Bojan Savic
email: b.savic@tu-berlin.de

Xunan Gao

Roland Baar

S Y Mpo

O Tatin

Turbocharger Heat Transfer Determination with a Power Based Phenomenological Approach and a CHT Validation

Keywords:

We present from our work on models to determine heat flows on turbochargers. A Conjugate-Heat-Transfer simulation has been carried out on the turbine for validation of the approach. Results have shown that isentropic efficiencies fit well for values of turbine inlet temperatures of °C. For other temperatures the differences between the determined values and CHT are greater. e differences rise with higher temperatures. Hence, an additional dependency of turbine inlet temperatures has been implemented in the model. e modification has shown be er results and smaller differences to CHT results. Especially at low speeds where the former approach has had big differences the modification improves the distribution. A wastegate turbocharger has been investigated from small gasoline engine applications. In future investigations the approach will be tested on the compressor with a modification, too, as well as on different turbochargers. e objective is to create a methodology for parameterization of a thermal network model based on the modified approach as a start point. Recently the experimental basis has been developed to determine heat flows on radial turbines and compressors. It has been validated for the compressor with experimental data under near adiabatic conditions. For the turbine a CFD simulation has been carried out. Both comparisons have shown good results for turbine inlet temperatures of °C.

INTRODUCTION

e conventional measurement of turbochargers on hot gas test benches is thermodynamically adapted on realizable options in terms of measuring devices, possibilities and costs. For todays high requirements of the application e.g. in automotive systems, this procedure leads to differences between reality and measurement because of the met assumptions and the considered operating range. Several studies including our own investigations have shown that the expansion and compression process is not adiabatic. Heat flows occur between the compressor and the turbine which lead to falsified isentropic efficiencies. Additionally heat flows occur between the turbocharger and the ambiance. Some of the heat flows are consumed by the lubrication oil and if available the cooling water. However, the heat flow phenomenon on a turbocharger is quite complex and it is illustrated in Figure .  One way to decrease the influence of heat transfers is measurements under adiabatic conditions.

is procedure is thermodynamically feasible on the compressor so that the isentropic efficiencies can be transferred to the hot gas measurements. On the turbine side, the operating points change significantly between these two experimental modes so that there is no possibility to assign and transfer the isentropic efficiencies. Another problem is the inhomogeneous flow at the turbine outlet, which needs special measurement equipment for reliable T data (temperature at turbine outlet)[ ].

is is the critical value for determining the turbine isentropic efficiency directly from measured data. Only in adiabatic measurements does this procedure lead to real isentropic efficiencies. In hot gas tests, the results are superimposed by heat flows and without the possibility to transfer the real isentropic efficiencies from adiabatic tests there is no knowledge about the quantity of outgoing heat. Another way to obtain the real turbine isentropic efficiencies is a CFD or a CHT simulation, requiring geometrical data and high computational costs. Other possibilities for heat flow determination are specific empirical approaches for the compressor, the turbine or the friction, to calculate the power balance. But these empirical approaches are commonly validated on specific turbochargers and have a limited prediction behavior on new ones. Please refer to [ ] for detailed literature information. erefore a powerbased approach has been developed in previous works to determine heat flows on compressors and turbines. It led to some advantages in relation to transfer issues on new turbochargers because the model does not have to be calibrated. It only needs measurement data. e basis for the approach was precisely described in It has been shown that the approach works without geometrical data and is applicable on various devices.

In [ ] the new phenomenological approach was expanded on the compressor. Furthermore the recalculation of the isentropic efficiencies of a wastegate turbocharger has been shown and validated for both the compressor and the turbine.

. METHODS

. POWER BASED APPROACH

e approach has been firstly described in [ ] and consists of a power based consideration. e reason for a power based consideration has been motivated by the following issue. For compressors it is possible to apply boundary conditions to minimize heat transfer. e real aerodynamic behavior is then described almost with conventionally used thermodynamic procedures. Because the inlet conditions of the compressor do not change significantly between adiabatic and hot gas tests the so obtained efficiencies can be transferred. At the turbine side the situation is different because of the changing inlet conditions. ere is no possibility of transition of the isentropic efficiencies from one test mode to the other. e reduced values e.g. massflows and speeds change significantly so that comparability is not available. By trying to find a way to solve this problem as well as to find a criterion to evaluate how adiabatic the experiments really are, the power based approach has been developed. In figure Figure the idea has been schematically emphasized. When plo ing measured effective turbine powers against isentropic compressor powers these lines occur. It has been visible that there are similarities between adiabatic lines in blue and hot gas lines in red. e exact motivation of the choice of axes are described in [ ],[ ] and [ ]. However, this approach showed that one difference between the two test modes is an offset of the measured effective turbine powers. In [ ] the procedure has been shown how to correct the measured powers with

η Tis corr = ∆h T corr ∆h Tis = P T corr m T P Tis m T = P T corr P Tis ( )
On the turbine side a CFD calculation has been applied and the comparison showed good agreement at T = °C. In both cases the largest differences were localized at lowest speeds. e explanation has been due to sensor accuracies which are very sensitive at low speeds. For a summary of the works please refer to [ ] and [ ].

In the present work the approach has been tested at four turbine inlet temperatures ( °C, °C, °C, °C). e main object of this investigation is the turbine. A CHT simulation has been carried out to prove the validity. For further information to the CHT setup and calculation please refer . CONJUGATE HEAT TRANSFER

. . Computational mesh e rotational region was meshed by a structural hexahedral grid in the fluid domain. e volute and diffuser have been meshed with unstructured tetrahedrals together with prism layers at the detached region. e solid domain has been meshed with a tetrahedral grid due to the geometric complexity. e dimensionless first grid wall distance (y + <) has been regulated below for all fluid domains to obtain a high resolution of boundary layers. A Grid independency study has been carried out to eliminate the grid-induced errors. An additional ambient domain has been built in order to model the convective heat transfer between turbine housing and ambiance which has been validated by a previous study. Figure shows the overview of the computational model. e complete model contains . Million elements.

e turbocharger mesh contains . Million elements.

. . Numerical Setup e commercial code ANSYS CFX . has been employed as the solver, which uses the element-based finite volume method. In the fluid domain, steady RANS (Reynolds Averaged Navier-stokes) equations have been solved with closure brought by the SST turbulence model(Shear Stress Transport). Energy equations were solved simultaneously for the solid domain.

e global timestep was set to /w, where w is the angular speed of the turbine. A specified "pseudo" timestep has been applied for solid domain to accelerate convergence because of the significant disparity in timescale between fluid flow and solid conductive heat transfer. e frozen-rotor method has been employed for the interfaces between rotational and stationary regions using GGI (General Grid Interface) mesh connection technology. e smooth . RESULTS AND DISCUSSION . CHT and Adiabatic Approach e experiments has been applied under various conditions such as inlet temperatures of lubrication oil, cooling water and exhaust gas. In the present work the influence of the turbine inlet temperatures has been investigated on the approach. e correction described in has been employed and the isentropic turbine efficiencies have been corrected.

e isentropic turbine efficiencies are plo ed against isentropic compressor powers to stay consistent with the power based approach.

e values for °C are plo ed in Figure .Standard hot gas measurement data is in red. ey are calculated with the temperature difference between T and T .

is shows the typical influence of heat transfer at low speeds due to the fact that the enthalpy differences decreases here stronger than the heat transfers. e lowest speed line reaches implausible values of . . e CHT results are in blue and show that the isentropic efficiencies stay in an approximately constant range of values over the speed lines. Although this scale is not appropriate to evaluate the values in detail, it is reasonable enough to make a statement of the qualitative trends. ere is a constant offset to lower values, especially for lowest speeds. e heat flows are overestimated and so isentropic efficiencies are underestimated.

e lines for T = °C are presented in Figure . ere are more speed lines than for T = °C. is is due to test bench limits of p . As the turbine inlet temperature decreases the massflow has to rise so that the turbine can provide bearings and compressor power. e temperature difference and the massflow are the only two parameters that can change the effective turbine power. temperatures and heat flows. ese results show that there is an obvious T impact on the heat transfer which is not fully considered by the correction method. ere is another parameter which has to be investigated additionally to the difference in the ordinate intercept between adiabatic and hot gas test data in Figure . e gradient of the connecting line of maximum isentropic compressor powers has a slightly different value from adiabatic line. It is always bigger than in the cold case. is difference increases with higher values so that there are actually two factors to be considered.

