Download A Framework for Assessing Middle School Students

1997, Vol. 9, No.1, 39-59
Mathematics Education Research Journal
A Framework for Assessing Middle School
Students' Thinking in Conditional Probability and
Independence
James E. Tarr
Graham A. Jones
Middle Tennessee State University
Illinois State University
Based on a synthesis of research and observations of middle school students, a
framework for assessing students' thinking on two constructs-eond.tional
probability and independence-was formulated, refined and validated. For both
constructs, four levels of thinking which reflected a continuum from subjective to
numerical reasoning were established.
The framework was validated from interview data with 15 students from Grades 4-8
who served as case studies. Student profiles revealed that levels of probabilistic
thinking were stable across the two constructs and were consistent with levels of
cognitive functioning postulated by some neo-Piagetians. The framework provides
valuable benchmarks for instruction and assessment.
Introduction
Recent world-wide curriculum reforms in school mathematics (e.g., Australian
Education Council, 1994; National Council of Teachers of Mathematics, 1989) have
advocated broadening the scope of probability in the middle school mathematics
curriculum. This emphasis on broader explorations of probability concepts in
problem settings has established the need for further research into the probabilistic
thinking of middle school students (Shaughnessy, 1992).
Although there has been substantial research on middle school students'
probabilistic thinking (e.g., Falk, 1983; Fischbein, Nello & Marino, 1991; Hawkins &
Kapadia, 1984; Piaget & Inhelder, 1975; Shaughnessy, 1992), little of the research
has focused on students' thinking in conditional probability and independence.
This lack of research on students' thinking in these two concepts is a matter of
concern given the fact that these concepts are increasingly being identified as
important ideas in probability instruction for the middle school (Australian
Education Council, 1994; Watson, 1995; Zawojewski, 1991).
Accordingly, this study addresses the need for a framework which
systematically describes and predicts middle school students' thinking in
conditional probability and independence. The need for such a framework is
critical if instructional programs and assessment targets in probability for the
middle school are to be informed by research-based knowledge of students'
thinking (Fennema, Carpenter & Peterson, 1989).
I
40
Tarr& Jones
Aims of the Research
Through a synthesis of relevant research, coupled with observation and
interviews with middle school children, this study set out: (a) to develop an initial
framework for describing how middle school children think about conditional
probability and independence; (b) to generate an assessment protocol for assessing
middle school students' thinking in independent and conditional probabilistic
situations; and(c) to refine and validate the framework.
Definitions
This study used the definition of conditional probability provided by Hogg and
Tanis (1993). The background for their definition is that, in some random
experiments, there is interest only in those outcomes that are elements of a subset B
of the sample space S. Within this context, they define the conditional probability of
an event A, given that event B has occurred, written P(A I B), as the probability of A
considering as possible outcomes only those outcomes of the random experiment
that are elements of B.
Hogg and Tanis (1993) noted that a special case of conditional probability
occurs in a random experiment carried out under "without replacement"
conditions. As an example, consider a bag with one red ball, one green ball, and
one blue ball, and an experiment where one ball is drawn and not replaced. The
sample space immediately prior to the second draw will be a subset of the original
sample space and the probability of "red," for example, will be conditional on the
outcome of the first draw. If a red is picked on the first draw, the probability of
"red" given the event "red" on the first draw will be O. On the other hand if a blue
ball is picked on the first draw, the probability of "red" given the event "blue" on
the first draw will be 0.5.
"Without replacement" situations, like those described above, not only
generate special cases of conditional probability, but are especially explicit because
the sample space is visually reduced. Because of their explicit nature "without
replacement" situations were used in this study to measure middle school
students' informal knowledge of conditional probability.
Hogg and Tanis (1993) also stated that two events are independent if the
occurrence of one of them does not change the probability of the occurrence of the
other. As an example, suppose a coin is flipped twice and the sequence of heads
and tails is observed. If A is the event, "head on the first flip," and B is the event,
"tails on the second flip, then P(B I A) = P(B). In other words the occurrence of A
has not changed the probability of B, and A and B are independent events.
Some mathematicians specify the events A and B in the above illustration as
independent trials (e.g., Larsen & Marx, 1986). In this study, we make no
distinction between independent events and independent trials, and we
consistently use settings involving repeated trials to assess middle school students'
thinking with respect to independence. In contrast with the conditional probability
settings, some of these repeated trials involve independent events arising from
"with replacement" situations. The use of random experiments involving repeated
trials is readily observable for these students and is consistent with the settings
used in this study for conditional probability.
Framework for Assessing Thinking in Conditional Probability
41
Although conditional probability and independence are clearly related both
mathematically and in the settings used in this study, there is no guarantee that
students will take cognisance of the relationship between these two concepts. In
essence this is one of the questions that this study sought to answer.
Theoretical Considerations
Conditional probability and independence, as defined above, are the key
constructs incorporated into the initial framework for describing and predicting
middle school students' thinking in probability (Table 1). The development of this
framework, which was built on an earlier framework Gones, Langrall, Thornton &
Mogill, 1997), is based on the assumption that'middle school students' thinking in
conditional probability and independence develops over time and can be described
across four levels. As is highlighted in Table I, Levell is associated with subjective
thinking, Level 2 is seen to be a transitional stage between subjective and naive
quantitative thinking, Level 3 involves the use of informal quantitative thinking
and Level 4 incorporates numerical re~soning. Furthermore it is posited that a
student's probabilistic thinking at any given time is stable over both constructs.
The notion of levels of thinking within specific knowledge domains is also in
concert with cognitive research which recognises developmental stages (Piaget &
Inhelder, 1975) and, more particularly, with neo-Piagetian theories which postulate
the existence of sub-stages or levels that recycle during stages (Biggs & Collis, 1991;
Case, 1985; Fischer, 1980; Mounoud, 1986). These neo-Piagetian researchers claim
that the sub-stages or levels, usually five, reflect shifts in the structural complexity
of students' thinking and that each level subsumes the previous one. For examp~e,
Biggs and Collis (1991) distinguish five basic levels in the learning cycle:
prestructural, unistructural, multistructural, relational, and extended abstract.
Prestructural responses belong in the previous stage or mode of representation,
extended abstract responses stretch into the next mode of representation and
unistructural, multistructural and relational levels fall within the present mode. In
addition, Biggs and Collis maintain that this learning cycle of five levels is
consistent across different stages and is applicable to school based tasks.
Each of the constructs, conditional probability and independence, is described
below and is amplified through reference to research on students' probabilistic
thinking. In addition, research on the two constructs is interpreted in the context of
the levels of the framework.
Conditional Probability
In this study, students' intuitive understanding of conditional probability was
measured by their ability to recognise and justify when the probability of an event
was and was not changed by the occurrence of another event. Students were also
expected to determine probabilities in response to the occurrence of another event.
