Case analysis

58

Table 4-2. The result of classification of the contexts within the problems Types of Context Japan（309 problems） New Zealand (262 problems）

Sports 69（22.3%） 13（5.0%）

Scientific 234（75.7%） 123（46.9%）

Social 3（1.0%） 75（28.6%）

Artistic 0（0%） 0（0%）

Economic 3（1.0%） 46（17.6%）

Political 0（0%） 5（1.9%）

and 80 problems had contexts related to tests. Therefore, about 80% of the total is occupied by the three contexts of sports, weather, and tests. In contrast, no problems in New Zealand have artistic context, but the remaining five kinds of contexts occur more evenly than in Japanese textbooks. In addition, although scientific contexts occur the most, a wide variety of subjects were included. Thus, in Table 4-1, the author found that problems have contexts at the roughly same proportion in Japan and New Zealand, but if we focus on the types of contexts, there is a difference in terms of types of context between two countries. In Japanese statistics education, there is large concentration of types of context such as sports, weather, and tests.

59

First, among the (Yes, Yes, Yes) problems, the only problem which requires decision making in Japan is the following (Matano et al., 2017, p. 184; translated by the author).

【Problem】

In a basketball club, you must select one among the three players A, B, and C to play the next game. You decided to analyse the characteristics of the three players using their performance data to appoint the player. The following table shows the scores of the three players in the last ten games, arranged in ascending order.

A 10 14 16 16 16 18 18 22 24 26

B 4 6 10 14 14 20 24 28 30 30

C 12 14 14 16 16 20 20 22 22 24

From these data, let’s draw a boxplot … for each of the three players A, B, and C. In addition, let’s consider what can be said about the characteristics of each player from the results. If you are the coach, which player will you select?

This problem is intended to elicit students’ inquiries based on the following three characteristics: the data of the three players A, B, and C, which have the same mean and almost the same median, the boxplot of player B is longer than the boxplots of players A and C, and the boxplot of player C is shorter than the boxplots of players A and B. Expected answers include ‘The average is the same for all players, but player C had the most stable score, so I will select player C’ or ‘The average is the same for all players, but player B’s probability of scoring 30 points is highest from this data, so I select player B’. There is no one correct answer regarding which player to choose and the emphasis is placed on the

60

explanation of why the player is selected. However, this problem contains the instruction that students calculate the mean, first quartile, median, third quartile, range, and interquartile range of each player’s data, draw each boxplot, and judge the decision ‘based on those calculation and figure’. Thus, at this stage, students’ decision making is based not on contextual knowledge but on statistical knowledge. In this sense, knowledge to be used in this problem is limited.

In the past, a problem which contained contextual knowledge in the inquiry and answer could be seen in the unit ‘Analysis of Data’ (Okamoto et al., 2014, p. 202; translated by the author).

【Problem】

When upper secondary school student A was at a lower secondary school, he was investigating, as his research project over summer vacation, the number of sea bathers in July in Fujisawa City, Kanagawa Prefecture, where he lived. The following bar graph shows the records for 2006 and 2007 based on the results of his survey.

He analysed this graph and concluded that the number of sea bathers in Fujisawa City decreased sharply in 2007 over 2006. Let us discuss the correctness of his results based on the graph.

61

There are three main answers expected for this problem. The first is to make decisions after numerical processing. By calculating the difference between the numbers of sea bathers in 2006 and 2007 and dividing the result by 31, it is possible to estimate the difference per day. The second answer is ‘There are only two data points for 2006 and 2007, and I cannot judge whether it is correct without checking the number of people in other years’. Finally, the third is an answer focusing on factors besides the number of people, such as ‘There may have been a special event in 2006, so I cannot make a decision without looking at the number of people per day’ or ‘There is a possibility that there were a lot of rainy and cold days in 2007, so I cannot make a decision without looking at data about the weather’. While both the first and second answers focus on the number of people, the third answer is related to an inquiry into the reason for the difference in the numbers of people between 2006 and 2007. Under this perspective, the consideration has shifted from numerical points to various contexts (events, weather, and so on) in each year. In this way, the author presented a problem which includes not only statistical knowledge but also contextual knowledge.

Next, the following problem is one of the (Yes, Yes, Yes) problems in New Zealand which include contextual knowledge within the inquiry or answer (Barton, 2010, pp. 409-410).

【Problem】

Read the following media article about a poll on who should succeed the Queen.

Charles and William evens for throne

by Kara Segedin

As Prince William prepares to leave New Zealand, a poll of the Herald reader panel shows the 27-year old in a neck-and-neck race with his father as the popular choice to succeed the Queen.

The survey – taken before the Prince’s recent three-day tour – found 33.3% wanted Prince Charles to be the next monarch, with 30.2% favouring William. But 29.4% of respondents preferred a republic in

62 the event Queen Elizabeth II died or abdicated.

Women aged 18 to 44 and men aged 18 to 29 would prefer Prince William as the next head of state, while men aged 45 to 59 were particularly keen on New Zealand becoming a republic.

Almost half of the supporters of a republic – 49.1% – had no opinion on who should be the first head of state. Former Governor-General Dame Silvia Cartwright was the top choice, with 15.5% backing.

The online survey of the Herald Readers’ Panel was conducted by the Nielsen Company between December 10 and 17.

(Source: The New Zealand Herald, 19 January 2010, p. A3)

Poll results

When Queen Elizabeth II dies or abdicates, who should replace her as New Zealand’s head of state?

・ Prince Charles 33.3%

・ Prince William 30.2%

・ Another royal 1%

・ NZ to be a republic 29.4%

・ No opinion 6%

If New Zealand became an independent republic, who should be the first head of state?

・ No opinion 49.1%

・ Dame Silvia Cartwright 15.5%

・ Sir Paul Reeves 11.7%

・ Jim Bolger 3%

・ Jonah Lomu 1%

・

63

Give two reasons why this survey was not representative of all New Zealander’s opinions.

There are two main answers expected to the above problem. The first is an answer using statistical knowledge, such as ‘Only proportions are shown because of the survey, so there is the possibility that only a small amount of data was collected’. The second is an answer using contextual knowledge such as ‘This is a survey of the readers of the Herald, so the data are likely to be biased’. As in ‘... the readers of the Herald, so ...’, the second answer says that the criteria of the judgment are contextual. There are answers based on numerical data like the first one and answers relying on non-statistical knowledge such as readers of the media article in the second answer, which is the same as for the Japanese problem. However, the New Zealand problem requires both contextual knowledge and statistical knowledge in the second answer, unlike the Japanese problem. Even in the above-mentioned Japanese problem about the number of sea bathers, the statistical knowledge to be used is at most that of calculating the increase or decrease from 2006 to 2007. On the other hand, in this New Zealand problem, it is possible for students to answer in such a form that contextual knowledge and statistical knowledge are synthesised. In other words, answers may combine statistical knowledge of sample survey with contextual knowledge of opinion polls.

From this comparison of the two countries, it can be said that the New Zealand textbook posed more realistic/authentic and more complex problems. In addition, the context of opinions is on the next king in New Zealand, and is a social and cultural issue in this country. It uses a genuine media article, and moreover, places the statistical content of the sample survey properly. On the other hand, problems in Japan tend to emphasise the use of statistical knowledge but not contextual knowledge. Thus, it is found that there are efforts to include some context in the problem itself, but these efforts are not enough to include context in the inquiry and answer when posing the problem, and that there is a bias in the types

64

of context within the problem itself. In addition, the synthesis of contextual knowledge and statistical knowledge is also an issue, as pointed out by Makar, Bakker, and Ben-Zvi (2011).

4.2 Educational practices in New Zealand