.

Modification of the Approach

Firstly, the approach has been extended with a second correction term for the gradient. Heat flows are increasing linearly with speed. e results show a be er fi ing with CHT isentropic efficiencies in relation to the overall offset. But they keep the trend at low speeds where the isentropic efficiencies tend to decrease intensely. is is obviously a be er approach but there are still unconsidered influences. A promising concept has been found in the process of investigating combinations of axes. e motivation of it has been the question from which axes can we get specific information because the approach itself is new and still not investigated from different perspectives. e best results came up with a change of the isentropic power as the abscissa to effective compressor power. is combination is schematically illustrated in Figure . Here the speed lines are much fla er and tend to appear as a line. In the former approach the speed lines rise with a polynomial that is slightly higher than linear.

e new approach has a clear and linear progress when comparing Figure and Figure . e angle bisector would be the hypothetical state of no friction and the turbine power equals the compressor power. e real gradients are slightly bigger than that. However, the advantage of this approach is that the question of how to consider different points of the same speed lines is less important because they lie on the same line. Nevertheless this aspect has to be investigated more detailed in future. is approach will be used as basis for further considerations and calculations. Only for comparability the corresponding isentropic compressor power of each operating point will be used for illustration. For calculation the modified approach has been used and for is is a good result because a simple correction method as proposed is possible to obtain values for the isentropic turbine efficiencies which are in the same range of a CHT simulation. It is important to understand that geometrical or material data here is not needed. e calculation takes a few seconds. For higher temperatures the results show good agreement with CHT. is statement can be comprehended in Figure . e alignment is much be er here than in the former approach. ere is a small difference at low speeds to CHT where the isentropic efficiencies are underestimated. e hot gas data has one speed line missing. e first line is intensely superimposed by heat flows so that it is outside the right scale. e results in Figure are quite promising. Two operating points at small isentropic compressor powers show bigger differences. e modified approach works here as well as for higher temperatures. It is obvious that the influence of turbine inlet temperatures now is considered.

ere is a small offset in Figure where the isentropic efficiencies are overestimated. Hence, the heat flows are slightly underestimated. Two operating points show underestimated efficiencies at small speeds.

e mean offset is about % points. ese results emphasize the consideration of the linearly increasing heat flows with higher speeds. e comparison shows be er results with a change from constant . CONCLUSION e approach showed in previous investigations its validity on different turbochargers in terms of the typical power based lines. e comparison of a wastegate turbocharger showed further good agreement in terms of isentropic turbine efficiencies for a certain turbine inlet temperature of °C. In the present work this investigations has been continued. A Conjugate-Heat-Transfer model has been set up and simulations of four different turbine inlet temperatures has been carried out. e comparison shows that the approach has a good agreement for °C and even be er for °C. In the first case the isentropic turbine efficiencies is underestimated. e results show disagreement with CHT for low speeds in both cases. e distribution has increasing differences with rising turbine inlet temperatures. e heat flows are underestimated and the isentropic efficiencies are overestimated for °C and °C. is trends lead to the conclusion that some T influences were still not considered, although the most obvious effect of the heat flows, the offset in the ordinate direction has been calculated. is has been shown in previous works. Two factors describe the connecting line between the points of maximum isentropic compressor power. e first is the ordinate intercept and the second the gradient. e la er has not been considered yet and has been in all cases greater in Investigations that has been carried out show be er alignment with CHT results by correcting additionally the slope. is improved the offset with higher turbine inlet temperatures.

Another abscissa solved the problem. Instead of isentropic, the effective compressor power has been applied. is approach also ensures a good alignment at low speeds as well as over the hole range of turbine inlet temperatures. Results show that only at °C the alignment to CHT data of the modified approach is similar to the former one. erefore the modified approach will be used in future works.

It is a big advantage to know the quantity of heat flows from experimental test data for various applications. e exact information of turbocharger aerodynamics is increasingly important with higher legislative expectations for T °C: Larger scale combustion engines. ermal management or turbocharger matching would be interesting fields e.g. that could profit from approaches with low computational cost and no calibration effort. On the other hand the approach showed good agreement with CHT simulations. e main challenge and motivation of the work is that turbochargers are fluid machines but they are described thermodynamically because it is simple to measure temperatures and pressures. In reality there are heat flows which superimpose the measurements. Technically it would be much more interesting to determine isentropic efficiencies aerodynamically but the measurement of these parameters is much more difficult.

It is important that the approaches developed in research are applicable in industrial boundary conditions because only then they can be used to optimize the existing processes.

Figure .

 . Figure . Heat flows in a turbocharger

Figure .

 . Figure . Turbine effective power against isentropic compressor power

Figure .

 . Figure . Turbinemesh structure without ambiance domain

  Figure . T = °C: Isentropic efficiencies against isentropic compressor power

Figure .

 . Figure .T °C: Isentropic efficiencies against isentropic compressor power

  Figure . T °C: Modification of the adiabatic approach at high temperatures

  Figure . T °C: Results of the modification at lower temperatures

  Figure . T °C: Results of the modification at standard temperatures

  Figure . T °C: Results of the modification at high temperatures

  Figure . T °C: Results of the modification at highest temperatures


Maria Cristina Costa

António Domingos

Promoting mathematics teaching in the framework of STEM integration

Keywords: Hands-on, mathematics education, primary school, professional development, STEM education

A growing number of studies and reports all over the world refer to the importance of integrating Science, Technology, Engineering and Mathematics (STEM), in order to meet the increasing challenges of the 21 st Century. In particular, STEM education can be an innovative way of learning and teaching mathematics. But, despite recommendations to relate these subjects, there is lack of research about STEM integration and there is the need of more empirical research about this subject. This paper presents an empirical study about the development and implementation of mathematical interdisciplinary tasks related to STEM integration in the context of primary school teachers Continuing Professional Development. With a qualitative methodology and an interpretative approach based on a case study, findings of our research show that it is possible to promote the teaching of mathematics in the framework of STEM integration by supporting teachers in the development and implementation of interdisciplinary tasks.

Introduction

A growing number of studies and reports, all over the world, refer to the importance of integrating Science, Technology, Engineering and Mathematics (STEM) in order to meet the increasing challenges of the 21 st Century [START_REF] Baker | Integrating STEM in elementary classrooms using model-eliciting activities: responsive professional development for mathematics coaches and teachers[END_REF][START_REF] Rocard | Science education now: A renewed pedagogy for the future of Europe[END_REF]. In addition, integrative approaches among STEM subjects have positive effects on student attainment, with better results in elementary school [START_REF] Becker | Effects of integrative approaches among science, technology, engineering, and mathematics (STEM) subjects on students' learning: A preliminary metaanalysis[END_REF]. In particular, "Integration of mathematics with Science, Technology and Engineering (STE) provides students with the context in which they can make meaningful connections between mathematics and STE subjects" (Becker & Park, 2011, p.25).

This paper aims to contribute to research by presenting an empirical study about the development and implementation of mathematical interdisciplinary tasks related to STEM integration, in the context of a collaborative Continuing Professional Development Program (CPDP) targeted to primary school teachers. In this regard, our research question is: how to promote the teaching of mathematics in the framework of STEM integration? In our STEM integration framework, we refer to tasks related to all the STEM subjects, i.e., tasks that integrate Science, Technology, Engineering and Mathematics. In these tasks, we intend to emphasize Mathematics. Based on the case study of a primary teacher, who participated in the CPDP, we analyse the STEM tasks designed and implemented by the teacher.

Literature Review

Despite the increasing calls for the promotion of STEM education, there is no agreement about a definition. In this regard, [START_REF] Baker | Integrating STEM in elementary classrooms using model-eliciting activities: responsive professional development for mathematics coaches and teachers[END_REF] refer that the academic community "struggle to define STEM integration" (p. 2). For example, several authors refer to STEM by only integrating two disciplines [START_REF] Ríordáin | Making mathematics and science integration happen: key aspects of practice[END_REF][START_REF] Treacy | Authentic integration: A model for integrating mathematics and science in the classroom[END_REF]. Concerning this matter, [START_REF] Stohlmann | A vision for future work to focus on the "M" in integrated STEM[END_REF] refers different interpretations of several authors and claims that mathematics should be more emphasized in STEM integration. In this sense, STEM education can be a form of innovation for teaching mathematics [START_REF] Fitzallen | STEM Education: What does mathematics have to offer?[END_REF] and to increase mathematical performance [START_REF] Stohlmann | A vision for future work to focus on the "M" in integrated STEM[END_REF].

