Abstract
Modelling activities serve to integrate real-life objects into STEM classes. This article investigates different settings for modelling with real objects and their representations both inside and outside the classroom. Using the example of mathematics, the settings working outside the classroom at the real object, inside the classroom with photos, and inside the classroom with a 3D model are considered and compared in an explorative study with 29 students. Questionnaire items provide information about the students’ perceptions of the different settings. The results report significant differences in the simplifying and structuring step as well as in the mathematising step when comparing the settings outside and inside with photos. The results are taken up for generating hypotheses concerning the role of the outdoors in interdisciplinary STEM modelling activities.
1 Introduction
In recent years, the physical space outside the classroom has gained in importance for education. The aim to connect the students’ real world and their everyday learning inside the classroom has led to outdoor learning activities throughout all school subjects. In particular, skills linked to STEM (Science, Technologies, Engineering and Mathematics) can be fostered through active engagement with their environment (Haas et al., 2021). It becomes particularly obvious that real world situations do not clearly divide related phenomena, questions and activities in single subjects – an issue that is clearly linked to an interdisciplinary STEM approach (Haas et al., 2021).
Up to now, research on outdoor STEM activities mostly refers to the observation and description of activities related to one of the STEM disciplines. Nevertheless, modelling activities can be seen as a link between these disciplines. Therefore, it seems legitimate to generate hypotheses for modelling in interdisciplinary outdoor STEM activities from observations on modelling in one discipline. In the article, this is done for an outdoor activity related to the subject of mathematics. Qualitative preliminary work highlights similarities and differences related to students’ modelling activities in the comparison of working indoors and outdoors (Jablonski, 2023). Based on these findings, a further research interest for this article emerges. In particular, it is the question of how far students perceive the outdoors as a setting for modelling activities differently from similar settings inside the classroom. This question will initially be narrowed down to the subject of mathematics and extended for interdisciplinary STEM activities in the course of the discussion.
2 Theoretical Background: Modelling in Outdoor STEM Education
“STEM education is an interdisciplinary approach to learning where rigorous academic concepts are coupled with real world lessons as students apply science, technology, engineering, and mathematics in contexts that make connections between school, community, work, and the global enterprise.” (Southwest Regional STEM Network, 2009, p. 3; cited in Bergsten & Frejd, 2019, p. 942).
Despite giving a definition of STEM education, the introduced quote emphasizes pedagogical approaches linking STEM disciplines: modelling and making connections (cf. Leung, 2018). For example, when following the STEM curriculum (UNESCO, 2020), the practice of Developing and Using Models is presented. In particular, it is pointed out that models are relevant in science (e.g. to explain natural phenomena), mathematics (e.g. to describe circumstances) and technology/engineering (e.g. to provide analogies in problem-solving). Thus, modelling is relevant for all STEM disciplines. Still, both similarities and differences in modelling in the different disciplines are evident: A shared aim of modelling practices is to integrate real-life situations and objects into STEM education (Hallströn & Schönborn, 2019). Precisely because of its clear relation to reality, modelling distinguishes itself from classical school problems. It is thus intended to arouse students’ interest (Hartmann & Schukajlow, 2021) and motivate sense-making (Holmlund et al., 2018). Further similarities aim to describe and simulate reality by choosing, extracting, and using relevant data. Also, the procedures of using different representations, e.g. diagrams and drawings, are part of all STEM disciplines. Differences can be seen in science modelling mainly aiming at explaining phenomena, whereas an additional focus in technology and engineering modelling is on problem-solving and abstraction. Mathematical modelling differs mainly through the role of mathematical structures and in the representations used, e.g. functions. Still, mathematical representations and patterns can be found in the science, technology and engineering modelling practices, too (cf. UNSECO, 2020).
