1 Introduction

The significance of incorporating instructional videos into teaching and learning mathematics has been widely acknowledged (Cevikbas & Kaiser, 2023; Engelbrecht et al., 2020). Many students use instructional videos autonomously, which reflects their positive attitude toward this instructional resource (Kay, 2012). Besides the positive perception that instructional videos receive, there are several other reasons why they should be used for teaching mathematics. The general advantages of videos include the opportunity they provide students to adapt the video to their own pace of learning (Kay & Kletskin, 2012). Regarding mathematical modeling, Stillman (2017) describes the potential of using instructional videos to facilitate critical thinking and to enhance meta-knowledge about modeling. Drawing on those aspects, instructional videos seem promising for supporting students in the challenging domain of mathematical modeling. Existing frameworks for instructional video design, however, relate to more algorithmic activities (Kay, 2014) or target the explanation of principles (Kulgemeyer, 2020). Thus, little is known about how to design instructional videos for non-algorithmic domains, which includes mathematical modeling. An example of text-based approaches to support students are heuristic worked examples that display a problem and a step-by-step solution while making the strategies that are used explicit (Reiss & Renkl, 2002). Instructional videos offer the opportunity to dynamically demonstrate step-by-step solutions by explaining how and why certain heuristics have been employed. We therefore want to engage the various benefits of instructional videos to provide heuristic worked examples on mathematical modeling.

On the downside, there are several concerns when using instructional videos in the mathematics classroom. Learning is not always enhanced and the design, learning context, and viewing patterns may play a crucial role (Kay, 2012). Moreover, passively watching instructional videos—receiving facts or procedures in tightly wrapped domains—does not reflect a constructive and meaningful mode for students engaging in mathematics. Using videos this way pushes the learning content onto the student, although Borba et al. (2016) described the need to adopt a pull approach when students use technology to learn mathematics. With a pull approach, students select their learning resources, thereby taking control of their learning process and initiating a change toward more personalized, social, open, and dynamic learning (Borba et al., 2016), which highlights the importance of carefully designed instructional videos that enable (collaborative) engagement. As an initial step toward investigating the effects of the use of instructional videos for learning mathematical modeling, considering the students’ perceptions of this learning resource seems fundamental. Referring to the pull approach, Engelbrecht et al. (2020) stated, “Our modern students want a bigger say in how they are taught and what they are taught” (p. 838).

In this paper, we begin by developing a framework for creating instructional videos on mathematical modeling by combining research on heuristic worked examples with research on instructional video design. We then investigate students’ perceptions of this kind of video to provide insights into how to refine the framework for creating instructional videos on mathematical modeling so that students will perceive this kind of video as helpful and feel supported in their learning process.

2 Mathematical modeling as a challenge for students

Starting with a problem from the real world, mathematical modeling involves first simplifying and structuring the problem. By developing a mathematical model and working mathematically with it, the mathematical results are interpreted and critically evaluated with regard to the real-world context (Niss et al., 2007). A modeling cycle idealizes these repeated sub-steps (e.g., Blum & Leiss, 2007). Being able to carry out the steps of the modeling cycle in different contexts is considered a component of students’ modeling competency (Greefrath & Vorhölter, 2016). Different activities, such as representing elements mathematically, are often essential for executing certain steps (Galbraith & Stillman, 2006), and for this, the application of strategies such as drawing may be helpful (Rellensmann et al., 2017). Moreover, as students orchestrate various sub-steps, metacognitive competencies are required (Kaiser, 2007). These include metacognitive strategies, which involve planning, monitoring, regulating, and evaluating the modeling process (Vorhölter et al., 2019). In summation and compared to applying algorithmic schemata in domains such as algebra, mathematical modeling is regarded as a non-algorithmic domain.

Mathematical modeling is a key component of many national school curricula and is considered important for citizen education (Kaiser, 2020). However, it is quite demanding for students. Besides the need to orchestrate various competencies, each step of the modeling process is a potential barrier (Galbraith & Stillman, 2006). Therefore, mathematics educators are looking into how to best support students in mathematical modeling. One promising approach, especially in initial skill acquisition, is to provide heuristic worked examples.

3 Heuristic worked examples as a measure of supporting mathematical modeling

As suggested by Reiss and Renkl (2002), heuristic worked examples combine the approach of worked examples in algorithmic domains used in the early studies of Sweller and Cooper (1985) with aspects of cognitive modeling (Collins et al., 1989). This is reflected in the way they present a problem and a step-by-step solution while making the strategies used explicit. Since solution processes in non-algorithmic domains, such as mathematical modeling, are rarely straightforward, heuristic worked examples include explorative steps, displaying an expert’s realistic solution process (Reiss & Renkl, 2002). Thus, heuristic worked examples on mathematical modeling demonstrate behavior that aids in executing the steps of a modeling process, serving as “models of modeling” (Tropper et al., 2015, p. 1228).

