Original Article
Overcoming challenges in learning probability
vocabulary
Randall E. Groth and Jaime Butler
Salisbury University, Salisbury, MD, USA.
e-mail: [email protected]
[email protected]
Delmar Nelson
University of Maryland-Eastern Shore, Princess Anne, MD, USA.
e-mail: [email protected]
Summary
Students can struggle to understand and use terms that describe probabilities. Such
struggles lead to difficulties comprehending classroom conversations. In this article,
we describe some specific misunderstandings a group of students (ages 11–12) held
in regard to vocabulary such as certain, likely and unlikely. We discuss our efforts to
help the students use such terms appropriately. In particular, we show how engaging
students in a game requiring the use of the terms helped them begin to develop their
vocabulary more fully. We also explain how a probability ladder visual organizer
helped students begin to organize their thinking about the meanings of terms relative to one another.
Keywords:
Teaching; Teaching statistics; Probability; Language; Vocabulary.
INTRODUCTION
What comes to mind when you hear words like
impossible, certain and improbable? Research
suggests that our students may not think about
these qualitative probability terms in the same
ways as we do. In one study, some students used
impossible to describe events that had small
probabilities, and some used certain to describe
events that had any chance of happening
(Fischbein et al. 1991). In another study, some
students used less probable to describe events
that cannot occur, and they used improbable to
describe events that occur frequently (Nacarato
and Grando 2014). Language difficulties can
persist as students study advanced ideas such as
conditional probability (Watson 2011). These vocabulary challenges contribute to some teachers’
beliefs that probability and statistics are difficult
to learn and impact their inclination to teach the
subjects (Leavy et al. 2013).
We recently experienced, firsthand, how challenging it can be to help students learn qualitative
probability vocabulary. During a 9-week summer
research project, we worked with four students:
Rebecca (age 12), Shonice (age 11), Joseph
(age 12) and DuJuan (age 11) (pseudonyms).
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We interviewed the students individually during
the first week of the study to become familiar with
their prior knowledge. During the interviews, we
found that they had considerable difficulties using
probability language. In this article, we describe
the student difficulties we discovered, our efforts
during the first two lessons to help students and
insights about teaching we gained along the way.
PRE-ASSESSMENT INTERVIEWS
The first interview task we posed to students is
shown in Figure 1. In it, they were asked to
choose items from a word bank to describe the
likelihoods of various situations. Students chose
appropriate terms for some of the situations, but
they struggled with others. In particular, there
were difficulties employing the terms certain,
likely and unlikely.
Joseph and Rebecca each had trouble using the
word certain. When asked to describe the chances
that he would have a test in math sometime this
year, Joseph replied ‘certain’. When asked why
he chose certain, he said, ‘Because I probably
will’. Joseph’s response implied that a certain
event is one that is just likely to happen. Rebecca
© 2016 The Authors
Teaching Statistics © 2016 Teaching Statistics Trust, 38, 3, pp 102–107
Learning probability vocabulary
103
Fig. 1. Qualitative probability language interview item (adapted from Romberg et al. 2003)
used certain appropriately when describing the
first two scenarios, but then surprisingly, did not
use it for the fifth. When examining the fifth scenario, she was the only one of the four students
to realize that in a room of 367 people, at least
two people must have the same birthday, stating,
‘Since there are 365 (days in a year) there must
be one person at least that shares a birthday.’ Despite this insight, she chose the word likely rather
than certain as her response.
Shonice and DuJuan also had difficulty using
the word likely when describing events. In describing her chances of having a test in math class
during the year, Shonice said ‘likely’, but then explained, ‘Because you have to take a placement
test’. So, she used likely to describe an event
she believed to be certain to happen. Similarly,
when DuJuan described the chances of rain within
the next month, he said ‘likely’ and explained, ‘It
is raining right now.’ Like Shonice, he used likely
to describe a certain event. Later in the interview,
DuJuan had a similar difficulty using the related
term unlikely. When asked to describe the
chances of rolling a 7 on a six-sided number cube,
he said that it could not be done because there
were only six sides. However, he chose to use unlikely rather than impossible. Shonice and DuJuan
both appeared to be hesitant to use terms such as
certain and impossible to describe probabilities.
Our students’ initial difficulties with probability
vocabulary might perhaps be partially attributed
to their school curriculum. The US Common Core
State Standards for Mathematics (US National
Governors Association Center for Best Practices
[NGA Center] and US Council of Chief State
School Officers [CCSSO] 2010) had been adopted
five years prior to the initial interviews. The
Common Core State Standards for Mathematics
include no standards pertaining directly to the development of qualitative probability vocabulary.
