Mobile learning in mathematics education

Over the past decade, interest in mobile technologies as emerging and innovative tools has grown (^{BENALI; ALLY, 2020}). Mobile learning broadly involves the integration of mobile devices into teaching and learning (^{GRANT, 2019}). There are different stances on the definition of mobile learning (e.g., ^{BRAZUELO; GALLEGO, 2014}; ^{UNESCO, 2013}) and mobile device (^{AGUILAR; PUGA, 2015}; CROMPTON, 2013a; ^{UNESCO, 2013}) becoming a topic of discussion among the research community.

Properly in this research, mobile learning is “[...] learning through multiple contexts, through social and content interactions, using personal electronic devices” (CROMPTON, 2013b, p. 4). This definition includes three important aspects of the term, on the one hand, it involves the diversity of contexts where learning can take place, for example, at home, school, or outdoors, as well as the interactions that can be achieved between students or between the student and the mathematical content. It also refers to the use of personal electronic devices, understanding a mobile device as any personal and portable technological device that could be used in the mathematics classroom, including devices such as laptops, smartphones, and tablets (^{AGUILAR; PUGA, 2015}).

Particularly, the use of mobile devices in the teaching-learning of Mathematics is gaining interest among researchers and education professionals (^{BORBA et al., 2016}), and in Primary Education three main results can be highlighted:

However, research that reports on uses or potentialities that could be the basis for the implementation of mobile devices is limited in Mathematics Education (^{BORBA et al., 2016}; SUNG; CHANG; LIU, 2016); in addition, in most researches, a specific mathematical concept is not identified (^{CROMPTON; BURKE, 2015}). In this regard, ^{Brazuelo and Gallego (2014)} raised the need to promote the design of mobile educational content and prepare teachers as a key element for the real integration of mobile technologies for educational purposes. Therefore, it is important to guide teachers on the design and implementation of mobile content for the teaching of Mathematics.

Training intervention

This research was based on the Ontosemiotic Approach (OSA) which is understood as a modular and inclusive theoretical system for Mathematics Education, consisting of principles and methodological tools to address aspects related to the teaching and learning processes of mathematics (^{GODINO et al., 2021}). It is considered that OSA can add elements that contribute to the previous problem, specifically there is research that reports training interventions related to “[...] the analysis of the didactic suitability of teaching experiences lived by the teachers themselves” (^{GODINO et al., 2017}, p. 103). In general terms, in these interventions “[...] it is about the teacher reflecting on the design, implementation, and evaluation of a teaching experience of a specific topic in a fixed educational context” (p. 107).

The above, through the facets and criteria of didactic suitability (^{GODINO, 2013}, ²⁰²¹), understood as the degree to which an instruction process or part of it meets characteristics that allow it to be qualified as optimal or adequate to achieve adaptation between personal meanings achieved (learning) and the meanings intended or implemented (teaching), taking into account the circumstances and resources available (^{GODINO et al., 2021}). Figure 1 presents the facets and criteria of didactic suitability.

Source: Created by the authors based on ^{Godino (2021)}.

Figure 1 Facets and criteria of didactic suitability 

This type of training action has been widely reported in research carried out by the OSA. For example, Morales, García, and Durán (2019) designed and implemented a training proposal for the development of the mathematical knowledge and competence profile of teachers, where they identified incidents or recurrences of each component of the didactic suitability. Pochulu, Font, and Rodríguez (2016) carried out a training cycle aimed at teacher trainers of mathematics, guided towards the design of tasks for the development of the didactic analysis competence, where the suitability criteria were used in the planning of the task sequences. Giacomone, Godino, and Beltrán-Pellicer (2018) analysed and evaluated the implementation of a training design for the development of the competence of analysis of didactic suitability. Meanwhile, Aké and other authors (2014) analysed a training experience of teachers, guided towards the development of knowledge to discriminate algebraic objects and levels of algebraization of mathematical activity.

In this case, the analysis of a training experience guided towards the design and implementation of tasks for mobile learning is presented, based on the experience lived by two in-service teachers, with the teaching of the meaning of numerical sequence and the use of Kahoot. To this end, didactic suitability, and other theoretical tools of the OSA were used, which are presented in the following section.

Participants and context

Two in-service Primary Education teachers participated in the training experience, whose information is shown in Table 1.