Apparently in the context of outdoor STEM activities, models are important to describe or experience natural phenomena actively. Outdoor education is defined “as a way to engage children with nature through educational contexts” (Bentsen et al., 2010, p. 235). It aims at linking educational contents to the students’ environment, potentially leading to improvements in performance, motivation and physical activity (Bentsen et al., 2010). Taking a look at recent outdoor STEM activities, especially the aim to contextualise concepts by connecting them to the real world becomes obvious (Crompton, 2020). For example, Haas et al. (2021) point out the potential of architecture that “needs mathematics, engineering, […] science and technology” (p. 205) in its construction. Further examples show that outdoor activities are often introduced for one discipline related to STEM (e.g. Gaio et al., 2020; Gross & Harmer, 2020; Wetzel et al., 2020).
In order to generate hypotheses for outdoor STEM modelling, the focus is placed on the example of mathematical modelling in the following. This choice can be grounded in recent discussions relating to the role of mathematics in the STEM approach. Gravemeijer et al. (2017) describe applying and modelling as a mathematical skill being required for the future society and Frejd and Bergsten (2019) highlight the role of the development and application of mathematical models in STEM-related work contexts. In particular, from the former outlined similarities and differences in the STEM modelling practices, the role of mathematical representations and patterns is to be highlighted here (cf. UNESCO, 2020). References to the interdisciplinary STEM approach will be made at appropriate places, particularly in the discussion section.
In Blum and Leiß (2007), the mathematical modelling process is ideally described as a cycle (cf. Fig. 1): Students understand a situation from reality, simplify and structure it through model assumptions, in order to transfer this model into mathematics through mathematisations. With the help of mathematical work, students achieve a result that is interpreted in reality, where it is subsequently validated and presented. Especially in terms of the second step, strong relations to STEM education can be seen since “models simplify aspects of reality” (Hallströn & Schönborn, 2019, p. 3).
The modelling cycle
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
Blum and Leiss (2007, P. 225)Real situations and objects are usually the basis to start a modelling task and can be integrated into it in different ways. In the following, the term setting is used to refer to the initiation of a modelling activity, i.e., with reference to Blum and Leiß (2007), the representation of the real situation. A literature review on modelling settings allows the distinctions of different settings of which three are further examined:
-
photos, if necessary with reference value (e.g., Hartmann & Schukajlow, 2021),
-
3D models, if necessary with reference value (e.g., Lavicza et al., 2020; Medina Herrera et al., 2019),
-
real objects (e.g., Ludwig & Jablonski, 2021).
These forms of presentation use different representations in the sense of replications of a real situation or a real object and already allow the starting point of a (not only mathematical) modelling activity to vary significantly: Photos give a visualization of the object/situation, whereas compared to the real object or the 3D model, they are “only” the two-dimensional representation of the three-dimensional reality, which does not allow touching the object. The comparison of the replicated 3D model in the classroom and the real 3D model in reality also offers further distinctions regarding the mode of representation, e.g. the perception of the real object in relation to one’s own body size. A detailed classification of all settings can be found in Jablonski (2023) and is summarized in Table 1. A theoretical basis for the distinction in Table 1 arises from the following work:
-
According to Kolb (1984), the Experiential Learning Theory states that learning is based on experiences, e.g. with the environment. Accordingly, it can be concluded that the Location of learning is crucial, e.g. indoors (in the sense of the classroom) and outdoors (in the sense of the environment outside the classroom).
-
The categories Representation, Artefacts, and Modality are distinguished according to Duijzer et al. (2019). It is concerned with the mode of representation and the use of materials.
-
In the Data Collection category, different ways of using required data to mathematise and solve the modelling task are considered. Greefrath (2009) provides the basis.
Comparison of the different task settings (cf. Jablonski, 2023)
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
The presentation of different settings shows that despite bringing representations of real situations and objects to the classroom, the outdoors can be used as a setting itself. The idea of so-called outdoor mathematics is to work on mathematical (modelling) tasks directly in real situations or at real objects. With regard to the basic intention of modelling, the connection between reality and mathematics seems almost obvious here. The representation is thus visual by nature and three-dimensional. Hereby, the real object becomes an artefact, which the students can see as well as touch (cf. Duijzer et al., 2019). Furthermore, data collection is possible directly on site and in the combination of estimating, measuring and comparing (cf. Greefrath, 2009) is almost unrestricted in terms of available data. Buchholtz (2021) describes this as a special feature of the outdoor setting and adapts the modelling cycle for tasks at the real object (see Fig. 2).
