and viewed and analyzed together YouTube videos of preschool children counting with and without objects. We discussed children’s ability to carry out a particular skill, as well as how a task may be designed and implemented to focus on a specific skill. No specific discussion regarding playfulness or the use of open and closed questions was carried out. During the third lesson, participants were introduced to the national mathematics curriculum for preschool (INMPC, 2010) which lists attainment goals for children ages 3–4, 4–5, and 5–6 years. At the end of the fourth lesson, participants were given the following home assignment, which is in essence the research tool of this study:

Plan an activity for a young child that can promote verbal counting and object counting competencies. State the competencies you wish to promote, what items you will use, how they will be placed, and how you will use them. State what questions you are likely to ask the child, how you expect the child to respond, and how you will react to the child.

Participants had two weeks to complete the assignment, were told that it would not be graded, but would be used as a basis for discussions and further learning in the coming weeks. The directions did not explicitly relate to playfulness, allowing us to analyze participants’ views on the types of activities they deemed appropriate for young children. We expected that requesting participants to consider how they would follow up on questions asked would lead to further interactions with the child. Thus, it could be said that the assignment had some parameters, but was sufficiently open to meet our research questions.

3.2 Data analysis

As per the first research question, we began by conducting a deductive analysis based on the verbal and object counting competencies mentioned in the background. In most cases, coding was straight forward, as the instructions for participants included a request for them to state the competencies they wished to promote. The only issue that arose was separating general object counting from specific competencies such as one-to-one correspondence (see Table 1).

Table 1 Frequencies of numerical competencies (N = 61)
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The main challenge was to characterize the playfulness of the adults’ planned activities. The first step, carried out independently by the three authors, was multiple readings of the activities to formulate how the activities could be decomposed into components that would assist in comparing the various activities. We began by categorizing the nature of the activity. The third author read and reread the data leading to an initial coding system of five categories, then shared with the first author: an activity within a story context, physical movement, a music context, a game-like activity, and an instructional activity. The most difficult decisions revolved around differentiating between game-like and instructional activities. In Table 2, the task proposed by A11 was considered game-like because it included throwing a die, a move that is associated with games. Another game-like activity was offered by A12 who wrote the following:

I will ask the child to turn around with his back towards me and count till 10. During that time, I will hide a piece of candy under one of two cups and when the child turns around, she has to guess under which cup lies the candy. If he guesses right, he can eat the candy.

Because the counting was embedded in the game of hiding and finding the candy, the task was categorized as game-like. Instructional tasks were missing this quality and had no playful elements.

Table 2 Materials used within different types of activities (N = 61)
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At this point, the second author was consulted and upon further analysis, it was decided that the music context involved either physical movement, such as banging on a drum, or included some imaginative context. After further discussion and consideration of the background literature (Li & Disney, 2023) we decided that what made a story playful was that it involved the children imagining themselves in different roles, where imagination is considered an important element of play. Thus, the category name was changed to make-believe situations. The final four activity categories were then: make-believe situations, physical movement, game-like activity, and instructional activity. Examples of each category may be found in Table 2. Since most participants included several small tasks within the activity, we categorized the activity based on its major playful characteristic. Thus, if an activity included both physical movement and instructional tasks, the whole activity was categorized as physical movement.

Regarding materials, deductive analysis was first carried out according to studies (e.g., Tudge & Doucet, 2004) which differentiated between items specifically designed for playing, such as toys and games, and those not designed for play, such as items found in the house. Further inductive analysis led to a third category – items specifically designed for the activity.

Finally, we looked to see in what ways adults allowed the child to make choices, increasing their involvement. This led to three categories: the child could choose the items to be counted and/or how they would be positioned, they could choose the number of objects to be counted or the order of those amounts presented, and they could choose when to end the activity.

