12th International Congress on Mathematical Education
Program Name XX-YY-zz (pp. abcde-fghij)
8 July – 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.)
TECHNOLOGY AND SIMPLE MATH IDEAS INSPIRE
TEACHING
Damjan Kobal
University of Ljubljana
damjan.kobal@fmf.uni-lj.si
Sharing group discussion would (partially as a workshop) present an alternative approach to
traditional teaching of several concepts. Ideas can be used as an inspiring motivation which
challenges and deepens the understanding of different math ideas (variable, equation, arithmetic
mean, function, parabola, ellipse, ...). Furthermore, math ideas are backed by simple and intuitive
technology solutions/inventions, which allow for a discussion to be presented at all educational
levels from primary to tertiary (and even advanced tertiary). For example, intuitive understanding of
‘car differential’ leads us from the understanding of the arithmetic mean all the way to the advanced
concept of ‘geodesics’. By modern dynamic geometry programs (like GeoGebra) we can (simplify
and) simulate particular technology functioning and visualize/understand the essence of the
mathematical ideas (behind technological solutions). The activity promotes the innovative use of
technology and smart application of ancient math ideas.
Function, steering, sound, geometry, IT.
INTRODUCTION
Several simple mathematical ideas, which have an incredibly intuitive and useful meaning in
modern technology, will be presented and discussed. An ancient idea of a parabola, with its
simple geometric insights can be used to explain the functioning of car lights and satellite
dishes. The discussion of ancient and modern (analytic) approach can deepen the
understanding of elementary properties of a parabola. Understanding of a parabola will be
discussed and connected to the basic concepts of an ellipse, which will be presented via
‘hands on’ folding paper, simultaneous computer simulation and ancient understanding of
geometric properties. Further, simple understanding of (discrete) functions will be discussed
to comprehend the ‘miracle’ of modern digital sound technology. Namely, elementary ideas
can explain digital transmission of ‘several phone conversations over one phone line’ and
more advanced ‘squeezing of information’ in modern digital technology. The transmission
(or storage) of sound will be initiated and motivated by ‘intuitive sounds’ that characterize
numbers (rational versus irrational) and other functions. We shall also discuss simple ideas of
(car) steering and the idea of arithmetic mean, which is essential to understand the powering
of modern cars. Namely, car differential is nothing but a mechanical realisation of the
mathematical idea of arithmetic mean. Driving a car, we are not even aware of this
marvellous device, which allows easy steering of a car and answers the question of how it is
possible to smoothly steer a car, which is powered by ...one (which one, left or right?) or by
two wheels (would make a car to go only straight!)? The idea can be connected with very
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interesting and also deep questions of practical definitions of straightness which give rise to
intuitive understanding of geodesics.
Discussions will be provoked and backed by smart dynamic geometry simulations, which will
help to intuitively visualize the ideas. Depending on the number, level and interests of
participants, we will also discuss the challenges of the smart use of technology.
APPLICABILITY OF SIMPLE MATH IDEAS
Participants will be encouraged to think and discuss some well known facts of modern life,
where smart math ideas enable technological achievements which we take for granted. For
example, what does happen when we drive a car at night and we dim flash lights? How is the
power from engine transmitted to the wheels of our cars? Can a mystery of modern digital
technology be explained in simple terms? How can dozens of TV channels, internet and
several phone lines be transmitted over a single old-fashioned telephone line? Can in
explaining ‘why...’ ancient geometry be even more powerful than ‘analytic mathematics’?
Presented and discussed ideas will try to promote the idea, that ‘being smartly simple’ is the
essence of good mathematics teaching.
The use of ancient geometric properties of a parabola
Simple geometric properties of a parabola, which were known already to the ancient Greeks
explain the use of parabola in car lights and satellite dishes. Teaching the geometric concept
of a parabola can start nicely and intuitively by computer simulation, where students can
interactively ‘play billiard’ by shooting at a ‘parabola shaped table’ and trying to hit the
(focus) point. Such a simulation will be used at WSG. An internet version can be found in
Kobal (2008). It is well known that any ‘horizontal hit’ is the answer and such a statement can
easily be proved by elementary geometry. Turning the direction of ‘ball travel’ and
exchanging it with a beam of light, we can explore and easily explain car headlights. It is
interesting to first explore how a single beam of light reflects from a parabola. Again, a
computer simulation will be perfect to experiment with. Putting the source of light in the
focus of a parabola, the beams reflected from the parabola travel a straight parallel path. This
explains car’s flash (long) headlights.