Although there has been relatively little research into middle school students'
42
Tarr & Jones
Table 1
Initial Framework for Assessing Middle School Students' Thinking in Conditional
Probability and Independence
LEVEL i
(Subjective)
LEVEL 2
(Transitional)
• Recognises when"certain" and
"impossible" events arise in
replacement and non-replacement
situations.
.
.Generally uses subjective reasoning
in considering the conditional
probability of any event in a "with"
or "without" replacement situation.
• Recognises that the probabilities
of some events change in a "without replacement" situation. Recognition is incomplete, however,
and is usually confined to events
that have previously occurred.
• May revert to subjective judgments or use inappropriate quantitative measures.
• Unaware that two events mayor
may not influence each other.
• Holds a pervasive belief that they
can control the outcome of an event.
• Uses subjective reasoning which
precludes any meaningful focus on
the independence or dependence of
events.
·Shows some recognition as to
whether consecutive events are
related or unrelated.
• Frequently uses a
"representativeness" strategy,
either a positive or negative
recency orientation.
• May also revert to subjective reasoning.
CONDITIONAL
PROBABILITY
• Keeps track of the complete composition of the sample space in judging the relatedness of two events in
both "with" and "without" replacement situations.
• Recognises that the probabilities of
all events change in a "without
replacement" situation, and that
none change in a "with
replacement" situation.
• Can quantify, albeit imprecisely,
changing probabilities in a "without
replacement" situation.
• Assigns numerical probabilities in
"with" and "without"
replacement situations.
• Uses numerical reasoning to
compare the probabilities of
events before and after each trial
in "with" and "without"
replacement situations.
INDEPENDENCE
• Recognises when the outcome of the
first event does or does not
influence the outcome of the second
event. In "with replacement"
situations, sees the sample space as
restored.
.Can differentiate, albeit
imprecisely, independent and
dependent events in "with" and
"without" replacement situations.
• May revert to the use of a representativeness strategy.
• Distinguishes dependent and
independent events in "with"
and "without" replacement
situations; using numerical
probabilities to justify their
reasoning.
CONDITIONAL·
PROBABILITY
INDEPENDENCE
thinking in conditional probability situations, two studies (Fischbein & Gazit, 1984;
Jones, Langrall, Thornton & Mogill, 1997) are seminal to the development of the
Framework for Assessing Thinking in Conditional Probability
43
framework for this research. These two studies also incorporate some earlier
research on conditional" probability reported by Piaget and Inhelder (1975),
Borovcnik and Bentz (1991), and Shaughnessy (1992).
Fischbein and Gazit (1984) carried out a teaching experiment involving 285
students from Grades 5, 6 and 7. They found that, when students were asked to
determine conditional probabilities in "with" and "without" replacement
situations, the performance of sixth and seventh graders dropped dramatically for
"without replacement" tasks when compared with "with replacement" tasks. The
percentage of correct responses for sixth graders fell from 63% to 43% and for
seventh graders from 89% to 71%. The fifth graders response rate was only 24% for
the "with replacement" tasks and remained approximately the same for the
"without replacement" tasks. Based on their analysis, Fischbein and Gazit
identified two misconceptions in students' thinking in conditional probability:
(a) they did not realise that the sample space had changed in a "without
replacement" situation; and (b) they found the probability of an event in a
"without replacement" situation by comparing the number of favourable outcomes
for the event before and after the first trial rather than making comparisons with
the total number of outcomes.
In a related study with Grade 3 children over a period of one year, Jones,
Langrall, Thornton and Mogill (1997) identified four levels of thinking in
conditional probability. Children at Levell (Subjective) could not list the complete
set of outcomes in either a "with" or "without" replacement situation and did not
recognise that the probability of any event changed in a "without replacement"
situation. Their reasoning was almost always subjective. By Level 2 (Transitional),
children were beginning to recognise that the conditional probability of some but
not all events changed in a "without replacement" situation. This recognition was
generally restricted to the event that had occurred in the previous trial. Level 3
(Informal Quantitative) thinkers recognised that the probability of all events
changed in a "without replacement" situation and had begun to use numbers in an
informal way to determine conditional probabilities. Students at Level 4
(Numerical) were able to find valid numerical measures for changing probabilities
in "without replacement" tasks, but the measures they used were ratios rather than
fractions. Jones, Langrall, Thornton and Mogill noted that, in their study, none of
the students reached Level 4 even after instruction.
With respect to conditional probability, our framework has built on all of these
findings. In particular, in describing students' probabilistic thinking from Levels 1
to 4 we have used similar descriptors to those identified by Jones, Langrall,
Thornton and Mogill (1997) and have amplified these descriptors with the findings
of Fischbein and Gazit (1984).
Independence
In this study, students' intuitive understanding of independence was measured
by their ability to recognise and justify when the occurrence of one event had no
influence on the occurrence of the other. More particularly, students had to
recognise that, when the probability of an event was not changed by the occurrence
of another event, the two events were independent. A key study in the area of
44
Tarr& Jones
independence was carried out by Fischbein, Nello and Marino (1991) with 618
students in Grades 4-8. In this study, the researchers asked students to determine
which event was more likely, obtaining three "heads" by tossing one coin three
times or by tossing three coins simultaneously. Thirty-eight percent of fourth and
fifth graders and 30% of junior high students, with no prior instruction in
probability, responded that the probabilities were not equal. By a ratio of nearly 2:1,
students at each grade level believed the probability of obtaining three heads, by
tossmg a single coin three times, was higher. Based on follow-up interviews,
~ Fischbein, Nello and Marino found that students harboured a pervasive belief that
the outcomes of a coin toss can be controlled by the individual. They concluded
that such a belief is incompatible with the notion of independence, given the
probability of obtaining a head on each trial remains constant at 0.5.
Similar misconceptions were evident in the National Assessment of Educational
Progress in Mathematics (Brown, Carpenter, Kouba, Lindquist, Silver &. Swafford,
1988) which asked students to state the most likely outcome on the next toss of a
fair coin which had landed "TITT" on four successive trials. Results indicated that
only 47% of the seventh graders selected the correct alternative-heads and tails
are equally likely. Slightly higher achievement was obtained in a study of 2930
British students aged 11 to 16 years (Green, 1983). In this study a fair coin was
flipped four times, each time landing heads up. When asked to name the most
likely outcome of the fifth toss, 75% of all students answered correctly, including
67% of 11-12 year olds.
In a third study using this same item (Konold, Pollatsek, Well, Lohmeier &
Lipson, 1~93), only 70% of the undergraduates in a remedial mathematics course
responded correctly. Moreover in extensions to this item, Konold et al. asked the
students to state which of the following sequences was most likely and which was
least likely to occur when a fair coin was tossed five times: (a) HHHTI;
(b) THHTH; (c) THTIT; (d) HTHTH; and (e) all four sequences are equally likely.
