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An International Journal for Teachers |
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Children's Understanding of Symmetry
John Truran, Goodwood, South Australia
(From The Best of Teaching Statistics)
Freudenthal (1982) has observed that:
"Symmetry as a source of stochastic understanding is a virtually unknown and badly neglected intuitive and didactic tool. Saying this should be a platitude, but didacticians appear to be no faster learners than their students."
In the teaching of probability three common aids which assume symmetry are
frequently used: coins, dice and urns. This paper will discuss some aspects
of children's understanding of the first two of these aids.
Kerslake (1974) drew attention to the fact that many primary school children
believe that when a die is tossed some numbers are easier to obtain than others.
She also observed that there was some "improvement' in year 4 when compared
with years 1 to 3. At least half the children in all four years believed that
six was the most difficult number to obtain. Kerslake attributed this
to the fact that many games with dice require a six to be thrown before a player
may start.
Green (1983) presented the results of a larger comprehensive survey of students
in years 7 to 1 1. He also found improvement with age, and concluded that "there
is a prejudice against 6 which persists for some right into adulthood".
This prejudice is most likely to be found in less able students. Green
suggested that the erratic response patterns of such students may reflect a
real fluctuation in belief about probabilistic situations.
Erratic response patterns may occur for many reasons. Clinical interviews
are able to examine these reasons in greater depth than is possible in a large
survey. This paper gives a preliminary analysis of a small number of clinical
interviews conducted in South Australian schools in 198 1.
Two sequences of questions relevant to this topic were asked near the beginning
of an interview which lasted between 20 and 35 minutes. The questions
were repeated near the end of the interview in those cases where the subject
had shown any sign of uncertainty or inconsistency. The following protocols
give the basic structure of the questioning, but this was varied according to
the responses received.
(a) When you toss a dice do you think that some numbers are easier to get than
others?
Why (not)?
(if appropriate)
Which number is easiest to get?
Which number is hardest to get? Why?
(b) When 1 toss this coin, is there any way 1 can make it come down heads?
Why (not)?
When 1 toss this coin, is there any way 1 can make it come down tails?
Why (not)?
The different types of responses received are classified and listed below, together with one example of each.
A. Misinterpretation of Question
AP (13:8, year 8)
I When you toss a dice are some numbers easier to get
than others?
S No, not really.
I Can you say why not?
S You don't really know which side it's going to land
on or anything, but ... yeah ... you don't really know which side it's going
to land on. It sort of keeps on rolling ... I can't describe it.
I That's all right, that's fine.
S Urn ... the most common number I've ever come across
is five.
I So some numbers are easier to get, are they, or not?
S Not easier, they just come up more often.
B. Lack of Appreciation of Symmetry
Bl. Belief that a Coin is Subject to Mental Powers
DR(8:9, year 4)
S 1 don't know, but it might.
I Have you ever tried "pleasing" a coin?
I If I toss this coin is there any way I can make it
come down heads?
S Not unless you tried.
I How could I make it come down heads?
S Just "please" it.
I Is there any way I can make it come down tails?
S The same.
I If I "please" it for a head, will it come
down heads every time?
S No. I tried "pleasing" a dice.
I Did that work?
S Sometimes it did and sometimes it didn't.
I Is that because you didn't "please" it hard
enough?
S I "pleased" it when 1 got it, 1 didn't really
"please" it enough when I didn't get it.
B2. Belief that a Coin is Subject to Physical Powers
MW (1 3:3, year 8)
I When 1 toss a coin, is there any way 1 can make it
come down heads?
S If you're pretty good at it 1 suppose you probably
could get a 60% chance.
I How would you do that?
S Just work out how far to throw it up and how much
spin to put on it and just keep on practising for a while.
I I see. Can I make it come down tails?
S Yeah, probably, if you do the same thing.
B3. Inconsistency between Symmetries of Coins and Dice
AM (1 4: 1 0, year 1 0)
I Are some numbers easier to get than others?
S Not really. Not really. It's got the same
chance of coming up.
I Quite sure?
S Sometimes 3 or 5 and 6 are hard, usually.
I What do you mean, "6 usually"?
S When you try for a 6 it never comes up till you don't
need it. 'Cos you throw the dice and you try and keep throwing it, a 6
hardly comes up.
I What number is easiest to get?
S 3.
I Why 3?
S I don't know. It usually comes up all the time.
Oh, not all the time, mainly 3 and 4.
I If I toss a coin, is there any way that I can make
it come down heads?
S No, not really, no.
I Is there any way 1 can make it come down tails?
S No, even chance, 1 out of 2.
I Why is it an even chance?
S It isn't a fiddle, or anything?
I No, it's a fair coin.
S ... It's just the way it is. It's either going
to come out heads or tails ...
I Yes?
S ??? about the same time.
B4. Failure to Appreciate that Easier and Harder are Complementary
PL (8:8, year 4)
I Which number is hardest to get?
S 6.
I Which number is easiest to get?
S I don't know.
I Is there an easiest number?
S They're all hard if you want them.
