Mathematics teachers' conceptions of how to promote decision-making while teaching statistics: The case of Japanese secondary school teachers

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Mathematics teachers’ conceptions of how to promote decision-making while teaching statistics: The case of

Japanese secondary school teachers

Orlando González

To cite this version:

Orlando González. Mathematics teachers’ conceptions of how to promote decision-making while teaching statistics: The case of Japanese secondary school teachers. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.665-671. �hal-01287067�

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of how to promote decision-making while teaching statistics: The case of Japanese secondary school teachers

Orlando González

Hiroshima University, Graduate School for International Development and Cooperation, Hiroshima, Japan, ogonzalez@hiroshima-u.ac.jp

The role of informed decision-making in today’s society is very important at personal, professional, societal, and even educational levels. Thus, mathematics teachers are required, now more than ever, to provide students with appropriate tasks having the potential to promote deci- sion-making skills. This article reports on the features and competence demands of tasks that are thought to promote decision-making skills by a purposeful sample of twelve Japanese secondary school mathematics teach- ers. A qualitative analysis on the collected data revealed some commonalities between the answers, as well as correspondences and discrepancies between what the participants and statistics educators think a task with the potential to promote decision-making is.

Keywords: Decision-making, teachers’ conceptions of decision-making, statistical tasks, Japanese secondary school mathematics teachers, statistics education.

INTRODUCTION

Statistics has become very important at all levels of citizenry in today’s society, in which large amounts of data are available to almost everyone. Thus, to be part of modern society in a competent and critical way, citizens need to be able to interpret such data in a broad sense, and understand the variability, dis- persion, and heterogeneity which cause uncertainty in interpreting information, in facing risks, and in making decisions. In the particular case of the latter, many statistics educators, curriculum developers and international agencies around the world agree on the increasing importance for students to gain competence in using, handling and interpreting data

to inform decision-making at personal, professional, and societal levels (Garfield & Ben-Zvi, 2008).

The last reform to the Japanese Course of Study echoes these ideas. In fact, the “difficulty … in description-type problems that require thinking, decision-making, and representation”, reported in the 2006 Programme for International Student Assessment (PISA) study report about Japanese students by the Organization for Economic Cooperation and Development (OECD), is the first motivation to that reform (MEXT, 2008, p. 1; MEXT, 2009, p. 1). As a result, in the particular case of secondary school level, the latest Japanese Course of Study emphasizes – in the mathematical domains “Practical Use of Data” at junior high school, and “Analysis of Data” at senior high school – nur- turing the attitude and ability to purposely process daily-life data, capture its trends and features, and make decisions based on such analysis (MEXT, 2008, 2009). Thus, on account of the significant place held by fostering decision-making skills in the Japanese secondary school mathematics curricula, teachers must be able to carry out themselves data-based decision-making, as well as to design instruction aimed to develop students’ decision-making skills.

Despite the attention given to students’ development of decision-making skills and attitude in the latest Japanese Courses of Study, teaching guides, and even the 2007 amendment to the School Education Law, what decision-making is, and how to promote the skills related to it, are not explained in any of those official documents. Therefore, teachers are left to determine by themselves how decision-making could be promoted in their students, which raises particular

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Mathematics teachers’ conceptions of how to promote decision-making while teaching statistics (Orlando González)

666 concern, due to the reported need for appropriate

training in statistics education in the preparation of mathematics teachers in Japan (Isoda & González, 2012; González, 2014).

Then, in order to shed light on how Japanese mathematics teachers conceptualize decision-making and its promotion, the following research questions are addressed:

(1) What kind of tasks do Japanese secondary school mathematics teachers regard as having the potential to promote decision-making?

(2) What knowledge and skills do Japanese secondary school mathematics teachers believe to be associated with the promotion of decision-making?

THEORETICAL BACKGROUND Decision – What is it?

A decision is defined as “the broader process within which a choice among specific options will be made”

(Brown, 2005, p. 1). Through this process, the decision-maker is ultimately able to determine what action to take, by identifying a choice – i.e., selecting among previously-identified options (Brown, 2005, pp. 1, 236–237).

