International Electronic Journal of Mathematics Education
2022, 17(3), em0690
e-ISSN: 1306-3030

Research Article

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Analyzing experimental and theoretical probabilities in Japanese
7th and 8th grade textbooks
Ippo Ishibashi 1*
Okayama University, Okayama, JAPAN*Corresponding Author: iishibashi@okayama-u.ac.jp
1

Citation: Ishibashi, I. (2022). Analyzing experimental and theoretical probabilities in Japanese 7th and 8th grade textbooks. International
Electronic Journal of Mathematics Education, 17(3), em0690. https://doi.org/10.29333/iejme/12061
ARTICLE INFOABSTRACT

Received: 10 Mar. 2022

Probability is a difficult concept, which is not always taught accurately. This study aims to clarify how experimental
and theoretical probabilities are taught in Japanese 7th and 8th grades through a textbook analysis. We analyzed
seven, government approved Japanese 7th and 8th grade textbooks each. Focusing on the definition andexplanation of experimental and theoretical probabilities and the law of large numbers, we identified eight
discrete perspectives. Findings revealed that, first, some textbooks aim to teach students to distinguish between
experimental and theoretical probabilities, whereas others aim to teach students to be aware of the connection
between the two. Second, after explaining theoretical probability, some textbooks asked questions to clarify thescope of theoretical probability, aiming to teach students to distinguish between the two probabilities. Third, in
explaining the law of large numbers, textbooks do not adopt the perspective that experimental probability
converges on true probability, but that it converges on theoretical probability.

Accepted: 20 Apr. 2022

Keywords: experimental and theoretical probability, true probability, law of large numbers, teaching probability
in Japan, 7th and 8th grade, textbook analysis

INTRODUCTION
As illustrated by the spread of COVID-19, modern society is uncertain. The importance of studying probability is increasing sothat people can quantify uncertainty and make decisions in all situations, including in everyday society and professional situations
where chance is present (Batanero & Chernoff, 2018; Batanero et al., 2016). However, Konold et al. (2011) and Stohl (2005) reported
that the experimental and theoretical probability pedagogy is not always accurate in school mathematics.
In Japan, experimental probability is taught in the 7th grade (1st grade in lower secondary school in Japan, from 12 to 13 years
old) and theoretical probability in the 8th grade (2nd grade in lower secondary school in Japan, from 13 to 14 years old) (Ministry of
Education Culture, Sports, Science and Technology [MEXT], 2018a). In Japan, a new course of study has been in effect since April
2021. It has been determined as the broad standard for all schools, from kindergarten through upper secondary, for organizing
their programs ensuring a fixed standard of education throughout the country. However, whether the course is being followed
accurately. Considering that understanding experimental and theoretical probabilities is difficult (e.g., Kazak et al., 2011;
Prodromou, 2012) and that effective intervention by teachers is essential for students to understand the relationship between
them (Ireland & Watson, 2009), it is important to clarify how experimental and theoretical probabilities are taught in Japan.
Thus, this study aims to clarify how experimental and theoretical probabilities are taught in Japanese 7th and 8th grades using
a textbook analysis approach as textbooks play an intermediate role between the intended and the implemented curricula(Valverde et al., 2002). If the Japanese course of study, which is the intended curriculum, were analyzed, it would not fully capture
the real-world practices of teaching probability because the level of abstraction is too high. Alternately, if we analyze a class, which
represents the implemented curriculum, it will not fully capture the actual teaching practices of probability because they arehighly dependent on the competence of individual teachers and students. Therefore, textbooks, which are a part of both intended
and implemented curricula, were used for analysis.