C. Appreciation of Symmetry for Wrong Reasons
Cl Reasoning Based on Indeterminacy of Outcomes
JM (13:7, year 8)
I Is there any way I can make (a coin) come down heads?
S I don't know.
I What do you mean, you don't know?
S There wouldn't be, because once it's up in the air
it'd just land.
I Is there any way you can make it come down tails?
S No, 'cause it depends. Whatever it comes down,
it comes down.
C2. Reasoning Based on Necessity for Change
CG (15:3, year 10)
I When you toss a dice are some numbers easier to get
than others?
S No.
I Can you say why not?
S 'Cause it's got six sides and you can't really take
the same every time. It varies.
I Are some numbers harder to get than others.
S No.
I What sort of thing do you toss coins for?
S Taking turns in washing the dishes and all that.
I Is that a fair way?
S Yeah!
I If I toss a coin, is there any way that I can make
it come down heads?
S No.
I Or tails?
S No.
I Can you explain why I can't?
S Same as the other one. It's got two sides and
you can't really have the same thing all the time. It has to change.
C3. Reasoning Based on Necessity to Obtain Marks in Class
AM (14:10, year 10)
I (second time of asking) If I toss a dice are some
numbers easier to get than others?
S Learning what we did in maths, no, not really.
When you do it some numbers come up.
I Oh, I see, you believe, you're sticking to you view
that, what did you say, threes and fours are easier?
S The middle numbers are easiest.
I And sixes are harder?
S Mm.
I How do you tie that up with what you do in maths,
as you put it?
S Oh, they tell you one thing, but when you go home
and do it, it doesn't seem to be the same.
I Have you gone home and done it, just for kicks, or
have you actually been set homework to go home and do it?
S Oh, no, I've just done it.
D. Correct Appreciation of Physical Symmetry
VS (15:10, year 10)
I When you throw a dice are some numbers easier to get
than others?
S Not really, there's a sixth chance, you have, when
you throw it ... no. No.
I No?
S No.
I Why not?
S Because it depends on the way that you throw.
Just because you throw it one way doesn't mean you could get a six. This
is hard to explain. You're not going to get the same, no number is easier
because they're all on a plane face and it kind of depends on how much it rolls
or how much you shake it and the way it falls depending on what side you get.
So no side's easier because they're all the same, except they've got different
dots on them.
This classification of responses is almost certainly incomplete. Nor are the types of response mutually exclusive. Undoubtedly the classification could be refined by the application of more rigorous research methods. But even in this state it may be of help in clarifying some immediate pedagogical problems.
Firstly, it may make it easier for teachers to assess their students' responses and to choose appropriate corrective experiences. For example, students responding with type Cl could be asked to toss drawing pins or to examine the frequency distribution of the gender of new-born babies in order to come to appreciate that indeterminacy does not imply equiprobability.
Secondly, transcript C3 shows that to teach a calculus of probabilities is not to guarantee that the calculus will be seen to be applicable to the real world. Hence, ways need to be found which will show that those using such a calculus will be at an advantage over those who are not. Herein lies the advantage of games.
Games are almost certain to be more effective than straightforward experiment. Transcripts B1, B2, and C3 show that experiment is unlikely to affect prejudice. To some extent, there is a sound mathematical reason for this. Freudenthal (1972) has observed that 100 tosses of a coin will produce a relative frequency between 0.4 and 0.6 with 95% probability, and that 2500 tosses are necessary for the 95% limits to be between 0.48 and 0.52. In other words, tossing a coin many times and recording the results is not sufficient to guarantee results sufficiently symmetrical to encourage a change of opinion. Sherwood (1 978) describes the work of Varga with asymmetrical equipment such as 'clumsy' dice. Experience with obviously asymmetrical material is likely to make it easier for a student to appreciate the special features which guarantee symmetry.
Thirdly, it is clear from Transcripts B1-B4 that naive students do not always have consistent views. Hawkins and Kapadia (1984) have argued that subjective probability is the most effective approach for teaching children. It remains to be seen at what stage children have a sufficient ability to observe inconsistency for this approach to be of most value.
Finally, the classification does not show the limitations of necessarily rigid
mass surveys. The development of the concept of probability in children
is still imperfectly understood. While a mass survey can give a good idea
of the status of a belief, it is less able to indicate the reasons for this
status or to give clues about how teaching methods might be refined.
References
Freudenthal, H. (1972). The Empirical Law of Large Numbers or "The
Stability of Frequencies". Educational Studies in Mathematics, 4,
484-490.
Freudenthal, H. (1982). Book Review. Educational Studies in Mathematics,
13, 227228.
Green, D.R. (1983). Shaking a Six. Mathematics in School, 12(5), 29-32.
Hawkins, A.S. & Kapadia, R. (1984). Children's Conceptions of Probability
-A Psychological and Pedagogical Review. Educational Studies in Mathematics,
15, 349-377.
Kerslake, D. (1974). Some Children's Views on Probability. Mathematics
in School, 5(4), 22.
Sherwood, P. (1 978). Introducing Dr Varga and some of his Ideas for
Probability in the Junior School. Mathematics in School, 7(3), 6-7
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