Adapted modelling cycle for tasks at real objects
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
Buchholtz, 2021, p. 145According to this, being outdoors at the real object plays a role especially in mathematising and validation. Compared to the classical modelling cycle, data collection is particularly highlighted here, which underlines the emphasis on the context and the real object. This idea can be applied to the interdisciplinary STEM approach, too: what has been described here as mathematisation can be seen under the broader concept of data collection regardless of discipline (cf. Kertil & Gurel, 2016). A general emphasis on this step seems to emerge when STEM tasks are solved outdoors at real objects.
3 State of the Art and Research Question
In preliminary work, qualitative differences between different settings were investigated based on observing students’ modelling activities (Jablonski, 2023). Outside at the real object, the students have more possibilities to take different perspectives and more intense discussions, e.g. about the relevance of inaccuracies. They intend to work as accurately as possible because they are aware of their possibilities of being on site of the object. Measurement is mainly done through comparison and estimation. When working indoors with photos, learners discuss inaccuracies, too, but differently. The students are aware of not being on site and accept that inaccuracies cannot be avoided – a circumstance that leads to a combination of estimating, comparing and measuring for data collection. The step of mathematising is mainly characterized by assumptions about perspective. When working with the 3D model in the classroom, the students do not focus on inaccuracies to the same extent as in the other settings – hypothetically, because 3D printing already simplifies the object. In terms of mathematising, this setting not only requires measuring, comparing and estimating, but also a comparison with a reference object. This leads to a main focus on scaling. These findings emerge from students’ observations, but do not further take into account the students’ perception of the different settings.
The students’ point of view concerning their enjoyment and interest in as well as value of modelling tasks in contrast to word problems and intra- mathematical tasks is investigated by Schukajlow et al. (2012). The authors do not find any significant differences concerning the students’ perceptions in these categories. The role of different modelling settings, i.e. the outdoors, is taken into consideration by Ayotte-Beaudet et al. (2017). On the basis of a meta study, the authors conclude that outdoor education could enhance interest in science activities in general. In addition, (primary) students do not seem to see clear connections between outdoor learning and its scientific value (Ayotte-Beaudet et al., 2017). Still more investigation is needed since the studies do not involve a focus on modelling and the results from different studies are contradictory. Hartmann and Schukajlow (2021), for example, compare the mathematical modelling settings inside and outside to see whether there are differences in the students’ interest. For this purpose, the experimental group solved modelling tasks outside on the real object and the control group solved modelling tasks using photos. The results show no significant differences, which leads the authors to conclude that the relevance of the task topic outweighs that of the setting. However, due to the different possibilities of representation in the classroom, the question arises whether representations other than the photo would lead to different results.
From the theoretical considerations and previous research findings, a further research interest arises. The focus lies on the students’ perception of different modelling settings. Based on Jablonski (2023), it remains unclear whether the observed differences, especially in relation to simplifying and structuring as well as mathematising, are perceived by the students, too. Secondly, this research interest ties in with the findings of Schukajlow et al. (2012) and Hartmann and Schukajlow (2021) by looking at the aspect of value (in the sense of a reference to reality), interest and enjoyment. Especially the results of Hartmann and Schukajlow (2021) will be taken up and extended with regard to the modelling setting indoors with a 3D print model. In combination, the article is based on the following research questions:
[RQ1] How do students perceive different task settings in terms of
-
interest and enjoyment,
-
reference to reality,
-
and the selected modelling steps of structuring and simplifying and mathematising?
[RQ2] Which implications can be drawn from mathematical modelling for modelling in an interdisciplinary STEM approach?
4 Methods
In order to identify the students’ perception of different modelling settings, a study was conducted in 2022 with 29 students in grades 6–8. At the time of the survey, the students attended the enrichment programme Junge Mathe-Adler Frankfurt for the promotion of mathematical giftedness (Jablonski, 2021). This sample was used because it could be assumed that the students would engage with new task formats and solve them with the necessary willingness to exert effort, so that a complete comparison would be possible.