Figure 1. Flash car headlights – the source of light is in the focus
What about dimmed car headlights? Exploring the interactive computer simulation, we will
see that putting the source of light ‘ahead’ of the focus of a parabola gives a ‘special
reflection’ which explains the dimmed headlights. Shading of the bottom part of beams
creates the functionality of dimmed car headlights.
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Figure 2. The source of light is ‘ahead’ of the focus
And that is exactly how the car headlights function. Just a couple of millimetres deference and
a ‘bellow-cap’ under the ‘dimmed headlight wire’ in a car headlight bulb makes all the
difference between the flash and dimmed car headlights.
Figure 3. Flash and dimmed headlights wires in a car headlight bulb
It is also easy to explain, why the car headlight bulb has the ‘front metal cap’.
The very similar ideas can be used to explain the functioning of a (parabola) satellite dish.
All used mathematical/geometry statements will be proved by elegant and elementary
reasoning.
An ellipse as a rendezvous of ancient geometry and modern technology
It is a well known ‘hands on activity’ to visualize an ellipse by folding paper. A circle, cut out
of paper, is folded many times so that a point on the edge coincides with a given point F
(different from the centre) in the interior of the circle.
Figure 4. ‘Folding lines generate’ an ellipse
In our discussion we will go further than that. By elementary reasoning we will prove that
‘folding lines’ are tangent lines of the ellipse, that the ‘convex hull’ of the ‘folding lines is the
ellipse, that, similarly to the parabola, a ball from one focus on the ‘ellipse billiard table’
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always bounces of (regardless of the direction of the hit) to the other focus. Furthermore,
besides classical folding paper ‘hands on’ activity, we will discuss an elegant computer
simulation (Kobal, 2012), which uses and explains all the essential geometric properties
involved. And many further interesting questions (like: what happens if point F is moved out
of the circle?) might be discussed.
Functions – an insight into modern digital (sound) technology
Several mathematical ideas can be given a meaningful intuitive insight by sounds. For
example, prescribing different tones to different digits we can 'listen' to many 'mathematical
meanings'. For example a finite decimal number can be presented as a simple (finite)
'melody'. A repeating (periodic) 'melody' nicely describes the 'rational nature' of a rational
number and the infinite decimal number is presented by infinite (though
monotonous/periodic) melody. How could one better explain the difference between rational
and irrational number as to play 'an infinite but periodic’ and 'an infinite but never repeating'
melody?
The essence of communication technology can be approached by visualizing/symbolizing
any sound, conversation, recording, ... as a function. We will not get into every details of how
sounds are presented as functions. For example musicians know, that a perfectly sounding A
tone can be described by a function Sin[Pi 440 t], or an A tone one octave higher is presented
by the function Sin[2 Pi 440 t]. As further examples one can present sounds expressed by
more complicated functions. For example, functions like
Sin(2π 440 t)
Sin(34+
2
Sin(950t))
Sin(700t+35t∙Sin(123t))
Sin(700t+Cos(150t)+45t∙Sin(350t))
can be (drawn or) played by program Mathematica. Obviously, simple sounds will be
presented by rather simple functions and more sophisticated sounds by more complicated
functions. Pure tones (of a single frequency) are of 'orderly shapes' presented by sin function,
while for example a simple human voice hello is a much more complicated function like the
one seen in the figure bellow.
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Figure 5. Recording of a human voice hello reveals a ‘complicated’ function
But whatever the sound, we can imagine, that it is presented by a function. A true shape of a
sound function will not be essential for our ideas. Thus, let us say that we could basically take
any function to present a sound. Different functions would present different sounds. Let us
start with a simple sin function.
Figure 6. Two sound presenting sin functions look very much the same
We drew two functions, which look very much the same. Imagine that the ‘slim’ function is a
sound that is recorded on one side of the phone line and the ‘thick’ function is a reproduced
sound on the other side of the phone line. Functions look the same and it seems just to say, that
phone line service provider is doing a good job, transmitting a perfect copy of the sound from
one side to the other. But let us take literally a closer look and let us focus on both graphs at
the point indicated by the arrow.