This is in sharp contrast to documents in place
in other nations that outline developmental
© 2016 The Authors
Teaching Statistics © 2016 Teaching Statistics Trust, 38, 3, pp 102–107
progressions of experiences to help build such vocabulary (Jones et al. 2007). So, during our first
two lessons, we focused mainly on development
of probability language in order to supplement
our students’ school-based experiences.
LESSON 1: DESCRIBING PROBABILITIES FOR
SPINNERS
At the beginning of the first lesson, students were
shown a spinner (Figure 2). Blue occupied half the
area, red and yellow each took up slightly less
than one-fourth, and green filled the remaining
portion. Students were to choose terms from the
interview word bank (Figure 1) to describe how
likely it would be for the arrow to land on each colour. They were also asked to describe how likely it
would be to land on pink (a colour not on the spinner). Students wrote individual responses. After
choosing descriptive terms, they were also asked
to write predictions about how often the spinner
would land on each colour in 12 spins. We then
had a class discussion, so they could share their
responses and reasoning.
Fig. 2. Spinner introduced to students at the beginning of the first lesson
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The most striking occurrence during the first
portion of the lesson was that Rebecca, Joseph
and DuJuan all used the word certain to describe
the chance of landing on blue. Shonice chose
likely. None of them chose evenly likely, even
though blue occupied half the spinner. Interestingly, the three who used certain did not choose
impossible to describe the likelihood of landing
on the remaining colours on the spinner. Instead,
they chose terms such as likely, rare, and others
implying there was a chance of landing on the
colours. Students were, however, unanimous in
using impossible for the probability of landing on
pink. This showed some understanding of the
meaning and use of impossible, even if they did
not immediately recognize that if one colour is certain, then the others must be impossible to obtain.
In several instances, students were actually
more successful assigning numerical probabilities
to events than they were in using qualitative probability language. For example, when asked to
predict how many times the spinner would land
on blue in 12 trials, Rebecca, Joseph and DuJuan
all predicted 6 of 12, and Shonice predicted 8 of
12. Also, when predicting how many reds would
be obtained in 12 trials, all four students predicted
3 reds. They disagreed, however, on the probability language that should be used to describe a
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probability of 12
. DuJuan and Rebecca called it
likely. Joseph chose evenly likely, and Shonice
opted for possible. This portion of the lesson made
it clear that students’ ability to assign numerical
probabilities to events was more developed than
their use of qualitative probability language.
Students’ success using numbers to describe
probabilities was surprising in light of the difficulties they had employing qualitative terms. Curriculum documents often recommend teaching
qualitative terms before students assign numbers
to probabilities (Jones et al. 2007). Our experiences with the students suggested that they had
followed a different learning pathway, developing
some aspects of qualitative language and some
aspects of numerical reasoning about probabilities simultaneously. Neither of these proficiencies
was developed completely. Going forward, we
looked for opportunities to take advantage of what
the students already knew about numerical probabilities to support their learning of vocabulary.
To continue to engage students in further use
and refinement of probability language, we played
and discussed a game using the spinner (Figure 2).
We assigned each student a colour on the spinner.
We then spun 12 times. A student received a point
when her or his colour was obtained. During
the game, students noticed that green was not
Randall E. Groth et al.
obtained in the first 12 spins. Shonice, who was
assigned green, said that it was impossible for
the spinner to land on green. Other students
disagreed. We then took the spinner and spun it
in front of students until a green was obtained.
This led Rebecca to comment, ‘It can happen, but
it doesn’t happen that often.’ Rebecca suggested
rare rather than impossible to describe the probability of spinning green, and others agreed. We
also used the game as an opportunity to revisit
the use of the word certain in relation to the blue
section of the spinner. As we continued to discuss
ways to describe the probability of obtaining blue,
students began to shift towards using almost
certain rather than certain as a description of the
probability, observing that a blue outcome was
not guaranteed.
Finally, we had students write and talk about the
fairness of a game in which each person was
assigned one colour on the spinner, and the
person accumulating the most points would win.
Students focused on the small probability of
obtaining green and remarked that it was ‘rare’
to obtain. All of them agreed that the game would
be unfair, since blue occupied a large portion of the
spinner. To encourage further thought and discussion of probability, we asked Shonice to create a
spinner that would make the game fair. Joseph
was asked to create a spinner that was unfair. Both
succeeded in these tasks. Shonice created a
spinner with four equal-size sections, and Joseph
created one where a single colour occupied half
of the spinner and several other colours occupied
the other half. This helped introduce fair and unfair
to the students’ conversations.