To answer the last research question, we focused on the questions participants planned for, and at what point were questions asked during the activity. Based on the background, we categorized questions as closed if they required one-word answers or if they were suggesting actions questions. Because requests for specific actions (e.g., count forwards till 30) are meant to elicit a specific response, we considered them as closed questions. Open questions were those asking for an explanation or asking a question that could potentially elicit different possible responses. We then analyzed if the plan included adult intervention only at certain points in the activity, such as if the child made a mistake, or were interactions planned for throughout the activity. The third author initially coded all questions and then discussed the coding with the second author. Disagreements were discussed with the first author, until full consensus was reached.

4 Results

4.1 Numerical competencies

Participants were requested to plan mathematical activities aimed at promoting verbal and object counting competencies. However, it was up to the participants which competencies they wished to promote. Some explicitly stated the aim or aims of the activity, such as A8 who wrote:

The aim of this activity is to count non-identical (in size and color) objects, and to count them when they are positioned in different ways. This will promote the abstraction principle and the fact that the way objects are set out does not affect the amount.

A8 has related to the abstraction and order-irrelevance principles (Gelman & Gallistel, 1978).

For others, we analyzed the tasks, including the questions that the participants planned to ask. For example, A6 wrote the following, “I will write with chalk the numbers from 0 to 10 on the floor… and then I will ask the child if he can jump on the numbers in the opposite order, starting from 10 and continuing to 0.” From this we surmised that the aim of the task was to promote counting backwards. Regarding object counting, some activities related to specific counting competencies, such as cardinality and one-to-one correspondence; others wrote tasks that only mentioned counting objects in general. For example, A37 wrote, “I will place each time a different amount of bottle caps on the table, and ask the child to count them, and when he finishes, I will ask, 'So, how many are there?' and so I emphasize the cardinality principle.” In contrast, A51 only wrote, “I will place the toys next to the basket and ask the boy to count them.” Most of the activities consisted of a series of small tasks, thus encompassing more than one competency (note the percents in Table 1 add up to more than 100). On average, participants’ activities encompassed 4 competencies.

Regarding counting forward, 11 participants used this task as a warm-up or introductory activity to ease the child into another, more challenging activity. As A(58) wrote, “First, I will ask the child to count till 20. Then I will take out Lego pieces, ask the child to count them, and then build from them a tower.”

4.2 Characterizing playfulness

Recall that participants’ activities were divided into four categories: make-believe situations, physical movement, game-like activity, and instructional activity. We were also interested in the types of materials participants would use for each type of activity. Table 2 reports on the frequencies of the types of materials used within each type of activity. Note that 55% of the participants planned activities that were only instructional and had no element of play. Regarding materials, 57% of the participants would use toys (or games or cards). Interestingly, using toys did not necessarily imply playfulness. As seen in Table 2, of the 33 participants who planned an instructional activity, 19 (58%) intended to use toys. In addition to the examples shown in Table 2, we offer additional examples of how participants used different materials.

Make-believe situation using house-hold items: “I will tell the child that we are having company for the holiday dinner. So, first she needs to count how many guests are coming and then the chairs we have, and of course the plates and silverware.” (A22)

Physical movement using objects created for the activity: “I will take 6 pieces of paper and draw on them different body movements and then glue the papers on a die. The child will roll one regular die and the one I made, and he will have to do the move shown on one die, the number of times showed on the other die.” (A19)

We now consider the three ways in which children were able to make choices: choosing the items to be counted and/or how they would be positioned, choosing the number of objects to be counted or the order of those amounts presented, and choosing when to end the activity. As seen in Table 3, most participants (66%) did not offer children any choice.

Table 3 Frequencies (%) of what children could choose (N = 61)
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Finally, we considered the number of times a child was given the opportunity to make a choice, in relation to the type of activity presented (see Table 4). For example, A20 in Table 5, offered the child a choice to choose the objects he/she would count within a game-like activity using yard items. Interestingly, game-like activities offered the children the most opportunities to make choices, probably because those activities often included the child rolling a die.

Table 4 Frequencies of number of choices given per activity type
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Table 5 Frequencies of activities that included open questions per activity type
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4.3 Questions during adult–child interactions

Recall that participants were requested to state what questions they were likely to ask the child. Of the 61 participants, 51 (84%) included only closed questions. These included questions that begin with “how many,” and/or requests such as, “count backwards from 10,” “put n items here,” “take n items from there,” or “find the card with n dots on it.” On average, those participants planned to ask 5 closed questions per activity.