Previous studies in the context of mathematical modeling have examined whether students are able to emulate the demonstrated behavior. After working with heuristic worked examples, students were found to imitate elements from the examples (e.g., estimating values for missing data) (Tropper et al., 2015). Moreover, the quality of those elements improved, and the students were able to better structure their solution process (Tropper, 2019, p. 354). In the same vein, after working with heuristic worked examples, knowledge about activities that help during each step of the modeling process increased (Hänze & Leiss, 2022). It should be acknowledged that while some studies report improved modeling competency, at least for technically oriented modeling problems (e.g., Zöttl et al., 2010), the change in behavior or knowledge did not always result in improved modeling competency (Hänze & Leiss, 2022) or more correctly solved problems (Tropper et al., 2015). In the latter cases, the results regarding outcomes may be due to relatively short interventions (about 45 min) or the fact that one comprehension issue prevented students from finding a correct solution. On the other hand, the fact that a short intervention enhanced integral components and knowledge of the modeling process gives rise to the hopeful outlook that longer-term support, including scaffolding after the intervention, will yield the intended effects.

Taken together, heuristic worked examples were found to enhance knowledge and activities that help within individual steps of the modeling process. So far, a text-based approach has been employed. One reported issue was that material becomes extensive, thereby imposing a high demand on students’ reading skills (Tropper, 2019, p. 137). In this case, combining spoken words and visualizations, as afforded by instructional videos, seems promising. A further potential of presenting heuristic worked examples in videos lies in the opportunity to externalize (cognitive) actions. Thus, using videos to synthesize the approach of (heuristic) worked examples and allowing students to observe the demonstrated behavior may be a powerful combination (Van Gog & Rummel, 2010). The first results supporting this claim in the domain of mathematical modeling were provided by Czocher et al. (2019). In a differential equation course for engineering majors, the instructor provided instructional videos thinking aloud while solving a modeling problem, with a focus on the decision-making process and modeling-specific strategies, thus presenting heuristic worked examples. The majority of students found these videos helpful for solving similar problems. However, the criteria for how to design videos that present heuristic worked examples in the context of mathematical modeling have not yet been developed.

4 Developing a framework for creating instructional videos on mathematical modeling

Existing frameworks for video design take up instructional approaches but, for instance, relate to algorithmic activities (Kay, 2014) or target the explanation of principles (Kulgemeyer, 2020). To meet the learner- and subject-related demands of mathematical modeling, we develop a framework building on the approach of text-based heuristic worked examples while considering instructional video guidelines. Heuristic worked examples on mathematical modeling consist of the following components (Reiss & Renkl, 2002; Zöttl et al., 2010): a modeling problem; the solution with the involved process; and learner support. Figure 1 specifies the relationship between these components when instructional videos are used as the medium of presentation. The modeling problem and the solution with the involved process are intertwined and accompanied by learner support. Moreover, when providing a heuristic worked example through an instructional video, the specifics of this medium need to be considered both within each component and regarding the overall presentation. Therefore, the multimedia presentation forms an overarching frame (see Fig. 1) which, like many approaches to designing instructional videos, is based on Mayer’s 15 guidelines from the cognitive theory of multimedia learning. These guidelines aim to reduce extraneous processing due to confusing material, manage essential processing resulting from the complexity of the material, and foster generative processing, so that students make sense of the material (Mayer, 2020, pp. 398–402). As some of the guidelines have been tested solely for non-video media (Fyfield et al., 2022), the framework focuses on those supported by instructional video studies.

Fig. 1
figure 1

A framework for creating instructional videos on mathematical modeling

Full size image

4.1 Modeling problem

Mathematical modeling involves a real-world context, in which a problem arises that can be worked on with the help of mathematics. According to Greefrath and Vos (2021), displaying the problem through video scenes may serve as an authentic context. Therefore, one advantage of using those scenes throughout the modeling process is to help students experience the connection between mathematics and everyday life (Stohlmann, 2012). During the solution process, video scenes may facilitate thinking about the constraints of the situation and provide natural decision points for checking (interim) results (Stillman, 2017). Thus, another advantage of using video scenes in the context of mathematical modeling is to help students make predictions and validate their solutions (Cevikbas et al., 2023). Therefore, using real-world scenes to display a problem within an instructional video on mathematical modeling seems promising.

4.2 Solution with the involved process

When working on mathematical modeling problems, various sub-steps need to be carried out and coordinated. Within each step, different activities and strategies are necessary. These characteristics are addressed through the components segmentation and modeling strategies.

