ICOTS10 (2018) Invited PaperIshibashi

EFFECTIVENESS OF USE OF DIAGRAMS FOR TEACHING CONDITIONALPROBABILITY FROM A SEMIOTICS VIEWPOINT
Ippo Ishibashi
Hiroshima University High School, Fukuyama
Graduate School of Education, Hiroshima University, Japanipishibashi@hiroshima-u.ac.jp
This study examines teaching approaches for learning conditional probability by using diagrams.
Toward this end, we first identify conventional knowledge about diagrammatic representations ofprobability from a semiotics viewpoint. Next, we present effective diagrammatic expressions forunderstanding conditional probability, and we provide a teaching approach to encourage efficient
diagrammatic reasoning. Finally, we evaluate teaching using the “effect of time axis.” As a result,the conventional knowledge of diagrammatic representations of probability was identified as “therectangle represents the denominator in the probability calculation and the part bounded by thecircle inside the rectangle represents the numerator.” Furthermore, it is suggested that thediagrammatic representation of conditional probability created based on conventional knowledge
is effective for understanding conditional probability.
BACKGROUND AND RESEARCH PURPOSE
Modern society is highly complex and therefore contains a lot of uncertain information. As
a result, probabilistic reasoning in decision-making is becoming increasingly important for citizens.
Probabilistic reasoning refers to a technique of reasoning based on conditional probability, such as“the probability that event B occurs when event A occurs is 50%.” This inference is suitable formodeling events with many uncertainties, unlike logic reasoning that is deterministically derived as“it is B if A.” Therefore, the importance of teaching conditional probability is increasing in schoolmathematics.
Many students find the reduction of the sample space in the calculation process ofconditional probability to be confusing. Ishibashi (2017) examined the difficulties faced by 80
Japanese second-year high school students in learning conditional probability. He found that23.75% of these students were confused by the notations
and. Students set thedenominator as
instead ofwithout paying attention to the condition that event A has
occurred (which corresponds to the denominator of). In this context, reduction of the sample
space means that the sample space is shrunk from whole event U to event A because it is alreadyknown that event A has occurred when obtaining the conditional probability of event B after event
A has occurred. However, it is difficult to reduce the sample space from a linguistic expression of
the conditional probability. It is important for students to understand this concept, and therefore, itis necessary to clarify the appropriate teaching approach for conditional probability.
Of course, some teaching approaches for conditional probability have been developedalready. In fact, diagrammatic expressions such as Euler diagrams are used in high schoolmathematics textbooks in Japan (Takahashi et al., 2011). Larkin and Simon (1987) noted that
diagrammatic representations are more effective than linguistic expressions because they mainly
use spatial position indications in the context of information search and recognition. We think thatthey are also effective in modeling linguistic expressions of conditional probabilities whenreducing the sample space. However, Ishibashi’s (2017) survey results clearly show that they are
not fully used in currently used teaching approaches for conditional probability.
Therefore, in this study, we consider the use of diagrammatic representations as a teachingapproach for conditional probability. We consider the “effect of time axis” that is often examinedin research on the difficulties faced in teaching probability to students. The reason for choosing the“effect of time axis” is that when answering this question, the event after time becomes thecondition for finding the probability of the previous event. However, modeling the linguistic
expression into a probabilistic model is particularly difficult for learners, and this remains adifficulty for this problem (Fischbein and Schnarch, 1997).

In M. A. Sorto, A. White, & L. Guyot (Eds.), Looking back, looking forward. Proceedings of theTenth International Conference on Teaching Statistics (ICOTS10, July, 2018), Kyoto, Japan.
Voorburg, The Netherlands: International Statistical Institute. iase-web.org [© 2018 ISI/IASE]ICOTS10 (2018) Invited PaperIshibashi