As Shonice and Joseph created their spinners,
we told Rebecca to create a spinner on which blue
was certain to win, and DuJuan to make a spinner
so that blue was almost certain to win. As she began to work, Rebecca started to divide her spinner
into sections and prepared to colour each one differently. As we saw her do this, we showed her the
spinner from our earlier game (Figure 2) and
asked if there was a chance of landing on a colour
other than blue. This was enough to prompt her to
realize that her entire spinner must be blue to
make blue certain to win. DuJuan had a more
difficult time with his task. He created a spinner
with yellow occupying the largest area, and blue
occupying the second largest (Figure 3). He reasoned that blue would be almost certain to win
because it occupied the second largest area.
When students saw and discussed DuJuan’s spinner, they remarked that it had much yellow and
that yellow would be more likely to win than the
other colours, so it appeared that they were
© 2016 The Authors
Teaching Statistics © 2016 Teaching Statistics Trust, 38, 3, pp 102–107
Learning probability vocabulary
Fig. 3. DuJuan’s Spinner
beginning to shift their thinking about what
makes an event certain. DuJuan, however, continued to insist that blue was almost certain to
win. When we asked him how likely it would be
for yellow to win on his spinner, DuJuan did start
to shift back and forth between using likely and
certain. His emerging language development
and that of his classmates convinced us that a
continued focus on probability language during
the next lesson would be worthwhile.
LESSON 2: A PROBABILITY LADDER
ORGANIZER
Our students’ developing but inconsistent use of
language led us to believe that they would benefit
from a structure to organize their thinking. In
considering structures that might be used, we decided to introduce a ‘probability ladder’ (Romberg
et al. 2003). Events that were certain to happen
would be placed at the top of the ladder,
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impossible events at the bottom and evenly likely
events in the middle. We also wanted students to
think about where almost certain, unlikely, likely
and almost impossible would fit in relation to the
benchmarks of certain, evenly likely and impossible. We used masking tape to create a picture of
an empty ladder on the whiteboard. Next to the
empty ladder, the seven qualitative terms and
spinners with different probabilities of obtaining
green were attached to the board with magnets,
as shown in Figure 4. The list of seven qualitative
terms differed slightly from the original word bank
(Figure 1) because we wanted to focus on the
complementary pairs almost certain/almost impossible and likely/unlikely to bring out symmetries within the probability ladder. The probability
ladder teaching device is similar to the technique
of arranging probabilities along a washing line.
In both models, students arrange qualitative
probability terms in order relative to one another
and eventually associate numerical probabilities
with the terms as well.
We started the lesson by trying to establish
consistent meaning for the word certain. We filled
a paper bag with 8 green cubes and asked students to draw one out without looking and then
replace it. After green had been drawn 10 times,
we asked students to predict the probability of
drawing green on the next trial. Joseph said there
would be a 100% chance. Shonice expressed this
as a 10/10 probability. Rebecca agreed with the
10/10 probability and also used the qualitative
term certain. We then showed students that the
paper bag contained 8 green cubes. They agreed
that certain and the fraction 8/8 described the
probability of obtaining green.
Next, we asked students to look at the spinners
on the board and choose one on which green was
Fig. 4. Empty probability ladder with qualitative terms and spinners
© 2016 The Authors
Teaching Statistics © 2016 Teaching Statistics Trust, 38, 3, pp 102–107
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certain to be spun. They all agreed upon the allgreen spinner. Rebecca explained, ‘Last time we
were here, we learned that certain meant that is
was like 100% going to happen. Even if you had
like a tiny sliver of a different colour you could still
land on it because it’s an option.’ Upon hearing
Rebecca’s comment, we moved the all-green
spinner to the top rung of the probability ladder
along with the term certain. We put the fraction
8/8 next to it and asked if there were any other
ways to express the probability. Students’ knowledge of equivalent fractions and percentages
led them to offer 100%, 4/4 and 1 as different
possibilities.
With certain in its place on the probability ladder, we did an activity to establish impossible as
the bottom rung of the ladder. We put eight red
cubes in a paper bag and had them draw several
times without looking. After they drew red each
time, we asked them to predict the probability of
drawing green on the next trial. We then showed
them that there were eight red cubes in the bag.
Students readily volunteered ‘impossible,’ ‘0 out
of 8’ and ‘0%’ as ways to describe the probability
of drawing green. Shonice came up to the board
and moved the red spinner and the word impossible to the bottom rung of the ladder. Once this had
been carried out, we recorded students’ 0/8 and
0% descriptors on the bottom rung of the ladder
as well.
