Amusement Park Mathematics in Teacher Education, O13/G03
Ann-Marie Pendrill, Department of Physics, Ann-Marie.Pendrill@physics.gu.se
Lisbeth Lindberg, IPD,
Thomas Weibull Dept. of Mathematics,

Göteborg University, Göteborg, SWEDEN

ABSTRACT

In this project, mathematics student teachers were given the opportunity make use of Scandinavia's largest amusement park, Liseberg, for mathematics learning. Initially, students interacted with a large number of school classes visiting the park. Later in the project, the park was used an environment for creating assessment problems for the students' school classes where Liseberg is a shared previous experience. The authenticity of the task, both involving the application of mathematics, and in the interaction with pupils and teachers, was found to inspire the students to creativitiy, but also to study the school curriculum, to use computers as a support, and to investigate mathematics in new situations.

Keywords: Mathematics Education, Preservice Teacher Education, Assessment, Informal Learning, Cooperative Learning

1. Introduction

1A. Rationale for change

The new teacher education (Utbildningsdepartementet, 2000) prescribes subject studies in situations outside the university classroom, typically in schools. This so-called "VFU", amounts to 25% of the initial subject studies. The VFU has the potential to deepen students' learning of the subject by exposing them to children's curiosity, thereby providing motivation for the students' own learning in a situation where they have easy access to specialists. The interpretation of the VFU intentions vary between different teacher educations, as well as the possibilities to integrate it into the subject. To fulfil the ambitious intention of the VFU as a fruitful component in subject learning is a serious challenge for all teacher educations.

This project addresses the VFU in mathematics studies and demonstrate how e.g. an amusement park, such as Liseberg, can be used as an extramural learning environment. Mathematics is important as a common first subject ("direction") studied by student teachers. Many of the students have then continued on to physics or science studies, whereas others have chosen to include, e.g., languages or sports in their teaching degree.

Extramural experiences are known to play an important role for learning and motivation of children (Woolnough, 1994). However, the outcome of a visit depends on the extent to which it is integrated with work in school. Teachers need preparation to ensure that activities outside school contribute to the intended learning. Including study visits during the initial teacher training will leave future teachers better prepared to use various possible external activities.

An amusement park provides an abundance of examples of mathematics and physics, although these are usually unnoticed by thrill-seeking visitors. The aim of this project was to develop, together with students, mathematics activites in extramural settings, in particular in the Liseberg amusement park. Many school classes from around the country visit Liseberg. We include amusement park activities for future teachers. This may help them to use future school journeys to Liseberg also for mathematics studies and is one way to prepare students for a richer repertoire of teaching activities.

An important aim of the project was to develop an inspiring platform where students can learn from the interaction with real school classes, and where the authenticity of the situation provides additional motivation, and contributes to internalization of the requirement on student assignments.

1B. Review of relevant literature

This project brings together a number of fields, including learning in informal settings. Whereas some areas are well studied, the context where student teachers assist children in informal settings, as part of their pre-service subject studies, seems unexplored: The subject "VFU" - subject learning in school for student teachers - is not yet well established. However, one of the participants in previous project (030/G02), Pernilla Nilsson, is now a PhD student in the national graduate school in Sciences and Technology Education, FoNTD, studying both learning of young children and student teacher's reflection on their own learning experiences (Nilsson et al, 2004, Nilsson 2005a, b).

Learing in informal settings

The amusement park, as an informal setting for learning, has much in common with science centers, where the literature, discusses the "Liseberg-effect" (Axelsson, 1997), i.e., that the children run around, having fun, and making sure they don't miss anytning. Thus, in both situation the pupils are easily distracted from learning, unless they are mentally prepared for the visit. This aspect is even more present in an amusement park than in a science center. The importance of preparation and follow-up for the learning outcome is documented e.g. by Rennie and McClafferty (1996). That this is often lacking has been observed in that and other studies (e.g. Kofod and Sørensen, 2002). That learning, can, indeed, take place in connection with an amusement park activity is documented e.g. by Bagge and Pendrill (2002) and Nilsson et al (2005). Learning in informal setting is in focus for the NSF-supported "Center for Informal Learning and Schools" (CILS, 2006).