However, despite relating STEM subjects has been widely advocated by several authors, there is lack of research about STEM integration and there is a need for more empirical research about this subject [START_REF] Becker | Effects of integrative approaches among science, technology, engineering, and mathematics (STEM) subjects on students' learning: A preliminary metaanalysis[END_REF]. Also, there is the need to develop research to understand how STEM integration can promote mathematics education [START_REF] Baker | Integrating STEM in elementary classrooms using model-eliciting activities: responsive professional development for mathematics coaches and teachers[END_REF]. In particular, there is lack of research about science and mathematics integration [START_REF] Treacy | Authentic integration: A model for integrating mathematics and science in the classroom[END_REF], mainly concerning in-service teachers [START_REF] Ríordáin | Making mathematics and science integration happen: key aspects of practice[END_REF]. [START_REF] Kim | Analysis of Korean elementary pre-service teachers' changing attitudes about integrated STEAM pedagogy through developing lesson plans[END_REF] sustain the creation of a curriculum that integrates STEM, being crucial to involve teachers into interdisciplinary lessons adequate to this approach. In this regard, there is the need to promote their Professional Development (PD) by providing mentoring and support to face the multiple challenges related to sustainable STEM pedagogy [START_REF] Baker | Integrating STEM in elementary classrooms using model-eliciting activities: responsive professional development for mathematics coaches and teachers[END_REF]. Also, resources are crucial for teachers and they shape them according to their individual preferences, this being a process of interpretation and design of the resources [START_REF] Pepin | Re-sourcing teachers' work and interactions: A collective perspective on resources, their use and transformation[END_REF]. For Gimeno [START_REF] Gimeno Sacristán | O Currículo: uma reflexão sobre a prática [The Curriculum: a reflection on practice[END_REF], tasks are teaching and learning activities, carried out in school environments, being inserted into the curriculum in action. The way how tasks are introduced and conducted by the teacher in the classroom is essential for teaching effectiveness [START_REF] Ponte | Gestão curricular em Matemática[END_REF].

To improve teaching and learning, it is crucial to develop a partnership between researchers and designers, in order to develop appropriate pedagogical approaches to integrate tasks in the classroom [START_REF] Geiger | Devising principles of design for numeracy tasks[END_REF]. Also, it is fundamental to create a network that motivates teachers and contributes to the sustainability of their professional development [START_REF] Hewson | Teacher professional development in science[END_REF][START_REF] Rocard | Science education now: A renewed pedagogy for the future of Europe[END_REF]. A PD program will only be successful if teachers can apply in their classrooms what they learned and experienced during training [START_REF] Buczynski | Impact of professional development on teacher practice: Uncovering connections[END_REF]. In fact, a PD only achieve real effects if innovation is appropriated by the teachers and transformed into their own practice [START_REF] Zehetmeier | Researching the impact of teacher professional development programmes based on action research, constructivism, and systems theory[END_REF].

Methodology

Based on the literature review recommendations, a partnership amongst university teacher's educators, local schools and a Continuing Training Centre was established, in 2015, in order to develop a CPDP that is adequate to local primary school teachers' needs [START_REF] Costa | Innovating teachers' practices: potentiate the teaching of mathematics through experimental activities[END_REF][START_REF] Stohlmann | A vision for future work to focus on the "M" in integrated STEM[END_REF]. With a total duration of 26 hours, the program lasts an entire school year (September to June) and includes STEM related workshops with a 2-3 hours duration. By the end of each school year teachers present a portfolio with a critical account on the CPDP and their proposals and implementation of innovative practices. In this context, teachers may choose a science theme to develop in the classroom with their students by integrating with the other STEM topics. So far, more than 70 teachers participated in the CPDP from 2015/2016 to 2017/2018 school year.

In this research, we use a qualitative methodology and an interpretative approach by resorting to a case study. According to [START_REF] Yin | Estudo de caso: Planejamento e métodos[END_REF], a case study is an empirical investigation that looks at a contemporary phenomenon within its real-life context, allowing a generalization of the obtained results. Data collected include participant observation (first author of the paper is a participant observer) and portfolios compiled by the teachers [START_REF] Cohen | Research methods in education[END_REF]. Participant observation takes place in the workshops with the teachers (to learn and practice what they are expected to implement) and at their classrooms (to support and observe them in action). When necessary, some semi-structured interviews are conducted to better interpret the case. In a first stage documental analysis was performed in teachers' portfolios in order to look for evidence of STEM contents. It was verified that most teachers chose astronomy or sound to implement in the classroom, and electricity is always at the bottom of their choices. This is the main reason why we decided to present the case study of Josefina (fictitious name) who chose electricity experiments to develop mathematical interdisciplinary tasks.

Teacher Josefina (42 years old, 18 years of service, in charge of a 3 rd and 4 th grade class) participated in the CPDP during the school year 2016/2017 and decided to conduct electricity experiments in her classroom with the educators help. The teacher's educators visited her class for two afternoons to observe and/or to help her implement some hands-on tasks. At the end of the CPDP, teacher Josefina (like all the other teachers) presented a portfolio with a critical account about the CPDP and documentary evidence of the activities that she performed with her students.

Data analysis, results and discussion

In this section, we begin by analyzing the case study of teacher Josefina that shows the development and implementation of mathematical tasks within the context of STEM integration, in particular related to electricity.

Josefina's case study

Based on classroom observations, semi-structured interviews and the teacher's portfolio, we realized that, in the first session, Josefina introduced the electrical current topic by calling students' attention to sustainable development using videos and information from the Internet. After, she asked the students to bring to class batteries that they had at home and that did not work.

In a second class, using the batteries, she asked the students to organise them according to their sizes and models (Figure 1). Next, they counted the different types of batteries and with this data, students produced graphics and diagrams, amongst other mathematical tasks.

In another class, with the educators' help, the teacher introduced concepts like potential difference (p.d.) and asked the students to verify if the batteries still had "energy" using multimeters. The students were organised in groups, and each group used a multimeter to measure the p.d. of each battery. After measuring they registered the p.d. (in volts) in a table. The batteries with more p.d. were saved to be used in future hands-on experiments. In a third session, teacher Josefina organised her students in groups of two or three, in order to implement several hands-on tasks. After training how to create electrical circuits to light one or two lamps (some of them used the batteries they saved in the previous class) the teacher decided to work mathematics. She gave a battery to each group and asked the students to measure and register the battery p.d. in volts (V). Then, she introduced biological batteries using fruit or vegetables and asked to measure their p.d., amongst other measurements (Figure 2). Josefina: Each group is going to choose different pieces of fruit or vegetable and measure its potential difference and give me the result! While students told the results, the teachers registered them on the black board. After having all the results registered, she continued the inquiry:

Josefina: Which is the fruit with the bigger potential difference?

Student:

The tomato! Josefina:

Which biological battery has the least potential difference?

Student:

The mushroom!

The teacher continued asking questions while introducing the concepts. For another section, she created a worksheet to perform more mathematical tasks such as organisation and processing of data from the measurements obtained in the last class. She also created problems to be solved by students. The above dialogue shows that she was able to perform mathematical tasks from the hands-on experiments, in particular mathematical tasks in the framework of STEM integration.

Indeed, Josefina developed hands-on tasks based on concepts and procedures from mathematics and science while incorporating the design methodology of engineering and using appropriate technology [START_REF] Shaughnessy | Mathematics in a STEM context[END_REF]. In particular, mathematics was worked in all the tasks, namely "Numbers and Operations" when students organized and counted the old batteries; "Geometry" when they organized the batteries according to their patterns and sizes and draw the different types of batteries; and "Data Organisation and Processing" when building tables to register and work the collected data from the batteries (Figure 1). Also, Josefina was able to introduce biological batteries and to teach the students to perform several measurements such as the p.d. in volts (V) and to develop the hands-on tasks using inquiry to lead the students to reflect on the performed experiments and discuss in order to obtain conclusions.