Teacher₁ was a teacher in a private school, where they taught Mathematics to two 3^rd grade groups, 3A (22 students) and 3B (24 students). At the time of the investigation, the health emergency due to COVID-19 occurred worldwide, and in accordance with the guidelines of the Ministry of Public Education, classes were taught virtually. Particularly, Teacher₁ used Zoom to teach classes by video call, synchronously using a computer, a tablet, and a smartphone. On the other hand, Teacher₂ taught in a public school, to the sole 3rd-grade group (21 students) virtually in an asynchronous format, using a smartphone and WhatsApp messaging.

Technological resource

For the design of tasks, Kahoot (https://kahoot.com/schools-u) was used, a technological resource that allows teachers to design interactive activities (tasks) that can be seen on a smartphone, tablet, and computer. This resource is interactive and intuitive and allows you to design tasks without the need for advanced knowledge about programming, the latter being the reason why it was selected.

In the premium version of Kahoot, the following tasks can be designed: a) quiz (Figure 3a); b) true or false (Figure 3b); c) write the answer (Figure 3c); d) puzzles (Figure 3d); and e) survey (Figure 3e). In addition, in the resource, each task is assigned a time limit to respond, which ranges anywhere from 5 seconds to 4 minutes.

Source: Activities available in the premium version of Kahoot.

Figure 3 Types of tasks in Kahoot 

Data collection

Both teachers received training in task design in the Designing interactive lessons in Kahoot workshop given over the course of eight weeks. In response to COVID-19 restrictions, the workshop was held synchronously via Google Meet. In total, there were eight sessions that were recorded (one session each week) with an approximate duration of 2 hours. Likewise, Google Classroom was used to work on activities that supplemented the information presented in the virtual sessions. Excerpts were taken from the recordings for analysis, and from Classroom activities, the planning of a class was taken as evidence. Table 2 shows the purpose of each workshop session.

Sessions 1 and 2 addressed aspects related to creating an account, the type of tasks, configuring tasks, among other functions of the resource. In sessions 3 and 4, the reference meaning of the natural number was discussed, emphasizing the type of situation-problems and the mathematical practices associated with each partial meaning of the mathematical object. Sessions 5 and 6 were devoted to the design of assignments and the planning of a class in which they were incorporated. Finally, in sessions 7 and 8 the tasks were implemented and validated; in the first moment, between the first author (researcher) and the participating teachers of the workshop, and in the second moment with the students of each teacher.

In general terms, the training experience consisted of providing two in-service teachers with specific knowledge about the use of Kahoot and the reference meaning of the natural number for the design of tasks. Subsequently, teachers were asked to plan and implement an instructional process that involved the tasks designed; and finally, the didactic suitability of the training experience was analysed.

Preliminary analysis

In this phase, the reference meaning of the natural number in the Decimal Numbering System (DNS) was reconstructed from its contexts of use (^{MORALES-GARCIA; NAVARRO, 2021}) and the system of practices involved, organised in epistemic configurations (EC) (MORALES-GARCIA; NAVARRO; DÍAZ-LEVICOY, 2021). In line with the above, six partial meanings of the natural number were established (Figure 4).

Source: Morales García and Navarro (2021, p. 1344).

Figure 4 Reference meaning of the natural number in the Decimal Number System 

Each meaning is described below.

Design of the didactic trajectory

In this phase, the teachers designed interactive lessons that were based on the tasks available in Kahoot (quiz, true or false, write the answer, puzzles, and survey) and the partial meanings of the natural number (cardinal, ordinal, numerical sequence, operational, measure, or symbolic). The designs were made after four sessions of the workshop, namely, based on the teachers’ knowledge of Kahoot management (Sessions 1 and 2), and knowledge of the reference meaning of the natural number (Sessions 3 and 4). Each interactive lesson was made up of 10 tasks and addressed one of the partial meanings of the natural number. Table 3 contains specific information.

In this research, and to show the potential of the theoretical tools of the OSA, the didactic trajectory involved in the design and implementation of the interactive lesson order and sequence created by Teacher₁ is analysed. In this sense, the objective pursued in this phase was to identify the actions that Teacher₁ considered before the implementation of the interactive lesson in a real context (teaching trajectory) and the effectively implemented meaning of the natural number as a numerical sequence (epistemic trajectory).