Mathematics in Amusement Parks and in Roller Coaster Design

Amusement park rides obviously involves motion, velocity and acceleration. The buildings, rides and park offer many different mathematical shapes: A number of investigations, suitable for amusement parks are presented e.g. by Bakken (2006), Coaster Quest (2006), The Mathematical Association of New South Wales (MANSW, 2006) and the project Slagkraft (2006).

During 2005, a new roller coaster, Kanonen, at Liseberg, which includes a loop, provided motivation to study more difficult mathematics. In order to provide safe riding experiences, with continuous higher derivatives of the velocity, a roller coaster loop is not purely circular, but has a smoothly changing radius of curvature. The loop shape in Kanonen is called a "clothoid", also known as a Cornu or Euler spiral, and invites the study of more advanced mathematics. It was introduced in amusement rides by the roller coaster designer Werner Stengel (Schutzmansky, 2001, Pendrill 2005), who was awarded an honorary doctorate at Göteborg university in 2005.

More advanced mathematical applications can also be introduced by using motion tracker measurements in a roller coaster, which provides data for acceleration and rotation around three axes (Pendrill and Rödjegård, 2005).

1C. Questions

The initial questions of this project focused on student and pupil learning in connection with joint extramural experiences. The later part of the project focused more on the students' development of assessment skills and experience, raising additional questions:

1D. Importance of the project

Mathematics is often viewed as a difficult subject, creating considerable anxiety among learners (e.g. Tobias, 1987, Ashcraft and Kirk, 2001). It has been found that much of today's mathematics education in Swedish schools consists of pupil's individual text-book based problem-solving (Skolverket, 2003). On the other hand, children are often curious about many aspects of the world around them. By bringing teacher students in contact with children's curiosity, we hope to inspire students to look beyond "Why do we need to know this? - It's not in the curriculum". This project also aims to help both children and student teachers develop a deeper understanding on mathematics by applications to new contexts, which invite observations, estimations, creativity and discussions.

Assessment of learner's knowledge and understanding is a complex ability, that takes a long time to develop. It benefits from discussions with colleagues and with teachers more experienced in the area. A focus on assessment in a course which involve teachers from different academic cultures, can make more explicit the different principles and traditions in examination and lead to a discussion among teachers about how examination affects learning outcomes.

2 Method

This project was run in collaboration between three departments and a number of schools, teachers, pupils and students. The students had to look for mathematics that is in line with a particular age group. The student teachers were given the opportunity to interact with school children visiting Scandinavia's largest amusement park, Liseberg. This gave them an opportunity to observe many teachers interacting with their classes and to reflect on the experience. The students had to prepare tasks in the subjects that would be relevant for the group of pupils and as authentic as possible. After the visit the student could immediately evaluate their own tasks and reflect upon how the pupils discuss mathematics in an extramural activity.

In the second part of the project, students visited the amusement park on their own, creating mathematics tasks for their classes, preparing assessment schemes and trying them in their own classes, as discussed in more detail below.

2A. Students

The students in this project studied mathematics as part of their teacher education. At Göteborg university, this education, in general, starts with one semester of more general courses, including education and interdisciplinary courses. For some students, it was the second and third semester in their education, whereas other had taken other subject before mathematics.

2B. Innovation

The project involves a number of innovations, both concerning the way to study mathematics and in the way students are given possibilities to interact with pupils and their teachers. An essential part of learning a concept is to develop different ways of experiencing it. A deep understanding requires many different representations of the concept and links to different representations, as well as to other contexts. Often, school mathematics is far removed from real world experiences, and also from mathematics as a discipline. As an introduction to the project, students were given some examples of possible math tasks and reminded of some of the steering documents' goals for mathematics (Skolverket 1994, 2000).

Throughout the project, we have tried ways to enhance the interaction between students and school as a way to enhance students' and pupils' understanding of math, by using authentic examples from shared experiences.

   
Figure 1 Geometry in the amusement park: Polygon flower box at Liseberg and a half-circular window, divided into several panes.


Figure 2: The classical Teacup ride with an example of a calculated track.