Decision-making situations demand from the decision-maker to determine a course of action, typically while considering uncertainty and data variability (Gal, 2004). In fact, a decision-maker operates, at once, in all the four dimensions of statistical thinking identified by Wild and Pfannkuch (1999).

Before choosing a particular course of action, a decision-maker has to engage in the many phases of the decision-making process, which are identified below (Wild & Pfannkuch, 1999; Gal, 2004, p. 43; Arvai, Campbell, Baird, & Rivers, 2004; Edelson, Tarnoff, Schwille, Bruozas, & Switzer, 2006):

Definition: here decision-makers define the spe- cific decision that has to be made, as well as a broad set of end objectives in the context of the impending decision. This phase is the equivalent to the “Problem” step of Wild and Pfannkuch’s (1999) statistical investigative cycle, known as the PPDAC cycle.

Planning: during this phase, the identification, design, and choice of optimal ways to use resourc- es – i.e., means to achieve end-objectives – are de- termined. The choices must be a set of appealing and purposeful alternatives from the objectives previously defined. Here, the identification of constraints and considerations as basic criteria for the decision is also fundamental, since some choices might be eliminated based on failure to meet constraints, while others may be ranked based on how they fare on considerations. This phase is not only connected to the “Plan” step of the PPDAC cycle, but also with strategic type of thinking, and the “Generate” step of Wild and Pfannkuch’s (1999) statistical interrogative cycle.

Data: the “Planning” phase tends to be followed by a recalling and seeking of information, as well as by collecting statistical data relevant to the achievement of end-objectives. This phase is connected to the “Data” step of the PPDAC cycle, to the “Seek” step of the statistical interrogative cycle, and to the recognition of need for data (Wild & Pfannkuch, 1999).

Evaluation: here, decision-makers must assess the implications of different choices for the decision, as well as the impact on the different stakeholders involved in the decision on hand.

This phase is connected to the “Criticize” step of the statistical interrogative cycle, as well as to the strategic type of thinking, and being logical.

Weighing impact: during this phase, decision-mak- ers weigh the impacts of the different options on stakeholders based on their own values. Thus, in this phase decision-makers bring in their values and see how different values can lead to different decisions. This phase is connected to both the

“Analysis” and “Criticize” steps of the PPDAC and statistical interrogative cycles, respectively.

Making and justifying a decision: during this phase, decision-makers finally select the course of action that better addresses their objectives, in the light of the decision-makers’ constraints, considerations, assumptions, value systems, judgment of probabilities, stakeholder impact, etc., and provide an informed justification for such a decision. This phase is connected to both

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the “Conclusions” and “Judge” steps of the PPDAC and statistical interrogative cycles, respectively.

Decisions – What types are there?

In the present study, decisions will be classified in four types: three of them – i.e., personal, professional and civic – were identified by Brown (2005, pp. 5–7), while the last type – i.e., object-related – is proposed by the author.

Personal decisions: those that decision-makers make on their own behalf; for example, about a career path or going on a date.

Professional decisions: those that decision-mak- ers, as professional individuals and specialist decision aiders, make on behalf of others in a work capacity, such as in business management or medical practice.

Civic decisions: these are decisions made on a pub- lic – i.e., not personal – issue, such as when decision-makers, as citizens, take a private position on someone else’s – e.g., government’s – choice, for which they have no direct responsibility.

Object-related decisions: those that decision-mak- ers make about parameters or particular features of the statistical objects – i.e., decisions regarding language situations, concepts, propositions, procedures and arguments (Godino, Ortiz, Roa,

& Wilhelmi, 2011, p. 277) – involved in a given statistical problem. An example could be a decision about what measure it would be appropriate to employ to decide what the best design for a paper plane is – e.g., decide whether to use the distance travelled, accuracy, or time spent airborne to determine which design is the best.

Decision-making while studying statistics – How to do it?