In the most likely case, approximately 61% of the undergraduates enrolled in the
remedial course responded correctly but only about 35% responded correctly in the
case of the least likely sequence. Clearly, a substantial number of students who
demonstrated some understanding of independence in the most likely case
abandoned this thinking in the least likely case. Konold et al.concluded that a
conflict existed between the belief that a coin has an equal chance of coming up
heads or tails and that roughly half heads and half tails are expected in a sample of
coin flips.
Two categories of misconceptions were identified in these· studies. The most
common misconception (gambler's fallacy or negative recency effect) was the
tendency to believe that, after a run of tails, heads should be more likely to come
up. A far less common but related misconception (positive recency effect) was the
tendency to believe that, after a run of tails, a tail is more likely to come up. Both of
these misconceptions illustrate a judgmental heuristic known as
"representativeness"-the belief that a sample or even a single outcome should
reflect the parent population (Kahneman & Tversky, 1972). Such a belief is
powerful and is generally in conflict with the concept of independence. Thus, even
when students seemingly exhibit an understanding of the concept of
independence, the representativeness heuristic may still prevail.
Framework for Assessing Thinking in Conditional Probability
45
In relation to independence, our framework attempts to capture the change in
thinking from subjective judgments and beliefs about control, to thinking that is
based on numerical judgments and resolves conflicts between beliefs about
distribution expectations and event probabilities. The framework descriptors also
try to recognise growth in thinking that results from the ability to make
connections between conditional probability and independence.
Methodology
Subjects
Students in Grades four and five in an ~lementary school in Bloomington,
Illinois, and students in Grades six through eight in a junior high school in Eureka,
Illinois, formed the population for this study~ Fifteen students, three from each
grade level, served as case studies in this investigation. The students represented
the full range of abilities in each grade, with one being randomly chosen from the
top third, one from the middle third and one from the lowest third in mathematics
achievement. None of these students had been exposed to instruction in
probability.
The Validation Process
In formulating the framework for this study, four levels of probabilistic
thinking were posited and initial descriptions were generated for each of these
levels over two constructs, conditional probability and independence (Table 1).
Adopting the validation process used by Jones, Thornton and Putt (1994) and
Jones, Langrall, Thornton and Mogill (1997), this study sought to: (a) refine the
initial descriptions of the four levels of probabilistic thinking; (b) examine the
profiles and stability of case-study students' thinking over the two constructs; and
(c) illuminate the distinguishing characteristics of each thinking level within the
framework. Qualitative analysis was used to address all three parts of the
validation.
Instrumentation and Data Collection
The framework (Table 1) guided the design of the structured interview protocol
used in this study. The interview protocol, based on the framework, was
administered by the first author to each of the students who, served as case studies.
All interviews were conducted within a two-week period and were audiotaped for
subsequent analysis.
The interview protocol comprised 14 tasks, of which 8 assessed thinking in
conditional probability and 6 assessed thinking in independence. The assessment
tasks in conditional probability focused on probability situations involving "with"
and "without" replacement conditions, and the tasks on independence assessed
whether students could recognise and justify independent or dependent trials
generated in various probability settings. Selected items from the interview
protocol are presented in Table 2. In generating the items for the interview protocol
the researchers used problem contexts such as gumball machines, locker
46
Tarr& Jones
Table 2
Selected Probability Tasks from the Assessment Protocol
CONDITIONAL PROBABILITY
CPl
Suppose you were to reach into the bag of
Halloween candy and draw out a ,candy bar. What
kind of candy bar do you have the most chance of
drawing? Why? A candy bar is drawn and not
replaced. Suppose you ate the candy bar and then
reached into the bag again. What kind of candy bar
do you have the most chance of drawing? Why?
Has your chance of drawing a Snickers bar
changed or is it the same chance as before? Explain.
CP2
Suppose your teacher is going to have a drawing to
determine who gets to go first to lunch today. If
your colour is drawn, then you get to go to lunch
first. Allow the student to observe that the cup contains
4 blue, 3 green, 2 red, 1 yellow chips. What colour do
you want to be? Why? Suppose you are assigned
green and your friend is assigned yellow. I'm going
to reach in and draw a chip. What colour do you
predict will be drawn? Why? A blue chip is drawn
and not replaced. Suppose your teacher has another
drawing on the following day. Has your chance of
winning changed or is it the same as before? Has
your friend's chance of winning changed or is it the
same as bef~re? Why?
CP3
Your class is going to elect a president and vice
president. There are five people rurining. Show the
INDEPENDENCE
INOI
Allow the student to examine a die. I'm going to roll
the die. If I gave you one chance to pick, do you
think you could predict what number will come
up? Why? Roll the die. I'm going to roll the die
again. Do you think the chance of rolling a (result
of first roll)has changed? Why? Suppose I wanted
·to roll a 3. How many rolls would it take to
guarantee that a 3 would come up? Why?
IN02
Allow the student to examine a chip that is coloured red
on one side, white on the other. Suppose I flipped the
chip five times and kept track of the result. Which
of the following sequences is the most likely result
of five flips?
a) ••• 00
b)
0.· 0 •
000
c) o.
d) • 0 • 0 •
e) All five sequences are equally likely.
Suppose I flipped the chip five times and kept
track of the results. Which of the above sequences
is the least likely result of five flips?
IND3
student five cards: Beth, Steve, Maria, Rick, and YOU.
All five students are considered to have an equal
chance of winning. After school the results are
announced. Is it more likely the class president will
be a boy or a girl? Why? Is it more likely your name
will be read or more likely it won't be read? Reveal
the name of the president. Is it more likely the vice
president will be a boy or a girl? Why? Compared
to last time, has the chance that your name will be
read changed or is it the same as before? Why?
CP4
Place the "One Away" game board in front of the
student. (see IND4) There is a secret three-digit
number. It is not 2-8-5, but each digit is one away
from 2-8-5. That is, the first digit can be 1 or 3, etc. If
I gave you one chance to pick, do you think you
could win? Is it more likely you would win, more
likely you would lose, or is it the same chance?
Allow the student to pick the first digit. (Regardless of
their choice) You are correct. Has your chance of
winning changed? Allow the student to pick the
second digit. (Regardless of their choice) You are
correct. Has your chance of winning changed?
Why?
I'm going to spin it with the needle on the line. Do
you think it's more likely for it to land on red, on
yellow, or is it the same chance? Why? Spin the
needle. I'm going to spin it again. Do you think it's
more likely for it to land on red, on yellow, or is it
the same chance? Why? (Repeat again) Suppose I
wanted it to land on red. Should I start the needle
on red, on yellow, on the line, or does it not make a
difference? Why?
lN04
174
L
L
:s::
396
Place the "One Away" game board in front of the
student. What is your chance· of picking the first
digit correctly? Allow the student to pick the first
digit. (Regardless of their choice) You are correct.