Only 10 (16%) participants included open questions, spread over all types of activities (see Table 5). For example, part of A52’s make-believe activity included the following:

I will place 12 beads on the table, two colors, 6 beads each. I will explain to the girl that we want to make bracelets from the beads, but before we begin, we need to see how many beads there are. So, I will ask her to count the beads and then I will ask her how many beads there are from each color. Then I will ask her: What colors are the beads? How many beads are there? How can we know which color has more?

The last question, “How can we know which color has more?” is considered an open question because the child is requested to explain her or his thought process. Other open questions included, “How do you know that your answer is correct?” and “How did you know where to begin to count?” On average, those participants asked two open questions per activity. Relative to the type of activity, make-believe situations had the most open questions.

In addition to investigating the types of questions adults planned to ask, we investigated when the adult planned to ask these questions. That is, did the adult ask a question, wait for a response, and end the task (no follow-up intervention), or was there a follow-up question or task, dependent on the child’s response. From Table 6, we see that about a quarter of participants did not plan to follow up after the child’s response (called, “no intervention”). It could be that participants just did not think of this option, two participants alluded to children they know, who they felt would not have any problems with the task. A third participant (A56) wrote, “During the activity, I will not correct the boy if he makes a mistake, but I will give him the freedom/possibility to act and to think.” As requested, most participants (72%) thought about how the child would react, and how they would respond in turn. However, nearly half only considered how to respond if the child showed some difficulty or made a mistake, while only a quarter thought of what to do if the child found the task simple or gave correct answers.

Table 6 Dependent interventions (N = 61)
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The types of intervention participants planned to carry out if the child showed some difficulty or made a mistake were varied. Some participants wrote merely that they would repeat the question or have the child repeat the same task. More often, participants thought how they could intervene in a more proactive way, albeit using general terms such as, “I will explain,” “I will teach,” “I will make the task easier,” “I will correct the mistake,” and “I will demonstrate.” Some participants were more explicit, such as A45, who planned a candle lighting activity related to an upcoming festival and wrote, “It is likely that the child will not know to match each number with one candle. I will explain to the boy that each candle is worth one number and therefore you say ‘one’ and then you touch that one candle.” Participants that wrote explicitly or alluded to making a task easier for a child showing difficulties, either wrote that they would use less objects to count (A43), or change the arrangement of the objects, for example, by taking a pile of objects and setting them out in a row (A52).

There were two types of reactions explicitly mentioned by participants in the case of a child finding the task easy. The first was to commend the child on a task completed well (A10 in Table 6). The other type was more of an intervention in that the participant wrote how the task could be made more challenging for that child, either by adding additional objects to count (A8 in Table 6) or by changing the task. For example, A14 wrote, “If I see that the child succeeds in the two previous tasks (counting out sticks from a container), I will ask the child to return the sticks one at a time to the container, while counting backwards.”

5 Discussion

The first question of this study was related to the counting competencies participants aimed to promote in their activities. The most frequent competencies mentioned were counting forwards and counting objects, specifically focusing on cardinality. In line with previous studies (Barkai et al., 2022; Levenson et al., 2022; Missall et al., 2017), participants paid less attention to additional competencies such as counting backwards and skip counting. This is surprising for two reasons. First, unlike those previous studies, participants in the current study had already participated in four, 90-min lessons dedicated to specific verbal and object counting competencies. Second, participants were teachers or prospective teachers who we thought would consider to a greater extent curriculum guidelines. Yet, this was not the case.