Mathematics in an amusement park

Triangles, squares, circles, angles, vectors, probabilities, and derivatives, parabolas, catenaries, splines and clothoides. An amusement park is filled by mathematics of all levels of difficulties, although most visitors don't think about this aspect. We found that it was important to provide students with a examples to initiate their creativity, and provide also a few examples here to help the reader see the possibilities.

The flower boxes as in Figure 1 have the shape of a regular polygon. How many corners does it have? How large is the angle between the sides? How long is each side for the largest box? How long is the distance across? Between two sides/ two corners? (Before the visit you can draw a similar polygon and measure in your picture.) On one side of the house you will find a half-circle shaped window. How many glass panes does it have? What fraction of the area is covered by the different panes?

Older pupils can study what is happening if two counter-directed circular motions with different radius and times of revolutions are combined, as in the classical Teacup ride (Figure 2). How does the relation between the periods of the two motions affect the shape of the track? In what position is the speed the highest? Where is the acceleration the largest? The tracks can be investigated with paper and pen, with Excel or matlab, as well as though observations in the park.

Student involvement.

During 2005, the students first had an introduction, which included examples of amusement park problems, and introduction to the structure of the national tests in mathematics. Their assignment was to construct one short problem and a larger problem, which would allow assessment of various aspects of mathematical knowledge. They should also write grading instructions and provide a matrix for assessment of the larger problem. During a seminar in groups of 8-10 students, they presented their problems, and discussed the formulation and the assessment instructions.

For the second term, groups of 3-4 students were formed, according to the age of the school classes where they had their VFU. Each group selected and refined one of the problems from the first term, and then all tested the problems in their own classes and graded their pupil's solutions. They then compared the solutions by the pupils, discussed their grading, and revised the assessment instructions. At the end of the project, every student discussed their task with 3-4 students having done different problems, giving each other feedback. They then returned to their initial group, comparing notes and writing a reflection on the response. They also wrote a longer individual reflection on the whole process. An important part of all seminars was to discuss the project itself, including difficulties, and possible improvements.

2C. Procedures

In May 2004, the students participated in a science day at Liseberg. Before the day, they had been given an introduction to the possibilities in an amusement park, and assigned one ride where they would spend one hour, with a chance to discuss with visiting school classes. During the rest of the time, their task was to go around the park and look for mathematics and possible problems.

For their next term, they were required to construct mathematics problems for their classes and then try them out. During the seminar at the end of the project, we agreed that it would be useful to let problem construction and assessment span both terms, and be initiated already in the spring. This is then the method adopted, as discussed above.

Workshops and networking

During August 2004, teachers and teacher educators from different schools and departments met for two days, sharing ideas and experiences, and making plans for future development. The idea to focus on assessment arose during this workshop.

Throughout the project, the students VFU supervisors have met with the university teachers, discussing the goals of the VFU, as well as aspects of this project.

The project was discussed at regional and national conferences, "Matematikbiennetten" in Göteborg (Jan 2005), The Development Conference for Higher Education in Karlstad (Nov 2005, http://hgur.hsv.se/utvecklingskonferensen/), and the Mathematics Biennial in Malmö (Jan 2006, http://www.mah.se/matematikbiennalen/).

In October 2005, Roller Coaster Designer Werner Stengel was awarded an honorary doctorate at Göteborg university. During his visit, he gave an open lecture, followed by a small workshop with teachers, students, teacher educators and administrators.

During March 2006, students, teachers and mathematics and physics faculty were invited for an "aftermath" workshop in the new Mathematics building at Göteborg university. During this workshop, the results of this project were presented as an introduction to discussion. The participants were distributed on 6 tables, each with 3 school teachers, 1 student and 2 faculty members. It was generally agreed that this type of workshop should be continued, 2-3 times per semester, preferrably on different days of the week.

In addition, AMP has visited upper secondary schools in Halmstad, Skövde and Göteborg to discuss amusement park applications in mathematics.

 

 

3. Results

3A. Student Teachers in the amusement park

Since 2002, we have been able to visit Liseberg with first-year teacher student who will major in mathematics. During 2004, the students visited Liseberg at the same time as pupils from different schools and school years did physics experiments in the attractions. The students had the possibilities to work with genuine questions from the pupils and have discussions with the pupils at and in the attractions.