EFFECT OF TIME AXIS
The problem of the “effect of time axis” is described as follows:“Yoav and Galit each receive two white and two black marbles.
(A) Yoav extracts a marble from his box and sees that it is white. Without replacing the firstmarble, he extracts a second marble. Is the likelihood that this second marble is also whitesmaller than, equal to, or greater than the likelihood that it is a black marble?
(B) Galit extracts a marble from her box and puts it aside without looking at it. She thenextracts a second marble and sees that it is white. Is the likelihood that the first marble she
extracted is white smaller than, equal to, or greater than the likelihood that it is a blackmarble?” (Fischbein and Schnarch, 1997, p. 99).
The correct answer to this question is “small” (i.e., “smaller than”) for both problems Aand B. A typical misrepresentation is to answer “small” for Problem A but “equal” for Problem B.
In Problem B, it is often noted that “the second result will not affect the first result” and that
because there are two white balls and two black balls initially, the answer given is “equal.”
The “effect of time axis” has been examined frequently in research on probability
difficulties and probability understanding as a problem to which people intuitively give wronganswers. For example, Fischbein and Schnarch (1997) conducted a survey of the “effect of timeaxis” problem with fifth-, seventh-, ninth-, and eleventh-graders in Israel to evaluate their judgmentof probability. We are investigating the actual situation. We found that the proportion who givewrong answers increases with school age, except for college students. Matsuura (2006) conducteda similar survey for fifth graders to university students in Japan and reported that only half of thestudents from any grade gave correct answers.
Fischbein and Schnarch (1997) believed that the difficulty faced with the “effect of timeaxis” arises from the tendency of human thought. They noted that such errors are “intuitively basedmisconceptions” and that it is assumed that later events will not affect previous events.
DIAGRAM
Semiotics based on Peirce’s epistemology is effective for considering the role of symbols
such as diagrams (Hoffmann, 2005). Therefore, we first follow Peirce’s semiotics and outline thediagram and diagrammatic reasoning. Then, we critically consider the present curriculum ofprobability and conditional probability in Japanese school mathematics from the viewpoint ofPeirce’s semiotics. We provide suggestions about diagrammatic expressions that encourage anunderstanding of the linguistic expressions of conditional probability.
Diagram and diagrammatic reasoning
Peirce defines a “diagram” as “a representation which is predominantly an icon of relations
and is aided to be so by conventions. Indices are also more or less used. It should be carried out
upon a perfectly consistent system of representation, founded upon a simple and easily intelligible
basic idea” (CP 4.418, 1903; quotations from Collected papers of Charles Sanders Peirce havebeen expressed as (CP volume number, paragraph number)). Finding and understanding newproperties and using them is called diagrammatic reasoning (Hoffmann, 2005), and it involves the“construction of a diagram” using familiar expressions, rewriting, thought experiments,“experiments,” and “observations” to recognize and understand new properties. This study aims toinfer mathematical models from the linguistic expressions of conditional probabilities throughdiagrams.
Kosslyn (1989) cited the use of conventional knowledge as a component of diagrams for
effective diagrammatic reasoning. Conventional knowledge refers to the rules for the abstraction of
expressive objects that are inherent to each diagrammatic representation. For example, a matrixtype table contains detailed information about combinations of two item sets (Hurley and Novick,2010). If it is necessary to express combinations of two types of item sets for problem solving, it isuseful to express the problem using a matrix-type table in a diagram. On the other hand, if some
information needs to be obtained from the matrix-type table, it is possible to efficiently show the-2-ICOTS10 (2018) Invited Paper

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relationship between the constituent elements by considering what type of conventional knowledge
the diagrammatic representation is based on.
A general diagrammatic representation shows individual components and theirrelationships. It is difficult to obtain accurate information from a diagrammatic representation
without knowing any convention, and therefore, the conventional knowledge of how abstracting
expressions are abstracted is the correct understanding of the diagrammatic representation. This isconsistent with “simple and easily intelligible basic idea” in the definition of the Peirce diagram.
Conventional knowledge on probabilistic diagram
As mentioned above, to efficiently obtain information from a diagrammatic representation,
it is necessary to be able to apply appropriate conventional knowledge for that diagrammaticrepresentation. Thus, effective diagrammatic reasoning is performed. Based on the above, thissection and the next section consider how conditional probability is currently taught in schoolmathematics. In this study, we hypothesize that the diagrammatic representation of conditionalprobability in textbooks does not work well, and therefore, it is important to understand thediagrammatic representation of conditional probability from a semiotics viewpoint.
In Japanese high school mathematics textbooks (mainly from primary school to highschool students of age 15–16 years), Euler diagrams are used for diagrammatic representation ofprobabilities (Takahashi et al., 2011, p. 39 (Figure. 1), p. 44 (Figure. 2), p. 46 (Figures. 3 and 4), p.
48 (Figure. 5)).