Concerning the way teacher Josefina introduced and implemented the tasks, we consider these are exploratory, investigative tasks. In this regard, students performed the tasks guided by the teacher who asked questions to lead them to search for answers. One example is related to the question "How much potential difference does the lamp need to light up?" and "Do you need three oranges to light the lamp?" To answer the questions students developed several activities in order to understand and gain knowledge about this theme.

Table 1 shows the contents of the STEM tasks developed and implemented by teacher Josefina. Planning, designing and performing electrical circuits.

Measuring the p.d. of the batteries in volts.

"Numbers and Operations" when students organized and counted batteries.

"Geometry" when they organized the batteries according to their patterns and sizes and draw the different types of batteries.

Data Organisation and Processing" when building tables to register and work the collected data from the batteries.

Next, we present some excerpts of the teacher's reflections that she wrote in her final report.

I was able to apply some new practices and methodologies in the context of the classroom, incorporating mathematics and science experiments (Josefina final report, June 2017).

The overall results of this training workshop are clearly positive: (…) enabled me to acquire / apply new knowledge in the context of the classroom (…); as well as collaboration including the sharing of good practices and the difficulties experienced (Josefina final report, June 2017).

In summary, teacher Josefina recognised that she gained skills to innovate her practices by highlighting the collaborative context of the CPDP. In addition, she developed exploratory, investigative tasks to work mathematics topics related to STEM.

Final considerations

This paper aims to contribute to existing literature by presenting an empirical study about the development and implementation of mathematical tasks in the framework of STEM integration. To face challenges related to STEM integration it is recommended to develop an adequate teachers' PD context [START_REF] Baker | Integrating STEM in elementary classrooms using model-eliciting activities: responsive professional development for mathematics coaches and teachers[END_REF][START_REF] Treacy | Authentic integration: A model for integrating mathematics and science in the classroom[END_REF], being crucial to provide a collaborative environment that supports the teachers to innovate their practices [START_REF] Capps | Inquiry-based professional development: What does it take to support teachers in learning about inquiry and nature of science[END_REF][START_REF] Costa | Innovating teachers' practices: potentiate the teaching of mathematics through experimental activities[END_REF]. According to [START_REF] Buczynski | Impact of professional development on teacher practice: Uncovering connections[END_REF], a PD program will only be successful if teachers can implement, in practice, with their students, what they learned and experienced during their training. Also, [START_REF] Zehetmeier | Researching the impact of teacher professional development programmes based on action research, constructivism, and systems theory[END_REF] sustain that innovations should be appropriated by those who implement them and transformed into their practice to have real effects. We believe this is what happened to teacher Josefina who designed and implemented interdisciplinary tasks that are not part of teachers' traditional practices. In fact, Josefina was able to introduce commercial and biological batteries and to teach the students to perform several measurements such as the p.d. in volts (V). She also developed several hands-on tasks using inquiry to lead the students to reflect on the performed experiments and obtain conclusions.

Based on Josefina's case study, it was verified that she implemented several mathematical tasks in the framework of STEM integration (Table 1). Indeed, Josefina developed hands-on tasks based on concepts and procedures from mathematics and science while incorporating the design methodology of engineering and using appropriate technology [START_REF] Shaughnessy | Mathematics in a STEM context[END_REF]).

Josefina's example shows how to develop exploratory and investigative mathematical tasks from hands-on STEM experiments. Mathematical tasks proposed by the teacher and performed by the students included problems and exercises related to several topics of the Portuguese curriculum such as "Numbers and Operations", "Geometry", and "Data Organisation and Processing", including tables, graphics and diagrams.

We observed that teacher Josefina promoted mathematics teaching in the framework of STEM integration by developing and implementing several interdisciplinary tasks in the classroom related to these subjects. Teacher Josefina's example shows that it is possible to work mathematics while performing several hands-on STEM tasks in class. We conclude that to promote mathematics' teaching in the framework of STEM integration it is necessary to provide teachers with a collaborative professional development context that supports them in the development and implementation of interdisciplinary tasks.

Figure 1 :

 1 Figure 1: Collection, organisation and processing of data from the batteries

Figure 2 :

 2 Figure 2: Potential difference and intensity measurements from fruit and vegetables

Table 1 .

 1 Contents of the tasks implemented by teacher Josefina

	Science	Technology	Engineering	Mathematics
	Electricity.	Computer
		Internet
		Video
		Multimeters,
		Lamps, interrupters,
		etc.

*This work is supported by national funds through FCT -Foundation for Science and Technology, I. P., in the context of the project PTDC/CED-EDG/32422/2017


Maren Hattebuhr
email: maren.hattebuhr@kit.edu

Martin Frank
email: martin.frank@kit.edu

Complex modeling: Does climate change really exist? -Perspectives of a project day with high school students

Keywords: High school mathematics, mathematical modelling, problem-solving, statistics

Introduction

Mathematics is at the core of many areas of everyday life in our society, in science, technology, and the economy. According to the PISA study, the aim of mathematical education is to develop an understanding of the relevance of mathematics in these areas. This should enable high school students to deal with and explain mathematical questions and problems with respect to real world life so that they can act as responsible citizens (cf. OECD, 2014). But process-oriented competences which belong to the educational standards of mathematics are often neglected due to the fact that mathematics lessons are often strongly content-oriented. Therefore, it is essential that the tasks are open, complex, realistic, authentic, problem-oriented and can be solved by carrying out a modeling process (cf. Maaß, 2006, p. 115). This is where the project Computational And Mathematical Modeling Program (CAMMP) intervenes: It offers extra-curricular possibilities for students to participate in modeling events which last between one day and one week. These events show the close connection of several subjects such as engineering, mathematics and computer science. By integrating such questions into interdisciplinary project teaching, mathematical modeling is extended by the dimension of independent experience and practical application. Students solve problems from everyday life or scientific research by using mathematical modeling. Thus, we understand CAMMP as a chance for students to solve real problems by mathematical modeling independently and to develop important future-oriented competencies. Our experiences show that by participating in CAMMP events students discover, among other things, the role of maths in our everyday life. Since the students should not use the modeling cycle as an abstract entity, but as a concrete help for the modeling process, CAMMP presents a simplified variant of Blum's modeling cycle with the four steps simplifying, describing mathematically, computing and interpreting (cf. Blum & Borromeo-Ferri , 2009, p. 46). As an example, a workshop lasting one day (CAMMP day) on the topic climate research is introduced.

Didactic reduction and realization of Climate Change Analysis in a CAMMP day

There are various simulations that predict a warming of the average temperature of the earth's surface by up to four degrees Celsius by 2100. Such warming would lead to further effects, from the emergence of deserts to rising sea levels and the disappearance of entire countries. However, many people ignore or deny climate change or the dramatic consequences that inaction would have. The report of the Intergovernmental Panel on Climate Change (IPCC) distinguishes various indicators that point to a climate change. Since climate change is a complex research field with many different forces affecting each other, a good didactical reduction is needed for teaching mathematical models representing earth climate to high school students. Therefore, the CAMMP daywhich we are currently designingfocuses on temperature analysis, an everyday parameter used to describe climate and often used in the media as a yardstick for climate change. The first step of modeling "Does a climate change exist?" we ask "Has the temperature increased significantly?" To answer this, the global average surface temperature from 1850 to 2018 is examined (see Figure 1). Furthermore, a linear trend is used to describe the trend since it is assumed in the IPCC report (see IPCC, 2013). For this purpose, the students first develop the methodology of regression analysis independently. One possible mathematical description is first to calculate the sum of least squares of the difference between the measured temperature function and the regression, called for each year . In a second step the sum of the least squares is minimized. The mathematical problem can be written as 2→min, where ( ) the measured temperature and ≔ + the modeled temperature value of the regression function with the unknown parameters and in the years are. Here the concept of differentiation is needed. It becomes obvious why the least square method is used: It is easier on high school level to differentiate a polynomial of grade 2; otherwise the absolute value function would be needed. Using real data this results in a linear trend function of which needs to be interpreted with respect to the real-world problem. Here the question of the quality of the adjustment for the data arises. The coefficient of determination is introduced and computed for the given data as . As an additional exercise, interested students can prove that in all cases holds and find out which indicates a good adjustment. These results lead to the presumption that approximates the data well and that the temperature rises by . In order to investigate whether the temperature rises significantly in the last century, the coefficients of the regression function are tested with a so-called t-test. The high school students formulate the null hypothesis ( : the temperature does not rise significantly) on the basis of the question as well as the test statistics. Inserting the corresponding values leads to the acceptance of . Further steps are to evaluate the influence of human being in this increase. These results do not only lead to a well-founded view on the topic of climate change, but also show the relevance of mathematics in society.