The aim for our students were to identify what mathematics the pupils need to master before the visit and what they can be able to work with after the visit.

The students identified tasks with among other things geometric concepts, to measure angles, length and time, and to use body measures to estimate heights and lengths. Generally it was possible to discuss different methods and accuracy of measuring with the pupils. The experiences from this visit were followed up in a seminar with the students and us.

In order for this visit to work well it is important to work with the attractions from a perspective of mathematics targeting each school year. The visit can be used to see mathematics out of the classroom. This was an important goal for the development of the project during the academic year 2004-2005, as discussed below.

3B. Students' construction of exam problems

During 2005, students were asked to construct exam problems suitable for their school classes, to discuss them with their supervisors before testing the problems on a group of pupils. Students are also asked to provide solutions, as well as a grading guide for pupil solutions. The task required students to analyse the curriculum requirements for different school years, and how they could construct problem to test various abilities. They also had to reflect on whether the problems were best suited for individual written solutions or for small-group discussions.

The response to the problems was often positive, from teachers, as well as from pupils, although a few of the students' supervisors commented that they, themselves, relied exclusively on ready-made tests, provided by the authors of the textbook.

Many students noted that the authenticity of the tasks provided extra incentive for the pupils to get involved. Still, a number of students used clearly unrealistic data. One of the students claimed that seats in a ride were 5 foot wide - with the intention to see if the pupils would note that this was a bit extravagant - and found that only one pupil in the class complained. We conclude that students, as well as pupils and often teachers, are too used to problems where the only interest in the result is a comparison with the answer in the end of the book. In real life, the ability to assess if a result is reasonable is important and sometimes lifesaving.

Many students discovered that the formulation of a problem sometimes demands more knowledge and work than solving the problem. The situation may need to be adapted and the problem given must be solvable. In some cases students backed off from problem ideas, because the formulation seemed to raise too large difficulties - and assumed that then also the solution would be too difficult for the pupils. The result was sometimes that the final problems were relatively trivial.

The authenticity of this student task involved not only the mathematics, but also the fact that the problems were discussed with active teachers and tried on real pupils. The work encouraged students' reflections over and investigations of their pupils level of mathematics skills. The motivations increased to study and interpret the steering documents. The task also opened for creativity.

3C. Problems constructed by students

As a preparation for the students' work, they were given a sheet with a few examples, as well as web-link to the Slagkraft WWW-page (Slagkraft, 2005). Many of the problems constructed were minor variations of these, whereas other showed considerable creativity. A large fraction (more than 85%) of the student problems could be classified into five categories. Nearly a quarter of the problems related to distance travelled during a ride, or average speed. About 20% of the problems related to time in que and capacities of a particular ride - which are clearly questions that a visitor would reflect on. Other popular problem types involved comparisons between the costs for various types of ride tickets and annual passes, geometrical questions, such as the circumference or area of circles, triangles or squares, and finally questions of probabilities to win on the various wheels-of-fortune. Some of the problems could be described as small variations of problems given on some of the web-pages assigned in the beginning of the project, but often they were adapted to the local school situation. An example was for the pupils to work out how many tours with the Balder roller coaster would be required to travel the same distance as the bus from their school to Liseberg, or how many turns the Ferris wheel would have to make in order to make it to their school.

One student assigned an open-ended problem discussing how many people you should expect to carry around the large chocolate box, which is the top prize for one of the wheels of fortune. The question to be answered was if there could be ground for a suspicion that the park would send out fake winners walking around the park.

3D. Problems avoided

The variation of possible mathematics task in an amusement park is considerably greater than indicated by the students' choices. Problems involving acceleration were avoided in most cases, possibly as a way to make sure that the problems were related to mathematics and not physics. Still, the definitions of acceleration as the second derivative of position clearly invites a mathematical treatment and accelerometer data from one-dimensional motion could be used for numerical integration.

Another rich problem which was avoided was the analysis the motion of two combined rotations, as in the classical Teacup ride, discussed above.