LITERATURE REVIEW
Borovcnik and Kapadia (2014) state that classical a priori theory, frequentist theory, and subjectivist theory are central to thenature of probability, and thus, are relevant for school mathematics. In the following literature review, first, we provide an
Copyright © 2022 by Author/s and Licensed by Modestum. This is an open access article distributed under the Creative Commons Attribution License which permits
unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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overview of probabilities of frequentist theory (experimental probability 1) and classical a priori theory (theoretical probability 2),which are taught in Japan. Next, we provide an overview of true probability which is estimated through experimental andtheoretical probabilities, followed by an overview of the relationship between the three probabilities.
Experimental Probability
The frequentist theory defines experimental probability. According to Gillies (2000), in frequentist theory, probability is
associated with events or a set of elements, is objective, and is independent of the individual who evaluates the probability, justas the mass of a thing is independent of the measurer in mechanics.
In the Japanese course of study, the aims and goals of probability in the first grade of lower secondary school (7 th grade) areset, as follows:
To teach students to acquire the following matters regarding the possibility of uncertain events through mathematicalactivities.
a. To acquire the following knowledge and skills:(i). To understand the necessity and meaning of experimental probability.
b. To acquire the ability to think, judge, and express the following:(ii). To read and express the tendency of the possibility of uncertain events based on the results of many observations
and trials (MEXT, 2018a, p. 68).
In the commentary on the Japanese course of study (MEXT, 2018b) prepared by lower secondary school teachers, university
teachers, and senior specialists for the curriculum, experimental probability is explained, as follows:
For example, consider the case where you throw a plastic bottle cap and find out how many times it lands face up. Let us
increase the number of times n throws the cap, find the number of times r lands face up, and calculate the value of the𝑟𝑟
relative frequency that it lands face up. As n is gradually increased, r also increases, but the value of gradually converges𝑛𝑛𝑟

on a certain value. This constant value that converges on is called the probability of throwing the plastic bottle cap and𝑛it landing face up. (MEXT, 2018b, p. 93)
Theoretical Probability
Theoretical probability has been defined in the classical a priori theory. According to Gillies (2000), in classical a priori theory,
probability cannot be inherent in objective properties but is rather correlated to human ignorance. With respect to three or more
events, we regard each as equally possible when there are no conditions to convince us that one of them occurs in preference tothe others. If there are n possible cases and m of them are favorable for outcome A to occur, then the probability of A occurring is𝑚defined as .
𝑛

In the Japanese course of study, the aims and goals of probability in the second grade of lower secondary school (8 th Grade)are set as follows:
To teach students to acquire the following matters regarding the possibility of uncertain events through mathematicalactivities.
a. To acquire the following knowledge and skills:(i). To understand the necessity and meaning of theoretical probability in relationship to experimental probability.
(ii). To calculate probability for simple cases.
b. To acquire the ability to think, judge, express, and so on, as follows:(i). To focus on equally likely events, and to consider and express how to calculate the theoretical probability.
(ii). To grasp uncertain events using probability, and to consider and express them (MEXT, 2018a, p. 71-72).
In the commentary on the Japanese course of study, theoretical probability is explained, as follows:
For example, if we assume that there are six possible ways to throw a die and that it is equally likely that the die will land1
on any side, we find that the theoretical probability that it will land on any side is (MEXT, 2018b, p. 123).
6

1
2

Experimental probability is sometimes referred to as empirical probability or objective probability (Chernoff, 2008).
Theoretical probability is sometimes referred to as model probability (Pfannkuch & Ziedins, 2014).

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True Probability
Experimental probability is defined as the ratio of the number of favorable trials to the total number of trials (Konold et al.,2011; MEXT, 2018b). Defined in this way, experimental probability is bound to change with each sample (Konold et al., 2011).
Theoretical probability is the probability assigned under a given model. For example, assuming the complexities of physics and1
the apparent (and probably imperfect) symmetry of a die, the probability of rolling a “four” is (Stohl, 2005). Therefore,
6
experimental and theoretical probabilities are estimates of something—of the actual but ultimately unknowable, probability ofthe outcome of interest (Konold et al., 2011; Stohl, 2005). Konold et al. (2011) refer to this probability as the “true probability.”3
The Connection Between Experimental and Theoretical Probability
The perspective that experimental probability converges on theoretical probability
Experimental and theoretical probabilities have been taught in many countries, including Japan (Batanero & Borovcnik, 2016),
and the relationship between them is often debated. Experimental and theoretical probabilities are connected by the law of large
numbers (Batanero et al., 2016). For example, in the commentary on the Japanese course of study, this has been described, as
follows:
For example, if we assume that there are six possible ways to throw a die and that it is equally likely that the die will land1
on any side, we find that the theoretical probability is . If we throw the die many times, the ratio of the number of times
61each side appears tends to stabilize at for each side. Thus, when all possible cases are equally likely, it is known that the6experimental probability converges on the theoretical probability as the number of trials increases (the law of large
numbers) (MEXT, 2018b, p. 123).
1

