Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 143
Curricular learning with MathCityMap: creating theme-based math
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 144
Within the concept of theme-based trails, math trails are directly linked to the mathematics curriculum. Therefore, theme-based trails can be seen as the adoption of the regular math lessons in outdoor settings. In this paper we introduce the idea of math trails and the MathCityMap app. Subsequently, we develop design principles for the creation of theme-based trails and illustrate them by focusing on the concept of linear functions. In this way we want to give a substantial contribution to implement MathCityMap math trails for the teaching and learning a specific curriculum topic.
CONCEPTUAL FRAMEWORK: MATH TRAILS & MATHCITYMAP
Mathematical outdoor education can be implemented by using the math trail method.
A math trail is a walking route consisting of several place-bound math tasks. These tasks treat mathematical questions about real existing objects in one’s environment, which allows people from all ages to perceive their own environment from a mathematical perspective (Shoaf, Pollak, & Schneider, 2004). The method generates
“an appreciation and enjoyment of mathematics in everyday situations, usually to complement work in the classroom” (Blane & Clarke, 1984, p. 1).
According to Cross (1997), the math trail method offers several advantages for mathematics education: By working on realistic and authentic tasks, students firstly experience relevance of mathematics in everyday life. Consequently, they learn to apply their theoretical knowledge in a wide variety of practical situations and can develop strategic problem-solving skills. Secondly, Cross stresses the value of group cooperating and communicating, which enables students to clarify and structure their mathematical knowledge. Thirdly, solving a math trail task usually requires to collect and record data – a worthwhile skill that is rarely fostered in regular math class. Finally, the method also promotes learning about the immediate environment, so math trails offer interdisciplinary and multi-faceted learning opportunities (ibid.).
MathCityMap: Digitalization of the Math Trail Method
The MathCityMap project revives the ‘old’ idea of math trails and supplements it with mobile learning (Ludwig & Jesberg, 2015), which can be defined as usage of mobile devices like tablets or smartphones in an educational context (Park, 2011). The computing power and portability of these devices as well as the possibilities of wireless communication and digital tools offer great potential for both traditional teaching and outdoor learning (Sung, Chang, & Liu, 2016).
MathCityMap is a two-component system consisting of a web portal and a freely available app. It provides a simplified, digital way to create, share and to run math trails (Ludwig & Jablonski, 2020). The first component, the MathCityMap web portal (www.mathcitymap.eu), is a database for finding and creating mathematical tasks and to combine them to math trails. Both, tasks and trails can be shared with members of the worldwide MathCityMap community. This aspect of sharing can be named as one of the core features of the web portal. For every math trail, a math trail guide is
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 145
available. It can be downloaded as PDF file or accessed via the MathCityMap app. This guide includes a map of the trail, all tasks and one picture of each task situation.
The MathCityMap app enables users to access math trails, which were created on the web portal. After downloading the trail guide to the mobile device, it is possible to run the math trail via app even without an internet connection. Only one smartphone or tablet with the installed app is necessary per group.
Features of the MathCityMap App
According to Ludwig and Jesberg (2015) mobile-supported math trails offer several advantages in comparison to ‘classic’ math trails: Firstly, the MathCityMap app eases the task localization by navigating students to the tasks via GPS. Secondly, the app provides hints to support students’ problem-solving progress. Thirdly, students receive a systemic feedback on their calculated solution (ibid.). Adding the value of pedagogical gamification, we will describe the listed benefits.
Figure 1: The Task “x-intercept” in the MathCityMap App: Task Formulation, a Hint, Validation of the Solution and Sample Solution (from left to right).
Stepped hints: For each task, the app displays up to three stepped hints to support students’ problem-solving progress (Fig 1, center-left). While students can call up those hints independently, they can determine the difficulty of the task. This enables learners to adapt task to their individual performance and motivation level.
Consequently, the hints allow both higher and lower performing students to work on the given tasks (Franke-Braun, Schmidt-Weigand, Stäudel, & Wodzinski, 2008).
Feedback: According to Reinhold (2018), a major potential of digital tools is the ability to give students immediate feedback on their work progress. In addition, through their feedback, digital tools should enable the recognition of possible errors and misconceptions in the solution process (ibid.) On one hand, the MathCityMap app gives students an immediate feedback regarding the numerical correctness of the solution entered (Fig 1, center-right). On the other hand, a look at the sample solution,
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which can be viewed after solving a task or after at least six wrong answers, allows the independent detection of errors in the solution process (Fig 1, right). All in all, the MathCityMap App can validate the calculated solutions of the students and enables the learners to identify potential errors or misconceptions by analyzing the sample solution.
