STUDENTS’ STRATEGIES FOR MODELLING A FERRIS WHEEL

Associate Prof. Mette Andresen

Department of Mathematics, University of Bergen, Norway
This study of students’ strategies for modelling was based on an episode where two
upper secondary students in an inquiry based setting model the movement of a Ferris
wheel with the use of GeoGebra. The aim of the study was to identify students’
strategies for creative inquiry and to learn about their beliefs.
Keywords: GeoGebra, Ferris wheel, creative mathematical reasoning, mathematical
inquiry
OBJECTIVES AND PURPOSE
This paper was based on an episode taking place in a mathematics classroom where
two students made a joint effort to model the movement of a Ferris wheel. The aim of
the paper’s study was to see what strategies students may unfold for modelling in a
setting designed for problem-solving. The episode was chosen as an example of
students’ modelling activity, picked out from data from a larger, running research and
development project (2013 – 2015). The larger project focused on students’ strategies
for inquiry and problem solving and was carried out by a group consisting of eight
mathematics teachers from five upper secondary schools in Bergen and one
university researcher (Mette Andresen) in mathematics education. The teachers
shared a wish to find ways to support and stimulate students’ creativity and students’
ability to think mathematically in an intellectually independent way. They wanted
the students to be able to do mathematics on their own, meaning to model, solve
problems, and to inquire and argue. As a researcher I wanted to conceptualize and
deepen the term ‘inquiry teaching and learning’ in accordance with research trends in
the whole Europe, where ‘inquiry’ and ‘creativity’ are supported and implemented in
a variety of understandings and conceptualizations (Andresen, 2013). In particular, I
was keen on getting insight into students’ inquiry strategies, to propose answers to
questions like: What does it mean to inquire, what signs of inquiry and creativity can
be seen and what are the outcomes?
This paper interprets two students’ activity in terms of modelling for concept
formation and seeks to identify their strategy. The aim of the paper’s study was to see
how these two students managed to reach a solution to the modelling task, in an
experimental setting with more open tasks and less strict guidance than they were
used to in the classroom. Insight into students’ problem solving strategies, in general,
should enable our group to direct teaching designs in problem solving towards the
realisation of the students’ learning potentials. The two students in the actual episode
were chosen because they, in advance, seemed to have good potentials for solving the
task within the limited time assigned for this particular observation.
THEORETICAL FRAMEWORK
The theoretical framework of this study was inherited from the larger project.
Inquiry based activity in the classroom
The overall theoretical framework was a sociocultural understanding of teaching and
learning. It takes as its basis the interpretative framework for analysing individual and
collective activity in classrooms developed by Paul Cobb et al. and described in
(Cobb, 1999). This framework coordinates both individual (psychological) and
collective (sociological) perspectives. According to (Yackel & Rasmussen, 2002)), p
316 the framework offers an interpretation of ‘inquiry based’ activities in the
classroom; ‘inquiry’ is indicated by the social norms that students are expected to
develop personally meaningful solutions. They are expected to explain and justify
their thinking, to listen to and attempt to make sense of the thinking of others, and to
raise questions and challenges when they disagree or do not understand. These norms
are, according to (Yackel & Rasmussen, 2002), indicative of the students’ beliefs. In
this interpretation, any change towards ‘inquiry’ in the mathematics classroom must
imply development of new or adjusted social and socio-mathematical norms, and it
must also imply some development of the students’ beliefs about what constitutes
mathematical activity, about their own role and about the teacher’s role.
Modeling
In general, analysis of students’ modeling activities can take place at either of the
following two levels depending on the research goals:
1) Modelling at functional level, which means expressive modelling, aiming at
problem solving and involving certain applications of mathematical concepts,
methods etc. This interpretation of mathematical modelling is in accordance with
(Blum, 1991). Modelling at functional level requests modelling competence in the
meaning described by (Niss, 2002)
2) Modelling at the level of concept formation following the ideas described in
(Gravemeijer, Cobb, Bowers, & Whitenack, 2000): the main design heuristics are the
horizontal and vertical mathematizing. In this interpretation, ‘horizontal
mathematizing’ happens by moving from situational to referential level with creation
of ‘emergent models’. Symbolizing is a main issue for these moves. The modelling
for concept formation continues during ‘vertical mathematising’, moving from
referential to general level.
In this study the modeling activity was interpreted mainly in accordance with 2)
whereas 1) was included in the paper’s final discussion of mathematical modelling as
a means for supporting students’ creative inquiry.
The teaching of mathematical problem solving
A central theoretical contribution of Alan Schoenfeld’s problem-solving research was
his framework for analysis of mathematical problem-solving behavior. Based on
discussions in our group of (Schoenfeld, 2011), the teaching experiments were
divided into two separate parts. In every classroom experiment, the first session
contained an introduction to an inquiry, problem solving working style. The other
part was the main problem solving session. The teaching of mathematical problem-
solving was planned to include explicit use of Polya’s scheme (Polya, 1985). The
teachers did not in advance see this as a major change in their classrooms because
they felt that problem-solving strategies would also be taught normally, although
implicitly. But they had the general impression that their students were in need for
elementary problem solving tools like for example strategies based on Polya’s
scheme. The teachers wanted to enable the students to make progress on their own
hand rather than call for help as soon as they felt lost. In particular, some of the
teachers also wanted to get rid of the students’ very close use of the textbook’s list of
answers to the tasks. The experiment aimed to widen the students’ picture of
mathematics in the direction of a subject open for ideas and including discussions
based on mathematical knowledge and imaginations. The teachers wanted to change
the students’ beliefs about mathematics and about their own roles, and the project
intended to contribute to a change of the classroom’s norms and practices.
The teachers felt comfortable with the preparation of materials for both parts of the
teaching experiment, supported by discussions in the group and in smaller meetings.
Students’ inquiry strategies
According to (Lithner, 2008) p 257, solving a task can be seen as carrying out four
steps: 1) A (sub) task is met, which is denoted problematic situation if it is not
obvious how to proceed. 2) A strategy choice is made. It can be supported by
predictive argumentation: Why will the strategy solve the task? 3) The strategy is
implemented, which can be supported by verificative argumentation: Why did the
strategy solve the task? 4) A conclusion is obtained. Further, Lithner discerns
between different types of reasoning involving strategy choice and strategy
implementation. The two main types of reasoning are IR (Imitative Reasoning) and
CMR (Creative Mathematically founded Reasoning). IR encompasses i) memorised
reasoning where the strategy choice is founded on recalling a complete answer and
the strategy implementation consists only of writing it down, and ii) three subtypes of
algorithmic reasoning where the strategy choice is to recall a solution algorithm
without creating a new solution; hereafter the remaining parts of the strategy
implementation are trivial. In contrast, CMR fulfils all of the following criteria
(Lithner, 2008) p 266:
1. Novelty. A new (to the reasoned) reasoning sequence is created, or a forgotten
one is re-created.
2. Plausability. There are arguments supporting the strategy choice and/or strategy
implementation motivating why the conclusions are true or plausible
 3. Mathematical foundation. The arguments are anchored in intrinsic
mathematical properties of the components involved in the reasoning.
Lithner’s studies were carried out at undergraduate level. Our group decided to take
students’ CMR as a goal for the teaching experiment meaning that we added CMR in
connection with problem solving as an indicative of ‘inquiry’.

