Teaching Children Mathematics / September 2005 83
Chance
Fair?
Eleven-year-old Paul reached into a paper bagthat he knew contained two pink plasticcubes and four blue plastic cubes. Without
looking, he stirred the contents before drawing
out a pink cube, a result that contradicted the
“blue” draw he had predicted. Three previous tri-
als with other color combinations had also failed
to support his predictions. When I asked him to
explain these results, Paul said, “It’s really
strange. Everything is playing tricks on us!”
Paul is learning to deal with probability, a field
of mathematics that, along with its close relation,
statistics, influences the decisions we make both as
individuals and as a society (Paulos 1988). Recog-
nizing the importance of this branch of mathemat-
ics, many countries around the world now include
probability theory in the elementary school cur-
riculum (Amir and Williams 1999). NCTM’s Prin-
ciples and Standards for School Mathematics
(2000, p. 181) recommends that probability
instruction be introduced informally in kinder-
garten and states that students in grades 3 to 5 “can
begin to learn how to quantify likelihood.” Accord-
ing to some researchers, however, the optimal tim-
ing and content of probability instruction in ele-
mentary and middle school is “still an open
Cynthia Pratt Nicolson, cpnicolson@yahoo.ca, teaches fourth and fifth grade at
Bowen Island Community School on Bowen Island, British Columbia. She also
works as an instructor of mathematics education courses at the University of
British Columbia. Cynthia is interested in finding ways to help children and
adults develop confidence in their ability to make sense of mathematics.
By Cynthia Pratt Nicolson
One Student’s
Thoughts on
Probability
Is
Copyright © 2005 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

question” (Kilpatrick, Swafford, and Findell 2001,
p. 294).
As an elementary teacher and researcher, I am
fascinated by children’s intuitive responses to
probability. Many fourth and fifth graders—with
Santa Claus and the Tooth Fairy still fresh in their
minds—maintain a magical outlook on life. How
well do they understand randomness and chance,
theoretical and experimental probability, and inde-
pendent and conditional events?
As part of an investigation into children’s proba-
bilistic thinking, I spent three, forty-minute after-
school sessions with Paul, one of my fifth-grade
students, videotaping his responses to a series of
activities using cubes, coins, dice, and spinners. The
sessions, which took place in our classroom, were
part of my university graduate work in mathematics
education and provided a pilot project for a master’s
thesis case study with my complete fourth- and
fifth-grade class. Paul, like most of the approxi-
mately three hundred students in our semi-rural
school, is from a white, middle-class, English-
speaking background. He had explored probability
briefly in fourth grade but had not encountered
probability in fifth grade at the time of the
interviews.
Like many elementary students, Paul is clear
about some aspects of probability and confused
about others. His responses demonstrate a mix of
sophisticated insights and surprising misconceptions.
Tossing Coins
After probing Paul’s general ideas about chance, I
introduced questions about flipping a penny. Many
commonly recommended activities for teaching
probability are based on the assumption that chil-
dren perceive heads and tails to be equally likely
outcomes of a coin toss. Paul’s comments show
that this assumption is not always valid.
Teacher. If I flip this penny, what are the possi-
ble things that could happen? What are the possible
outcomes?
Paul. Well, it could land on heads, it could land
on tails, or—it would be very improbable that it
could land on its side like this. (He demonstrates
by holding the penny on its edge.) I’ve only had
that happen to me once before.
Teacher. You’ve actually had that happen?
Paul. Yeah, but it was with a bit of a thick coin.
Teacher. Mmm, it would have to be, I think. So,
what are the chances that we’re going to toss heads?
Paul. Well, of all the times I’ve tossed a coin,
mostly it’s been heads.
I accepted without comment Paul’s statement
that he usually tosses heads and asked him to con-
duct a trial of ten tosses, which resulted in six
tails and four heads. When asked to explain this
result, Paul launched into a complicated but pre-
cise explanation.
Paul. I’m not sure, but I think the reason they
land on a certain side is because of the weight of
one side. When it’s flipping, one side becomes
heavier so it drops on that side and, if you look,
heads has more on it. So it’s heavier on that side,
so when it comes down like this [on the table]
it’s tails and when we put it in our hand and flip
it over (flips coin from right palm to back of left
hand) it’s heads—because it turns to the oppo-
site side.
Clearly, Paul’s comments show that he is used
to seeking and finding rational explanations for
real-world events. His thoughts about coin tossing
are based on deterministic, rather than probabilis-
tic, thinking. They also reveal a creative ability to
interpret events to fit preconceived notions. In this
case, Paul managed to turn tails into heads (with
the classic hand flip) in order to support his self-
professed tendency to toss heads!
Next, I wanted to check Paul’s thoughts about
the connection between theoretical probability and
the results of a particular experiment. In order to
work successfully with probability, children must
recognize that a single trial of a random process
such as coin tossing is essentially unpredictable. At
the same time, they must keep in mind that the
overall outcome of many trials can be predicted
using probability theory. This concept of “predict-
ing the unpredictable” is a major challenge for any-
one studying probability.
Teacher. If we did one hundred tosses, what
would you expect to have happen? How many
heads and tails?
Paul. About sixty tails, because with this test we
got six out of ten. But to get a really accurate esti-
mation you would have to do about two or three 10
ones [trials of ten tosses] to be really accurate. But
if you just do it once, it could be anything, right?
But if it keeps going in one way then you know.
Coins, dice, and spinners are used to create ran-
84 Teaching Children Mathematics / September 2005