Figure 7. A closer look at the two functions
Phone provider can 'cheat' and only transmit a discrete function, which consists of points at a
certain distance. Of course, points have to be dense enough for customers, talking on the
phone, not to notice any 'empty spaces'. Certainly, if the provider would only 'transmit' a point
every five minutes, we would hear nothing. But if one imagines a point every millionth of a
second, what we get is 'very smooth' looking function.
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This is the main idea which we will discuss further. From here on it will be easy and intuitive
to understand the essence of digital revolution. As technology has long ago won the race with
human sensitivity and it can 'split the time' to far tinier bits than a human ear could notice, the
phone provider gains 'lots of free time'. The necessary density of points is determined by
human ear sensitivity, and as technology is fast enough to be programmed to 'listen' only one
tenth of a time ... it becomes obvious, how ten conversations can be squeezed into one phone
line. Basically, we see, that if we imagine the above 'dots' as discrete values of a function, we
can squeeze many other points (information) in-between.
The ideas will be discussed while observing very intuitive computer simulations (Kobal,
2007) and compared to everyday communication technology experiences.
Finally, the idea can be given a funny but meaningful parallel. Imagine a class of students
taking a test and a teacher attending the students not to cheat. If a teacher leaves the classroom
unattended, students might be tempted to start communicating and cheating. So it is hard to
imagine, how the same teacher could take care of two different classes of students in two
different classrooms at the same time. But that is because a teacher would be forced to leave
one class unattended while attending the second class. But for how long? Imagine the
teacher's strict eye browsing around the classroom every split of a second... Theoretically,
imagine a speedy teacher who could shift its full presence and attention from one class to the
other in tenth of a second. Is it not obvious that in such circumstances one such a speedy
teacher could attend not only two but ten classes simultaneously?
From arithmetic mean to ‘social behaviour’
How is the power transferred from car’s engine to car’s wheels? Which wheel is powered?
Left? Right? Or both? Would not simultaneous powering of left and right wheels make the car
move (only!) straight? When a car is turning left the right wheel(s) rotate faster than the left
wheel(s). How can this be true if both wheels are powered?
Discussion about these simple and intuitive technical issues will bring us to understand a
simple device – car differential, which is an ingenious mechanical realisation of the idea of
arithmetic mean. The question of how to appropriately distribute power (P) to the left (L) and
right (R) wheel can be nicely summarized by ‘formula’
.
Discussion about these issues and about above formula will lead us to interesting intuitive
(and deeper cognitive) understanding of simple (but basic) mathematical concepts of variable
and equation. The idea can be developed into an intuitive understanding of sophisticated
mathematical concepts of straightness and geodesics.
Discussed problems, technical solutions and mathematical ideas will be backed by
experimenting with Lego models and computer simulations (Kobal, 2006).
An interesting view on the question of ‘synchronising the speed of left and right wheel’ while
turning (which is so nicely summarized in the mathematical concept of arithmetic mean), can
be given by analysing subconscious ‘social behaviour’ of two promenaders walking in the
park...
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‘Smart’ as a prerequisite of ‘Smart use of technology’
Depending on the expertise of participants, we will also discuss the challenges of the smart
use of technology. We will advocate the idea, that teaching is a complex endeavour where
sharp and comprehensive understanding is necessary in order to efficiently present simple
(cognitively sound) ideas. On a couple of examples (GeoGebra applets) we will discuss these
issues. By analysing some (cognitively common) mistakes based on 'shallow approach', we
will advocate the power of technology to promote smart simplicity, which is a prerequisite of
good teaching.
References
Kobal, D. (2006). Car Differential
http://uc.fmf.uni-lj.si/com/dif/cdif.html.
[html].
Retrieved
(March
2012)
from
Kobal, D. (2007). Sound Functions [html]. Retrieved
http://uc.fmf.uni-lj.si/com/Digimusic/digimusic.html.
(March
2012)
from
Kobal, D. (2008). Parabola and Car Lights [html]. Retrieved (March 2012) from
http://uc.fmf.uni-lj.si/com/Parabola/parabola.html.
Kobal, D. (2012). Folding Ellipse [html].
http://uc.fmf.uni-lj.si/RaInt/FoEllipse.html.
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Retrieved
(March
2012)
from
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