4.2.1 Segmentation

Within heuristic worked examples, it is recommended to structure the solution along a process model of the underlying domain (Reiss & Renkl, 2002). In mathematical modeling, a process model often represents a student-oriented modification of a modeling cycle. Process models, realized as “solution plans,” have been shown to be beneficial in supporting students in autonomous modeling as modeling (sub-)competencies and self-reported use of (meta-)cognitive strategies have developed (Beckschulte, 2020; Schukajlow et al., 2015). A solution plan can serve as a structure to divide a video into chunks, including pauses after each segment. This may contribute to students being able to recognize meaningful steps and may support students in handling the challenge of transient information in videos (Spanjers et al., 2012). Altogether, the solution with the involved process should be structured based on a modeling-specific solution plan with integrated pauses after each segment.

4.2.2 Modeling strategies

To successfully model, a variety of activities needs to be carried out (Galbraith & Stillman, 2006). Different strategies applied in certain steps support the modeling process, such as making a drawing of the mathematical structure (Rellensmann et al., 2017). Explicating those strategies through cognitive modeling, in which experts think aloud when solving problems (Collins et al., 1989), is recommended when presenting heuristic worked examples (Reiss & Renkl, 2002). This includes demonstrating metacognitive behavior (Schoenfeld, 1983) which is seen as one potential of presenting the solution of modeling problems through instructional videos because it may enhance students’ critical thinking and meta-knowledge about modeling (Stillman, 2017). Therefore, it is suggested that an instructor dynamically demonstrates the application of modeling strategies while reflecting out loud on how and why they are used. This is one of the differences to creating instructional videos in algorithmic domains (e.g., Kay, 2014) but is deemed necessary due to the complex processes mathematical modeling involves.

4.3 Learner support

In order to facilitate learning from examples, it is important that students engage in knowledge-generating activities (Chi & Wylie, 2014) and then be given the chance to apply their understanding in practice problems (Renkl, 2021). Therefore, learner support should be provided, including self-explanation prompts and subsequent related modeling problems.

4.3.1 Self-explanation prompts

When students study (heuristic) worked examples, a common scaffold to support processing involves self-explanation prompts (e.g., Zöttl et al., 2010). The goals include students explaining underlying principles to themselves or anticipating the next step of the example (Renkl, 1997). Prompts can be seen as a generative activity, and it has been recommended that they be included in instructional videos so that students actively engage with the presented material (Fiorella, 2021). Having students construct knowledge collaboratively might be particularly beneficial when watching a video (Chi & Wylie, 2014). When including self-explanation prompts in heuristic worked examples on mathematical modeling, the number of example elaborations and knowledge about activities that help within each step of the modeling process has increased (Tropper, 2019, pp. 350–360). On the other hand, the study by Hänze and Leiss (2022) showed no effects of prompts on modeling competency and knowledge about activities that help within each step of the modeling process. However, the quality and quantity of self-explanations were not investigated, while being able to generate correct self-explanations may be decisive for students to learn successfully (Rittle-Johnson et al., 2017). Overall, it seems promising to include self-explanation prompts to support students in (collaboratively) constructing knowledge. However, a careful design of self-explanation prompts adapted to the students’ level of learning is of great importance, including scaffolds such as on-demand instructional explanations (Rittle-Johnson et al., 2017).

4.3.2 Subsequent related modeling problems

Since the goal of heuristic worked examples is for students to eventually solve problems autonomously, students need to engage in subsequent problems when they have acquired a certain level of comprehension (Renkl, 2021). This may be especially important when learning from instructional videos because videos tend to be perceived as “easy,” compared to printed material (Salomon, 1984). Hence, students do not necessarily invest the required effort to make sense of the content and when working on subsequent problems, students become aware of their (insufficient) understanding. As in other studies on heuristic worked examples on mathematical modeling (e.g., Hänze & Leiss, 2022), subsequent problems are related concerning the mathematical content. Moreover, related problems would involve the aspects that should be learned from the video. Therefore, in the context of mathematical modeling, subsequent problems should enable the application of those modeling-specific activities and strategies that the instructional video seeks to enhance in a related mathematical domain.

4.4 Multimedia presentation

What is known from research on instructional videos has already been considered in each of the previous components (e.g., the generative activity principle (Fiorella, 2021) in the category of “self-explanation prompts”). We specify here further criteria for multimedia presentation regarding the overall presentation. An instructional video should be kept as short as possible and exclude facts not related to the learning goal. It is important to note though, that explanations should always be adapted to the specific target groups’ previous knowledge so that students are not overwhelmed. On the other hand, certain strategies students may be familiar with do not need to be explicated in detail. One objective is to avoid the expertise reversal effect which states that learning can be hindered if students are confronted with information of which they are already aware of Kalyuga (2021). Taken together, including points where learners can skip explanations may keep the video length for learners with much prior knowledge short while learners with less prior knowledge have the chance to listen to those explanations. Finally, learners should be given control over pausing, fast-forwarding, and rewinding (Fyfield et al., 2022).

