Teaching Mathematics and Its Applications (2011) 30, 1^9
doi:10.1093/teamat/hrq016
Bridging the divideçseeing
mathematics in the world through
dynamic geometry
HATICE AYDIN AND JOHN MONAGHAN*z
†
z
Mathematics Education Department, Faculty of Education, Gazi University, Ankara, Turkey and
Centre for Studies in Science and Mathematics Education, School of Education, University of Leeds, Leeds, UK
*Email: [email protected]
[Submitted July 2010; accepted September 2010]
InTMA, Oldknow (2009,TEAMAT, 28, 180^195) called for ways to unlock students’ skills
so that they increase learning about the world of mathematics and the objects in the
world around them. This article examines one way in which we may unlock the student
skills. We are currently exploring the potential for students to ‘see’ mathematics in the
real world through ‘marking’ mathematical features of digital images using a dynamic
geometry system (GeoGebra). In this article we present, as a partial response to
Oldknow, preliminary results.
1. Introduction
In a recent issue of TMA Oldknow wrote:
We have the technology. Much of it is already in the hands of the learner—more soon will. They
have the skill to put it to any use. As educators we need to find ways to unlock those skills so that
they increase students’ learning about the world of mathematics, science and technology; their
curiosity about how the objects around them work; and the excitement, challenge and pleasure
they get from relating the first with the second. (Oldknow 2009: p. 194)
This article examines one way in which we may unlock the student skills to which Oldknow refers.
We, like many others, are keenly interested in finding ways in which students can be helped to make
links between mathematics and the real world. But linking school and real world mathematics is
notoriously difficult: Magajna & Monaghan (2003) examine this difficulty from the point of view
of (a specific group of) workers relating what they do to school mathematics, where school mathematics is viewed as a ‘frozen’ body of knowledge; Monaghan (2007) examines this difficulty from the
point of view of school students relating what they do to a world of work problem and argues that
the fact that students do their work in a mathematics classroom leads students to interpretations of the
problem that are not relevant to the work-related goals of the problem. The issue, however, is so very
important that we cannot let difficulties put us off finding ways to make links between mathematics
and the real world. We are currently exploring the potential for students to ‘see’ mathematics in the
real world through ‘marking’ mathematical features of digital images using a dynamic geometry
ß The Author 2011. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications.
All rights reserved. For permissions, please email: [email protected]
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BRIDGING THE DIVIDE
system (GeoGebra). In this article we present, as a partial response to Oldknow, preliminary results.
We first introduce our ongoing research and then present and discuss results.
2. Our research
What mathematics can you see in this picture?
FIG. 1. Mathematics in nature? (This figure appears in colour in the online version of Teaching Mathematics
and its Applications.)
Readers of this article are likely to be mathematicians, so you probably see real world approximations to circles, ellipses, spheres, line segments and fractals, as well as reflection and rotational
symmetry. But what mathematics can, say, a secondary student see? We tried this on four 15- to
16-year-old female students in England. We showed each student 37 pictures on a computer, mostly
from nature but also artefacts such as houses and bridges, and asked them what mathematics they could
see in these pictures. We present some of the results below but, overall, the students could see very
little mathematics (basically just comments on symmetry). We then addressed how we could lead
students to ‘see’ more mathematics and our thoughts were similar to Oldknow’s—get the students
familiar with dynamic geometry software and then get them to use this software to analyse the pictures. This is ongoing research, and our results are initial results, but they are, nevertheless, quite
promising in terms of students seeing more mathematics in pictures with (and after training) dynamic
geometry software.
In this article, we report on one of the four students, Semra (not her real name). We report on only
one of the students for reasons of space—there was really very little difference between the students in
terms of the results we report on here. Semra was 16 years old and in Year 11 at the time of the work
we report on. She loves mathematics and is a high-achieving student. All the work (and extracts) was
done in English but English is Semra’s second language.