The 29 students were divided into nine groups. Each group worked on three tasks related to real objects in a 90-minute session (see Fig. 3):
-
Determine the height of the Body of Knowledge (sculpture of a seated person with legs drawn up) if it were standing. Give the result in meters.
-
Determine the volume of the stone. Give the result in m³.
-
Determine the surface area of the Rotazione sculpture. Give the result in m².
Overview of the task objects (from left to right: Body of Knowledge, stone, Rotazione sculpture)
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
For each object, three different task settings were defined:
-
Outside at the real object: The students solved the task outside directly at the real object. As tools they had measuring materials with them.
-
Indoors with photos: The students solved the task using a series of photos of the real object with a person as a possible reference size (see Fig. 4). Next to the photos, they had a ruler to measure sizes.
-
Indoors with a 3D print: To solve the task, the students were given a 3D representation of the real object, which was previously printed to scale (see Fig. 5). Due to time and costs, the 3D printer was chosen. This setting is representative of the use of 3D models regardless of the material used. A LEGO figure and a ruler were provided as a reference value.
Photo setting for the stone task
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
3D print model setting for the stone task
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
To ensure that each group worked on each object and setting exactly once, they were arranged in the Latin Square design (cf. Field & Hole, 2002). Thus, each group worked on each setting with an alternative task object (see Table 2). This procedure should ensure that an effect from one setting on another was minimized, i.e. each setting was solved firstly, secondly and thirdly by an equal number of students each.
Group arrangement according to the Latin Square Design
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
After each task completion, a questionnaire was filled out by the 29 students to record their agreement with statements related to each task setting. Agreement for all items were mapped by a five-point Likert scale from 0: strongly disagree to 4: strongly agree. For each task setting, the same questionnaire with four categories and 13 items was used. The + and – markings indicate whether the statement is formulated positively or negatively in terms of the category.
1) Task-specific interest and enjoyment (according to Hartmann & Schukaljow, 2017)
a) IE_1_+: I found solving the task exciting.
b) IE_2_+: I enjoyed solving the task.
c) IE_3_+: I would like to solve such tasks more often.
2) Relevance and reference to reality (based on Wess et al., 2021)
a) R_1_+: In this task, I recognised a reference to reality.
b) R_2_+: I think that such a task has a practical use.
c) R_1_–: I find it difficult to see the benefit of the task for my everyday life.
3) Selected modelling steps – simplifying and structuring (based on Blum & Leiß, 2007)
a) S_1_+: I was able to distinguish well between important and unimportant information.
b) S_1_–: I found it difficult to distinguish between important and unimportant information.
c) S_2_–: Structuring this task was very difficult for me.
4) Selected modelling steps – mathematisation (based on Blum & Leiß, 2007)
a) M_1_–: I could hardly see how I could use mathematical knowledge to solve the task.
b) M_2_–: The selection and collection of data (e.g. measuring/estimating lengths) seemed arbitrary to me.
c) M_1_+: I had no problems in selecting and collecting data (e.g. measurements/estimates).
d) M_2_+: It was easy for me to decide which mathematics I needed to solve the task.
The first two categories can be seen as generally relevant to outdoor STEM activities since they focus on broad concepts that STEM modelling aims at, namely interest and relevance (cf. Bentsen et al., 2010). Categories three and four are formulated based on steps of the mathematical modelling cycle. For simplifying and structuring, the step can easily be transferred since STEM modelling practices generally require the choice of relevant data from reality which requires particularly structuring (cf. UNESCO, 2020). Concerning mathematisation, a major focus on mathematics is evident, especially in terms of M_1_– and M_2_+. Still, the items concerning data collection are transferable in the sense that data extraction is required in STEM modelling independent of the discipline (cf. UNESCO, 2020).