The next portion of the lesson aimed to draw
students’ attention to the middle of the probability
ladder. We put four green cubes and four red cubes
in the paper bag. After students had drawn several
times without looking and then were shown the
colour composition of the bag, we asked them to
describe the probability of drawing green. They
initially used quantitative descriptions such as
‘fifty-fifty’, ‘one-half’ and ‘50%.’ We had to explicitly prompt them to use a qualitative term from
among those on the board. They then quickly selected evenly likely as a descriptor. When asked
to locate this term on the probability ladder,
Joseph explained that it should be in the middle
because a one-half probability is halfway between
the top and bottom rungs. He moved the term
evenly likely and the half-green spinner to the
middle rung. We added the students’ suggested
representations of ‘1/2’ and ‘50%’ there as well.
With the three benchmark terms certain,
evenly likely and impossible in place, we set out
to help students establish meanings for terms
falling in-between. We showed the students 6
green and 2 red cubes and asked them to describe
the probability of drawing green without looking.
Shonice and Joseph volunteered ‘almost certain,’
Randall E. Groth et al.
‘6/8’ and ‘3/4.’ Shonice moved the spinner with
6 green spaces and 2 red spaces to the rung immediately above evenly likely along with the term
almost certain. At that point, Rebecca said she
would have chosen differently. She came to the
board and moved almost certain up to the rung
immediately beneath certain. She also moved
the spinner with 7 green sections and 1 red section next to almost certain. The others agreed
with her choice and moved likely next to the spinner with 6 green sections and 2 red. Students
then used the symmetry they noticed in the ladder to place almost impossible one rung above
the bottom and unlikely one rung below evenly
likely. Their final placement of spinners and terms
on the ladder is shown in Figure 5.
To conclude the second lesson, we asked students to write a letter to DuJuan explaining the
meanings of probability terms from the lesson,
since he had been absent that day. All three
Fig. 5. Students’ final placement of spinners and
qualitative terms on the probability ladder
© 2016 The Authors
Teaching Statistics © 2016 Teaching Statistics Trust, 38, 3, pp 102–107
Learning probability vocabulary
students included correct explanations for the
benchmark terms certain, impossible and evenly
likely. In her explanation of these terms, Rebecca
also included the numerical probabilities of 100%,
0% and 50%, respectively. Rebecca and Shonice
also associated the numerical probabilities of
25% with ‘unlikely’ and 20% with ‘almost impossible.’ Although these numerical probabilities
place the terms in correct order relative to one
another, it is not correct to say that a precise
numerical probability can be associated with
these two qualitative probability terms. The three
benchmark terms certain, impossible and evenly
likely have fixed numerical probabilities, but
those in-between the benchmarks can fluctuate
while staying in the same relative positions. The
presence of this pattern in the students’ writing
made us aware of the need to emphasize this
distinction between terms in future lessons.
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so, to what extent are they more successful?
These are open questions for research, worthy of
further investigation, because they have the potential to help curriculum designers around the
world be more purposeful in designing intended
learning sequences.
Acknowledgements
This material is based upon work supported by the
US National Science Foundation under Grant
Number DRL-1356001. Any opinions, findings
and conclusions or recommendations expressed
in this material are those of the authors and do
not necessarily reflect the views of the US National
Science Foundation. We would like to thank the
anonymous reviewer and the editor for their
helpful comments.
CONCLUSION
References
By the end of the second lesson focusing on
qualitative probability, we felt that students had
developed enough probability language to have
productive classroom conversations. However,
we also learned that it would be necessary to
re-visit the precise meanings of terms such as
unlikely and almost impossible and the ranges of
numerical probabilities that can be associated
with them. As students dealt with topics such as
compound probabilities and probability experiments, we continued to look for opportunities to
develop and refine their understanding and use
of probability language. In being alert to the need
to continuously attend to students’ use of probability language, teachers can help students
comprehend and contribute to discourses about
probability they encounter in the classroom and
in everyday life.
Although our experience involved only four
students, it highlights an important issue for
broader consideration: What is an optimal curricular progression for learning probability vocabulary? Some curriculum documents recommend
teaching probability language before numerical
probabilities (Jones et al. 2007). Other documents, such as the one influencing the school experiences of the students we taught (NGA Center
and CCSSO 2010 in the US), do not take a stand
on the issue. If students experience a curricular
progression that encourages the learning of probability vocabulary in the early grades, do they become more successful probabilistic reasoners? If
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Teaching Statistics © 2016 Teaching Statistics Trust, 38, 3, pp 102–107