The context of the research we report on is:
Early November 2009, first activity, individual activity. Thirty-seven pictures shown and Semra is
repeatedly asked, ‘What mathematics can you see in this picture?’ The activity lasted 19 min. Each
student activity was audio and video recorded. Immediately after the activity, students wrote a short
account of their feelings about doing the activity.
H. AYDIN AND J. MONAGHAN
3
One week later, Semra took part in a group interview with the other three students. The interview
focused on what they thought and felt about the activity. The interview, which was audio and video
recorded, lasted almost 30 min.
Late November, 2009, GeoGebra training. Each student was given an individual 15 min introduction
to GeoGebra. After this, they were encouraged to use it on their home computers, which all of them
did. We were very impressed with their enthusiasm and technical competence. One student taught her
younger brother how to use GeoGebra and later reported ‘we are both addicted to it now’.
In December 2009, mathematics concepts training was done. Two group sessions, each of about
40 min, in which students were shown a large number of pictures and mathematical features were
highlighted, e.g. symmetry (line and rotational), mathematical shapes (polygons, circles, ellipses), the
golden ratio, Fibonacci numbers, fractals. In each session, there was a great deal of interaction (discussion on mathematics) between the four students.
In late January 2010, second activity, individual activity was done. Semra is seated at a computer in
her empty mathematics class. GeoGebra is loaded onto the computer and Semra has a file of 22
pictures (very similar but not identical to the 37 pictures in the first activity). As in the first activity
she is asked ‘What mathematics can you see in this picture?’ but this time she is encouraged to place
each picture into GeoGebra and demonstrate the mathematics. For example, in the first activity she
could simply say ‘that is a circle’ but in this second activity, she would have to construct the circle in
the photograph using GeoGebra. The activity lasted 59 min. As for the first activity, each student
activity was audio and video recorded but in this activity we also used screen capture software, which
recorded all computer interactions. Immediately after the activity, students wrote a short account of
their feelings about doing the activity.
One week later, Semra took part in a group interview with the other three students. The interview,
which was audio and video recorded, lasted almost 30 min. As with the first interview, the focus was
on what they thought and felt about the activity, but in this interview students were encouraged to
comment on the use of GeoGebra.
3. Results
We present selected extracts from Semra’s two activities, her short written account of her feelings
following each activity and her comments in the interviews. We present the activities first. We present
selected episodes, student comment on and interaction with a picture (text in square brackets are
explanatory notes for the reader), to illustrate differences between the first and second activity. Six
episodes in three pairs (before and after) are presented. Each pair presents episodes from similar
pictures. We limit presentation to three pairs for reason of space only, but the comparisons we comment on were also present in the episodes not presented. Each pair represents a category of pictures:
nature—plants (a flower); artefacts (houses); nature—animals. We then present extracts from the
group interviews following the first and second individual activities, with a focus on Semra’s contributions.
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BRIDGING THE DIVIDE
FIG. 2. Mathematics in nature (plants), (a) first activity and (b) second activity. (This figure appears in colour
in the online version of Teaching Mathematics and its Applications.)
First activity
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Er, are the petals symmetrical?
Petals.
Symmetrical.
OK, just talk, so the petals are symmetric you’re saying.
Yeah, you have to say right?
You can say anything.
Erm, because if you split in half its symmetrical both sides.
OK.
Look at each petal.
Hmm.
Can I do next?
Second activity
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
There is a line of symmetry. [Semra constructs a line of symmetry]
Yeah.
Also. [Semra constructs another line of symmetry]
Yeah.
and
Talk aloud.
And here erm, I can see a circle.
You see a circle.
Yeah.
Can you show me that circle for me?
Yes, this is a circle. [Semra constructs a circle]
Yeah, and
Yeah there is another circle.
Another circle, where?
This flower. [Semra constructs another circle]
Yes, just make it bigger, yeah a little bit larger, yes this is another circle.
And the whole shape can be embedded into a circle. [Semra constructs the larger circle]
And erm, there is fractals, ‘cos the shape’s repeating, erm.