Gamification: The study of Lieberoth (2014) indicates a positive motivational impact of the gamification of activities. By solving MathCityMap tasks each group receive up to 100 points per task, depending on the quality of their solution. By using the optional function Leaderboard, the groups are listed in a local ranking. Those gamification elements lead to both an increase in the number of completed tasks per hour and a reduction of blind guessing (Gurjanow, Olivera, Zender, Santos, & Ludwig, 2019).
RESEARCH INTEREST: CREATION OF THEME-BASED MATH TRAILS Although recent studies about MathCityMap math trails have shown positive motivational effects (Cahyono, 2018; Gurjanow et al., 2019) as well as positive learning effects (Zender, 2019), math trails have so far mainly been used for a methodical variation of math teaching: They often aim at a wider revision of already learned topics. Consequently, the math trail method is – until now – rarely used for practice lessons with an explicit connection to a topic of the math curriculum.
This lack could be caused by the low amount of so-called theme-based math trails.
Even though it is possible to develop curriculum-related math trails for many mathematical topics (Cross, 1997; Zender, 2019), the creation of such trails has an inherit challenge: As the tasks of the ‘classic’ math trail arise from local conditions (and with respect to the curriculum), the possible task types of a theme-based trail are predetermined by the curriculum. Therefore, a suitable object for a predefined task has to be found at the chosen location. At the same time, those topic-related tasks should not only allow the learning of a specific topic, but still remain an authentic problem concerning a real existing object a given place. In conclusion, the difficulty of creating theme-based trails is to identify objects in one’s environment that raise realistic questions related to a particular curriculum topic. Within this paper, we aim to show possibilities for the curriculum-based use of the math trail method.
DESIGNING A THEME-BASED TRAIL
By taking up the concept of generic tasks, we identify design principles for the creation of theme-based trails linking the math trail method and curriculum-based topics in arithmetic, algebra, analysis and stochastic. By following these principles, teachers should be enabled to create their own theme-based trails on current classroom topic.
Generic Tasks and Theme-based Trails
A generic task is a mathematical task which can be applied to frequently occurring objects, e.g. the slope of a ramp or a handrail. These objects offer the possibility to easily transfer existing tasks to other locations (Ludwig & Jablonski, 2020). Within the Erasmus+ project MoMaTrE (Mobile Math Trails in Europe), a catalogue of generic
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tasks was created and translated in six languages, including English, French and German. Those tasks can be considered as best-practice examples for MathCityMap tasks with a specific reference to the mathematics curriculum.
A theme-based trail is a collection of generics tasks of one common topic, e.g. fractions in arithmetic, percentage calculation in algebra and linear functions in analysis as well as combinatorics in stochastic theory. Within a theme-based trail, a specific curriculum topic is addressed and can therefore be directly connected to regular math class. Zender (2019) already followed this approach during his study about stereometry for ninth graders by examining math schoolbooks for recurring topic-related tasks.
Design Principles for Theme-based Trails
The development of theme-based math trails requires design principles which are presented in the following. First, a detailed look into the school curriculum is necessary to identify the sub-areas of the current mathematics topic, e.g. the concept of slope as part of linear functions. Based on these concepts, a suitable method is to examine math textbooks for common task types. This ensures that all major concept-related task types used in the school are covered by the theme-based trail. After studying the curriculum and identifying the characteristic tasks, the required data and results have to be defined for all these task types: Which data is given in the task formulation? How is the result calculated? What is the expected solution process?
Finally, the author can search for suitable objects outdoors. An object must fulfil several conditions in order to be usable. First, it must be suitable to collect the required data, e.g. by measuring and counting. It is important to recognizes which data are directly measurable for the students and which data must be obtained by calculations.
Furthermore, the task objects must be publicly available and clearly identifiable.
Otherwise it is possible that the students will neither find nor reach the object during the lesson. In some cases, reference sizes are helpful or even necessary, e.g. auxiliary lines in the task picture, or reference objects as a lantern if the intercept theorem is used to calculate the height of a building.
Regarding to Zender (2019), we recommend the creation of filling tasks (non-topic-related tasks), in order to offer students a variety during the theme-based math trail. If students realize that they work on several similar tasks in a row, e.g. measuring the diameter of a circle for calculating the circumference, they first lose their motivation and then remain unfocused. Consequently, Zender (2019) suggests the creation of filling tasks to interrupt the sequence of constantly recurring tasks so that students do not blindly work through algorithms. To ensure that the topic-based focus is still maintained, every third or fourth task should be designed as a filling task.