For all categories, the internal consistency of the items was determined in advance with Cronbach’s alpha. Six items (positive and negative) were selected respectively formulated for each of the mentioned categories. The resulting questionnaire was completed by 57 pre-service teacher students for mathematics after they had worked on the stone task indoors with photos. Using the data, three or four items were selected for the main study. Due to test economics and the risk of students getting test fatigue from answering the questions multiple times, it was decided to not use the full set of 24 items tested. Table 3 summarizes the internal consistency results for the final selected items presented earlier.
Results of internal consistency with Cronbach’s alpha
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
According to Blanz (2015), the categories of task-specific interest and enjoyment, relevance and reference to reality as well as simplifying and structuring show a high internal consistency with the previously presented items. In the category of mathematising, the value is acceptable, whereupon one of the items (M_2_+) was linguistically specified. The list already contains the linguistically revised version of the item.
To evaluate the agreement in each category, the agreements of the category’s items were added (if necessary, after prior inversion in the case of negative items), so that a scale from 0 to 12 was formed for categories 1–3 in each case, and a scale from 0 to 16 for category four. In order to record the differences concerning the students’ perception between the settings 3D printing, photos and outdoors in each of the four categories, a one-factor ANOVA was carried out (cf. Field & Hole, 2002). It was chosen since all 29 students evaluated each setting. Therefore, the sample for each setting was invariant. Prior to this, Levene tests were used to check for equality of variance as a prerequisite, which can be assumed for all scales and groups. If the ANOVA showed significant differences, post-hoc tests were then conducted between the three task settings for each category. Significant differences were quantified with Cohen’s d effect size (cf. Field & Hole, 2002). The normal distribution of the data was examined using a Q-Q plot. Based on this graphical analysis, it had to be assumed that the data were not normally distributed. In principle, it can be assumed that a one-factor ANOVA is robust against the violation of the normal distribution assumption (Blanca et al., 2017). Consequently, in addition to the one-factor ANOVA, the non-parametric Kruskal-Wallis test was performed in each case. Both tests always showed the same results with regard to the significance of the ANOVA and subsequent post-hoc tests.
5 Results
In the following, the results of the questionnaire evaluation are presented in the categories task-specific interest and enjoyment (IE), relevance and reference to reality (R), simplifying and structuring (S) as well as mathematising (M). Table 4 first gives a descriptive overview of the responses of the sample. The results of the one-factor ANOVA are presented subsequently.
Descriptive overview of the responses of the sample divided by the settings 3D printing (3D), photo (P) and outdoors (O)
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
5.1 Interest and Enjoyment
For the category interest and enjoyment, no significant differences can be found between the settings 3D printing, photo and outdoors, F(2,84) = .843, p = .434. When comparing the mean values of the sum of all items, it becomes apparent that these are basically very high in all three task settings, i.e. the students showed a high level of interest in the tasks and had fun working on them, regardless of the task setting.
5.2 Relevance and Reference to Reality
Also in the category relevance and reference to reality no significant differences between the task settings can be described, F(2,84) = .742, p = .479. Although the mean values are on average one point below those of the category interest and enjoyment, the tasks with reference to a real object still seem to be relevant for the students.
5.3 Simplifying and Structuring
In the category simplifying and structuring, significant differences between the three task settings can be seen, F(2,84) = 3.611, p = .031. The comparisons using post hoc tests show a significant difference in favour of the outdoor setting compared to the setting indoors with photos. A medium effect can be assumed, p = .032, d = .687. The other comparisons (3D–outdoor; 3D–photos) showed no significant differences (cf. Fig. 6).
Boxplots for the category simplifying and structuring
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
5.4 Mathematising
With regard to the modelling step of mathematising, there are also significant differences between the task settings, F(2,84) = 4.140, p = .019. The only significant post hoc comparison indicated a significant difference between the indoors with photos and outdoors settings with a medium effect in favour of the outdoor setting, p = 0.15, d = .755. The further comparisons did not reveal any significant differences between the task settings (cf. Fig. 7).