And what kind of, what else, which mathematical concepts, you see circles and
Yeah.
Those are symmetry lines.
Yeah and these shapes are ellipses.
Ellipses, can you show me those?
Petals are in the shape of ellipse.
H. AYDIN AND J. MONAGHAN
Interviewer:
Semra:
Interviewer:
Semra:
5
Yeah, you are saying the petals are in ellipse shape. [Semra constructs an ellipse]
Yeah erm, erm. [Semra is looking at the picture]
Talk aloud, anything else.
Eee, I can’t see anything else.
FIG. 3. Mathematics in artefacts (houses), (a) first activity and (b) second activity. (This figure appears in
colour in the online version of Teaching Mathematics and its Applications.)
First activity
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Erm, it has e, e, e, each windows.
Each windows?
Erm, the windows are, there is two squares.
Two squares.
Yeah, erm, like each window, this has two.
Are they square?
This is rectangle, erm. That’s it. Yeah,
Second activity
Semra:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
This is stairs and it looks symmetrical from here going down symmetrical, also these lines are parallel the
fence lines, and the stair lines and these lines parallel as well. Erm, the stairs are like rectangular shape,
rectangular. Erm, the fences are also symmetrical, each fences are symmetrical. The tree looks like an ellipse.
The fractals ‘cos the shapes are repeating like the rectangle over here. And the fences are also rectangular
fences, are rectangular.
[During this talk Semra constructed all the lines and rectangles that can be seen in Figure 3(b) as well as the
ellipse]
And I can see semicircle again.
Is it a semicircle then?
Erm, also these are the same.
Can you show me one of them? [Semra constructs both semi-circles]
These like triangular shapes over here, yeah triangular, also this triangular shape. [Semra constructs both
triangles]
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BRIDGING THE DIVIDE
FIG. 4. Mathematics in nature (animals), (a) first activity and (b) second activity. (This figure appears in colour
in the online version of Teaching Mathematics and its Applications.)
First activity
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Erm,
What are you thinking?
Nothing, I am thinking nothing, erm, I don’t know. Is that paper?
Sorry?
Is that paper?
Is that?
Paper.
No, it is just nature.
Oohhh, erm, ee, body of the rock, I don’t know.
You don’t know. OK.
Erm.
Second activity
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
Interviewer:
Semra:
This is a turtle, and I see the line of symmetry, through the vertically.
So you are saying the body is symmetric. [Semra constructs a vertical line]
Yeap, I see this is a hexagonal shape. [Semra constructs a hexagon]
How many sides?
Six sides, it is a hexagon, and fractals ‘cos they are repeating each other and there is also like square kind of
rectangular shape, they are also fractals ‘cos they are repeating each other. [Semra constructs the quadrilateral
which looks like a rectangle] And, erm, 1, 2, 3, 4, 5, 6, 7. [Semra constructs a heptagon]
So it is 7 sided polygon, yeah?
Yes, heptagon.
Heptagon.
Yeah and this, this like leg like ellipse shape. [Semra constructs an ellipse]
The leg is in ellipse shape you are saying.
And, the head is like kind of triangle. It is in triangular shape, yeap. [Semra constructs triangle] And the lines
are parallel.
Which lines?
These lines are parallel. [Semra constructs parallel lines] Erm also these lines the shapes [referring to two
sides of the rectangle] of the rectangles. [Semra constructs lines with negative gradients]
H. AYDIN AND J. MONAGHAN
7
We now present selected extracts from Semra’s short written account of her feelings following each
activity. After the first activity she wrote:
I felt like I was saying the same things for each . . . , I didn’t know that I had to say mathematical
things . . . , I didn’t know if half of what I was saying was mathematical at all.
After the second activity she wrote:
I feel that the second experiment was much easier than the first because I had taken the training
therefore I was able to recognize the mathematical concepts in the pictures . . . I found it fun because
I recognised the concepts and was able to show them on GeoGebra . . . Now I recognize these
concepts outside of class and I am glad that I took the opportunity to broaden my mathematical
mind.