We should pay attention to the meanings of “convergence” and “close” (Stohl, 2005). For example, given a sequence 𝑥𝑛 = ,𝑛there is only one value it can take. The convergence point is determined to be 𝑥∞ = 0. However, when we consider the example
of repeated coin tossing, this is not easy. If we place the random variable corresponding to the 𝑖-th coin toss as 𝑋𝑖 , the random
𝑋 +⋯𝑋𝑛variable 𝑋𝑖 can take two values, 0 or 1. In this case, the average percentage of n coin tosses that result in heads puts 𝑌𝑛 = 1.
𝑛1

When n is sufficiently large, the ratio of the number of tosses that results in heads 𝑌𝑛 seems to converge on , but there is still the2possibility that all coin tosses result in heads and 𝑌𝑛 = 1. The implication of the law of large numbers is that experimental
probability rarely differs significantly from theoretical probability if several trials are conducted (Stohl, 2005). Stohl (2005) argues
that we should pay attention to the teachers’ and the textbooks’ explanations. He picks examples of explanations from the
textbooks to make his case: “With a large number of trials, the experimental probabilities will converge on the theoreticalprobability” and “the experimental probability approaches, or gets closer and closer to, the theoretical probability” (Stohl, 2005,
p. 348). Ireland and Watson (2009) report that many students continue to believe that fairness means obtaining the expecteddistribution of outcomes in the short term.
The perspective that experimental probability converges on true probability
Konold et al. (2011) and Pfannkuch and Ziedins (2014) argue against the perspective that experimental probability converges
on theoretical probability. Konold et al. (2011, p. 83) argue that: “From this perspective, it is misleading and incorrect to claim thatdata from throwing a real die will converge on the theoretical probability, which is how many curricula currently portray it.” This
perspective maintains that experimental and theoretical probabilities are positively considered to be estimates of true probability.
We explain these three probabilities using a fair coin as an example with reference to Pfannkuch and Ziedins (2014). First, the1
theoretical probability of a coin coming up as heads is . However, because there is a slight bias in the actual coin, the true
2

1

probability is not exactly . If we assume that each trial is an independent trial and that whenever a coin is tossed, it has the same2
probability of showing up heads, the true probability of tossing the coin and it coming up heads can be estimated by theexperimental probability, which is the proportion of the coin coming up heads over many trials. Thus, if we can relate the1theoretical probability to the experimental probability, we can correctly estimate the true probability (which converge on but
2

1

not exactly ). However, if one fails to properly recognize the relationship between the three probabilities and confuses, for2
example, the theoretical probability with the true probability, one might estimate the true probability incorrectly, such as1
considering the true probability to be exactly .
2

Accordingly, Konold et al. (2011) state that empirical probability converges only on true probability and converges ontheoretical probability only if it happens to be a perfect model of the coin. Furthermore, they argue that activities that require
students to explore both experimental and theoretical probabilities of throwing a die or tossing a coin should be avoided because
they obscure what they really want to know, which is true probability.
Research Questions
Experimental and theoretical probabilities are different. They are connected by the law of large numbers. In this case, we needto pay attention to the meanings of “convergence” and “close.” In addition, if we consider that experimental and theoreticalprobabilities are estimates of true probability, then experimental probability converges on true probability, and the discourse that3

Konold et al. (2011) and Pfannkuch and Ziedins (2014) call it true probability, while Stohl (2005), for example, calls it the actual probability.

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Table 1. Textbooks analyzed in this paper
Textbook
Fujii et al. (2021a)
Fujii et al. (2021b)
Ikeda et al. (2021a)
Ikeda et al. (2021b)
Okabe et al. (2021a)
Okabe et al. (2021b)
Okamoto et al. (2021a)
Okamoto et al. (2021b)
Sakai et al. (2021a)
Sakai et al. (2021b)
Shigematsu et al. (2021a)
Shigematsu et al. (2021b)
Yoshida et al. (2021a)
Yoshida et al. (2021b)

Grade
7
8
7
8
7
8
7
8
7
8
7
8
7
8

Label in this paper
Textbook 7a
Textbook 8a
Textbook 7b
Textbook 8b
Textbook 7c
Textbook 8c
Textbook 7d
Textbook 8d
Textbook 7e
Textbook 8e
Textbook 7f
Textbook 8f
Textbook 7g
Textbook 8g

experimental probability converges on theoretical probability is incorrect. Based on the above, our research questions (RQs) are
as follows:(RQ1) In Japan, are experimental and theoretical probabilities taught with the awareness that they have different
probabilities?
(RQ2) Which perspective is employed in teaching probability in Japan, “experimental probability converges on theoreticalprobability” or “experimental probability converges on true probability”?