5 Students’ perception of using instructional videos as a resource for learning

Students’ perception when using instructional videos plays an important role because it may affect their will to engage with a learning resource (Engelbrecht et al., 2020). Research in STEM education suggests that a feature students perceive as being important is a clear structure of the video (Beautemps & Bresges, 2021). In worked example videos in algorithmic mathematical domains, students appreciated the step-by-step development of the explanation. Additionally, an explanation using dynamic visualizations was an important component (Kay & Kletskin, 2012). However, the explanation should be efficient and the video length should be appropriate (Shoufan, 2019). In addition, in the study by Kay and Kletskin (2012), students liked the opportunity to actively practice the presented content, working through a related problem while watching the video. Students perceive videos as being a convenient opportunity for learning since they can be watched anywhere and at any time whenever support for learning is required (Kay, 2012; Kay & Kletskin, 2012).

Besides their positive perception, students may find not being able to ask questions to be a challenge. Moreover, some students struggle with the self-discipline required to watch a video as opposed to following in-class instruction (Kay, 2012).

6 Research questions

The design criteria for creating instructional videos on mathematical modeling in Sect. 4 were developed from the approach of heuristic worked examples and design guidelines for instructional videos. Promising results from the study by Czocher et al. (2019) suggest that instructional videos presenting heuristic worked examples on mathematical modeling are perceived as helpful by students. However, a detailed analysis of students’ perceptions of instructional videos on mathematical modeling against the backdrop of systematically developed design criteria has not been conducted yet. So far, research in STEM education reveals several aspects that may contribute to students’ perception of a video (see Sect. 5). It is not clear, however, if these aspects apply to the proposed instructional videos on mathematical modeling. For example, making the decision-making process transparent may contradict an explanation perceived as efficient by students.

To refine the developed framework based on the students’ perceptions, the following research questions are addressed:

RQ1

What advantages do students perceive when they work with an instructional video on mathematical modeling and solve a subsequent related modeling problem?

RQ2

What challenges do students perceive when they work with an instructional video on mathematical modeling and solve a subsequent related modeling problem?

It is important to acknowledge that fulfilling all design criteria in their entirety may not always be feasible. An example is the potential conflict between displaying a realistic solution process and keeping the video length short. Therefore, a third research question is investigated:

RQ3

How are the perceived advantages and challenges of working with an instructional video on mathematical modeling and solving a subsequent related modeling problem connected?

7 Method

7.1 Participants

The study was conducted at secondary schools in North Rhine-Westphalia (Germany). A total of 28 upper-secondary students divided into 14 pairs participated in the study (17 female and 11 male students, Mage = 17.46, SDage = 1.14). The students participated in pairs to enable a discussion about the self-explanation prompts included in the video. To form the pairs, we followed the mathematics teacher’s recommendation concerning social compatibility. The students were then selected following the purposeful sampling strategy of maximum variation concerning their mathematics grades (Patton, 2015, p. 283) to gain a detailed understanding of the advantages and challenges perceived by students with different mathematical abilities. Their mathematics grades ranged from “very good” to “poor”. Participation in the study was voluntary and written consent was obtained from all the students and the parents of underaged students.

7.2 Design and procedure

The student pairs participated during regular school days outside classroom lessons. In alignment with research on heuristic worked examples, this study was directed at learners with little experience regarding mathematical modeling because they need more guidance while learners with more experience may benefit more from less-guided instruction, including autonomously solving problems (Kalyuga, 2021). We checked student pairs’ experience by having them work on a modeling problem in the beginning. In the second phase, the student pairs watched an instructional video on a modeling problem; they then worked on a subsequent related modeling problem and were allowed to return to the video. The final stage of the study comprised a semi-structured interview that included the following questions corresponding to RQ1 and RQ2: What advantages do you see in using such an instructional video? What challenges did you face while working with the video? Are there aspects that we didn’t mention in the interview but that you would like to add? The student pairs reflected on those questions together and the interviewer asked students to elaborate on unclear or general statements, for example: “I thought the video was cool (laughs). With the swimming pool.” Moreover, if students did not relate to the video at hand (see Student A’s excerpt in Sect. 7.4), the interviewer encouraged students to refer to the video used in the study. Either an instructor who had been briefed about the study or the first author conducted the interview and explained the procedure, including the interactive video features that were employed (see Sect. 7.3); this was done following a script to ensure comparability of the sessions. The instructor did not answer content-related questions during the working process. By attending the sessions, their role was to ensure that the session went smoothly and to better comprehend what students were referring to in the interview. To investigate the connection between the perceived advantages and challenges of working with an instructional video on mathematical modeling and solving a subsequent related modeling problem (RQ3), we used the interviews as well (see Sect. 7.4). Qualitative methods were chosen because they offer an effective way of understanding students’ perspectives and unanticipated consequences can be taken into account (Patton, 2015, p. 13). For each phase, the students were given as much time as they needed. The complete procedure took 130 min on average and was videotaped.