Boxplots for the category mathematising
Citation: Research in Integrated STEM Education 1, 3 (2023) ; 10.1163/27726673-bja00016
6 Discussion and Conclusion
With regard to the question How do students perceive different modelling settings in terms of interest and enjoyment, reference to reality as well as the selected modelling steps structuring and simplifying as well as mathematising? the following findings emerge. In the categories interest and enjoyment as well as relevance and reference to reality, no significant differences between the modelling settings can be observed. Especially in the category interest and enjoyment, there was a fundamentally high level of agreement, so that a ceiling effect can be assumed. The new tasks seem to have aroused the students’ interest and they enjoyed the tasks regardless of the setting. This result confirms the results of Hartmann and Schukaljow (2021) and furthermore extends them for the task setting with 3D printing. From the category relevance and reference to reality, it can be concluded that a task setting involving a real object shows a certain reference to reality. However, the actual representation of this real object does not lead to a significant difference in the experience of reality and relevance which contributes to the results of Ayotte-Beaudet et al. (2017).
Significant differences are found when comparing the modelling steps simplify and structuring as well as mathematising for the settings outdoors and indoors with photos. Based on the items, it can be concluded that the students find it clearer to distinguish between important and unimportant information outdoors. Furthermore, the mathematisation process seems to be easier outside. In particular, data collection on the real object seems to be clearer and less arbitrary compared to working indoors with photos. These results can extend the qualitative results on differences between the settings in Jablonski (2023) as follows. For structuring and simplifying, it was observed that students have more opportunities to look at and discuss about the object from different perspectives outside. The agreement with the items at this point allows the hypothesis that this facilitates the modelling step. In comparison, the acceptance of not being on site and therefore accepting inaccuracies due to the representation of the real object was evident in the photo setting, a circumstance that may make simplification and structuring more complex. The measured differences in mathematisation can also be reconciled with the qualitative analysis. Outdoors, measurements and the question of how to measure as precisely as possible are the focus of the discussions. Since the students can usually measure directly at the object, they probably perceive the data collection as less arbitrary compared to working with photos. Because the photo depicts a three-dimensional object in two dimensions, perspective is the primary focus in the discussions of mathematising – nevertheless, or precisely because of this, they seem to perceive the data collection as more difficult and more arbitrary compared to the work outside. This result presents modelling outside, with real objects, as an opportunity to enrich modelling in the classroom concerning the steps simplifying and structuring and mathematising. The agreement to the items in combination with the qualitative observations show that these steps require different activities than working in the classroom and that the students notice these differences compared to working with photos.
Concerning the second research question Which implications can be drawn from mathematical modelling for modelling in an interdisciplinary STEM approach? the following statements can be made. Especially for simplifying and structuring, it can be assumed that different perceptions show up in interdisciplinary STEM modelling activities, too. In order to model a phenomenon of nature or to find a generalizable approach, simplification and structuring activities are indispensable. Since the students find it clearer to solve the tasks outside and are able to make their own simplifications in reality, it is quite conceivable that similar effects will be seen in interdisciplinary STEM approaches. In contrast, mathematisation initially seems to be strongly related to the subject of mathematics. Nevertheless, as pointed out before, the data extraction and collection can be seen as relevant for all STEM disciplines – and in particular for interdisciplinary questions. Although the type of data collection varies greatly within STEM disciplines, it seems legitimate to assume that similar differences in students’ perception could be reported when the data is collected directly in the environment compared to the work with photos inside the classroom. Especially in the assessment of arbitrariness, this seems to be of relevance independent of the discipline. This hypothesis is supported by mathematical structures and representations – and therefore mathematisations – being relevant for science, technology and engineering modelling, too (cf. UNESCO, 2020). Of course, the differences between the disciplines, e.g. in terms of explanation, abstraction and problem-solving, require further investigation.
Due to the positive selection of mathematically gifted students as a sample, the transferability to heterogeneous learning groups remains to be examined and could be the subject of future research. Further limitations result from the test instrument used. In particular, the described ceiling effects in the categories interest and enjoyment as well as reference to reality suggest that the five-point Likert scale used was not able to differentiate sufficiently between the response options. Furthermore, the test instrument was tested with university students in only one of the settings. Possible limitations in the quality due to Cronbach’s Alpha cannot be ruled out when transferring the test to students, as well as the danger that the response behaviour could change when the questionnaire is filled out three times. Through the Latin Square design used, both the tasks and the settings were systematically varied, so that it can be assumed that these potential influencing factors do not falsify the results.