Figure 1 :

 1 Figure 1: Global average surface temperature


Eduard Krause
email: krause@mathematik.uni-siegen.de

Nguyen Van Bien

Tran Ngoc Chat

Phuong Nguyen

Frederik Chi

Jochen Dilling

Kathrin Geppert

Tuan Anh Holten

Simon Le

Chu Cam Kraus

Tho

Dilling

Nguyen Phuong Chi

Frederik Dilling

Jochen Geppert

Kathrin Holten

Tuan Anh Le

Simon Kraus

Chu Cam Tho

Inter TeTra -Interdisciplinary teacher training with mathematics and physics

Keywords: Developing country project, interdisciplinarity, relationship of physics and mathematics, teacher training curriculum development

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Motivation of the Inter TeTra-project

A primary goal of the OECD and the educational capacities in Vietnam is to reform their educational policy by developing a more competency-oriented curriculum (Communist Party of Vietnam, 2013). Germany has two decades of experiences with these types of reforms. The adjustments of the curricula since the mediocre results of German students in the TIMSS and PISA study show the opportunities and challenges such reforms can present [START_REF] Kmk | Standards für die Lehrerbildung: Bildungswissenschaften[END_REF] and the importance of teacher training for the success of school reforms [START_REF] Hattie | Lernen sichtbar machen[END_REF]. The cooperation between German and Vietnamese educators offers the opportunity to avoid well-known problems in implementing such reforms. In addition, it can generate new knowledge out of the cultural and structural differences between the two societies, allowing the further development and implementation of novel and distinctive curricula. Consequently, the DAAD (German Academic Exchange Service) endorses the Inter Tetra-project (the name of the project is a short form for Interdisciplinary Teacher Training), a subject-related partnership established between the University of Siegen and the Hanoi National University of Education (HNUE). The outcomes of this project are the design of a permanent module for the subjects mathematics and physics at the HNUE and the performance of an interdisciplinary pre-service and in-service teacher training. One of the distinguishing features of this course is the view beyond the subject boundaries of mathematics and physics. Since 2000, the advantages of interdisciplinary work in teaching and learning in schools have been emphasized, mostly in the form of problem-based and cross-disciplinary teaching that is organized in projects [START_REF] Labudde | Fächerübergreifender naturwissenschaftlicher Unterricht -Mythen, Definitionen, Fakten[END_REF][START_REF] Moegling | Kompetenzaufbau im fächerübergreifenden Unterricht[END_REF]. In order to achieve the greatest educational benefits, numerous variants of these demanding and complex learning concepts have been developed [START_REF] Caviola | Wie Fächer miteinander ins Gespräch kommen: Modelle der Fächervernetzung und ihre Lernziele[END_REF][START_REF] Labudde | Naturwissenschaften vernetzen -Horizonte erweitern: Fächerübergreifender Unterricht konkret[END_REF]. The ability to systematically combine, apply and reflect knowledge from different disciplines is the goal of this project. However, in spite of all the educational advantages, interdisciplinary work also reveals many challenges. The combination of different methods of the participating subjects, overcoming communication difficulties, the identification of common research subjects, the handling of prejudices as well as the handling of group dynamic processes are only some of the problems of interdisciplinary cooperation (Defila & Di Giulio, 2002, 24). The didactic potential of subject-linking teaching and learning has been discussed for some time, especially in mathematics, natural sciences and technology [START_REF] English | Advancing elementary and middle school STEM education[END_REF][START_REF] Kelley | A conceptual framework for integrated STEM education[END_REF]La Force, 2016;[START_REF] Michelsen | Expanding context and domain: A cross-curricular activity in mathematics andphysics[END_REF].

Despite the many benefits of interdisciplinary teaching and learning at school, the teacher training at German as well as Vietnamese universities is still largely organized in a discipline-oriented manner. The first phase of the teacher training should clarify the advantages and tackle the challenges of an interdisciplinary education. To make interdisciplinary teaching and learning in schools be more than the additive juxtaposition of elements of knowledge from different disciplines [START_REF] Wellensiek | Beltz Wissenschaft: Bd. 38. Interdisziplinäres Lehren und Lernen in der Lehrerbildung: Perspektiven für innovative Ausbildungskonzepte[END_REF], the universities should implement holistic concepts for integrative teacher education. The inadequate fit of non-integrated teacher education with the requirements of interdisciplinary teaching [START_REF] Bröll | Zur Qualifikation der Lehrkräfte für den NWA-Unterricht -eine Bestandsaufnahme in Baden-Württemberg[END_REF] is repeatedly cited by teachers in schools as an objection to integrated instruction [START_REF] Jürgensen | Das integrierte Fach Naturwissenschaften und seine Beliebtheit bei Lehrern und Schülern[END_REF][START_REF] Rehm | Legitimationen und Fundamente eines integrierten Unterrichtsfachs Science[END_REF]. If interdisciplinary teaching at school is to succeed, teacher training must also be adequately designed [START_REF] Brown | Professional development through STEM integration: how early career math and science teachers respond to experiencing integrated STEM tasks[END_REF][START_REF] Cormas | Preservice teachers` reconciliation on an epistemological issue in an integrated mathematics/ science methods course[END_REF]. Studies show the importance of the preservice teachers' knowledge of interdisciplinary pedagogy (An, 2017). However, it has become apparent that experienced teachers in particular have reservations about interdisciplinary teaching [START_REF] Thibaut | Teachers' attitudes toward teaching integrated STEM: the Impact of personal background characteristics and school context[END_REF]. For this reason, the Inter TeTra project is planning an interdisciplinary master course and in-service teacher training. Appropriate didactic concepts have already been developed at some universities (Krause & Witzke, 2017;[START_REF] Witzke | Fachdidaktischverbindendes Lernen und Lehren im MINT-Bereich[END_REF]. The University of Siegen works currently on interdisciplinary education projects such as MINTUS, FäMaPDi and InForM PLUS [START_REF] Holten | Chancen und Herausforderungen fachdidaktischverbindender Elemente in der Lehramtsausbildung. Beiträge zum Mathematikunterricht[END_REF]Krause, 2017). For this purpose, the subjects of mathematics and physics seem to be the most appropriate, since these subjects have numerous epistemological parallels [START_REF] Krause | Erkenntnistheoretische Parallelen zwischen Schulmathematik und -physik[END_REF]. First approaches for establishing subject-linking lessons already exist in Vietnam [START_REF] Nguyen | Entwicklung interdisziplinärer Themen im Naturwissenschaftsunterricht[END_REF]. In contrast, the present project does not initially focus on interdisciplinary teaching in Vietnamese schools but starts earlier by adding an interdisciplinary module to teacher education. In this way, the connecting element in the subject-linking lessons of the future is not only the common subject of instruction (which is viewed from the perspective of different subjects) but rather the embedding of lessons in the comparative discussion of didactic theories in the participating subjects from the very beginning, with the goal of concretization and implementation in the Vietnamese curricula. This approach is an innovation especially for Vietnam, where teacher training is currently very isolated and compartmentalized. While the modern application-oriented teaching of mathematics looks to physics didactic concepts for experimentation, modern physics teaching also requires mathematical didactic knowledge to deal with technical problems via mathematics [START_REF] Schwarz | Vorwort zur zweiten Auflage[END_REF]. Consequently, the repertoire of future teachers will become richer by incorporating the didactics of neighboring subjects. So this project aims to make a meaningful and lasting impact through teacher training, combining classroom teaching with practical instructions. The aim is the development of a competence-oriented curriculum for teacher training in Hanoi. The Vietnamese partner university is the authoritative body of educational policy reforms within the country, ensuring later dissemination of the developed module. The previously mentioned projects at the University of Siegen have demonstrated that interdisciplinary teaching in teacher education provides a deeper insight into the didactics [START_REF] Witzke | Fachdidaktischverbindendes Lernen und Lehren im MINT-Bereich[END_REF]. The combination of didactical theory and teaching practice is particularly important to our approach.