7.3 Materials

For the instructional video, the “lifeguard problem” presented in Fig. 2 was chosen because finding the optimal path from A to B with parts of the path being differently weighted can be considered a typical optimization problem in calculus (Pennings, 2003). Moreover, activities that are characteristic of mathematical modeling, such as making assumptions (see Sect. 2), need to be carried out. An instructional video based on the developed framework (see Sect. 4) was created. Real-world scenes from a swimming pool were used to display the situation (see Fig. 3). The step-by-step solution was segmented based on the solution plan by Beckschulte (2020) which involves the steps of understanding & simplifying, mathematizing, working mathematically, interpreting, and checking. Following a realistic solution process, it was run through twice, since checking the initial solution showed the need for improvement. The H5P video editor enabled the inclusion of interactive features, such as pre-determined pauses after each segment. Modeling strategies were modeled by the instructor thinking aloud. Each step of the solution plan was addressed with a principle-based and an anticipative prompt. Students were prompted to either discuss a step (see Fig. 4) or to note a solution on a corresponding worksheet (see Fig. 5). When the students continued watching the video, one possible solution was outlined as feedback. To minimize the expertise reversal effect (Kalyuga, 2021), the students were able to skip the instructional explanations at two points with a clear-cut result. A subsequent related optimization modeling problem (see Fig. 6) contained different surface features (context), but the same activities could be applied (e.g., using a reference value). The mathematical model needed to be modified because the boundary values had to be treated differently. As learner control, the students were given the opportunity to pause, fast-forward, and rewind the video. In addition, a table of contents enabled switching between segments. Since the solution for the lifeguard problem can be quite complex and a realistic rather than an idealistic solution process was displayed, keeping the video length as short as possible still resulted in a video length of 29 min and 32 s. The length of the longest segment was 5 min and 12 s.

Fig. 2
figure 2

Overview of the lifeguard problem presented in the instructional video. The problem is based on Pennings (2003), describing a similar situation for a dog

Full size image
Fig. 3
figure 3

A frozen image from the real-world scenes which was used throughout the solution process

Full size image
Fig. 4
figure 4

Assignment 2 as an example of a principle-based prompt included in the video

Full size image
Fig. 5
figure 5

Assignment 8 as an example of an anticipative prompt included in the video

Full size image
Fig. 6
figure 6

Overview of the tunnel problem used as the subsequent related modeling problem. The idea stems from a long-term discussion concerning a neighborhood in Munich (Germany) (e.g., Schubert (2022)). Map data from OpenStreetMap, license CC-BY-SA 2.0

Full size image

7.4 Evaluation method

To systematically describe the aspects that the students named as perceived advantages and challenges, we followed the principles of thematic qualitative text analysis (Kuckartz, 2014, pp. 69–88). The category frame built in this cyclic process aims to structure and describe the material in depth. First, the interviews were transcribed. In the second step, the main categories perceived advantages (RQ1) and perceived challenges (RQ2) which correspond to the interview questions were formed. Third, the interviews were coded with these two main categories. Coding units were defined through thematic criteria which means that one unit ends when the topic changes. Creating those units while coding is typical for qualitative text analysis and coding units may overlap so they are comprehensible out of context (see Student E’s excerpt in Sect. 8.3 which was later on coded as structure, self-explanation prompts, and duration). Moreover, the material is not necessarily coded seamlessly if components of the text are not meaningful with regard to the research questions, for example:

I often watch videos from the internet like Daniel Jung [German YouTuber] or something like that/ these are in principle already super famous personalities. So actually every German student knows them. And um/ that is of course super helpful. (Student A)

The fourth step encompassed the data-driven development of categories applied to the compiled text passages of the main categories (Kuckartz, 2014, pp. 58–60): If a relevant passage did not fit into an existing subcategory, a new subcategory was created. Afterwards, all text passages regarding each subcategory were compiled to collapse similar categories and to split categories that ended up containing different themes. Because perceived advantages and challenges were either directly linked to the design of the video or to the processes of students’ self-regulated use of the video (self-regulated learning), the sub-categories were grouped accordingly to increase the internal consistency of these categories. To ensure the dependability of the process, the category frame was discussed with researchers experienced in the method and in conducting research on digital technology. This resulted in clarifying some of the category definitions. Using the revised and final coding scheme (see Table 1), all data were coded again. Sixty percent of the data were coded by a second well-trained rater and the interrater reliability was very good (Cohen’s Kappa κ = 0.83) (Landis & Koch, 1977).