Has your chance of correctly picking the second
digit changed? Allow the student to pick the second
digit. (Regardless of their choice) You are correct.
Has your chance of correctly picking the third
changed? Why?
Framework for Assessing Thinking in Conditional Probability
47
combination numbers, Halloween candy drawings, first to lunch drawings, and
games involving coin flips and die rolls, all of which were very much part of these
students' real world. The design of each of the conditional probability and
independence items enabled the researchers to explore students' thinking across all
four levels of the framework. In each case-study interview, tasks on conditional
probability and tasks on independence were· administered in an alternating
sequence. Items were structured so that students would respond orally, but they
were also encouraged to use probability materials to demonstrate and explain their
respons~s.
, A double-coding procedure described by Miles and Huberman (1984) was
used to code the interview protocols. Initially, both researchers independently
coded each item. of each student's interview assessment. Using the framework
criteria (Table I), items were coded according to the level of thinking exhibited by
the student. These codings were then used to determine the dominant (modal)
probabilistic thinking level for each student on both conditional probability and
independence. Agreement between the two researchers was achieved on the
coding of 28 out of 30 levels, that is, a reliability of 93%. Variations were then
clarified until consensus was reached on each student's dominant level of thinking
on both conditional probability and independence.
During the coding process described above, both researchers used a groundedtheory approach (Bogdan & Biklen, 1992) to discern the key thinking patterns
exhibited by students at each level of the framework and across both constructs.
These patterns were used to refine the framework descriptors and to generate
summaries that illuminated the descriptors.
Data Analysis and Results: Validating the Framework
The results of the validation process are presented in three parts. The first part
describes refinements made to. the framework following the collection of data. In
the second part, the profiles and stability of students' thinking across the two
constructs of the revised framework are examined. Finally, based on an analysis of
students' thinking, summaries and exemplars are presented to illuminate the four
levels of probabilistic thinking in the revised framework.
Refinements to the Framework
The refined framework (Table 3) was developed following the analysis of the
case-study data. Although only the first descriptor under Level I, Independence
(Table 1) was rejected in refining the initial framework (Table 3), several additional
descriptors were added to the refined framework to enhance the initial descriptors
for conditional probability and independence. These additional descriptors have
been italicised in Table 3.
The descriptor, "Unaware whether two events influence each other or not" (see
Table 1: Levell, Independence, first descriptor) was found to be inappropriate and
was omitted from the refined framework. This action was taken because neither of
the case studies at Level 1 exhibited. ambivalence concerning the influence of
consecutive events on each other. On the contrary, both case study students
maintained that consecutive events were always dependent; that is, they rejected
the concept of independence. In order to underscore Level 1 students'
48
Tarr& Jones
Table 3
Refined Framework for Assessing Middle School Students' Thinking in Conditional
Probability and Independence
LEVEL 1
(Subjective)
CONDITIONAL
PROBABILITY
-Recognises when "certain" and
"impossible" events arise in
replacement and nonreplacement situations.
-Generally uses subjective
reasoning in considering the
conditional probability of any
event in a "with" or "without"
replacement situation.
-Ignores given numerical information
in formulating predictions.
-Predisposition to consider that consecutive events are always related.
INDEPENDENCE
- Pervasive belief that they can control the outcome of an event.
- Uses subjective reasoning which
precludes any meaningful focus
on the independence.
-Exhibits unwarranted confidence in
predicting successive outcomes.
CONDITIONAL
PROBABILITY
INDEPENDENCE
- Recognises that the probabilities of
all events change in a "without
"replacement" situation, and that none
change in a "with replacement"
situation.
-Keeps track of the complete composition of the sample space in judging
the relatedness of two events in both
"with" and "without" replacement situations.
-Can quantify, albeit imprecisely,
changing probabilities in a "without
replacement" situation.
- Recognises when the outcome of the
first event does or does not influence
the outcome of the second event. In
"with replacement" situations, sees
the sample space as restored.
-Can differentiate, albeit imprecisely,
independent and dependent events in
"with" and "without" replacement
situations.
-May revert to the use of a representativeness strategy.
LEVEL 2
(Transitional)
- Recognises that the probabilities
of some events change in a "without replacement" situation. Recognition is incomplete, however,
and is usually confined to events
that have previously occurred.
-Inappropriate use of numbers in determining conditional probabilities. For
example, when the sample space contains two outcomes, always assumes
that the two outcomes are equally
likely.
-Representativeness acts as a confounding effect when making decisions about conditional probability.
- May revert to subjective judgments.
-Shows some recognition as to
whether consecutive events are
related or unrelated.
- Frequently uses a
"representativeness" strategy,
either a positive or negative
recency orientation.
- May also revert to subjective reasoning.
- Assigns numerical probabilities in
"with" and "without" replacement
situations.
- Uses numerical reasoning to
compare the probabilities of events
before and after each trial in "with"
and "without" replacement situations.
-States the necessary conditions under
which two events are related.
- Distinguishes dependent and
independent events in "with" and
"without" replacement situations,
using numerical probabilities to
justify their reasoning.
-Observes outcomes ofsuccessive trials
but rejects a representativeness strategy.
-Reluctance or refusal to predict outcomes
when events are equally likely.
Framework for Assessing Thinking in Conditional Probability
49
preoccupation with dependence, an additional descriptor, "Predisposition to
consider that consecutive events are always related," was included (see Table 3:
Levell, Independence, first descriptor).
A somewhat related addition was necessitated by Levell students' tendency to
ignore quantitative information inherent in the sample space and to display
unwarranted confidence in predicting successive outcomes in replacement
situations (see Table 3: Levell, Independence, fourth descriptor). By way of
contrast, both students exhibiting level 4 thinking were clearly reluctant to predict
outcomes when events were equally likely, and often refused to pick altogether
unless the odds were in their favour (see Table 3: Levell, Independence, fourth
descriptor).
Two other additions to the framework (see Table 3: Level 2, Conditional
Probability, third descriptor; and Level 4, Independence, second descriptor) were
incorporated to build a more comprehensive picture of students' thinking with
respect to the representativeness strategy. When these additional descriptors were
cOlnbined with the original descriptors pertaining to representativeness they
resulted in a more ordered description of student's use of the representativeness
strategy across the four thinking levels. The remaining additions (see Table 3:
Levell, Conditional Probability, third descriptor; Level 2, Conditional Probability,
second descriptor; and Level 4, Conditional Probability, third descriptor) were seen
to add greater precision and amplification to the original descriptions and are
illustrated in the next section through the thinking of Alicia (Levell), Brian
(Level 2) and Denise (Level 4).
Profiles and Stability Across the Two Constructs
From a validation perspective, the stability of students' thinking across the two
constructs is of prime importance. In particular, if the framework is to be used in
curriculum development and assessment it is essential that descriptors provide
parallel benchmarks at each level across both constructs, conditional probability
and independence.