Figure. 1: Probability of throwing a single die to give an even number of rolls

Figure. 2: Probability of product event

Figure. 3: Probability of sum events

Figure. 4: Probability of exclusive events

Figure. 5: Probability of complementaryevent

In school mathematics in Japan, Euler diagrams are first presented in “Set andProposition.” We call the set of all things the “whole set,” denote it as “U,” and display it as arectangle. Next, we consider the appropriate set in the whole set. For example, if the whole set
contains all natural numbers, consider a set of multiples of 2, denote it as “A,” and display it as acircle. Hereafter, unless otherwise noted, it is assumed that the whole set is displayed as a rectangle
and all sets displayed as circles show a subset of the whole set (Hashimoto, 2016). Hashimoto(2016) noted that Euler diagrams are considered a type of Venn diagram. Such diagrams provide
students with a simple and easily intelligible representation for conventional knowledge.

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IshibashiStudents can learn probability by using this conventional knowledge. Takahashi et al.
(2011) introduced probability using Figure. 1:

In Figure. 1, the rectangle represents the whole set for the throw of a single die, and thecircle represents the event that an even number of rolls appears. This is consistent with theconventional knowledge mentioned above. Meanwhile, new conventional knowledge that was notin the set is added. That is, “the rectangle represents the denominator in the probability calculation,and the part bounded by the circle inside the rectangle represents the numerator.” Figures. 2–5
show some more diagrammatic expressions. If students use these repeatedly during learning forproblems such as that shown in Figure. 1, they can acquire these diagrammatic expressions asconventional knowledge.
Critical consideration of diagrammatic representation of conditional probability
Figures. 1–5 show conventional diagrammatic representations. However, Takahashi et al.
(2011) used the diagrammatic representation shown below for expressing the conditional
probability (Figure. 6).

Figure. 6: Conditional probability (Takahashi et al., 2011, p. 58)
In Figure. 6, the denominator and numerator in the probability calculation are represented
by the part bounded by the circle inside the rectangle. This is contrary to the definition of Peirce’sdiagrammatic representation that it must be performed on a consistent representation scheme.
Furthermore, considering that conventional knowledge is essential for understanding thediagrammatic representation (CP 4.418; Kosslyn, 1989), this unconventional diagrammaticrepresentation (Figure. 6) may hinder students’ understanding of conditional probability. The
above example suggests that the diagrammatic representation of conditional probability in
textbooks is not one that sufficiently considers conventional knowledge, implying that the diagramused for representing conditional probability in the textbook does not work well. This proves thehypothesis of this study.
TEACHING CONDITIONAL PROBABILITY USING DIAGRAMS
Diagrammatic representation of conditional probability based on conventional knowledge
As described above, the diagrammatic representation of conditional probability in thetextbook does not sufficiently consider the conventional knowledge of probabilistic diagrams.
Therefore, in this section, we develop a diagram of conditional probabilities based on conventional
knowledge and verify its validity through class practice.
Based on conventional knowledge, the conditional probability diagram can be constructed
as follows (Figure. 7).