According to Godino and colleagues (2011, p. 274), a basic statistical problem concerning decision-making is contextualized in a real situation; is driven by uncertainty; involves specific statistical practic- es – such as randomization, collecting sample data, transnumeration, data reduction, and using statistical models – ; and leads to the emergence of specific representations, concepts, procedures, properties and arguments. Several statistics educators – e.g., Makar &

Fielding-Wells, 2011 – recommend to engage students

in decision-making through making them experience the complete statistical investigation cycle. This makes absolute sense, because it is possible to match every phase of the decision-making process to steps of the PPDAC cycle.

METHODOLOGY

Data-collection instrument and participants In order to address the research questions of this empirical study, an assignment-like survey was designed, asking respondents the following open questions:

1) From a textbook, teacher’s guide, student work- book, internet, academic journal, or other type of source, choose a task or activity that, in your opinion, would promote decision-making skills in your students in secondary school mathematics when you teach contents in the mathematical domains “Practical Use of Data” or “Analysis of Data”. You may also develop a task or activity by yourself.

2) Attach a copied or printed version of the chosen task or activity, and write down its source.

3) Briefly explain why, in your opinion, the chosen task or activity has the potential to promote decision-making skills.

At the time of writing this article (September 2014), the data-collection process – which began in mid-July 2014 – was ongoing. Thus, in this paper it is reported a preliminary analysis of the data gathered from the first twelve teachers who voluntarily and anonymous- ly responded and mailed back the survey booklets.

The respondents are part of a purposeful sample of 25 Japanese secondary school mathematics teachers who participated in a national academic meeting on mathematics education in Japan. Five of the respondents were working at junior high school, while the remain- ing seven were working at senior high school. The respondents were between 24 and 63 years old, and they had between two and forty-one years of teaching experience – with seven of them with at least 13.

Data analysis

During the initial phase, all the questionnaire answers were translated from Japanese into English by the author of this paper. Also, they were read repeat- edly in order to gain an overall impression, as well

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668 as to identify commonalities among the participants.

A “bottom up” approach to coding was initially used to analyze the tasks’ features and the participant’s reasons for choosing such tasks. This grounded form of analysis ensures that the themes or categories ex- tracted were, in fact, grounded in the data and hence reflected the participants’ own knowledge base and conceptions regarding decision-making. The author reviewed all the given answers to the three questions and identified answers that occurred frequently in the data. Such answers appearing to contain similar content were initially given the same code, and each code was further analyzed to find true meanings within their text. A process of reduction and clustering of categories, which were refitted and refined, followed, resulting in summary groupings (or “clusters”) of themes sharing common meaning.

RESULTS

Tasks’ features – What kinds of tasks are thought to promote decision-making?

The participants used a variety of sources to select their tasks. The most commonly used source was the

Internet, used by four teachers – i.e., by T2, T7, T11 and T12. Three teachers – i.e., T3, T5 and T6 – referred to research reports or academic journals; used activity books, mathematics textbooks or exercise books – i.e., T4, T7 and T11 – ; and developed the chosen task by themselves – i.e., T1, T9 and T11. Two teachers – i.e., T1 and T8 – chose problems from the National Assessment of Academic Ability and Learning Situation, conducted yearly by the National Institute for Educational Policy Research of Japan. Finally, one teacher – i.e., T10 – selected a task he learnt at the prefectural training program.

From the qualitative analysis performed on the collected data, several task features were identified, in- cluding the number of solution strategies, number and kind of representations displayed, or if the task was set in a real-life context. The result of sorting and clustering such task-based features is shown in Table 1.

From Table 1, it seems the all the participants agree that a task intended to promote decision-making should engage students in statistical investigations, connect different statistical concepts, be set in a re-

TEACHER

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12

FEATURES

Number of choices offered by the task 2 0 0 0 0 2 0 0 2 0 0, 2 2

The task explicitly requests students to think of several possible solutions / to solve the problem in different ways