In the next step, the results concerning students’ perception in simplifying and structuring as well as data collection should be empirically evaluated in the context of interdisciplinary STEM tasks. By extending the tasks, e.g. taking into account the stone’s density or modelling the stone by means of 3D print technology, the study’s methodology could be reused and bring added value to the research and creation of an interdisciplinary STEM approach using the outdoors. Especially the enrichment through qualitative data could give further insights in the interdisciplinary STEM approach and involved modelling practices from its disciplines.
Acknowledgements
I thank Dr. Hans Messer Foundation for funding the study “Modelling, Argumentation and Problem Solving in the Context of Outdoor Mathematics” (MAP). In addition, I thank Tim Läufer for creating the 3D print models and Simon Barlovits for his support in the data analysis.
References
Ayotte-Beaudet, J.-P-, Potvin, P., Lapierre, H., & Glackin, M. (2017). Teaching and learning science outdoors in schools’ immediate surroundings at K-12 levels: A meta-synthesis. EURASIA Journal of Mathematics Science and Technology Education, 13 (8), 5343–5363. www.doi.org/10.12973/eurasia.2017.00833a.
Bentsen, P., Søndergaard Jensen, F., Mygind, E., & Barfoed Randrup, T. (2010). The extent and dissemination of udeskole in Danish school. Urban Forestry & Urban Greening, 9, 235–243.
Bergsten, C., & Frejd, P. (2019). Preparing pre-service mathematics teachers for STEM education: An analysis of lesson proposals. ZDM, 51, 941–953.
Blanca, M. J., Alarcón, R., Arnau, J., Bono, R., & Bendayan, R. (2017). Non-normal data: Is ANOVA still a valid option? Psicothema, 29 (4), 552–557. www.doi.org/10.7334/psicothema2016.383.
Blanz, M. (2015). Forschungsmethoden und Statistik für die Soziale Arbeit: Grundlagen und Anwendungen. Kohlhammer.
Blum, W., & Leiß, D. (2007). How do students and teachers deal with mathematical modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (pp. 222–231). Woodhead Publishing. www.doi.org/10.1533/9780857099419.5.221.
Buchholtz, N. (2021). Students’ modelling proceses when working with math trails. Quadrante, 30(1), 140–157. www.doi.org/10.48489/quadrante.23699.
Crompton, H. (2020). Contextualizing STEM learning: Frameworks & strategies. In M. Ludwig, S. Jablonski, A. Caldeira, & A. Moura (Eds.), Research on Outdoor STEM Education in the digiTal Age. Proceedings of the ROSETA Online Conference in June 2020 (pp.13–22). WTM. www.doi.org/10.37626/GA9783959871440.0.02.
Duijzer, A. C. G., Van den Heuvel-Panhuizen, M., Veldhuis, M., Doorman, L. M., & Leseman, P. P. M. (2019). Embodied learning environments for graphing motion: A systematic literature review. Educational Psychology Review, 31(3), 597–629. www.doi.org/10.1007/s10648-019-09471-7.
Field, A., & Hole, G. (2002). How to Design and Report Experiments. Sage.
Gravemeijer, K., Stephan, M., Julie, C., Lin, F. L., & Ohtani, M. (2017). What mathematics education may prepare students for the society of the future? International Journal of Science and Mathematics Education, 15, 105–123. www.doi.org/10.1007/s10763-017-9814-6.
Greefrath, G. (2009). Schwierigkeiten bei der Bearbeitung von Modellierungsaufgaben. In M. Neubrand (Ed.), Beiträge zum Mathematikunterricht 2009 (pp. 137–140). WTM.
Gaio, A., Branchetti, L., & Capone, R. (2020). Learning math outdoors: Graph theory using maps. In M. Ludwig, S. Jablonski, A. Caldeira, & A. Moura (Eds.), Research on Outdoor STEM Education in the digiTal Age. Proceedings of the ROSETA Online Conference in June 2020 (pp. 95–102). WTM. www.doi.org/10.37626/GA9783959871440.0.12.