Profiles of the students' thinking levels (Figure 1) revealed strong internal
consistency in that thinking patterns across the two constructs were at the same
level for 11 of the 15 students. In the case of the four, exceptions, it was noteworthy
that none of the differences were more than one level apart and that two of the
differences were associated with higher thinking levels on independence and two
with higher levels on conditional probability. These observations support the
"stability" hypothesis for the framework and this position is further endorsed by a
Pearson correlation coefficient of r = 0.83, indicating 69% of shared variance
between thinking levels for independence and conditional probability.
Notwithstanding the constraints associated with a sample of 15 students, the
individual profiles yield further findings. Given the fact that the two students who
exhibited Level 1 thinking were in Grades 4 and 5 and one of the students who
exhibited Level 4 thinking was in Grade 8, it is compelling to conclude that
thinking becomes more sophisticated with increased age. However, such a
conclusion may be oversimplifying the situation because within each grade there
was a diversity of probabilistic thinking (see Figure 1). Moreover, two fourth-grade
50
Tarr& Jones
students exhibited Level 3 thinking while one eighth-grade student was at Level 2.
These extremes and the diversity of probabilistic thinking within grades appear to
be just as explicit as differences between grades.
Conditional
Probability
f:A
Jana
CI}
~
Grade 4
Mary
4
CI}
~
~
._
~ I
i5
0
~
CP
0
IND
Ashley
CI}
Q)
:>
Grade 5
~
4
Q)
:>
i5
I
CP
Q)
._
~
._
~ I
I
0
Grade 7
gf2
CI}
Q)
CP
0
IND
4
i5
3
bJ>
] 2
c
.- I
~
CP
0
IND
CI}
4
Q)
:>
3
Q)
:>
3
~
CP
IND
~
IND
gf2
~
.-
1
0
CP
Denise
Sergio
gf2
I
0
Q)
~
4
~
.-
IND
4
CI}
~
3
CP
Kurt
CI}
~I
gf2
~
IND
~ 2
Q)
Grade 8
0
CP
4
Melissa
~
~
.3bJ> 3
I
0
~
.- J
Vanessa
4
~ 2
CI}
Q)
.3 3
IND
IND
4
gf2
0
CP
Daniel
CI}
~I
.3bJ> 3
i§
0
IND
4
Brian
Q)
CP
~
CP
CI}
~
~ 3
....:l
0
IND
Kiko
Q)
1
CP
4
CI}
gf2
CI}
3
gf2
~
0
IND
Christina
4
.3
i§
CP
gf2
IND
Q)
Grade 6
~
4
Neil
CI}
._
~ I
.3 3
~
0
bJ>2
c
3
~
gf2
3
~
Brad
CI}
3
~
CI}
bJ>2
c
I
4
~
3
~
gf2
Independence
Carlos
4
~
3
l1li
~
CP
IND
I
0
CP
IND
Figure 1. Probabilistic thinking patterns: Profiles of case-study students.
Framework for Assessing Thinking in Conditional Probability
51
Analysis of the Probabilistic Thinking at Each Level
The responses and thinking of each of the students who served as case studies
were analysed, and summaries and exemplars were produced to illuminate the
thinking patterns outlined in the refined' framework (Table 3). Although all
students at a particular level on a construct tended to generate similar probabilistic
thinking and misconceptions, we have, for ease of reading, used case-study
students who were at the same level on both constructs.
Level 1 Summary. In interpreting probability situations, students exhibiting
Levell thinking tend to rely on subjective judgments, ignore relevant quantitative
information and generally believe that they can control the outcome of an event.
Given' this void of quantitative referents, Level 1 students form conditional
probability judgments by constructing their own reality, either searching for
patterns which do not exist or by imposing their own system of regularity. They
predict outcomes with unwarranted confidence and often use their own recent
experiences in playing games (availability heuristic-see Shaughnessy (1992)) to
predict or estimate the chance of an event. Level 1 students tend to believe that
previous outcomes always influence future outcomes and they essentially deny the
existence of independence. In essence, Levell students' proclivity to use subjective
judgments, to impose regularity or to rely on readily available experiences
involving chance precludes any meaningful focus on independence and
conditional probability.
Alicia, a fifth-grade student, demonstrated a reliance on subjective reasoning
that exemplifies Level 1 thinking. When asked to predict what colour would be
drawn in First to Lunch (Table 2, CP2), Alicia immediately chose red because, "it ~s
my favourite colour." Her response typically ignored the numerical composition of
the cup which contained fewer red chips than either blue or green. The distinctive
feature of Alicia's conditional thinking was her lack of awareness of the role that
numbers play in making probability judgments-to her, numbers were simply not
relevant.
With respect to the concept of independence, Alicia sought to impose her own
system of regularity when interpreting random experiments. After subsequent
rolls of 2,3, and 4 (Table 2, INO 1) she believed that the chance of rolling a 6 had
increased, saying, "it's gone 2, 3, 4, so it's laying a little pattern." Her response is
indicative of the common belief among Levell students that previous results
influence outcomes in "with replacement" situations. Typically she also exhibited
unwarranted confidence in predicting outcomes. When predicting the initial roll of
a die (Table 2, INO 1), Alicia was emphatic that the result would be a 6 and vividly
recalled how she had rolled consecutive 6's when playing a game in second grade.
Although she appeared to use an availability heuristic (Shaughnessy, 1992) in
recalling an earlier success in rolling 6's, her strategy may well be subsumed under
the more general rubric of "imposing regularity where randomness prevails."
These observations demonstrate that Alicia was not receptive to the existence of
independence.
Level 2 Summary. Students who demonstrate Level 2 probabilistic thinking are
in transition between subjective and informal quantitative thinking. They
sometimes make appropriate use of quantitative information in making
52
Tarr& Jones
conditional probability judgments, but are easily distracted by irrelevant features.
For example, they tend to put too much faith in the distribution of previous
outcomes when forming predictions; as a result, they often use a representativeness
heuristic (Shaughnessy, 1992), incorporating either a positive or negative recency
orientation. When considering conditional probabilities, some Level 2 students
may also start from a subjective perspective or may even revert to subjective
judgments when outcomes do not occur as they expected. Others faced with a
probability situation containing two outcomes are prone to assuming that the two
outcomes are equally likely. Their vulnerability to irrelevant features or imprecise
reasoning during this transitional stage seems to explain the vacillations they
exhibit when faced with tasks which' involve conditional probability and
independence. Even when they do use informal quantitative reasoning, their
thinking is limited. They are able to recognise that the probabilities of some events
change in "without replacement" situations, but recognition is incomplete and is
usually confined to events that have previously occurred.