METHODOLOGY
To analyze the explanations in textbooks, this study used content analysis—one of the textbook analysis approaches—because
it is the primary research method employed in mathematics education research for the analysis of textbook explanations (Rezat& Sträßer, 2015). Krippendorff (2004) defined this method as “a research technique for making replicable and valid inferences fromtexts (or other meaningful matter) to the contexts of their use” (p. 18).
In Japan, the course of study is revised about once every 10 years. The current Japanese course of study for lower secondary
schools was implemented in April 2021. It stipulates considerations for the overall curriculum and the number of class hours, while
also broadly specifying the goals, content, and handling of content for each subject. Textbooks are created based on them. For 7th
and 8th grade textbooks, publishers collaborate with university teachers and lower secondary school teachers to produce
textbooks. The textbooks are then submitted to the MEXT for approval, and only those that are accepted can be published as
MEXT-approved textbooks4. In 2021, seven MEXT-approved textbooks were published for 7th and 8th grade mathematics each5. In
this study, all 14 textbooks (Fujii et al., 2021a, 2021b; Ikeda et al., 2021a, 2021b; Okabe et al., 2021a, 2021b; Okamoto et al., 2021a,
2021b; Sakai et al., 2021a, 2021b; Shigematsu et al., 2021a, 2021b; Yoshida et al., 2021a, 2021b) that have passed the MEXT
assessment were analyzed (Table 1).
We are interested in the explanation of experimental and theoretical probabilities and the law of large numbers. Therefore, wefirst examined the relevant explanations in all textbooks for each of these concepts. To examine experimental probability, theapplicable sections in which experimental probability is first explained in the 7 th grade textbooks were examined. In addition,
because Japanese textbooks require students to reflect on previous, related, learning content before starting new content, we
also examined applicable sections in which students reflect on experimental probability learned in the 7 th grade when studying it
again in the 8th grade textbooks. Next, we examined applicable sections in which theoretical probability was first explained in the
8th grade textbooks. These findings correspond to RQ1. Then, we examined applicable sections in which the law of large numbers
was explained after the explanation of theoretical probability in the 8th grade textbooks; this corresponds to RQ2.
As a result, eight analytical perspectives were created (Table 2). Krippendorff (2004, p. 180) mentions the “application ofstandards” in applying the analytical structure. In this case, our standards for creating the textbook analysis perspectives andjudging their relevance were based on our professional mathematical knowledge and their consistency with the RQs, as seen in
Stacey and Vincent (2009).

4
5

For information on courses of study and textbook certification in Japan, see MEXT (n. da, n. db).
The author of this paper was not involved in the preparation of any textbook.

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Table 2. Eight perspectives on textbook analysis
No
1
2
3
4
5
6
7
8

Perspective
Subject of explanation of experimental probability
What to call experimental probability
Teaching materials to explain theoretical probability (i.e., die, coin, playing card)
What to call theoretical probability
Question about the probability of a fair die
Explanation of the law of large numbers
Question on the law of large numbers
Teaching true probability

Table 3. Results of throwing a plastic bottle cap (Textbook 7f, p. 249)
Number of times thrown (times)
Number of times facing forward (times)
Number of times turned sideways (times)
Number of times facing backward (times)
Relative frequency of face-up

20
7
5
8
0.350

40
16
7
17
0.400

60
21
7
32
0.350

80
26
8
46
0.325

100
30
11
59
0.300

200
57
19
124
0.285

400
98
36
266
0.245

600
151
50
399
0.252

800
205
65
530
0.256

1,000
260
77
663
0.260

Figure 1. Example of a graph of relative frequency (The vertical axis represents its value and the horizontal axis represents thenumber of times the plastic bottle cap was thrown. The experimental probability of landing face up is approximately 0.2)