Gross, K., & Harmer, S. P. (2020). “CSI VIENNA” – Design and evaluation of the out-of- school lab ELKE. In M. Ludwig, S. Jablonski, A. Caldeira, & A. Moura (Eds.), Research on Outdoor STEM Education in the digiTal Age. Proceedings of the ROSETA Online Conference in June 2020 (pp. 205–208). WTM. www.doi.org/10.37626/GA9783959871440.0.26.
Haas, B., Kreis, Y., & Lavicza, Z. (2021). Integrated STEAM approach in outdoor trails with elementary school pre-service teachers. Educational Technology & Society, 24 (4), 205–219. www.jstor.org/stable/48629256.
Hartmann, L.-M., & Schukajlow, S. (2021). Interest and emotions while solving real- world problems inside and outside the classroom. In F. K. S. Leung, G. A. Stillman, G. Kaiser, & K. L. Wong (Eds.), Mathematical Modelling Education in East and West. International Perspectives on the Teaching and Learning of Mathematical Modelling (pp. 153–163). Springer. www.doi.org/10.1007/978-3-030-66996-6_13.
Holmlund, T. D., Lesseig, K., & Slavit, D. (2018). Making sense of “STEM education” in K-12 contexts. International Journal of STEM Education, 5 (32). www.doi.org/10.1186/s40594-018-0127-2.
Jablonski, S. (2021). Wie potenziell mathematisch begabte Kinder argumentieren. Eine Längsschnittstudie mit Kindern der Klassen 3 bis 6 im Rahmen des Enrichmentprogramms “Junge Mathe-Adler Frankfurt”. Springer Spektrum.
Jablonski, S. (2023). Is it all about the setting? A comparison of mathematical modelling with real objects and their representation. Educational Studies in Mathematics. www.doi.org/10.1007/s10649-023-10215-2.
Kertil, M., & Gurel, C. (2016). Mathematical modeling: A bridge to STEM education. International Journal of Education in Mathematics, Sciences and Technology, 4 (1), 44–55.
Kolb, D. (1984). Experiential learning: Experience as the source of learning and development. Prentice-Hall, Inc.
Lavicza, Z., Haas, B., & Kreis, Y. (2020). Discovering everyday mathematical situations outside the classroom with MathCityMap and GeoGebra 3D. In M. Ludwig, S. Jablonski, A. Caldeira, & A. Moura (Eds.), Research on Outdoor STEM Education in the digiTal Age. Proceedings of the ROSETA Online Conference in June 2020 (pp. 23– 30). WTM. www.doi.org/10.37626/GA9783959871440.0.03.
Ludwig, M., & Jablonski, S. (2021). Step by step – Simplifying and mathematising the real world with MathCityMap. Quadrante, 30(2), 242–268. https://quadrante.apm.pt/article/view/23604/18998.
Schukajlow, S., Leiss, D., Pekrun, R. et al. (2012). Teaching methods for modelling problems and students’ task-specific enjoyment, value, interest and self-efficacy expectations. Educational Studies in Mathematics, 79, 215–237. www.doi.org/10.1007/s10649-011-9341-2.
UNESCO (2020). Designing a contemporary STEM curriculum. Current and Critical Issues in Curriculum, Teaching, Learning and Assessment, 39, 1–23.
Wess, R., Klock, H., Siller, H.-S., & Greefrath, G. (2021). Measuring professional competence for the teaching of mathematical education. Springer.
Wetzel, S., Milicic, G. & Ludwig, M. (2020). Experiencing computational thinking and the concept of loops in an outdoor CS unplugged approach. In M. Ludwig, S. Jablonski, A. Caldeira, & A. Moura (Eds.), Research on Outdoor STEM Education in the digiTal Age. Proceedings of the ROSETA Online Conference in June 2020 (pp. 187–194). WTM. www.doi.org/10.37626/GA9783959871440.0.23.