Level 2 thinking was exemplified by Brad, a fifth grader, and by Brian, a
seventh grader. At times, both students were able to use quantitative information
appropriately, but they were also prone to subjective judgments and imprecise
quantitative reasoning. Although Brad and Brian chose blue in the drawing to be
first to lunch (Table 2, CP2), Brian had some reservations, "Red is my favourite
colour, but I'll go with blue because there's more of them." In the same conditional
probability task, after a blue chip was drawn and not replaced, both thought the
probability that green would be drawn had increased because there was an equal
number of blue and green chips. Brad explained, "You already have one blue out so
that makes three blues, three greens, and that's the same amount." However,
typical of Level 2 students, their conditional thinking was not complete as each
maintained in the same problem that the probability that yellow would be drawn
was "still the same," because "there's still only one yellow in the cup."
A further illustration of Brian's imprecision in using quantitative information
occurred when the sample space contained two outcomes. He consistently
assumed that both events were equally likely as the following exemplars show.
When asked the probability of selecting the three-digit number in one pick (Table 2,
CP4, IND4), Brian said, "There's a 50-50 chance I could or could not." After
learning that he had correctly selected the first digit, Brian still maintained that his
chance of winning versus losing was "still 50-50" even though it had increased
from 1/8 to 1/4.
Both students exhibited some recognition of independent events but were
easily distracted by the outcome distribution of successive trials. When asked to
predict if a spinner was more likely to land·on red or yellow (Table 2, IND 3), Brad
said, "It's the same chance it'll be either one of those." After the spinner landed on
red, Brad maintained this stance. However, when it stopped on red a second time,
Brad predicted it was now more likely to land on red, "Because the last time you
spinned it, it was red so it might be red this time." Neil, a sixth grader, who had a
dominant thinking level of 3, actually shifted his thinking in the opposite direction
to Brad on this task. He initially predicted that the needle would land on red
because it started on red. After the needle stopped on yellow he abandoned his
subjective reasoning in favour of a more quantitative argument saying, "half ofthe
circle is still red and half of it is still yellow."
Framework for Assessing Thinking in Conditional Probability
53
The pervasiveness of a representativeness strategy confounds Level two
students' ability to make proper judgments in conditional probability and
independence. In selecting the most likely outcome of five flips of a chip (Table 2,
IND2), Brian chose B "because it looks like the one that's most common-it's
mixed." When asked which result was the least likely result of five flips, Brian
chose C "because there are more whites than red, and if anything it would
probably come up even-well, it can't come up even, but pretty close to even."
Level 3 Summary. Students exhibiting Level 3 thinking are aware of the role that
quantities play in forming conditional probability judgments. Although these
students do not usually assign precise numerical p~obabilities they often use
relative frequencies, ratios or some form otodds in an appropriate way to
determine conditional probabilities after each trial of an experiment "with" or
"without" replacement. They keep track of the complete composition of the sample
space and usually recognise that the conditional probabilities of all events change
in "without replacement" situations. Level 3 students' predisposition to monitor
the sample space composition also enables them to recognise independent events
in "without replacement" situations; however they sometimes revert to
representativeness strategies (Shaughnessy, 1992) in dealing with a series of
independent trials.
Levei 3 thinking was exemplified by Carlos, a fourth grader, and Christina, a
sixth grader. Both students kept track of the composition of the sample space
before and after each trial and consequently recognised that the conditional
probabilities of all events changed in "without replacement" situations. For
example, after a Milky Way was drawn from a bag of Halloween candy and not
replaced (Table 2, CPl), Carlos explained that the chance of being selected had
increased for both Butterfingers and Snickers. He said, "Butterfingers is tied with
the most and it was behind ... Snickers is one away from the top and it was two
away from the top." A similar approach, for which we have coined the term
"competition strategy/' was utilised by Christina in predicting the results of the
class election (Table 2, CP3). After the name of the president was revealed, she
compared her chance of being named vice president to her prior chance of being
named president as follows: "It [vice president] is better-now there's three people
against me instead of four."
By monitoring the composition of the sample space, both students were able to
determine whether two outcomes were independent. In replacement situations,
they recognised when the sample space had been restored and linked this
phenomena to the concept of independence. For example, after a spinner had twice
landed on yellow (Table 2, IND 3), Christina stated that it was equally likely for red
or yellow; she said, "It's the same chance .. '. because half of the circle is still red and
half of it is still yellow (authors' italics)." Similar thinking was generated by
Melissa, an eighth grader, when she considered whether probabilities had changed
with successive rolls of the die (Table 2, IND 1). After the first roll yielded a I, she
stated that the chance of rolling a 1 had not changed: "That number hasn't been
eliminated ... so it just starts all over again."
54
Tarr& Jones
Neither student at Level 3 assigned numerical probabilities in forming
probability judgments. This limitation in making precise probability judgments
may have produced some reversion to a representativeness strategy for both
students. For example, when asked to determine the most likely result of five flips
of a chip (Table 2, IND 2), Carlos initially said, "1 think they're all equally likely."
However, when pressed to compare the relative likelihood of [A] and [B], Carlos
hedged and said, "No, not [A], three times in a row. Maybe not." Only after
comparing all four sequences did Carlos reaffirm his position: "all have the exact
same chance."
Level 4 Summary. Students who demonstrate Level 4 thinking use numerical
reasoning to interpret probability situations. They are keenly aware of the
composition of the sample space, recognise its importance in determining
conditional probabilities, and can assign numerical probabilities spontaneously
and with explanation. Using numerical thinking, Level 4 students consistently
distinguish between independent and dependent events and can even identify the
conditions under which two events are related. Their reliance on numerical
reasoning enables them to hold strong convictions when making conditional
probability judgments. In "without replacement" probability settings, Level 4
students can determine the minimum number of picks before an event is certain,
and in "with replacement" situations, they realise that there is no maximum
number of trials that will produce a certain event. With respect to replacement
situations they are aware of the distribution of outcomes in an on-going series of
trials but they reject a representativeness strategy in favour of using more precise
conditional probability measures. As further instance of their strong numericallybased convictions, these students are reluctant to make predictions when all
outcomes are equally likely. However, they point out when an outcome occurs with
the odds, and acknowledge the possibility that events can occur against the odds.
Level 4 thinking was demonstrated by Daniel, a sixth grader, and Denise, an
eighth grader. Both students consistently and pervasively used numerical
reasoning to form probability judgments. For example, as the following interaction
shows, Denise was able to use her numerical precision to focus on and interrelate
multiple tasks involving conditional probability and independence. When asked if
she could correctly select the three-digit number in one pick (Table 2, CP4, IND4),
Denise said it was unlikely because there were eight possibilities and only one
correct combination. Then, after learning that the first digit was 1, she said, "Now I
have a better chance [of guessing the three-digit number] because I got to eliminate
four choices-396, 394, 376, 374-so now there's only four for me to choose from."