One important factor for the problem selection was the classes where they were assigned for their VFU - most of them were in primary or lower secondary school, which often did not let them use the university mathematics in constructing problems. In authentic situations, the real conditions must be taken into account.

 

 

4. Discussion

4A. Analysis

Views on mathematics

Do real-life applications add value to mathematics? Mathematics, in itself, does not need applications, but can be cultivated as a pure and beautiful art in its own right. However, by experiencing a variety of applications of a particular equation or concept, a learner is likely to develop a deeper understanding of the concept (Marton and Booth, 1997). The steering documents for the compulsory school state that "the math teaching shall give the pupil possibilities to practice and communicate mathematics in meaningful and relevant situations, in an active and open search for understanding, new insights and solutions to various problems." (Skolverket, 1994) This approach is emphasized also in the recent report by Skolverket (2003).

In this project, we have aimed to help students get used to discovering mathematical concepts and applying mathematical skills to the world around them. The mathematical physicist, Eugene Wigner, has said that mathematics is fantastic because it gives such a wholesale return of results, without the need for understanding. However in order to benefit from the results, training is needed to connect a mathematical description to reality. In most cases, the students also found that their pupils are excited by problems relating to a positive, familiar environment. They also found that many pupils resisted making their own estimates of various quantities, which is another requirement in the steering documents (Skolverket, 1994).

The university mathematicians that the students encounter during their education may have a more puristic view of the subject. The exam problems in the mathematics courses are often decontextualized. The university mathematicians may have difficulties in finding Liseberg applications relating to the university mathematics. In this way, students may be deprived of possibilities to discover the use of mathematics outside the classroom.

Some qualities are particularly important for student teachers, i.e. the ability to identify relevant mathematics in a particular situation and to see how a given mathematical concept can be applied to a large number of different situations, and also the ability to judge qualities in pupils' solutions and to discern their difficulties and skills. Although the points of view differ between teachers in various disciplines, they may agree on desired qualities in the student understanding of mathematics. Differences are more likely to occur in the views on how these abilities are best reached. These desired qualities, present, as well as missing, can be a starting point for all teachers involved to agree on.

4B. Implications

This project and its participants are embedded in an environment involving other courses, in teacher education and in other subjects. Ideas and experiences are shared, both concerning the use of extramural activities and examination.

The assessment system used in the national tests in mathematics is generally unkown to university teachers outside the educational department. Through this project, the methods of assessment has been discussed among many teachers in both the mathematics and science departments at the university. The students studying in both environments become more aware of the different assessment cultures in use, and are more prepared to discuss assessment, both in schools and at the university. The students find that they often inspire the school environments by discussing these issues. During this project, assessment has got a more central role also for the science teacher education.

Already during the first semester in the Göteborg university teacher education, students using extramural learning settings as a way to experience many different teachers. During the second educational semester, they visit other museums and suggest ways to use them in education. The teacher educators involved in these courses are now discussing ways to make these learning experiences more authentic by adding student-led VFU class visits.

Student response

The student project finished with a joint seminar, where students shared problems and experiences. The students also wrote a reflexion on different aspects of this assignment. Below follow some of their comments:

4C. Conclusions

The authenticity in the experiences with real school classes was found to be a critical quality aspect for the students. In trying our on their classes problems they had formulated, themselves, and then discussing the grading of authentic solution, they truly appreciated the complexity in the teacher's task. In the project we have experimented with different ways to organize an amusement park experience so that it bests captures the students and the pupils' experiences and observations. This project has given us many experiences and thoughts for further development of the content of mathematics. We see many possibilities and different perspectives with joint ventures for students and pupils that can be seen as positive and that can give the desire to learn mathematics.