Concept of the courses in the Inter TeTra-project

General information

The duration of the Inter TeTra-project will be four years. In the first year (2018) the colleagues from Vietnam have joined the referring projects in Siegen -the FäMaPDi and InForM PLUS projects [START_REF] Holten | InForM PLUS vor der Praxisphase -Zwischenbericht eines interdisziplinären Elements in der Lehramtsausbildung an der Universität Siegen[END_REF] and both sides discussed the theoretical framework of the project. The courses for the HNUE will be designed in the year 2019 and will take place in Hanoi in 2020 and will be repeated in 2021. The idea is the implementation of an interdisciplinary course for pre-service teachers in the teacher training curriculum at the HNUE and the offering of an interdisciplinary course for in-service teachers with the subjects mathematics and physics. These courses will be composed of a theoretical and a practical part.

Theoretical part

The aim of the courses is to enable students to compare the didactical theories of their own discipline, which they have come to know during their studies, with the didactical theories of the other subject. Since only a limited number of sessions are available for the theoretical part, only a selection of topics can be found that is relevant for both subjects. In order to combine subjects in a meaningful way, one should compare the common "Big Ideas" of the subjects (c.f. Chalmers, 2017). One research desideratum that the Inter TeTra Project focuses on is to explicate topics, which are relevant for mathematics and physics education. For this reason, handbooks and conferenceproceedings of mathematics and physics education will be compared systematically to identify intersections. Even if this research is still in process an exemplary selection of topics, which are suitable for an exchange between mathematics and physics education, can be listed:

Nature of science vs. beliefs of mathematics: What is physics? What role does the experiment play in physics?

To what extent does a physical theory depict reality? Because questions of this kind are crucial for teaching-learning processes, the didactics of physics has been dealing with them for some time under the heading "nature of sciences" (Aydeniz et al., 2013;[START_REF] Dass | Understanding the nature of science enterprise (nose) through a discourse with its history: The influence of an undergraduate `history of science` course[END_REF][START_REF] Hötticke | Rekonstruktion der Vorstellungen von Physikstudierenden über die Natur der Naturwissenschaften -eine explorative Studie[END_REF][START_REF] Kahana | Understanding of high-achieving science students on the nature of science[END_REF][START_REF] Kartal | Improving science teachers' nature of science views through an innovative continuing professional development program[END_REF]. Likewise also the mathematics didactics researches on the different views on mathematics (e.g. [START_REF] Grigutsch | Einstellung gegenüber Mathematik bei Mathematiklehrern[END_REF]Witzke & Spieß, 2016), because the individual conceptions determine our concrete activities in science. What similarities can be identified between perceptual research in mathematics didactics and NoS research in physics didactics? What are the differences? Interesting approaches can be found in the literature: Lawson (2008) talks about Nature of Science and mathematics. [START_REF] Rolka | Researching young students` mathematical world views[END_REF] talk about mathematical worldviews of students.

Modelling:

Modeling is considered in some publications to be a linking element between disciplines. [START_REF] Blum | Applied mathematical problem solving, modelling applications, and links to other subjects -State, trends and issues in mathematics instruction[END_REF][START_REF] English | Promoting interdisciplinary through mathematical modelling[END_REF][START_REF] Michelsen | Functions: A modelling tool in mathematics and science[END_REF]. In physics, models (such as atomic models) are an integral part of the theory canon. Accordingly physics education has to clarify how to deal with models in the teaching-learning process [START_REF] Gilbert | Models and modelling: Routines to more authentic science education[END_REF]Oh & Oh, 2017). It is not so common to stress that models should not just be taken over, but rather to emphasize the process of creating models. In mathematics education modeling has been discussed for many years [START_REF] Burkhardt | Modelling in mathematics classrooms: Reflections on past developments and the future[END_REF][START_REF] Frejd | Mathematical modelling as a professional task[END_REF]Guerro-Ortiz et al., 2018;[START_REF] Kaiser | Mathematical modelling as bridge between school and university[END_REF]. Several models for modeling -so-called modeling cycles -have been developed. So it seems that physics education focuses on the general product while mathematics education is more interested in the process of individual modelling. It is obvious that both sides can learn from each other in this matter, even if it is more laborious than it seems [START_REF] Neumann | Modellieren aus mathematischer und physikalischer Perspektive[END_REF].

Preconceptions: In physics didactics preconceptions are treated to different physical contents. They are often negatively connoted as misconceptions. In mathematics education one rarely speaks about misconceptions, trying to highlight positive pre-theories among learners. The term "Grundvorstellung" (vom Hofe, 1992) is certainly prominent in this context in Germany. At the same time, mathematics educators have developed more general theories on context-related learning, which can also be applied to other subjects. For example, the approach of Heinrich Bauersfeld to "Subjektive Erfahrungsbereiche" should be mentioned [START_REF] Bauersfeld | Subjektive Erfahrungsbereiche als Grundlage einer interaktionstheorie des Mathematiklernens und -lehrens[END_REF]. Krause has sketched the transfer of this theory to physics and discussed the question of how on the other hand mathematical didactics can be fertilized by approaches of physics didactics [START_REF] Krause | Nutzen von fächerverbindender Didaktik am Beispiel von subjektiven Lernvoraussetzungen im Mathematik-und Physikunterricht[END_REF].

Practical part

In addition to the theoretical sections the courses mentioned and planned in this project will also include lessons at schools. The fact that the didactics of a neighboring subject can be relevant to one's own subject should be clarified by the conception, implementation, and reflection of lessons that try to combine school mathematics and physics in a meaningful way. For this purpose, the interdisciplinary comparison of theories on learning and teaching mathematics and physics in schools will be used to develop research questions in the seminar, which will be examined in the course of the lesson designs developed in the seminar and tested at the cooperation school. These lessons will be videotaped and incorporated into the previously discussed theory. This combination of theoretical lessons and review is established in North Rhine-Westphalia during the practical semester in teacher training [START_REF] Hoffart | In der Situation ist mir das gar nicht aufgefallen!" Reflexionsanlässe in der Lehrerbildung als Bindeglied zwischen Theorie und Praxis[END_REF]. Such a theory-based classroom reflection research is not yet part of teacher education in Vietnam. However, evaluation criteria do not refer primarily to how the lessons themselves succeeded, but to how the teaching process can be analyzed and classified on the basis of the theoretical sessions. At this stage of the seminar, students should be able to evaluate and justify didactic decisions based on theory.

Research Explication of relevant content

The selection of topics in the theoretical part should not be made ad hoc but should reflect the intersection of mathematics and physics education research in terms of content and points of contact. The contents presented in Chapter 2 were determined by the lecturers themselves during the pilot phase in Germany. For the realization of the teaching interventions in Vietnam (which is to take place for the first time in 2020), the theoretical topics are to be systematically selected. For this purpose, the project participants will carry out a qualitative content analysis (Mayring, 2010) of current manuals and proceedings with the question: Which contents are relevant for an interdisciplinary exchange of mathematics and physics didactics? Results of this analysis are still pending.

Evaluation of the teaching interventions

The evaluation of the Master's course and the teacher training should contribute to answering the following research question: How aware are the participants of the relevance of didactic theory of the other subject for their subject? Especially in Vietnam, where teachers are only trained in one subject, it is important to raise the awareness that mathematics didactics is important for physics teaching and that physics didactics is also important for teaching mathematics. The participants will be interviewed at the beginning and end of the theoretical part with an open questionnaire (which is still in the conception phase). What is special about the Inter TeTra project is, that in the practical part the participants are to hold a lesson which will be be evaluated video-based. This evaluation also takes place exclusively under the question at which points of the lesson the didactics of the other subject are relevant. This approach was also used in the pilot implementation in Germany. First results may be found in [START_REF] Holten | InForM PLUS vor der Praxisphase -Zwischenbericht eines interdisziplinären Elements in der Lehramtsausbildung an der Universität Siegen[END_REF].