Moreover even though she correctly selected the first digit, Denise indicated that
her chances of picking the second digit had not changed. She explained, "If you got
the first one wrong, you'd still have the same chance of getting the second one right
or wrong." Upon learning that the second digit was 7 she replied, "1 have a better
chance now because now I know that it's not 194 or 196; it's either 174 or 176." She
reiterated that her chances of correctly picking the next digit had not changed and
stated, "It's the same because it doesn't matter what you get the first two times."
Her ability to keep everything in perspective during this exchange is a tribute to
her strong sense of numerical precision.
Framework for Assessing Thinking in Conditional Probability
55
The use of numerical reasoning was also evident when Denise interpreted
activities involving independent events. When asked if a spinner was more likely
to land on red or yellow (Table 2, IND3) she replied, "It's the same because each
side is equal ... there's the same amount of red as there is yellow." In seven
successive trials the spinner landed alternately on red and yellow. Denise noted the
pattern but rejected a representativeness strategy. She articulated, "It went red,
yellow, red, yellow, red, yellow, but even though it's going in a pattern, there's still
the same chance because both sides are equal." With her adherence to numerical
reasoning, Denise could even identify the conditions under which two events are
related or unrelated. For example, after a candy bar was drawn and not replaced
(Table 2, CP1), she said the probability of selecting a Snickers bar changes "unless
you'put it back in." .
Daniel also realised that probabilities do not change in "with replacement"
situations. He was reluctant to predict the outcome in the roll of a die (Table 2,
IND1) because the chance was the same for each trial. Daniel indicated that if he
had to predict he would choose 4 but added, "If I do happen to get it, it would be
defying the odds." After a 4 was rolled, he said the chance of rolling a 4 had not
changed because "all the numbers are still there ... it's still the same because it's still
one out of six." By realising that a non-changing sample space in "with
replacement" situations is tantamount to probabilities remaining constant, he was
able to distinguish independent and dependent events. When asked how many
rolls would be required to guarantee that a 1 would come up Daniel said, "There is
no real maximum number of hitting that I."
Discussion
There is a growing consensus that knowledge of learners is an important
component of teacher knowledge (Shulman, 1986). Accordingly, the lack of
research on middle school students' thinking in conditional probability and
independence is problematic given the emerging emphasis on these probability
topics in the middle school curriculum. This research sought to develop and
validate a framework for describing and predicting students' thinking in
conditional probability and independence that might inform instructional
programs in probability.
The initial framework, based largely on previous research, identified four
levels of probabilistic thinking which ranged from subjective judgments to
numerical reasoning. Following analysis of case-study data, a refined framework
was developed and profiles of case-study students' thinking levels for each
construct were generated. These profiles of students' thinking revealed a general
pattern of growth in probabilistic thinking with age; however, broad ranges of
probabilistic thinking existed within each grade level. The profiles also suggest that
levels of probabilistic thinking, as described in the refined framework, appeared to
be stable across both conditional probability and independence. Such stability is
important from a validation perspective because it implies that the framework not
only presents a coherent picture of middle school students' thinking in conditional
probability and independence but also provides parallel benchmarks for the two
constructs which can be used in instruction and assessment.
56
Tarr& Jones
Notwithstanding the apparent stability discussedaboye, some caution should
be used in claiming that the levels of probabilistic thinking across both conditional
probability and independence are highly stable for.1al'geuum bers of students.
Earlier research (Jones, Langrall, Thornton & Mogill,1997) revealed that the
internal consistency of young children's probabjIi§tiCthinking across four
constructs was more erratic than the probabilistic thihl<it1gexhi1:>ited by students in
this study. Such instability is also consistent withth~fin.qit1gs of Dawson and
Rowell (1995) who concluded that "during the process~fconcept development
there will be periods of transition within which leamerswilluot have a full and
stable belief in anyone explanation" (p. 90).
. .... .......>
In addition to the value these framework descri~~()~§hay¢jn instruction and
assessment, they also extend and illuminate the resear~~;lit.efflture on students'
thinking in conditional probability and independ~J.'l~~~\':fJ1~iian.alysis of data
revealed that students exhibiting Levell thinking we~i80J.'l~i~t~n.t1y restricted to
subjective judgments when interpreting tasks incg~~jti~:J.'l~lprobability and
independence. In essence, they appeared to demonstrati?;tli~sl1~J.'acteristicsof what
Biggs and Collis (1991) describe as the prestructurallever~fi~'el~arningcycle; that
is, they engage in the task but are easily distracted by featittest!iatareirrelevant to
probabilistic thinking. Note how Alicia typically ignorecixeleyant quantitative
information and, instead, fixated on an irrelevant featureoftl'ietClsk-for example,
her favourite colour.
Students exhibiting Level 2 thinking were charact~risediby their limited
quantitative reasoning in exploring probability tasks. At linles,they were able to
make appropriate-albeit primitive-use of numbers to form j'\ldgments, but at
other times, they made inappropriate and imprecise useo! •.quantities. Such
characteristics serve to illustrate Level 2 as a period of transition in which there is a
naive attempt to quantify probabilities. At this level, sh.J.q.ellts' thinking is
indicative of what Biggs and Collis (1991) term the unistructurallevel in the sense
that the learner engages in the task in a relevant way butQI'llY. one aspect is
pursued. For example, Brian recognised that the probabilities of blue and green
had changed (Table 2, CP2), but maintained that the probability of yellow had not
changed. In essence, he focused on just one comparison, betweellthe target colour
and the event that had just occurred, rather than making severalcomparisons with
. the changed sample space including those relevant to the event"yellow."
Students demonstrating Level 3 thinking effectively used quantitative
reasoning when interpreting tasks based on the two constructs. ·They kept track of
the composition of the sample space to determine conditional probabilities in
"with" and "without" replacement situations and to decide whether two events
were dependent or independent. Students at Level 3 seemed to exhibit
characteristics of the multistructurallevel of the learning cycle (Biggs & Collis, 1991)
in that they focused on more than one relevant feature of a task. For example, in
contrast to the Level 2 thinkers who focused on only one aspect of the learning
task, note how Carlos, in a similar task (see Table 2, CP1), monitored the changing
composition of the sample space for all events. The ability to consider several
events simultaneously and to use more than one strategy are features of Level 3
thinkers which distinguish them from Level 2 thinkers.
Framework for Assessing Thinking in Conditional Probability
57
Students exhibiting Level 4 thinking use numerical reasoning to interpret
probability situations. They assign numerical probabilities to determine
conditional probabilities in "with" and "without" replacement situations and to
decide whether two events are related or unrelated. Level 4 students appeared to
exhibit thinking at what Biggs and Collis (1991) term the relational level of the
learning cycle; that is, they integrate numerous relevant aspects of the task in a
comprehensive and coherent manner. Denise's management of the dependent and
independent features in the "One Away" problem (See Table 2, CP4, IND4) is a
poignant illustration of the ability of students at Level 4 to integrate multiple
aspects of the learning task. For example~ after learning that she had correctly
picked the first digit, she realised that although the number of equally,.likely
outcomes in the sample space for the three-digit combination had decreased from 8
to 4, ,the sample space for the second digit had been maintained. As a consequence,
Denise was able to articulate that her chances of winning the game had increased
but her chance of picking the second digit had not changed. Consistent with
relational level thinkers, Denise was not only able to monitor, within the same
context, different sample spaces and their associated probabilities, but she was also,
able to integrate and isolate elements of the problem when the need arose.