RESULTS
Explanation of Experimental and Theoretical Probabilities in Japanese Textbooks (RQ 1)
In this subsection, we analyze the explanation of experimental and theoretical probabilities in Japanese textbooks to examine
RQ 1.
Experimental probability
In five 7th grade textbooks, the experiment of throwing a plastic bottle cap is used as teaching material. For the other two, an
experiment of throwing a Shogi piece is used in Textbook 7d and an experiment of throwing a die is used in Textbook 7g. In all
textbooks, the process of explaining experimental probability begins with a table summarizing the results of the experiment (seeTable 3). Next, the results are shown in a graph, where the horizontal axis represents the number of throws and the vertical axis
represents the relative frequency (see Figure 1). Based on the table and graph, it is explained that when the number of throws issmall, the variation in the relative frequency is large; however, when the number of throws is large, the variation becomes small,and the relative frequency converges on a certain value. Although it differs from textbook to textbook, the number of throws fell
between 1,000 and 3,000. Subsequently, experimental probability is defined. For example, the explanation written in Textbook 7c
is as follows:
As the number of times the plastic bottle cap is thrown increases, we find that the relative frequency of it landing face-up
converges on a constant value of 0.190. This value is thought to indicate the degree to which the event of landing face-up
is likely to occur. When making an experiment or observation, the number that represents the degree of possibility of acertain thing occurring is called the probability of that thing occurring (Textbook 7c, p. 246) (emphasis in original).
Textbook 7g, which illustrates experimental probability through the number of times a die is thrown and a 3 is obtained, showsa table similar to Table 3, summarizing the results of the experiment, and a figure similar to Figure 1, summarizing the change inthe relative frequencies. The textbook states,6 / 11Ishibashi / International Electronic Journal of Mathematics Education, 17(3), em0690

Table 4. Results of throwing a plastic bottle cap (Textbook 8a, p. 249)
Number of times thrown
50
100
200
400
600
800
1,000
1,200
1,400
1,600
1,800
2,000

Number of times obtaining a 1
7
13
32
70
89
125
165
202
239
269
299
334

Relative frequency of obtaining a 1
0.140
0.130
0.160
0.175
0.148
0.156
0.165
0.168
0.171
0.168
0.166
0.167

As the number of times a die is thrown increases, the relative frequency at which a 3 is obtained gradually converges on acertain value. The following table shows the results. The constant value can be thought of as a number that expresses thedegree of possibility of obtaining a 3. In this case, the value is approximately 0.167. Thus, when conducting an experimentor making an observation, the number that represents the degree of possibility of a certain thing occurring is called theprobability of that thing occurring (Textbook 7g, p. 255) (emphasis in original).
As observed from the explanations in Textbook 7c and 7g, experimental probability is defined as probability, not experimental
probability. The same is true for the other textbooks.
After experimental probability is defined, questions evaluating students’ understanding of it are presented. In Textbook 7c,
7d, 7e, 7f, and 7g, for example, a table showing the number of births by sex per year in Japan is presented, and questions are asked
to find the probability that a “girl is born” in Japan based on this table. In Textbook 7c, an experiment wherein the crown cap of abottle is thrown 1,000 times is presented, with a table presenting the number of times the head of the crown appears for each 100
times; the question is to find the probability that a head appears. In Textbook 7a, a problem wherein students find the probability
that a certain direction of a plastic bottle cap will appear is presented.
In all 8th grade textbooks, experimental probability is reviewed with the experiment of throwing a die. In Textbooks 8a, 8f, and8g, a graph and a table of the relative frequencies resulting from many throws of a die are shown. Students are asked to determine
the experimental probability of a certain outcome. For example, based on Table 4, Textbook 8a asks about the probability ofobtaining a 1 when throwing a die. In other textbooks, a graph and a table of relative frequency, similar to Figure 1, Table 3, andTable 4, and sample results of calculating the experimental probability are shown, such as: “Therefore, the probability of obtaininga 3 is considered to be 0.17” (Textbook 8b, p. 180); “Therefore, the probability of obtaining a 1 is about 0.167” (Textbook 8c, p.
188); “From the experiment on the previous page, we can find that the probability of obtaining a 1 is about 0.167, which converge
1on ” (Textbook 8d, p. 160); and “For example, we can assume that the probability of obtaining a 1 is about 0.167” (Textbook 8e,
6p. 184).
Theoretical probability
As mentioned already, in all 8th grade textbooks, a die-throwing experiment is used as teaching material to explain
experimental probability. After determining the experimental probability, the textbooks illustrate students saying: “Isn’t itpossible to find the probability from the fact that there are six faces without making an experiment?” (Textbook 8c, p. 188) or “It’s
hard to find the probability by making an experiment” (Textbook 8d, p. 159). Next, the motive for finding the probability without
an experiment is presented, and theoretical probability is explained. For example, in Textbook 8c, the following explanation is
given:
When a correctly made die is thrown, the results are 1, 2, 3, 4, 5, and 6. What occurs in each case is equally certain. In suchcases, each case is said to be equally likely to occur. When each case is equally likely to occur, the probability can be found,as follows: Suppose there are n possible cases in an experiment or observation, wherein each case is equally likely to occur.
𝑎
If there are a cases in which event A occurs, then the probability p that A occurs is 𝑝 = (Textbook 8c, p. 189) (emphasis in𝑛original).
Textbook 8e explains as follows:
When a correctly made die is thrown, a certain number is not considered particularly likely to appear. Therefore, equal
certainty can be expected for any of the numbers to appear. In such cases, that a number from 1 to 6 will appear on the die
is said to be equally likely to occur. Based on this idea, we can find the probability that the number 1 will appear, as follows.
[1] There are six possible outcomes and it is equally likely that any of them will occur.
[2] There is only one case in which the number 1 appears.