On the teaching trajectory, and in accordance with the planning of Teacher₁, this teacher organises their class in three moments: beginning, where they present an introduction to the topic being studied, usually with some video about the content of the class; later in the development they work with activities included in their textbook, and finally, in the closing, some elements presented throughout the class that will not be clear, are discussed, and it is at this time where Teacher₁ incorporated the interactive lesson order and sequence that in the words of Teacher₁ “[...] will serve to reinforce the content worked on in the class sessions.” In that sense, the students had previous knowledge on how to solve the situation-problems established in the interactive lesson, which somehow allows greater success in their resolution.

Teacher₁: [...] on Friday we write down in the notebook, how we can compare two numbers [natural numbers]; let’s start by comparing the thousands [the absolute value of the thousands] if they are equal, we go to the hundreds [we compare the absolute value of the hundreds]; if they are equal, we go to the tens [the absolute value of the tens], if they are equal, we go to the ones [the absolute value of the ones]. That’s why I put these activities [tasks in the interactive lesson].

The procedure suggested by Teacher₁ to determine when one number is greater than or less than another is to make comparisons between the absolute value of the digits of the compared numbers, starting with the digit of greater order (Figure 5).

Source: Own creation.

Figure 5 Procedure to compare natural numbers 

In summary, Teacher₁ considered as important actions to address the procedure beforehand, during the beginning and development of the class. In that sense, the application of the interactive lesson was considered as a kind of evaluation of the learning achieved by the students on the comparison of natural numbers with the intention of reinforcing the content taught, hence the importance of including it in the closing of the class.

On the other hand, in the epistemic trajectory of the interactive lesson, the problem situations, the sequence of expected practices in their resolution, and the primary objects involved in each task are shown. This information was obtained by considering the speech of Teacher₁ when presenting the 10 tasks (session 7) that made up the interactive lesson and the description of each task, included in the planning of the class. Table 4 presents the ontosemiotic analysis (^{MORALES-GARCIA; DÍAZ-LEVICOY, 2022}; MORALES-GARCIA; NAVARRO; DÍAZ-LEVICOY, 2021) of the tasks, identifying in the situation-problem the expected response and the time allocated for its response, the sequence of operational and discursive practices necessary to solve the task, and the primary objects involved in the resolution of each situation-problem.