We now present selected extracts from Semra’s comments in the group interviews following each
activity. We format the presentation by the interview questions.
Question: How did you use your mathematical knowledge while you are looking to those pictures,
or how did you use your self knowledge?
First interview: I used some mathematical knowledge and . . . like we get all about symmetry and so
shapes in school.
Second interview: I did use that knowledge that we’ve learned from the training and erm, . . . I
learned some new things during the training as well.
In the second interview, the interviewer followed this question with ‘what kinds of differences did the
training make in your life?’ Semra replied:
I think it has, like, affected, just, not just using GeoGebra, but, like, in every year, like, ‘cos, like,
just now, I’ve seen [looking at the wall] lines in the wall they are parallel . . . you just remember the
training is not something, you know, you’re gonna forget ‘cos you see every day you can relate back
from what you’ve seen in the pictures.
Question: Which factors were motivating you while you were doing these activities?
First interview: . . . not really, I just like I was seeing first time to see any mathematical concepts in it
because in school we don’t, like, have pictures and seeing mathematical things, like, talk about
maths in plants and stuff.
Second interview: The information motivating me as well because, like, before I just, I think the
only thing I was saying that is symmetrical and then after the training session like, I picked out
different things . . . and also my interest in maths motivating me . . . different concepts trying to find
out different things using GeoGebra as well.
Question: How do you feel when you are looking and studying with these pictures? Was it
school-life or real-life?
First interview: Like usually like well I have never been asked to look at a plant and see something
that like if it has got any mathematical concepts in it, yeah it is real life.
Second interview: Erm, I think is also both, school life and real life. Before the training, erm, maths
was just about you know algebra and stuff like but then after the training you realize that erm, like
after the GeoGebra training you see the other stuff like in even in houses, in nature as well, erm,
there is mathematical concepts like everywhere in. I think you realize maths is not just about school
and shapes like you know all these stuff that we learn in school is got is basically everywhere and I
think after the training yeah so it is both real life and school life.
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BRIDGING THE DIVIDE
In the second interview, the interviewer followed this question with, ‘Was there any difficulty when
you were doing the second experiment?’ and Semra replied:
OK, erm, I don’t think there is much difficulty in the second experiment ‘cos like even when you
look at picture and you first realise ‘‘ohhh, nooo, I haven’t seen anything’’ . . . but then you see one
thing you see so many other things as well . . . and GeoGebra was easy to use . . . I think the second
was easier.
4. Comments on the results
As mentioned in Section 1, this research is ongoing; we are developing data analysis tools to examine
student activity and discourse, but this development is not sufficiently refined to qualify for report in
this article. We can, however, provide comment on the above results. We begin with rather matter of
fact (but still, to us, interesting) comments.
The second activity, which used fewer pictures (22) than the first (37), took more time (59 min) than
the first (19 min). Related to this, Semra’s second activity comments following the pictures in Figures
2(b), 3(b) and 4(b) were all substantially longer than her first activity comments following the pictures
in Figure 2(a), 3(a) and 4(a). The time factor may, of course, be related to the fact that, in the second
activity, she was marking mathematical features of the pictures using GeoGebra and this took more
time than just commenting on the pictures. But we believe that this is because she ‘saw’ more mathematics in the second activity and so had more to say.
There are more mathematical words used in Semra’s remarks following the first set of pictures
shown above, Figures 2(b), 3(b) and 4(b), than there were in her remarks following the second set of
pictures above, Figure 2(a), 3(a) and 4(a). In the first set of pictures the only mathematical words used
are symmetrical, square and rectangle. In the second set of pictures mathematical words used are line
of symmetry, circle, fractal (though incorrect), ellipse, symmetrical, parallel, rectangular, semi-circle,
vertical, triangular, hexagon(al) and heptagon.