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1[3] From [1] and [2], the probability that the number 1 appears is (Textbook 8e, p. 184).
6

As observed from the explanations in Textbooks 8c and 8e, theoretical probability is defined as probability, not theoreticalprobability. The same is true for the other textbooks.
After defining theoretical probability, in some textbooks (Textbook 8c, 8d, and 8f), cards or die are used to find probability, but
not in others. In the former, for example, “Find the following probabilities when throwing a single die. (i) the probability of getting
a 4, (ii) the probability of getting an even number, and (iii) the probability of getting a number less than or equal to 2” (Textbook
7d, p. 190) is asked. In the latter, a picture of an unfair die is shown and the following question is asked: “For a die shaped like the1
one on the right, can you find that the probability of obtaining a 1 is ? Also, explain why you think so” (Textbook 8a, p. 163).
6

Explanation of the Law of Large Numbers and True Probability in Japanese Textbooks (RQ 2)
In this subsection, we analyze the explanation of the law of large numbers and true probability in Japanese textbooks for
examining RQ 2.
The explanations for the law of large numbers can be divided into three patterns. First, there are explanations about theconvergence of experimental probability on theoretical probability. Second, there are questions regarding the law of large
numbers. Third, there are neither explanations nor questions.
In Textbooks 8b and 8e, the convergence of experimental probability on theoretical probability is explained. In Textbook 8b,
1
it is explained that the probability of obtaining any number from 1 to 6 on a die is for each of them. Furthermore, in Textbook 8b,
61

it is explained that “0.17, which is the probability of obtaining a 3 in the experiment on the previous page is almost equal to ”
1

6

(Textbook 8b, p. 181). In Textbook 8e, it is explained that the probability of getting a 1 on a die is . In Textbook 8e, it is explained
61

that: “This value of is almost identical to the experimental probability of 0.167” (Textbook 8e, p. 184).
6

Next, Textbooks 8a, 8b, 8d, and 8f include questions about the law of large numbers. For example, Textbook 8a has thefollowing question:
When throwing a die, Mr. A thought about the probability of obtaining a 1 as follows: Do you think this idea is correct?
1

The probability of getting a 1 is , so, if I throw a die six times, at least one of them will be 1 (Textbook 8a, p. 164).
6

Textbook 8b has the same question. In Textbook 8f, students are also asked to select the correct statement out of five1
sentences describing how to roll a die with a probability of obtaining a 1:
6