One possibility for this outcome might be that the participants believed that counting backwards and skip counting are more advanced competencies. This is in line with Tsamir et al. (2014) who found that preschool teachers underestimated children’s ability to count backwards, as well as say the number that comes after or before some other number. Evidence of this in the current study may be seen among participants who mentioned those competencies in a follow-up activity they planned for cases where the child easily solved the first task. It is also possible that participants were viewing the practice of counting from the child’s perspective, which, like other number concepts (e.g., the meaning of numerals) might not necessarily be in line with curricula guidelines (Stott &Voutsina, 2024). In the case of counting, participants (many of whom were parents or grandparents) might have considered the home environment and what might be most practical or natural in an everyday context, where skip counting might not be as relevant as object counting. This result suggests that merely gaining knowledge of what competencies are advocated by policy makers is not enough, and that we also need to raise awareness of how these advanced counting competencies can serve the children as they develop additional number competencies (Ramani et al., 2015) and how they are tied in to everyday occurrences, such as counting backwards as each cookie on the plate gets eaten.

The second aim of this study was characterizing the potential playfulness of participants’ planned activities. While previous studies discussed the benefits of play-based mathematical activities (Hirsh-Pasek, 2009), we did not find in those studies ways of evaluating the playfulness embedded in an activity. While studies have seen play-based contexts as a compromise between pure instruction and free play (Weisberg et al., 2013), it seems that the continuum between instruction and play-based approaches was not considered (Vogt et al., 2018; Weisberg et al., 2013). Thus, another contribution of this study is explicating elements of an activity that can contribute to its playfulness (such as physical movement, imaginative or make-believe situations, games, and affording children opportunities to make choices). Studies have found that some parents and caregivers of young children believe that play is for amusement only (Fisher et al., 2008; Mao et al., 2024). Considering the importance of play to learning mathematics at an early age (Ginsburg, 2006), workshops for adults might want to stress this aspect of learning. For example, in the current study, participants hardly planned to offer children any choices during the activity, regardless of the type of activity. This is an element that can be added to any type of activity, offering the children a feeling of playfulness (King & Howard, 2016).

Previous studies mentioned the materials used during informal mathematical activities, but in those studies, participants were usually given the materials to be used (Anderson, 1997; Missal et al., 2017). The current study investigated materials specifically chosen by the participants to be used and how they were to be used within the various types of activities. Most participants sought to use readily available toys and games, even in instructional activities. Adults may have assumed that by using children’s toys, the activity would automatically be playful, but this was not the case. Just as objects are not inherently mathematical (Brandt, 2013), objects are not inherently playful.

The last research question related to the questions and their timing during the activities. Nearly all participants planned to ask closed, rather than open-ended questions, and most participants planned to intervene in the case of children having difficulties, but not in the case of children finding the task simple. These results are in line with previous observations of parent–child mathematics interactions that reported how parents mostly focus on eliciting knowledge and prompting behaviours (Andeson, 1997; Bjorklund et al., 2004). However, in the current study, participants were mathematics teachers, and so we thought that these participants, teachers who presumably request students to explain and justify their solutions, and who adapt tasks to meet the learning levels of students, would plan to act similarly with young children. Yet, this was not the case.

Recall that participants mostly aimed to promote forward verbal counting and hardly any advanced counting competencies. Similarly, participants might have thought that high cognitive demands such as asking children to explain their strategies or justify their solutions (Stein et al., 1996), would be too difficult for young children. Yet, research has shown that children can reason and explain their mathematical thinking, even within the context of verbal and object counting (Björklund & Palmér, 2020; Nergård, 2023). Reflecting on the course content, we acknowledge that children’s reasoning abilities were not in focus.

One might argue that evaluating tasks does not really tell us what would happen in real time. Nevertheless, previous studies have focused on the potential of tasks to occasion mathematical creativity (Klein & Leikin, 2020), the potential of tasks to elicit argumentation (Prusak et al., 2012), as well as tasks that may demand high cognitive thinking (Stein et al., 1996). Such studies then help us to evaluate task choices, task designs, and task implementation. Although planning for an activity does not guarantee how that activity will be implemented (Henningsen & Stein, 1997), we argue that examining those planned mathematical activities opens a window into how adults think about mathematical activities for children. This is a start. Results of this study can help plan workshops that might further address different ways of interacting with mathematics in a playful way.