The framework formulated and validated in this study has implications for
instruction and assessment. It offers comprehensive and internally consistent
benchmarks for conditional probability and independence which teachers can use:
(a) to identify key elements and tasks in an instructional program; (b) to provide
the initial assessments of student's thinking that will inform instruction; and
(c) provide on-going assessment once instruction has begun. Moreover, the
framework can be used to establish targets and tasks for state and national
assessment.
More is known about how students learn mathematics than is known about
how to apply this knowledge to mathematics instruction (Romberg & Carpenter,
1986) and this is certainly true with respect to instruction in conditional probability
and independence for middle school students. Accordingly, there is a need for
future research to use a framework, such as the one generated in this study, to
develop and evaluate middle school instructional programs in conditional
probability and independence that are informed by knowledge of students'
probabilistic thinking.
References
Australian Education Council. (1994). Mathematics: A curriculum profile for Australian schools.
Carlton, Victoria: Curriculum Corporation.
Biggs, J. B., & Collis, K. F. (1991). Multimodallearning and the quality of intelligent behavior.
In H. A. H. Rowe (Ed.), Intelligence: Reconceptualisation and measurement (pp. 57-66).
Hillsdale, NJ: Erlbaum.
Bogdan, R. c., & Biklen, S. K. (1992). Qualitative research for education: An introduction to theory
and method. Boston: Allyn and Bacon.
Borovcnik, M. G., & Bentz, H. J. (1991). Empirical research in understanding probability. In
R. Kapadia & M. Borovcnik (Eds.), Chance encounters: Probability in education (pp. 73105). Dordrecht: Kluwer.
58
Tarr& Jones
Brown, C. A, Carpenter, T. P., Kouba, V. L., Lindquist, M. M., Silver, E. A, & Swafford, J. O.
(1988). Secondary school results for the fourth NAEP mathematics assessment: Discrete
mathematics, data organization and interpretation, measurement, number and
operations. Mathematics Teacher, 81, 241-248.
Case, R. (1985). Cognitive development. New York: Academic Press.
Dawson, C. J., & Rowell, J. A (1995). First-year university physics: Who succeeds? Research
in Science Education, 25, 89-100.
Falk, R. (1983). Children's choice behaviour in probabilistic situations. In D. R. Grey, P.
Holmes, V. Barnett & G. M. Constable (Eds.), Proceedings of the First International
Conference on Teaching Statistics (pp. 714-716). Sheffield, UK:Teaching Statistics Trust.
Fennema, E., Carpenter, T. P., & Peterson,' P. (1989). Teachers' decision making and
cognitively guided instruction: A new paradigm for curriculum development. In N. F.
Ellerton & M. A Clements (Eds.), School mathematics: The challenge to change (pp. 174187). Geelong, VIC: Deakin University.
Fischbein, E., & Gazit, A (1984). Does the teaching of probability improve probabilistic
intuitions? Educational Studies in Mathematics, 15, 1-24.
Fischbein, E., Nello, M. S.,& Marino, M. S. (1991). Factors affecting probabilistic judgments
in children and adolescents. Educational Studies in Mathematics, 22, 523-549.
Fischer, K. (1980). A theory of cognitive development: The control and construction of
hierarchies of skills. Psychological Review, 57, 477-531.
Green, D. R. (1983). A survey of probability concepts in 3000 pupils aged 11-16. In D. R. Grey,
. P. Holmes, V. Barnett & G. M. Constable (Eds.), Proceedings of the First International
Conference on Teaching Statistics (pp. 784-801). Sheffield, UK: Teaching Statistics Trust.
Hawkins, A., & Kapadia, R. (1984). Children's conceptions of probability: A psychological
and pedagogical review. Educational Studies in Mathematics, 15,349-377.
Hogg, R. V., & Tanis, E. A. (1993). Probability and statistical inference (4th ed.). New York:
Macmillan.
Jones, G. A, Langrall, C. W., Thornton, C. A., & Mogill, A. T. (1997). A framework for
assessing and nurturing young children's thinking in probability. Educational Studies in
Mathematics, 32(2), 101-125.
Jones, G. A, Thornton, C. A, & Putt, I. J. (1994). A model for nurturing and assessing
multidigit number sense among first grade children. Educational Studies in Mathematics,
27, 117-143.
Kahneman, D., & Tversky, A (1972).' Subject probability: A judgment of representativeness.
Cognitive Psychology, 3, 430-454.
Konold, c., Pollatsek, A, Well, A, Lohmeier, J., & Lipson, A (1993). Inconsistencies in
students' reasoning about probability. Journal for Research in Mathematics Education, 24,
392-414.
Larsen, R. J., & Marx, M. L. (1986). An introduction to mathematical statistics and its applications
(2nd ed.). Englewood Cliffs, NJ: Prentice-Hall.
Miles, M. 8., & Huberman, A. M. (1984). Qualitative data analysis: A sourcebookfor new methods.
Beverly Hills, CA: Sage Publications.
Mounoud, P. (1986). Similarities between developmental sequences at different age periods.
In I. Levin (Ed.), Stage and structure (pp. 40-58). Norwood, NJ: Ablex.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for
school mathematics. Reston, VA: Author.
Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children (L. Leake, Jr., P.
Burrell, & H. D. Fischbein, Trans.). New York: W. W. Norton.
Framework for Assessing Thinking in Conditional Probability
59
Romberg, T. A, & Carpenter, T. P. (1986). Research on teaching and learning mathematics:
Two disciplines of scientific inquiry. In M. C. Wittrock (Ed.), Handbook of research on
teaching (3rd ed., pp. 850-873). New York: Macmillan.
Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and directions.
In D. A. Grouws (Ed.), Handbook ofresearch on mathematics teaching and learning (pp. 465.
494). New York: Macmillan.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational
Researcher, 15(2),4-14.
.
Watson, J. (1995). Conditional probability: Its place in the mathematics curriculum.
Mathematics Teacher, 88, 12-17.
Zawojewski, J. S. (1991). Curriculum and evaluation standards for school mathematics, Addenda
series, Grades 5-8: Dealing with data and chance. Reston, VA: Author.
Authors
James E. Tarr, Middle Tennessee State University, Department of Mathematical Sciences,
.
Email: jetarr@mtsu.edu
Murfreesboro, Tennessee 37132, USA
Graham A Jones, Illinois State University, 4520 Mathematics Department, Normal, Illinois
61790-4520, USA.
Email: jones@math.ilstu.edu