1

There is a die for which the probability of obtaining a 1 is .
6

Choose the correct answer from [a] to [e] that describes how this die rolls.
[a] If you throw the die six times, one of the trials will always roll a 1.
[b] If 1 has not appeared even once after 5 throws, it will always appear on the 6th throw.
[c] After 6 throws, numbers from 1 to 6 will always appear once.
[d] If you throw 30 times, 1 will always appear 5 times.
[e] If you throw the dice 6,000 times, 1 will appear about 1,000 times (correct answer; Textbook 8f, p. 177).
In Textbook 8d, it is asked whether the following statement is correct or not: “When taking out a ball from a box containing
four red balls, two yellow balls, and three blue balls, and putting it back 900 times, you will always get a red ball 400 times.”
Textbooks 8c and 8g did not explain the law of large numbers.
None of the textbooks included an explanation that experimental probability may not converge on the value of theoreticalprobability even when the number of trials is large, as noted by Stohl (2005). According to Stohl’s (2005) assertion about the lawof large numbers, if a coin is tossed 10,000 times, it would be extremely rare for it to land face up each time, but not entirely
impossible. Moreover, none of the textbooks included explanations about true probability, such as experimental and theoreticalprobabilities are estimates of true probability.

DISCUSSION
In this section, based on the results of the textbook analysis in the previous section, we will answer the RQs.

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The Relationship Between Experimental and Theoretical Probabilities
RQ1 is “In Japan, are experimental and theoretical probabilities taught with the awareness that they have different
probabilities?”
First, in the explanation of experimental probability in all textbooks for the 7 th and 8th grade, except Textbook 7g, teachingmaterials for which theoretical probability could not be found, such as a plastic bottle cap, are used. Thus, it is being taught thatthe probability of an event can only be determined through experimental probability, which implies that experimental andtheoretical probabilities are taught as different probabilities (Konold et al., 2011).
Next, in the explanations of experimental probability in textbooks for 8 th grade, a die is used as teaching material, with anawareness of the connection with theoretical probability that follows in all textbooks. The experimental results are showngraphically, and then the theoretical probability is explained. If we can have students perform this activity, they can conduct alearning activity that is consistent with the results of Aspinwall and Tarr (2001) and Stohl and Tarr (2002). Accordingly, students
can deepen their understanding of the role of sample space and number of trials in the relationship between experimental andtheoretical probabilities by conducting their own trials and expressing the results graphically. However, the explanation ofexperimental probability was different in one textbook, which is, as follows: “From the experiment on the previous page, we can1find that the probability of obtaining a 1 is about 0.167, which converge on ” (Textbook 8d, p. 160). We believe this is not an
6appropriate explanation because students who have only learned experimental probability do not necessarily think that the1
probability of obtaining a 1 converge on after they find the probability to be approximately 0.167. This can be considered an
6
explanation that will lead to the learning of theoretical probability later on. However, before learning theoretical probability, it is1
preemptive for students to think that the probability of obtaining a 1 on a die converge on after they found the probability to be6

approximately 0.167. Therefore, we believe that the explanation in Textbook 8d may cause students to confuse experimental
probability with theoretical probability.
In the explanation of theoretical probability for the 8th grade, after defining theoretical probability, a picture of a die that was
not fair is presented, and the question of whether theoretical probability could be used to determine probability is asked in some
textbooks. This question is intended to clarify the scope of theoretical probability; therefore, it can be said that textbooks havebeen written with the awareness that experimental and theoretical probabilities are different.
The Perspective About Converges of Experimental Probability
RQ2 is “Which perspective is employed in teaching probability in Japan, “experimental probability converges on theoreticalprobability” or “experimental probability converges on true probability”?
The results of the analysis showed that true probability was not taught in teaching probability in Japan. We believe there are
two reasons for the same. First, it is not explicitly stated that experimental and theoretical probabilities are estimates of trueprobability in textbooks. If textbook authors are actively considering this, they are expected to clarify this in writing. Second, all
textbooks have developed teaching that converges on the probability of a certain value (e.g., 1) on a die from both experimental
and theoretical probabilities. Konold et al. (2011) state that such teaching is not suitable for teaching true probability. It causes
students to equate theoretical probability with true probability, obscuring the idea that what they really wish and strive to know
is the true probability (Konold et al. 2011).
The explanation of the law of large numbers differed among textbooks. First, although the explanations in Textbooks 8b and8e are not incorrect, they are somewhat oversimplified as pointed out by Stohl (2005). Based on Stohl’s (2005) point, it is suggested
that the explanation of “almost” “almost equal” (Textbook 8b, p. 181) and “almost identical” (Textbook 8e, p. 184) needs to beenhanced.
Next, Textbooks 8a, 8d, and 8f have questions about the law of large numbers, although they do not explain “converging,”“closing,” or “corresponding” as Textbooks 8b and 8e did. In these questions, it can be considered that the goal is for students tounderstand that relative frequency does not converge on theoretical probability in a small number of trials and that even in a large
number of trials, relative frequency does not exactly match theoretical probability. In light of this, we believe that problems related
to the law of large numbers provided in the textbooks aim to help students understand the meanings of “closing” and “close”
through problem solving. Thus, we can consider that Textbook 8b is attempting to help students understand the meaning of“almost” through problem solving in relation to the law of large numbers.
However, no explanation was given for the fact that experimental probability may not converge on the value of theoreticalprobability even when the number of trials is large. This implies that the textbook does not correctly explain the law of large
numbers. Alternatively, it is possible that rigorous explanations of something that rarely happens could interfere with students’
understanding of the law of large numbers, and thus, the textbooks do not strictly explain it.
The discussion is summarized as shown in Table 5.