Table 1 Description of the categories related to the perceived advantages and challenges
Full size table

Finally, to investigate how the perceived advantages and challenges were related (RQ3), we determined and analyzed the co-occurrence of codes.

8 Results

An overview of all the categories regarding the perceived advantages and challenges with the number of groups that referred to each category is provided in Table 2. We began by analyzing the perceived advantages of working with an instructional video on mathematical modeling.

Table 2 Number of groups for which a category related to the perceived advantages and challenges was coded
Full size table

8.1 Perceived advantages (RQ1)

The students reported advantages relating to the design of the instructional video. Referring to the overall structure of the video, they liked how the video was based on an example (lifeguard problem). They found the step-by-step explanations structured by the solution plan to be particularly helpful for planning and monitoring the solution process, as the following excerpt underlines:

I thought that [proceeding along the solution plan] was good because you always worked through the individual steps. You always knew which step you were at and how to proceed. (Student B)

Another positively perceived feature of the video was the explanations, which were regarded as an alternative approach to those of the teacher (not necessarily better or worse) that could be re-watched if necessary. The explanations clarified how each step was linked to the next and were illustrative through their connection to real-world scenes. Using visualizations, such as the dynamic display of calculations, aided understanding. The real-world scenes that were included made it more realistic to work on a modeling problem and emphasized the importance of considering the context when making assumptions. As one student stated:

What I also liked is that the paths were drawn on the picture of the pool, so one could visualize that—not that there was a sketch first, but that one could keep an eye on different things. Like when it was said “you have to consider the starting platforms.” With a sketch, you wouldn’t have seen that at all. (Student C)

The advantages described above relate to the design of the video, whereas the following advantages refer to how the students described making use of certain features for their self-regulated learning. The students liked having control over technical features such as pausing, fast-forwarding, rewinding, and skipping parts of the video (adaption to the level of learning). In particular, they saw the option of rewinding as an advantage over classroom instruction because some felt that they could not ask the teacher to repeat the same thing over and over again; in addition, they would avoid asking questions during class when they thought they were expected to know about a topic. The students also took the opportunity to skip explanations after two self-explanation prompts to omit information that was unnecessary for them. Some still liked having the opportunity to listen to explanations even though their answer had been verified. Moreover, the integrated self-explanation prompts supported concentration and were used as starting points for the students to create their own parts of the solution to the lifeguard problem. Additionally, discussing the prompts with a peer helped to monitor the students’ understanding of what they had seen so far. Some students mentioned that the explanations related to the prompts and those in the video in general served as a direct control of their thinking process.

Regarding the perceived advantages, the structure, the explanations, and the adaption to the level of learning were the ones that were most frequently mentioned (see Table 2).

8.2 Perceived challenges (RQ2)

The students reported challenges that were either related to the video’s design or to the processes of self-regulated learning with the video (see Table 2).

A perceived challenge that resulted from the design of the video was the duration (29 min and 32 s). Seven groups referred to this issue, making it the most frequently mentioned challenge.

Concerning the processes of self-regulated learning, the challenge of transfer was mentioned by two groups. They found it hard to transfer what they had seen to a related problem, referring to the proposed approach to modeling and the mathematical content. Since there was no teacher present, one group missed the teacher–student bond and found it challenging that they could not turn to the teacher when questions arose.

Like I said, I also have private tutoring. It’s just that not everyone can afford it but it’s not all that bad. But then I also have this teacher connection, so to speak, where I can directly ask how it works or “can you explain it again?” (Student D)

One group stated that they would struggle with self-discipline if they watched the video by themselves. They would skip the prompts and watch the video in its entirety if there were no peer present.

8.3 Connection between perceived advantages and challenges (RQ3)

In the following, we take a closer look at co-occurrences of the mentioned challenges and advantages, analyzing the extent to which the perceived challenges were put into perspective by the advantages.

The duration of the video was mentioned by seven groups. Three groups thought that the structure of the video (i.e., the step-by-step approach) and the integrated pauses, combined with the self-explanation prompts, helped them stay focused.