In summary, Table 1 presents the developed design principles for theme-based trails:
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Step Object
1 Analyze the curriculum to identify the sub-area of the chosen topic.
2 Examine textbooks searching for common task types.
3 Define required data and results.
4 Search for suitable objects outdoors.
5 Create theme-based tasks and filling tasks in a 3:1 or 4:1 ratio.
Table 1: Design Principles for Theme-based Math Trails.
An Example: Theme-based Trail on Linear Functions
In order to illustrate the described design principles, we introduce the concept of linear function as a possible topic for a theme-based math trail in eighth grade. By examining math textbooks, we have identified five characteristic tasks and defined their required data (Tab. 2). Since all these task types can be applied on frequently occurring objects, they can be considered as generic tasks for linear functions. Therefore, a theme-based trail on linear functions can be easily created almost at any place by using these five generic tasks. In the following, we present exemplary tasks for all five identified sub-areas (Fig. 2) for the object “handrail”.
Table 2: Generic Tasks for the Concept of Linear Functions.
Focusing on the task types “slope” and “x-intercept”, we describe students’ solution process and possible hints given by the MathCityMap app.
Sub-Area Object Task Type Required Data Proportional
Relationship.
e.g. Price list. Calculate the costs of z pieces, e.g. balls of ice-cream.
Change in x, y.
Slope. e.g. Ramp. Calculate the slope of the ramp. Give the result in percentage.
Change in x, y.
Slope-intercept form.
e.g. Slide. Define the linear function given by the slide.
Change in x, y
& y-intercept b.
x-intercept. e.g. Handrail. Calculate the root of the linear function given by the handrail.
Change in x, y
& y-intercept b.
Point of intersection.
e.g. Gable roof. Find the point of
intersection of two lines given by the gable roof.
Equations of two lines.
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Figure 2: Selection of Tasks for Linear Functions within the MathCityMap App.
Slope (Fig. 2, center-left): To identify the slope of the handrail in percent, students have to measure the its length x and its maximum height difference y. Subsequently, they have to divide y by x and convert the fraction into percentage. This process could be guided by stepped hints: (i) Use a gradient triangle. Hint (ii) could include a picture of a gradient triangle, whereas hint (iii) aims at the percentage notion: The result should be given in percentage. For example, m = 0,21 equals 21 percent.
x-intercept (Fig. 2, center-right): In order to identify the x-intercept, the handrail function has first to be defined by calculating the slope m and measuring the y-intercept b. To structure students’ solution process, the general term of the linear function and the required data could be given in the first two hints. Hint (iii) could clarify the further proceeding: Students have to equate the handrail function with zero and thereby identify its x-intercept.
CONCLUSION
The math trail method enables learners and teachers to explore their environment in a mathematical way. The usage of the MathCityMap system for creating and working on theme-based math trails adds the benefits of mobile learning to the ‘classic’ math trail idea: The app guides learners through a math trail, providing GPS, the task formulation, hints, feedback and a sample solution. Teachers can create their own math trails within the MCM web portal or use public available math trails.
Considering the positive effects on motivation (Cahyono, 2018; Gurjanow et al., 2019) as well as learning growth (Zender, 2019), the math trail method is underrepresented in regular school lessons. So far, trails are mainly used for a broader repetition of already learned topics, but not for the targeted work on current math content. We explain this by the lack of curriculum-related math trails, which – so far – prevents a more frequent embedding of the math trail method in the regular teaching units. To address this issue, we have introduced the idea of a theme-based math trail that covers the content of a regular math lesson through curriculum-related tasks. For this purpose, we have developed design principles to support teachers in creating theme-based trails.
These include the analysis of specific mathematical sub-areas in textbooks and the
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identification of generic tasks that can be applied on common objects in one’s own environment. In this way, we offer a possibility to make the math trail method usable for many, if not all, contents of the math curriculum. Enabling teachers to create their own theme-based trails fulfils an important requirement for the long-term integration of the math trail method into regular math lessons.
REFERENCES
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Cross, R. (1997). Developing mathematics trails. Mathematics Teaching, pp. 38-39.
Franke-Braun, G., Schmidt-Weigand, F., Stäudel, L., & Wodzinski, R. (2008).
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Sung, Y.-T., Chang, K.-E., & Liu, T.-C. (2016). The effects of integrating mobile devices with teaching and learning on students' learning performance: A meta-analysis and research synthesis. Computers & Education, (94), pp. 252-275.
Zender, J. (2019). Mathtrails in der Sekundarstufe I. Der Einsatz von MathCityMap bei Zylinderproblemen in der neunten Klasse. Münster: WTM-Verlag.
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