THE STUDY OF AN EPISODE

This study was carried out as analysis of an episode cut out of data from the larger
research and development project. The episode was chosen as an example of
students’ modeling for problem solving. This paper only brings a summary of the
episode, with excerpts from the transcripts etc. First, the students’ activity was
interpreted in terms of modeling for concept formation. Then it was analyzed with the
aim to identify students’ strategies for modeling the movement of the Ferris wheel,
which was the task they had to solve. Finally, mathematical modelling was evaluated
as a scene for students’ creative inquiry.
DATA
The episode
The task (in the episode) was to make a graph and create a function which models the
distance above the ground as a function of time, task e) - f), based on data about the
wheel (Figure 1) and on the results from tasks a) to d).
Facts about the Ferris wheel
Height limits: Accompanied: 0 – 129cm
Alone over 130 cm
Tickets: 1
Type of item: Rotating vertically
Made by: Anton Schwarzkopf, Germany
Year: 1967
Name: Reisenrad
Number of passengers: 120 persons, 20 gondolas with 6 persons each
Main rotation: 2,4 rounds per minute
Main dimensions: Height: 25 meters

Figure 1: The Ferris wheel data

The tasks
Imagine that you insert a coordinate system on the Ferris wheel with origo in
the centre of the wheel and positive x-axis horizontally (and positive y-axis
upwards). You enter the gondola when it is on the bottom, meaning at - 90˚
a) You enter gondola number 8. After you come 12 gondolas which must
be filled. What is the angel between you and the positive x-axis when
they load gondola number 20? How high above the ground are you?
 The Ferris wheel starts to move with 2,4 rounds per minute. It stops after 3
minutes.
b) How high are you after 1 minute?
c) When will you be 20 meters above the ground?
d) Outline (by hand) a graph which describes you distance to the ground as
a function of time
e) Use ICT and trace for making a graph showing distance to the ground as
a function of time.
f) Can you create a function which models the distance above the ground
as a function of time? (…)