So, if I had really watched this video for half an hour and there had not been any breaks, I would not have had to do anything. Then I think I would not have paid attention so much. Because it does help/ it’s hard to just completely focus on listening to it. Okay, half an hour might even work. If you sometimes do something on the blackboard with teachers, it can take that long but (-) um, it helps. Especially when you try to explain something step-by-step. That you then do tasks in between. As long as they don’t take longer than the video. But that was okay here. (Student E)

That it was possible to skip explanations (adaption to the level of learning) was mentioned by two other groups as a mitigating factor for the video duration being perceived as too long. Two groups, however, still wished that the video had been shorter.

Although the transfer of what had been seen to the subsequent related modeling problem was perceived as a challenge by two groups, both groups still mentioned that they felt supported by the structure of the video. Specifically, they referred to the solution plan as a guide for how to approach modeling problems and to the easy-to-follow explanations based on an example.

One group missed the teacher–student bond and not being able to ask the teacher questions (category no teacher). They saw the adaption to the level of learning as an advantage of the video only if they were not complete beginners to the topic and wanted to repeat certain aspects. Accordingly, being introduced to a new topic would require a teacher that they can turn to on demand.

Regarding the challenge of self-discipline, we did not find any co-occurrences of advantages when this challenge was mentioned.

9 Discussion

Instructional videos are frequently used for mathematics learning and offer various advantages. This motivated the development of a framework for creating instructional videos on mathematical modeling. However, it is important to consider the students’ perceptions of a digital technology so that they would feel supported in their learning process (Engelbrecht et al., 2020). Therefore, we conducted interviews on the advantages and challenges the students perceived after they had collaboratively worked with an instructional video on mathematical modeling and solved a subsequent related modeling problem. On the one hand, the advantages point to the strengths of the framework for creating videos on mathematical modeling; on the other hand, the challenges the students faced help to elaborate the adaption of the framework to further support students.

9.1 Perceived advantages (RQ1)

Previous findings have indicated that the structure of an instructional video is important to students (Beautemps & Bresges, 2021). The students’ positive impression of the video structure was one of the most frequent advantages and suggests that, as in the study by Czocher et al. (2019), using the heuristic worked example approach within a video on mathematical modeling is helpful. Segmenting the video according to Beckschulte’s (2020) five-step solution plan made the steps for solving a modeling problem transparent. Thus, from the students’ perspective, using such a strategic instrument supports the identification of meaningful chunks in videos on mathematical modeling.

Another frequent advantage in this study was the explanations. Students noted that the explanations clarified the connections between different steps of the modeling process. The explanations in the video did not only model the applied strategies but also made the decision-making process transparent. Because previous studies found that an efficient explanation and a video length as short as possible are important to students (Shoufan, 2019), these findings suggest that the efficiency of an explanation within a video needs to be considered against the backdrop of the domain. While an idealized solution could also be displayed, shortening explanations on the decision-making process may be the wrong end to save at, pointing to the importance of a realistic solution process within instructional videos on mathematical modeling from the students’ perspective.

Prior studies have indicated that students like the use of dynamic visualizations in videos (Kay & Kletskin, 2012). Implementation of visualizations was also seen as advantageous in the case of the provided video. Specifically, the students felt that visualizing the modeling problem through scenes from the real world—using the video and a screenshot throughout the solution—made the process more realistic. Hence, using video scenes to visualize the modeling problem in instructional videos could be worth the effort.

Regarding the students’ self-regulated use of videos, the results support that including elements of adaptivity, such as giving students the option to skip certain explanations, is not only important from a theoretical perspective in order to minimize the expertise reversal effect (Kalyuga, 2021) and to keep the video length short but also one of the most frequently reported advantages by students. Thus, the elements of adaptivity included in the developed framework (learner control and opportunity to skip explanations) can be seen as essential.

Moreover, some students appreciate it when they are supposed to practice the video content (Kay & Kletskin, 2012). That the students positively perceived the interactive feature of the self-explanation prompts is encouraging, since they targeted active construction and communication when collaboratively watching a video on mathematical modeling. This mode of communication may contribute to changing the use of videos as a somewhat one-way communication, which has been seen to be problematic in learning mathematics with technology toward more social learning (Borba et al., 2016). Finally, the students reported that they liked using the explanations to directly control their understanding which underlines the importance of well-considered explanations in instructional videos on mathematical modeling.

Overall, the framework was designed for creating instructional videos that are not just received passively. The students’ perceptions of their involvement as an advantage is an encouraging step toward them engaging in a constructive mode and potentially leading to the acquisition of mathematical modeling competencies (Kaiser, 2007; Stillman, 2017).