Conclusion

The project's research goals are to clarify relevant topics for an interdisciplinary exchange between mathematics and physics didactics, to design and implement a corresponding course in the teacher training curriculum and a course for in-service teachers at HNUE. The performance of the courses will be evaluated in order to assess the effectiveness of interdisciplinary teaching in the subjects mathematics and physics during the training of preservice teachers.

In this project, an intellectual exchange between four institutes is taking place. Each institute will be focusing primarily on the mathematics and physics disciplines in Hanoi and Siegen as well as four components of teacher training and further education, hence the derivation of the name Inter-Tetra from the Latin "Inter" (between) and Greek "Tetra" (four) and its dual connotation for Interdisciplinary Teacher Training.


Kirsten Wohak
email: wohak@kit.edu

Martin Frank
email: 2martin.frank@kit.edu

Complex Modeling: Insights into our body through computer tomographyperspectives of a project day on inverse problems

Keywords: High school mathematics, inverse problems, mathematical modeling, MATLAB, problem-solving

How does an autonomous vehicle recognize a traffic sign? How can seismic measurements be used to determine the origin of an earthquake? How do you get a three-dimensional tomographic reconstruction of a body part from a medical scanner? The methods used to answer all these questions come from the field of so-called inverse problems and are central to our society. They are omnipresent in technology, science and in everyday life and include mathematics without any people being aware. The process of mathematical modeling is used to solve inverse problems. Complex problems are simplified in such a way that mathematical models are developed and then accurate solutions can be calculated. The question now arises as to whether high school students are able to independently and accurately solve these kind of questions while only using their mathematical school knowledge and the computer. An approach to show how this could be done is given in this paper.

Introduction

When taking a look at maths classes in school you often find students asking: "Why do I need to study this?" Schoolbooks are very method-oriented which has the effect that students can use certain algorithms and formulas but do not recognize the usefulness of mathematics for everyday life. This is why the development of interactive, problem-oriented material for high school students should be increased. Using this kind of material as an addition to the existing units students see the potential of mathematics to solve real life problems1 .

The project presented here will be part of the CAMMP (Computational and mathematical modeling program) project of the Karlsruhe Institute of Technology2 and the RWTH in Aachen3 . For several years, similar projects have been carried out in one-week and one-day formats with students of different ages in Aachen (One example can be found in Greefrath & Siller, 2018, p. 137-163). Until now over 400 students have participated in a modeling week and over 2900 in a modeling day. Since the beginning of 2018 the project is also being realized in Karlsruhe.

CAMMP wants to offer ways to innovative mathematics teaching in terms of content, didactics and methodology. Concepts for this were developed and tested. CAMMP is not only involved in mathematical modelling projects with pupils and teachers, but also in research and teaching in the field of mathematical modelling didactics. The goal is a didactics of problem-oriented mathematical research, in particular a creative handling of mathematics. Tasks and projects are always as problem-oriented as possible and actively dealt with by the students. Adapted to the given framework conditions (e.g. modeling week, -day, regular school lessons), innovative concepts are developed to balance the dimensions active-passive and method-oriented4 problem-oriented.

These projects are needed because a change in teaching mathematics must take place. Many graduated students do not know a lot about mathematical modeling even though it is one of the process-related competencies of the core curriculum. And according to the educational standards of the Conference of Education Ministers, all students should be able to model mathematically by the end of their school career. That means they should be able to go through the elementary steps of the mathematical modeling cycle independently (Kultusministerkonferenz der Länder in der Bundesrepublik, 2012). Real life is not about calculating as much as possible to exercise a certain algorithm (Maaß, 2015, p. 70). Mathematics should not be used for its own sake, but rather as a tool for solving problems (Consortium for Mathematics and Its Applications, 2016). It is clear that the necessary competence for the creation of models can only be built up if the pupils carry out their own modeling independently and regularly during their school years (Maaß, 2011, p. 8). However, modeling is new and unknown to many teachers and learners. Each step of the modeling cycle represents a cognitive hurdle for the students [START_REF] Heiliö | Introduction[END_REF]). On the one hand they find it difficult to understand the real situation, on the other they expect teachers to evaluate the correctness of their solution, neglecting the validation step. If teachers face each of these challenges, studies have shown that the treatment of authentic modelling has a significant positive impact on their opinion of mathematics (Borromeo Ferri, Greefrath & Kaiser (Hrsg.), 2013, p. 30-34).

The project presented in this paper makes an important contribution to developing interactive, problem-oriented material by presenting a way to include computer tomography as an example of inverse problems into maths classes. It shows that the abilities of students should not be underestimated. The basics needed to understand inverse problems are already taught during school mathematics.

Inverse problems in school and everyday life

When having a look at cause-effect relationships one can either calculate the effect with given causes or turn the problem around and draw conclusions from given effects on the cause. If you have a look at image processing you can ask yourself what a clear image would look like if the camera were focused incorrectly. When turning the problem around you get the inverse problem, which means trying to deblur an image that was taken by an incorrectly focused camera. In both cases you have to solve a system of equations, but in mathematics there is a difference between the equations you use for the two problems. In real life you never have flawless data, which means that by using some kind of measuring tool you get errors in your data. If the error stays in the same magnitude while computing the solution, everything is perfectly fine. However, what happens if a tiny error in the given data leads to enormous errors in the solution?

These problems are so called ill-posed inverse problems. Jaques Hadamard described three conditions through which one can determine whether the problem at hand is well or ill-posed (Engl, Hanke & Neubauer, 2000, p. 31): There must be at least one solution (existence). There may be no more than one solution (uniqueness). The solution must depend continuously on the data (stability).

At the moment schools mainly deal with tasks in which direct problems occur. Data is presented and a function must be applied to it. This means that you start with the cause and calculate the consequence [START_REF] Lasanen | Introduction to inverse problems[END_REF]. This type of task only requires students to perform prescribed operations. Students are therefore often limited in their mathematical thinking [START_REF] Gardiner | Teaching mathematics at secondary level[END_REF]. In the long term, however, the inverse problems are important, since through them students learn to work more freely, flexibly, playfully and application-oriented with the mathematics they have learned [START_REF] Gardiner | Teaching mathematics at secondary level[END_REF]. From the teacher's point of view it is more challenging to discuss inverse problems in school because the approaches of students to inverse problems are far more open and are therefore not well predictable. These open tasks make it more difficult to have a joint discussion in class since there are many different calculations that lead to a correct solution. On top inverse problems are a lot more complex to grade in an exam. Eventually, due to the lack of discussion, confrontation and exercise, students achieve results a lot worse when having to solve inverse problems than when dealing with direct problems [START_REF] Gardiner | Teaching mathematics at secondary level[END_REF]. Thus to prepare their students and give them the opportunity to think freely in terms of mathematics, teachers should include more inverse problems into their classes especially ones that are similar to the one presented in the following.

Perspectives of a project day about computer tomography

The aim of the project day for one is to show the working principle of computer tomography, but moreover it should also show the relevance of mathematics and simulation sciences in society in general. The students should realize that already with their knowledge they can understand and compute the basic methods behind complex processes. The material is formulated in a way that different approaches are possible and welcome. By using the software MATLAB the requirements regarding programming are set very low. The students are given a prepared code in the form of a gap text in which they only have to enter missing formulas or equations, which are then checked as they run the code. This means that students receive feedback on their considerations at all times. On top it is possible to include worksheets into MATLAB, so that the surface looks rather like a digital working sheet with code lines in between than a programming software. Through all of this, one hope of the developer is that students become more involved with mathematics.

Computer tomography

For computer tomography, the internal structure of an object is determined depending on the entry and exit intensity of the X-ray radiation and conclusively depicted [START_REF] Grumme | Zerebrale und spinale Computertomographie[END_REF]. The objects are irradiated with parallel beams at different fixed angle settings to determine the absorption coefficients of the different materials of the object [START_REF] Mueller | Inverse problems with practical applications[END_REF]. Depending on the material the intensity of the incoming X-ray beam decreases differently. Thus, the out coming intensity depends on the absorption property but also on the layer thickness of the material through which the X-ray beam passes. If a single layer is penetrated by a beam with the intensity , the output intensity is calculated by the following equation:

With the help of the a transformation, the parallel beams passing through the object are described mathematically as straight lines. By doing so, the function that was originally dependent on the Cartesian coordinates and is transformed so that the function is now described along a straight line with the angle and the distance to the origin (see Figure 1). The points lying on the beams are described with , with .