Figure. 7: Diagrammatic representation of conditional probability based on conventional
knowledge-4-ICOTS10 (2018) Invited Paper

Ishibashi

In Figure. 7, a rectangle represents the denominator () in a probability calculation, anda circle represents the numerator (). In Figure. 6, it is represented as a rectangle for whole
event U, which is a diagrammatic representation of the denominator based on conventional
knowledge. Because whole event U was found to be the subject of consideration as event A wasfound to occur, this was shown in Figure. 7 by a dotted line.
Teaching for use of diagrammatic reasoning
To clarify the role of Figure. 7 (as an external representation) in learning processes, it iseffective to follow diagrammatic reasoning, by which one can explain the development ofknowledge on the basis of a three-stage activity: constructing representations, experimenting withthem, and observing the results (Hoffmann, 2005). If three stages of diagrammatic reasoning areintended, teaching can be designed with the following flow (Figure. 8).

Figure. 8: Process of diagrammatic reasoning based on example shown in Figure. 7
First, because familiar expressions in the composition stage of the diagram refer toexpressions based on conventional knowledge in this paper, we use Euler diagrams to represent theproduct phenomenon (Figure. 2). Here, the sample space refers to event U. In the second
experimental stage, given the condition that, for example, event A occurs, we manipulate thediagram to reduce the sample space. At this time, Figure. 2 is drawn and expressed as shown inFigure. 7. This emphasizes that the sample space has been reduced owing to the information thathas been obtained. After the operation as an experiment, in the third observation stage, based onconventional knowledge, an understanding of conditional probability is encouraged by using therewritten diagram.
Survey summary and results
The survey was conducted with 39 first-grade high school students (age 15–16 years) who
learned conditional probability by using the teaching approach shown in Figure 8. The survey
problem was the “effect of time axis.”
In this class, first, we shared the conventional knowledge of the diagrammatic
representation of probability (the rectangle represents the denominator in the probability
calculation and the part bounded by the circle inside the rectangle represents the numerator). Next,
we presented the following conditional probability problem from the textbook: “Take 1 ball out ofa box containing 9 balls of which five balls are numbered 1, 2, 3, 4, and 5 and four balls arenumbered 6, 7, 8, and 9. What is the probability of getting an odd-numbered ball given theremoved ball is red?” Then, we let the students think about the difference from the probability
problems they have been thinking about thus far, and we demonstrated how the sample space isreduced based on the process shown in Figure 8.
The survey result showed that the ratio of the number of people who correctly answered
both problems A and B was 46%. By contrast, Fischbein and Schnarch (1997) and Matsuura(2006) conducted surveys and reported correct answer rates of 30% (with 20 people) and 36%(with 38 people), respectively. Because Fischbein and Schnarch (1997) did not cover the tenthgrade, which is the first grade in Japanese high schools, it is compared with the survey result forthe 11th grade here. As shown in Figure. 9, in addition to students’ intuition, modeling andprocessing using diagrammatic reasoning helped students process the “effect of time axis,”
resulting in a high correct answer rate.

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Event A is “Thesecond marble is a whitemarble.”
Event B is “The firstmarble is a white marble.”
The probability thatthe first marble is a whitemarble when the second
marble is a white marble is.
The probability thatthe first marble is a black
marble when the second
marble is a white marble is.
Figure. 9: A description given by a student who solved the “effect of time axis” using a diagram(The English sentence on the right in Figure. 9 is an English translation of the student'sdescription.)
CONCLUSION
Under the hypothesis that the diagrammatic representation of conditional probability in thetextbook is not sufficient, this study uses Peirce’s semiotics definition for diagrams to consider adiagrammatic expression that is effective for understanding conditional probability. As a result,first, because the diagrammatic representation in textbooks is not based on conventional knowledgeof probabilistic diagrammatic formulation, it was revealed that it is insufficient for understanding
conditional probability. Second, we developed a diagrammatic representation of conditional
probability based on conventional knowledge and showed that students could use it effectively forunderstanding conditional probability by considering the “effect of time axis” problem as anexample.
This paper focused on conditional probability. In the future, I would like to develop asystematic approach for teaching probability through diagrammatic representations.
REFERENCES
Fischbein, E., and Schnarch, D. (1997). The Evolution with Age of Probabilistic, Intuitively Based
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Hashimoto, K. (2016). Set Operations and Venn Diagrams. Studies in Philosophy, 41, 1-26.
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