9.2 Perceived challenges and their connection to perceived advantages (RQ2 and RQ3)

The most frequently perceived challenge that resulted from the design of the video was the video’s duration. The pauses, the included prompts, and the opportunity to skip certain explanations were seen as mitigating factors by some of the students in this respect. However, the issue of extensive material, which is evident in text-based research on heuristic worked examples (Tropper, 2019), cannot be completely resolved by using a combination of visualizations and verbal explanations. With videos, it may be even harder for students to identify parts to skip compared to text, where information can often be scanned for an overview. Therefore, integrating more positions for skipping information, as determined by the instructor, could reduce the video length for more experienced learners. However, this also leads to a limitation of using an instructional video on mathematical modeling. A video can only be fragmented to a certain extent, and when most of the explanations provided can be skipped, an extensive video might not be an appropriate approach for learning. In this case, several shorter videos—each explicating one strategy—could be used on demand during autonomous modeling.

A challenge concerning the processes of self-regulated learning the students faced after working with the instructional video was transferring what they had seen to the subsequent related modeling problem even though they had felt supported by the structure of the video to some extent. That students struggle with transfer in the sense of correctly solving a modeling problem as a whole has also been evident in studies of text-based heuristic worked examples (e.g., Hänze & Leiss, 2022). This challenge may be one reason why the students found it difficult that there was no teacher they could ask. Therefore, further learner support needs to be considered. One consequence might be the teacher’s adaptive support, scaffolding the transition from heuristic worked examples to autonomous modeling (Tropper et al., 2015). Moreover, an overview page displaying the steps of the solution plan could guide this transition. The students were able to return to the instructional video any time when solving the related problem, assisted by a table of contents, but this may have been too complex. One potential explanation is that the students lacked adequate strategies for using this offer, so training them on how to use the interactive video features might be necessary. Another challenge described in the literature is that self-discipline can be difficult when students watch a video in a self-regulated way (Kay, 2012). One student mentioned that she would skip the prompts without a peer present. In line with what is known about self-explanation prompts (Rittle-Johnson et al., 2017), this again highlights that prompts need to be designed carefully and that prompts for individual learners may need to differ from prompts for collaborative learning from videos. For example, individual learners could be asked to write down all answers, potentially guided by a structure. In sum, while instructional videos in previous studies have often been used autonomously at home (Kay, 2012), the findings in this study suggest that this usage scenario may need reconsideration, especially in a challenging domain such as mathematical modeling.

All in all, Fig. 7 refines the developed framework for creating instructional videos on mathematical modeling based on the students’ perceptions, thereby furthering the knowledge of using instructional videos on mathematical modeling.

Fig. 7
figure 7

Refinement of the developed framework for creating instructional videos on mathematical modeling based on the students’ perceptions

Full size image

9.3 Limitations

Using qualitative methods allowed for us to gain insight into students’ perspectives and identify which video features they liked or why they found certain aspects difficult. Before participating, the students were assured that their participation would not affect their grades and encouraged to answer honestly. It is possible, though, that some students still answered in a way that they thought would be socially acceptable and might have focused more on the advantages than the challenges.

Moreover, the scope of the study was limited to a laboratory setting without a teacher present. The need for improving the learner support makes it even more important to implement this video format in an in-class setting with a teacher present and further scaffolds. This illustrates that merely providing instructional videos and a corresponding problem may not be sufficient in a complex domain such as mathematical modeling.

Since our findings are limited to the age group of upper-secondary students, further research is required to explore younger students’ perceptions by using an instructional video adjusted for this age group. It is possible that additional challenges arise for younger students since they are often less used to working autonomously.

All in all, when interpreting the data, it is important to bear in mind that the framework was implemented for one instructional video only. This allowed us to investigate students’ perceptions in detail based on a video covering a characteristic modeling problem in upper-secondary calculus. To develop a full picture of the advantages and challenges students perceive when working with an instructional video on mathematical modeling, studies will be needed using further videos based on the developed framework.

10 Conclusion

Overall, this paper contributes to a growing body of research on instructional videos by developing design criteria for creating instructional videos on mathematical modeling based on the approach of heuristic worked examples (Reiss & Renkl, 2002) and on instructional video design criteria. It also addresses the need for further research on heuristic worked examples in the domain of modeling (Renkl, 2017). The results suggest that there was for the most part a positive perception of the developed video format, which gives rise to the hopeful outlook that students will engage in it. Consequently, studying the effects of the use of this video format on students’ learning about modeling is a necessary next step. The reported challenges resulted in a discussion of suggestions for improving the learner support and for further measures to adapt the video format to the students’ level of learning. As a result, the developed and refined framework may serve as a starting point for creating and investigating instructional videos on mathematical modeling. Based on the design criteria elaborated here, this video format could be concretized for other age groups or other non-algorithmic mathematical domains, such as proving or problem solving. Heuristic worked examples in those domains have much in common with the ones used in mathematical modeling but the underlying process model and strategies would have to differ. This would expand the research on using instructional videos for learning mathematics.