The episode
The episode consisted of two boys, B1 and B2, working on task e) and f). It started
under the group work following the presentation of results from previous lessons
including answers to tasks a) to d). Prior to the previous lesson’s group work, the
teacher had introduced Polya’s problem solving scheme to the students and
encouraged them to make use of its principles. In this way, and with his open
questions, inquiry reflections and comments during the presentation of results and
answers, the teacher intended to set the scene for students’ inquiry based activities
under the subsequent group work.
When the episode started, the boys had sketched the curve with paper and pencil. The
teacher T encouraged the boys to open GeoGebra; they agreed and graphed cosine.
The graph was fitting the extreme points horizontally but not vertically.
195 T: Little too fast...
196 B1: (Repeats) Little too fast (stretches the graph horisontally)
198 B1: Okay what is happening here? (points to the graph in his paper) we
used this (points to the midpoint of the curve on the paper) but what
time is it here?
B1 referred to the answers to task a) - d). The boys continued, focusing on time as the
parameter.
231 B2: It is like if you start in the bottom .. if you start on the top, I mean, like
I did here, if you start on the top, time is zero, and then it will be 90
times this, okay? Because, then you start on the top (moves his hands
like turning around) and 90 times as much then you reach zero. (refers
to his paper)(…)
248 B1: (referring to the graph on the screen) too fast! We try again.
249 B1: what if we substitute one... x (writes)... divided by..
251 B1: Oi! (see Figure 2, picture P1020784)Maybe, maybe... (…)
Figure 2: (P1020784, reconstruction)
326 B1: (stretches the graph vertically) what do we have to find?
328 B2: (points to the screen) we probably have to find the distance between
these two (points to the distance between two top points on the curve)
332 B2: we can start with writing the distance between two wave tops
334 B1: yes, then we need... (draws the line y=28 and uses intersection
between curves and the line, marks two top points A (0,28) and
B(39.48,28), (see Figure 3 Picture P1020789)

Figure 3: (P1020789, reconstruction)

340 B2 must it take 25
341 B1 it should take 25
342 B2 yes because that is the full rotation from the top (points to the top,
bottom and next top of the curve on the screen) (...)
The two boys spent the next 20 minutes on attempts to fit the curve horizontally.
Suddenly:
506 B2: (counts) one, two three four, four
507 B1: we had like 19, like..
508 B2: it is 4 times as fast as the other..
509 B1: 4 times as fast?
510 B2: yes, because there is one, no, there is one, two, three, four (points to
the screen and counts) (they count together)
 513 B1: what do we multiply with 4 – here, maybe
514 B2: yes, try (see Figure 4 Photo P1020795)

Figure 4: (P1020795, reconstruction)

515 B1: maybe
516 B2: good to find it exactly
517 B1: yes good (both) Hahahaha! Yes! Yahoo! (see Figure 5 Photo
P1020796) (end of episode)

Figure 5: (P1020796, reconstruction)