References

Ashcraft, M.A: and Kirk, E P (2001)
The relationships among working memory, math anxiety, and performance, J. Exp. Psychology, 130 224
Axelsson, B (1997)
Science Center med elevers och lärares ögon - En observations- och intervjustudie kring kunskaper, attityder och undervisning i Pedagogisk forskning i Uppsala, nr 130, Uppsala Universitet
Bagge, S. and Pendrill, A-M (2002)
Classical physics experiments in the amusement park, Physics Education, 37, 507-511
Bakken C (2006)
Physics/Science/Math Days at Paramount's Great America, http://physicsday.org.
CILS (2006)
Center for Informal Learning and Schools, http://www.exploratorium.edu/cils
Coaster Quest (2006)
Coaster Quest- The Physics and Math of Thrill Rides, Kutztown University, http://www.coasterquest.cc/
Kofod L and Sørensen H. (2002)
Experimentarium og skole, Conference Proceeding, Det 7 Nordiske forskersymposiet om undervisning i naturfag i skolen, Kristiansand, Norway, June 2002.
Lindberg L and Pendrill A-M (2005a)
Mathematics and Physics in an Amusement Park, MAV Annual Conference, Melbourne, http://www.mav.vic.edu.au/pd/confs/2005/
Lindberg L and Pendrill A-M (2005b)
Prov, bedömning och examination inom lärarutbildning i matematik, Utvecklingskonferensen för högre utbildning, Karlstad, Nov 2005, http://rhu.se/utvecklingskonferensen/
Lindberg L and Pendrill A-M (2006)
Prov, bedömning och examination inom lärarutbildning i matematik, Matematikbiennalen, Malmö, Jan 2006, http://www.mah.se/matematikbiennalen
MANSW (The Mathematical Association of New South Wales) (2006)
Mathematics in Luna Park http://www.mansw.nsw.edu.au/studentservices/luna-park.htm
Marton, F and Booth, S (1997)
Learning and Awareness, Erlbaum
Mårtensson-Pendrill, A-M and Axelsson, M (2000)
Science at the Amusement Park, CAL-laborate, 5
Nilsson P, Pendrill A-M and Pettersson, H (2004)
Learning Physics with the Body, Contribution to XI IOSTE; Lublin, Poland
Nilsson P (2005)
Barns kommunikation och lärande i fysik genom praktiska experiment, (Children's communication and learning in physics through practical experiments), NorDiNa 1, 58-69 (2005) and contribution to the 8 Nordic Research Conference on Science Education, Aalborg, Denmark, May 2005
Pendrill A-M (2005)
Roller Coaster Loop Shapes, Physics Edcuation, 40 517-521, also in proceeding for the Mathematics Biennial, Malmö 2006
Pendrill A-M and Rödjegård H (2005)
A Roller Coaster Viewed through Motion Tracker Data, Physics Edcuation 40 522-526
Rennie L J and McClafferty T P (1996)
Science Centres and Science Learning, Studies in Science Education, 27 53-98
Skolverket (1994)
Läroplan för det obligatoriska skoläsendet, Lpo 94, and
Läroplan för de frivilliga skolformerna, Lpf94. http://www.skolverket.se
Skolverket (2000)
Kursplaner och betygskriterier.
Skolverket (2003)
Lusten att lära - med fokus på matematik, Rapport 221
Slagkraft (2006)
Project "Impact" Science in the Liseberg amusement park, http://physics.gu.se/LISEBERG/
Schutzmannsky K (2001)
Roller Coaster. Der Achterbahn-Designer Werner Stengel, See also http://www.rcstengel.com
Utbildningsdepartementet (Department of Education), (2000)
En förnyad lärarutbildning Regeringens proposition 1999/2000:135
Woolnough, B E (1994)
Effective Science Teaching, Open University Press

Author Note

This project has been funded by the Swedish Council for Higher Education (RHU) under grant O13/G03. Liseberg has generously opened its amusement park for student investigations.

The project "Slagkraft" received initial funding from FRN in 1999. Additional funding was also provided by Göteborg university. RHU funded an earlier project, O30/G02 on "Extramural learning in teacher education". A related project, involving engineering education was funded by Chalmers, through the programme CSELT - Chalmers Strategic Initiative on Learning and Teaching.

Early collaborators in the project have been Sara Bagge, now att Navet Science Center in Borås as well as Pernilla Nilsson, Högskolan i Halmstad and Roger Andersson, Högskolan i Karlstad, and both PhD students at FoNTD - the National Graduate School for Science and Technology Education. Experiences are shared within the NNORSC - Nordic Network of Researchers in Science Communication. In addition, many other around the country, and around the world, have shared their experiences, insights and ideas.