If you now look at different distances from the line to the origin, the straight line moves to the right for (see Figure 1, blue line) and to the left for (see Figure 1, red line). This is of great importance since an object gets scanned by parallel beams. So the mathematical model also has to include several beams with a constant distance between one another. In addition, the X-ray tube is rotated around the object so that the beams are sent through the object at different angles. Due to the symmetry, it is sufficient to only look at the angles between and and therefore only to run a semicircle. Considering both factors one gets beams and therefore equations in total when having parallel beams and angle positions.

Assumptions and simplifications

Before the students establish a system of equations and have a look at its solubility and the reconstructed inner structure of the scanned object, simplifications must be made. Strictly speaking, an object, like a human body, consists of an infinite number of different absorption coefficients. For a first model however, it firstly is sufficient to approximate the cross-section of an object as a square and secondly to allow only a small number of different absorption coefficients. The idea is to cut the square into smaller squares called pixels and to assume that each pixel only consists of one particular material and therefore only has one absorption coefficient , see Figure 2. In addition, the absorption law is simplified. Instead of looking at the influences of the absorption coefficients, where the integral would have to be used, the object is described by gray values and these are weighted and added along one beam. Thus, the measured values, depicted as with in Figure 2, do not represent intensities but simply weighted and added gray values , , which must be correctly distributed to the respective pixels for the reconstruction of the object. With the first worksheet the students get familiar with what actually happens during computer tomography. The students start into the task, where they try to reconstruct the gray values of the pixels for the first time. Because of the simplifications and assumptions that were made beforehand the students have a first impression of how the gray values of the pixels can be reconstructed, namely simply to solve a linear system of equations. So the students are given four values for with

Worksheet 1: First radiography of a simplified object

. By doing so a lot of questions arise: Can numerical values for the pixels be found so that the row and column sums are satisfied? If there are such entries, how are they determined and can they be determined in a unique way? If they do not exist, why not?

With the help of these questions, the students can independently work out the conditions of the concepts of existence and uniqueness and work out connections for solubility. They notice that with these values they do not have enough information to uniquely determine the gray values for the reconstruction of the object even though the students have four equations and four variables. The solution is not unique because the equations are not linearly independent. This first approximation to the mathematical description of the problem is done without formulas but instead using examples of possible gray values that the pixels could have with the given values . Students should find a gray scale distribution and then either by comparing their own solution with other groups of students or by realizing mathematical considerations recognize that their distribution is not a unique solution. After this realization, the students are faced with the challenge of extending the problem in such a way that the gray values of the pixels can be explicitly solved. To do this, they must transmit a further beam through the object and determine the measured values of these new beams. For example it is possible to add a diagonal beam (see Figure 2), which leads to the system of equations (1) which has a unique solution.

When you increase the number of pixels you keep having the same problem. Looking at the object subdivided into nine pixels (see Figure 3) it is again not enough to send nine beams through the object. The problem why this system of equations also is not uniquely solvable is that you only have eight linearly independent equations. So even more beams have to be used. Since the system of equations keeps growing it makes sense to write an algorithm that describes the beams mathematically and creates the matrix one needs to solve the system of equations. At this point of the project the algorithm can be developed, as the students have now completely penetrated the problem. In addition, the number of equations is still manageable, which also means that drawings of the object and the beams can be made without being confusing. This way errors and new solutions can be found more easily. The main goal of this worksheet is to find a mathematical model to describe the beams more generally for any number of parallel beams and angle settings, since so far the students only had the situation shown in Figure 3. The idea is to describe the beams as straight lines in and then come up with an algorithm that calculates the length that a beam spends in each pixel. The last important step then is to not only get the individual lengths but to know where in the matrix the value has to be written.

Worksheet 2: Mathematical description of the beams

For this task the students get the starting vectors of the points where the beams enter the object including the value of the associated angle for a given number of parallel beams and different angles. The standardized direction of the beam is described through the vector . Therefore the beam itself can be described as . Students know this way of writing straight line equations from school. Nevertheless, help cards are also available to the students as support, since they may have difficulties, for example, with the use of the cosine and sine.

The points of interest are always those where either the or coordinates are equal to one of the pixel borders, namely or (see Figure 3). Thus to determine the interceptions the equation for the beam needs to be separated coordinate wise and solved for :

Again at this point the students have the possibility of using help cards, since they have to think of separating the coordinates and solving the equations. Some students have difficulties with deformation calculation steps. As feedback and an output of MATLAB the students get a depiction of their description of the beams through the object, so that they themselves can see whether the result seems to be consistent.

Once the students achieved this task they can easily calculate the length between two neighboring interceptions which is the length a beam spends in a pixel. The hard part is to figure out in which column and row of the matrix the length has to be written. One approach is to calculate the distance between the center of the pixels and the middle of the considered distance (see Figure 4). The center to which the distance is smallest is the pixel in which the beam spends the particular length. A second approach would be to have a look at the the or coordinates of both interception points. After having numbered the pixels and defined the possible values for the and one can check to which both points belong and therefore determine the pixel number. While defining the pixels you have to include the border of the pixels each time. After this mathematical model for the beams the students get the chance to reconstruct an image such as the first one in Figure 5. To get this first reconstruction the students have to find the gray values so that the difference between and the measured values get minimized. To make it a little less complicated for the students they calculate with a small matrix and small vectors and :

From school they know that they have to differentiate after and which leads the students to the gradient of . They then have to set the gradient to zero to find the that minimizes the equation:

The formula that results for the gray values that is, when multiplied with , closest to the measured data, is the following:

If one uses data without errors the reconstruction is perfect. However, using flawless data does not depict reality. Every measuring device creates some kind of error so that you have to use data with a small error. Besides that in reality the scanned objects consist of more than nine different materials and therefore more than nine different absorption coefficients or in our case gray values. So on top of adding noise one has to increase the number of rows and columns in the object, which also leads to a higher number of parallel beams and more angles. If one adds noise one can already no longer recognize the original object (see Figure 5, picture 3). This is due to the fact that the problem is not stable since the matrices are singular and therefore badly conditioned. Thus the problem has to be slightly modified, so that one gets a more stable problem. This can be achieved by different mathematical methods which deliver results that come very close to the original picture (see Figure 5, picture 4 and 5). The aim is that the students should be guided as little as possible during the project. It would be optimal if they could independently develop a method to improve the reconstruction. At the moment this seems very hard to achieve but will be improved once the project has been realized with students.

Outlook

In the future, the existing material will be realized with high school students. Furthermore the workshop will be evaluated in order to assess what the students learned and how the material could be changed. The material will then be iteratively improved. Moreover material will be developed so that the implementation of the project can be adapted to school hours and thus be carried out in units. In addition, a draft of the described problem for an exam task will be developed.

In conclusion this project shows a possibility to treat authentic and relevant problems with the help of mathematical modeling in such a way that students are not only able to understand them but can also solve these. This should show that it is also possible to treat modelling competence in such open questions at school. The students' knowledge of mathematics is sufficient to understand the basics of these problems from their everyday life. This example should encourage teachers to address more complex problems in class in order to train and further develop their students' mathematical modeling skills. For example, parts of this workshop can be incorporated into the classroom, allowing students to directly see an authentic application of mathematics.

Figure

  Figure 1: Illustration of the radon transformation

Figure

  Figure 3: Nine beams through nine pixels

Figure

  Figure 4: Illustration of the pixel number determination

Figure 5 :

 5 Figure 5: The object and different forms of reconstructions with 5 % noise

We understand the word problems in the sence of[START_REF] Heinrich | Handbuch der Mathematikdidaktik[END_REF].

http://www.scc.kit.edu/forschung/CAMMP

https://blog.rwth-aachen.de/cammp/

By method-oriented we mean the application and practice of mathematical methods to solve purely mathematical problems without reference to application. In the problem-oriented treatment of tasks, certain mathematical methods are only applied when they are needed to solve a real extra-mathematical problem[START_REF] Heinrich | Handbuch der Mathematikdidaktik[END_REF].