RESULTS
Students’ modelling activity
Before the experimental lessons on the Ferris wheel, the two boys knew about sine
and cosine but they knew nothing about the harmonic oscillator or its expression f(x)
= r sin(cx)+h. From a concept-formation point of view, the goal was to reach and
understand an expression of harmonic oscillator. When the episode started, the
students had decided to use cosine and determined the extremes 12.5 and -12.5 of the
curve. They intended to determine the time when the curve intersects the axis but did
not succeed. By the use of GeoGebra they fixed the graph starting with (0,28), and
extremes y=3 and y=28.
During the episode they aimed to introduce time as the variable x, and connect time
with periodicity. The teacher gave a hint for connection between the graph and the
speed by saying ‘Little too fast’(L195) but the students did not react immediately
(L196 – 198). B2 tried to somehow intuitively express the relation between number
of rotations and angular velocity (L 231ff), referring to his paper where the graph was
symmetric around the x-axis. He might suggest (in my interpretation) that the period
of time from the wheel started (called zero) until the wheel had turned 90 degrees
would be equal to the x-value of graph’s intersection point with the x-axis (denoted
zero too!). In that case, that x-value could be determined by multiplying the angular
velocity by 90, according to B2, in my interpretation. Neither of the boys, though,
followed this idea. Instead, they went back to GeoGebra and graphed 12,5
cos(x/6,25) + 15,5. (L248-249, Figure 2, P1020784). They could not explain why
they choose 6,25. Apparently, this was a guess supported by the calculation of the
half height of the wheel (12,5/2) and one fourth of the period (25/4), successively. A
little later they changed the value into 2π (not in this except).
After a while they seemed to start over again, maybe inspired by Polya’s scheme
(L326-338, Figure 3, P1020789). They decided to take the distance between top
points as their starting point and used GeoGebra to measure this distance by the
construction of B, the intersection point between the curve and the line y=28. They
were aware that the distance might fit with the period but they did not manage to fit
the graph by moving B. Suddenly, more than twenty minutes later, B2 counted the
curve’s tops, and grasped a new connection between curve and speed (L 506 – 517,
Figures 4 and 5, P1020795 and P1020796). Neither the students’ dialogue nor their
experiments with the curve on the screen reveals any signs of emergent
understanding or inspiration for this idea prior to B2’s sudden statement.
At the end of the episode, B1 and B2 had reached the expression f(x) = 12,5 cos(x/4)
+ 15,5 of harmonic oscillator. They seemed to understand the meaning of 12,5 and
15,5 in relation to the curve, as well as the physical meaning of these constants
(obvious from their dialogue not quoted in the excerpts here). The two students were
convinced that they had found a constant ¼ which made the curve fit with its fixed
values. It was obvious from the preceding trials (see the four reconstructions) that
they had grasped the connection between values of the coefficient to x and stretching
the curve horizontally. Their struggle with fitting the curve revealed that they did not
understand the physical meaning of this coefficient, or its relation with, for example,
the rotational speed.
Students’ strategies in the episode
The main question for the students in the episode was how to apply the main rotation,
2,4 rounds per minute, to the model of the harmonic oscillator.
They chose a strategy which, according to common experience, is often seen amongst
experienced GeoGebra users: They switched between the graphical interpretation and
the algebraic expression, supported by formulas or examples from the textbook.
During the episode B1 and B2 took only little notice of the textbook, though. By
using this strategy the two students automatically merged the tasks e) and f).
Even if their work resampled guessing and trying, they did not make a systematically
try out. None of them suggested, for example, introducing a slider for the
expression’s constants. The value of the coefficient of x was not determined by
deduction, only by ‘luck’ under the visual inspection of the graph which counted as a
sufficient argument. Neither did they give arguments for the value ¼ after it was
determined. The two boys did not follow the instructions from Polya’s scheme
strictly. For example, they never mentioned making a plan. Neither did they discuss
the meaning of their results or give arguments. But when they got stuck during their
work they several times asked questions like: What are we supposed to find? (L
326). So, they might still be inspired by the lesson on Polya.
Mathematical modelling evaluated as a scene for students’ creative inquiry.
The task and the setup intended to invite for inquiry. Modelling the Feris wheel at a
functional level gave good opportunities for concept formation thanks to the
imaginative character of the wheel and because the two boys had appropriate tools in
the form of GeoGebra and, maybe, of knowledge about Polya’s scheme.
Mathematically, they were well prepared for the task by knowing about trigonometry
whereas their complete lack of knowledge about harmonic oscillator gave room for
novelty in their reasoning, which was one criteria for CMR. The plausibility rested on
the sociomathematical norm that visual inspection might play a major role as
argument for a solution or strategy choice. This also pointed to the close connection
between norms and the individual student’s beliefs about mathematics and
mathematical relations, and to the strong influence of GeoGebra as the student’s tool.
The mathematical foundation was based on the tight connection between curve and
algebraic expression. During the episode none of the boys, for example, gave
arguments based on knowledge about transformations or change of variables for
functions which would have counted for relevant, intrinsic mathematical properties. It
was revealed during their initial work (not in the excerpt) that they were aware of the
similarity between sine and cosine which would both be useful. It was apparent that
they were capable of interpreting the task’s data but they did not know about or apply
basic physics, i.e. mechanics.
All in all, the episode demonstrated a case of creative inquiry in our study’s meaning
of the term.

CONCLUSIONS AND PERSPECTIVES

This study aimed to bring new knowledge about students’ strategies when they
worked in an inquiry based learning environment, and about their learning outcome
and beliefs. It appears manifest from the results that one important factor of influence
is the tools. The students’ strategy choices are determined by the tools they have at
their disposal. This means that if we want students to inquire and create they must
have the opportunity to build appropriate tools, for example like in this study in the
form of GeoGebra and Polya’s scheme. The fact that students will not be able to be
inquiry and creative just because they are told not to do things in the usual way points
to new research in strategies and to development of ‘toolboxes’ useful for inquiry.

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