Table 4 Ontosemiotic analysis of the interactive lesson order and sequence 

Task (situation-problem)	Sequence of practices	Primary objects
Expected response: 4028 Time: 20 seconds	1. A look at the two images is given to identify the numbers. 2. The thousands digit is the same in both numbers. 3. While the hundreds digit in the number 4028 is 0; and 1 in the number 4120. 4. Therefore, the number 4028 is less than 4120, because the hundreds digit is lesser.	Language: 4028, 4120. Concepts: place value, less than. Procedure: comparison of the digits of two numbers. Proposition: the number 4028 is less than 4120. Argument: the hundreds digit is lesser.
Expected Response: False Time: 20 seconds	5. The thousands digit is 3 in both numbers. 6. The hundreds digit is 0 in the number 3050 and 3 in the number 3300. 7. Therefore, the number 3300 is greater, because the hundreds digit is greater. 8. The answer is False.	Language: 3050, 3300. Concepts: thousands, hundreds, greater than. Procedure: comparison of the absolute value of the hundreds digit. Proposition: The number 3050 is not greater than 3300.
Expected response: 1230 Time: 20 seconds	9. In both numbers the thousands digit is 1. 10. The number of the hundreds is 2 in the number 1230, and 3 in the number 1320. 11. Therefore the lesser number is 1230, because the hundreds digit is lesser.	Language: 1230, 1320. Concepts: digit, thousands, hundreds, less than. Procedure: comparison of the hundreds digits. Proposition: the number 1230 is lesser. Argument: the hundreds digit in 1230 is lesser.
Expected response: 5234, 5199, 4609, and 3210. Time: 20 seconds	12. Between the numbers 5199 and 5234, the greater number is 5234, because the hundreds digit is greater. 13. The next greatest number is 5199, next is 4609, because the thousands digit is less than in 5199. 14. The next one is 3210 because the thousands digit is less than in 4609. 15. The correct order is: 5234, 5199, 4609, and 3210.	Language: 3210, 5199, 5234 and 4609. Concepts: digit, thousands, hundreds, greater than. Procedure: Identify the greater number and make comparisons to determine the order of the remaining numbers.
Expected response: 1239, 1456, 4100, and 4250. Time: 20 seconds	16. Between the numbers 1456 and 1239, the lesser number is 1239, because the hundreds digit is lesser. 17. The next is 1456. 18. Between the numbers 4250 and 4100, the lesser is the number 4100 because the hundreds number is lesser. 19. Therefore the order is: 1239, 1456, 4100, and 4250.	Language: 4250, 1456, 1239, 4100. Concepts: place value, less than. Procedures: Identify the lesser number and make comparisons to determine the order of the remaining numbers.
Expected answer: True. Time: 20 seconds	20. When comparing the numbers 6543 and 3999, the number 6543 is greater, because the thousands digit is greater. 21. So the answer is True.	Language: 6543, 3999. Concepts: thousands, greater than. Procedures: comparison of the digits of a number.
Expected response: 7818 and 5090 Time: 20 seconds	22. Between the numbers 7547 and 7818, the greater number is 7818, because the hundreds digit is greater. 23. The number 7547 follows because the thousands digit is greater compared to the remaining two numbers. 24. Between the numbers 5900 and 5090, the number 5090 is less than, because the hundreds digit is lesser. 25. Therefore the greater number is 7818 and the lesser is 5090.	Language: 7547, 7818, 5900, 5090. Concepts: thousands, hundreds, greater than, less than. Procedure: identify the greater number to sort the remaining numbers, until you identify the lesser number.
Expected response: 4532, 4440, 3412 and 2999. Time: 20 seconds	26. The sequence of numbers 4532, 4440, 3412, and 2999 is correctly ordered from greater to lesser. 27. While 5674, 3432, 9123, and 2000, it is not ordered from greater to lesser, because 9123 is greater than 5674. 28. The sequence of numbers 1234, 2342, 5320, and 5500 is not ordered from greater to lesser. 29. The answer is therefore: 4532, 4440, 3412, and 2999.	Language: four-digit numbers. Concepts: number sequence, greater than, less than. Procedure: Apply the number comparison to each number sequence.
Expected response: 2300 Time: 20 seconds	30. Identify that the sequence is ascending by 100 at a time. 31. The missing number in the sequence is 2300, since, if the sequence advances by 100 at a time, after 2200 the next is 2300.	Language: four-digit numbers. Concepts: numerical sequence. Procedure: counting by 100 at a time.
Expected Response: False Time: 20 seconds	32. The number 999 is less than the number 1300 because it has fewer digits. 33. Therefore the answer is False.	Language: 999, 1300. Concepts: digit, greater than. Procedure: comparison of two numbers based on the number of digits.

Source: Own creation.

The analysis of the epistemic trajectory (Figure 6) of the interactive lesson is important to show the effectively implemented meaning of the studied concept, in this case, the numerical sequence meaning of the natural number. For its analysis, the sequence of practices and the primary objects identified in each task, organised in epistemic configurations (EC), were considered. For example, the primary objects identified in Task 1 make up epistemic configuration 1 (EC1), the primary objects of Task 2, EC2, and so on for each task. It should be noted that each epistemic configuration has a function in the instruction process, for example, in EC1, EC2, EC3, and EC6 it was intended to put into practice the procedure taught for the comparison of two, four-digit natural numbers. Whereas, in EC4 and EC5 the procedure taught was applied, to put in order (ascending or descending) four numbers. On the other hand, in EC7 the level of difficulty was increased in some way since it is necessary to select among four numbers: the greatest and the least. In EC8 it was selected among three number sequences, which was correctly ordered from greater to lesser. In EC9 it was asked to indicate which is the number that completes the number sequence; here the student’s ability to recognise how much a number sequence increases and consequently identify which number completes it correctly is evaluated. Finally, in EC10 another procedure is presented to distinguish when one number is greater than another, considering the number of digits.

Source: Own creation.

Figure 6 Epistemic trajectory of the interactive lesson order and sequence. 