GeoGebra appeared to have been quickly mastered and used appropriately. Furthermore, the use of
GeoGebra and the second activity appeared to be something that Semra enjoyed doing.
At a deeper level the sequence <activity 1, GeoGebra training, mathematics concepts training,
activity 2> appears to have been a learning sequence. We do not wish to introduce various theoretical
accounts of what learning is into this short article but, whatever social and psychological factors are
involved in learning, the product of learning is doing something one could not do before. Semra clearly
learnt about GeoGebra but, and this is more important to us as mathematics teachers, she learnt about
recognizing and marking mathematics in real world situations (our comment above about Semra’s
remarks following the first and the second set of pictures provide evidence for this claim). A question
which could be raised regarding this learning sequence is ‘could any parts be omitted?’, e.g. was the
mathematics concepts training essential for what we view as a positive outcome? We are considering
this as we plan for future experiments in this research. Semra’s remarks, in the short written account
and in the group interviews, refer to both the GeoGebra training and the mathematics concepts
training, suggesting that both are important.
We were pleased with the (GeoGebra enabled) mathematical constructions Semra produced in the
second set of pictures shown above. The reason for this is that there is, to us as mathematicians, an
important difference between, for example, saying that a round shape is a circle and showing something to be a circle by marking a centre and a circumference which is mathematically related to this
centre point; we agree with Noss & Hoyles (1996): ‘Mathematics is more than action in the sense of
H. AYDIN AND J. MONAGHAN
9
activity-with-objects. It is activity-with-relationships’ (p. 124). The use of GeoGebra (or any dynamic
geometry software) enables, even forces, actions with mathematical relationships.
But there may be a sense in which Semra ‘saw through GeoGebra’, that she only remarked on
features which she could mark using GeoGebra. For example, in a second activity response to a picture
of a bridge (not shown in this article) she referred to it as a semi-circle, though it appeared much more
parabolic than circular to us. Semra knew what a parabola was but parabolas are more difficult to
construct in GeoGebra than circles or ellipses. Did the relative difficulty of constructing parabolas in
GeoGebra (by marking a point and a line rather than simply marking points as one does with an
ellipse) prevent her from seeing (or commenting on) a parabola in the bridge?
5. The future
We, like Oldknow (2009), are very excited about a future in which technology can be used to bridge
the divide between mathematics and the real world. Oldknow presents many possibilities, and we have
explored one of these. We feel our research to date is very promising, but more research and then work
on how positive features can be disseminated to students and teachers needs to be done. In our future
research, we need to look at the components of our current learning sequence and evaluate their
importance for learning. We also need to explore the possible phemonena of ‘seeing through
GeoGebra’ discussed in the last paragraph. If this, as we expect, is successful, then we need to develop
ways in which our research can lead to the design of teaching sequences based on the needs and
practices of students, teachers and schools.
REFERENCES
MAGAJNA, Z. & MONAGHAN, J. (2003) Advanced mathematical thinking in a technological workplace. Edu. Stud.
Math., 52, 101–122.
MONAGHAN, J. (2007). Linking school mathematics to out-of-school mathematical activities: student interpretation of task, understandings and goals. IEJME, 2, 50–71.
NOSS, R. & HOYLES, C. (1996) Windows on Mathematical Meanings: Learning Cultures and Computers.
Dordrecht: Kluwer.
Oldknow, A. (2009) Their world, our world – bridging the divide. Teach. Math. Appl., 28, 180–195.
Hatice Aydin has a BSc in mathematics from Middle East Technical University and an MSc mathematics
education from Gazi University. She taught mathematics in Turkish secondary schools 17 years and in
England, 2005–2010. She worked with John on the research reported here as preparation for starting a PhD
in mathematics education at Gazi University.
John Monaghan is Professor of Mathematics Education in the Centre for Studies in Science and
Mathematics Education, School of Education, University of Leeds where he enjoys teaching and supervising
students and conducting research. His work with Hatice was a prelude to Hatice’s research at the University
of Gazi. John will continue as an advisor to this research.