CONCLUSION
The purpose of this study was to clarify how experimental and theoretical probabilities are taught in Japanese 7th and 8th
grades by employing textbook analysis. We analyzed all Japanese 7th and 8th grade mathematics textbooks that were written basedon the Japanese course of study implemented in April 2021 and approved by the MEXT. Regarding the explanation of experimental
probability, some textbooks are written to teach students to distinguish between experimental and theoretical probabilities, whileIshibashi / International Electronic Journal of Mathematics Education, 17(3), em0690

9 / 11

Table 5. Summary of textbook analysis and discussion
Textbook
(1)
(2)
(3)
(4)
7a, 8aPlastic bottle cap Probability
Die
Probability
7b, 8b
Plastic bottle cap Probability
Die
Probability
7c, 8cPlastic bottle cap Probability
Die
Probability
7d, 8dShogi piece
Probability
Die
Probability
7e, 8ePlastic bottle cap Probability
Die
Probability
7f, 8fPlastic bottle cap Probability
Die
Probability
7g, 8gDie
Probability
Die
Probability
Note. From (1) to (8) are perspectives on textbook analysis (Table 2)(5)
Included
Included
Not included
Not included
Included
Not included
Included(6)
Not included
Included
Not included
Not included
Included
Not included
Not included(7)
Included
Included
Not included
Included
Not included
Included
Not included(8)
Not included
Not included
Not included
Not included
Not included
Not included
Not included

others taught them to be aware of experimental probability converges on theoretical probability. Regarding the explanation oftheoretical probability, some textbooks have questions intended to clarify the scope of theoretical probability and are writtenwith the goal of teaching students to distinguish between experimental and theoretical probabilities. Regarding the explanationof the law of large numbers, the perspectives of Konold et al. (2011) and Pfannkuch and Ziedins (2014) that experimental
probability converges on true probability were not taught. Those textbooks that do have explanations and questions about thelaw of large numbers based on the perspective that experimental probability converges on theoretical probability, however, haveinadequate explanations of the law of large numbers.
The significance of this study is twofold. First, we discussed the Japanese course of study and the textbooks that were
implemented in April 2021. Since the course of study is revised every 10 years, this study can serve as a basis for teachingprobability in Japan for the next nine years. It is also expected to make an impact on the next revision of the Japanese course of
study and textbooks. Second, we discussed the teaching of probability in Japan, in English. Batanero et al. (2016) reviewed theteaching of probability worldwide; however, Japan was not included in their review. We believe one of the reasons is that research
published in English regarding the teaching of probability in Japanese schools is scarce. Therefore, given that this study will be
published in English, there is a chance that the teaching of probability in Japan will now be included in the global probability
teaching research.
A future task is to clarify how the differences in textbooks revealed in this study affect teachers and students. We plan to
conduct classroom observations at lower secondary schools using different textbooks and compare the results. Finally, it should
be noted that this study was not written with the intent of recommending any particular textbook to learners or teachers; instead,they were analyzed and discussed from a neutral standpoint
Funding: This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant (No. 20K22188).
Acknowledgements: The author would like to thank Editage (www.editage.com) for English language editing.
Declaration of interest: No conflict of interest is declared by the author.

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