Implementation of the didactic trajectory

The tasks designed by Teacher₁ and analysed above make up Prototype 0 of the interactive lesson order and sequence. This was implemented twice; each implementation cycle was intended to validate and refine the included tasks by identifying Meaningful Didactic Facts (WILHELMI; FONT; GODINO, 2005). Figure 7 shows the deployment cycles.

Source: Own creation.

Figure 7 Interactive lesson implementation cycles 

The following describes the Significant Didactic Facts (SDF) identified in the implementation cycles.

Cycle 1 of the application was carried out with the participants of the workshop, in this case, the person in charge of the workshop and first author (I) and Teacher₂ answered the 10 tasks that made up the lesson of Teacher₁, to discuss their relevance (session 7). Based on this implementation, some adjustments were made to prototype 0. The first modification was made to Task 5 since the response options were not ordered from lesser to greater (SDF1).

Teacher₂: [...] I think that Task 5, is wrong because it was to order from lesser to greater the four numbers, and so I did, but it turns out that my answer is wrong.

Teacher₁: you are right I think that in this task I was wrong when indicating the order in the numbers I must edit the answer in Kahoot, to resolve it.

The other modification was guided towards the time assigned to Task 8, where it was asked to select the numerical sequence that was correctly ordered from greater to lesser. Here initially Teacher₁ had assigned 20 seconds to respond, but according to the demand of the task, it is concluded that the time should be increased (SDF2) and that it should be increased to 40 seconds.

Teacher₂: you do not think that, in this task, you would have to give it more time, 20 seconds is very little.

Teacher₁: I think so, here I see that 20 seconds is very little, and the task demands a lot of activity [comparisons between numbers] on the part of the students because they must analyse each of the sequences.

I: You are correct, because there are three sequences, and the time must be increased [...]

The modifications made to prototype 0 were applied to the tasks, now generating Prototype 1 of the interactive lesson. In Cycle 2, Teacher₁ applied Prototype 1, but now to their 46 students in the 3rd grade of Primary Education. The results according to the Kahoot report (shows the percentage of correct answers), allowed to identify three difficult tasks; the first was Task 5 of the puzzle type. Here Teacher₁ pointed out that the students presented difficulties to drag the answer options on the computer (SDF3), since these returned to the initial place, and in the end only 8% of the answers were correct.

I: And how did the students do?

Teacher₁: At first, they found it difficult to sort numbers [Kahoot’s puzzle task] [...] because as you move them on the computer [the answer options] they move again [it was not as they wanted to order them], but I told them not to worry that they could try again.

The second task is Task 7, where it was asked to select the greatest and least number, and you had 20 seconds to respond. Here only 27% of the students had correct answers, with the allotted time being insufficient to respond, so it was increased to 60 seconds (SDF2). The third task is 8, where it was asked to select the sequence that was correctly ordered from greater to lesser; for this, 40 seconds were available. In this case, only 31% of the answers were correct; for this reason, the modifications were focused on increasing the time to 60 seconds (SDF2) and/or changing the type of task (SDF1), trying to use fewer number sequences to decrease the process and reach the solution. The modifications made to Prototype 1 were applied to the tasks, allowing to generate Prototype 2 of the interactive lesson order and sequence.

Teacher₁: [...] the most difficult tasks were Tasks 5, 7, and 8.

I: What happened in Task 7?

Teacher₁: I think that there the time was not enough, I will increase the time to 60 seconds.

I: And what can you say about Task 8?

Teacher₁: [...] well here I think it was the type of task [include three number sequences], although in the notebook we had already seen some examples, at the beginning [...] of the sessions, but still the understanding or I do not know [...].

Teacher₂: ... or time.

Teacher₁: yes, perhaps time...

Teacher₂: How much time did you assign?

Teacher₁: 40 seconds.

I: If you think it wasn’t enough, you can increase it to 60 seconds.

Teacher₁: [...] I still must work more on this type of task.

Retrospective analysis of the training experience

In this last phase, the retrospective analysis was carried out using the components of didactic suitability (Figure 8). In line with the above, Table 5 presents the assessment of the epistemic, cognitive, affective, mediational, interactional, and ecological facets of the formative experience (^{GODINO, 2013}).

Source: Created by the authors based on ^{Godino (2013)}.

Figure 8 Components of didactic suitability