Y Y N N N Y Y N Y Y N N

The task invites students to engage in open

inquiry and investigation Y Y Y Y Y Y Y Y Y Y Y&N Y

The task required to connect different sta-

tistical concepts Y Y Y Y Y Y Y Y Y Y Y Y

The task is a multi-step one, comprised of

several mini-tasks Y N N N N Y Y Y N Y Y N

The task explicitly asks students to commu-

nicate / justify their procedures Y N Y N Y Y Y Y Y Y Y N

Different types of statistical representa-

tions in the task 1 0 1 1 1 0 1 1 2 1 4 0

The task includes the use of manipulatives N N N N N Y N N N Y N Y

The task is set in a real-life context Y Y Y Y Y Y Y Y Y Y Y&N Y

The task can be solved in several ways Y Y Y Y Y Y Y Y Y Y Y&N Y

Type of decision requested by the task (Pe = personal; Pr = professional; C = civic; O

= object-related)

O C C Pe C Pe O O Pe O O Pe

Environment in which the task is supposed

to take place (I = indoors, O = outdoors) I I&O I I I I I I I I I I Table 1: Features of a task with potential to promote decision-making, according to the information provided by the participants on the questionnaire

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al-life context, and have multiple ways to be solved.

This is in line with many previous studies on the instruction of decision-making at school level – e.g., Arvai et al., 2004; Edelson et al., 2006; Edwards &

Chelst, 2007; Garfield & Ben-Zvi, 2008; Godino et al., 2011; Pfannkuch & Ben-Zvi, 2011.

Since 11 out of 12 teachers chose a task to be carried out completely inside the classroom, it seems that the majority of participants consider that promot- ingdecision-making should be done within the limits of the mathematics classroom. By doing so, students might miss the opportunity to extend their knowledge beyond the limits of the classroom, into the real world, which is needed to be statistically literate – cf. Garfield

& Ben-Zvi, 2008.

Nine participants shared the idea that a task with the potential to promote decision-making must explicitly ask students to provide arguments justifying their decisions. In fact, this feature is the ultimate phase of the decision-making process. Many researchers – e.g., Edelson et al., 2006; Edwards & Chelst, 2007 – recommend teachers to give substantial guidance when introducing the decision-making process to students, in order to provide scaffolding for them to learn how to do well-informed and justified decisions by themselves.

Half of the respondents seem to consider that a task intended to promote decision-making should be a multi-step problem, or should explicitly request finding multiple solving strategies. Thus, regarding these pair of task features, this sample of respondents did not lead toward a particular one. In relation to being a multi-step problem, some teachers added questions to the original tasks in order to assure students of considering multiple solving strategies. For example, after challenging students with a problem from the 2012 National Assessment of Academic Ability and Learning Situation for Grade 9, T1 added the following step: “Listen to your friends’ ideas. At the end, whose idea do you agree with? (Do not mind if it is your own idea.) Please, also write down the reason.” In this way, T1 is not only handling classroom communication, but also facilitating discussion and negotiation, which is fundamental in any statistical investigation (cf. Makar

& Fielding-Wells, 2011).

Also, it is also noticeable from Table 1 the lack of an overwhelming consensus about which type of deci-

sions should be requested in a problem intended to promote decision-making. Five teachers chose tasks requiring an object-related decision; personal decisions were required in the tasks selected by four teachers; and three respondents demanded civic decisions in their problems. In this regard, specialists say that, although decision-making skills are not specific to any particular type of decision, developing such skills has been done mainly in the context of personal and civic ones, to which students can most immediately relate (cf. Brown, 2005, p. 155; Edelson et al., 2006).

Moreover, by just requesting object-related decisions, the completion of the task seems to depend more on students’ statistical content knowledge, and hence it is missed an opportunity to create and strengthen a link between the learning of statistics and students’

actions in real situations.

Reasons for choice – What competence aspects are associated to decision-making?

From a grounded analysis of the reasons given by teachers about why their chosen tasks have the potential to promote decision-making, six category clusters of competence aspects that participants seem to associate with decision-making were identified. Such results are shown in Table 2.

Many trends in teachers’ answers were noted from the results in Table 2. For example, 10 out of 12 participants associated decision-making with skills related to mathematical and statistical literacy, by using ex- pressions such as “mathematical grounds”, “handling of information and data”, “knowledge about statistical ideas”, “ability to read data properly”, “ability to grasp data trends”, and “practical use of multiple statistical representations”. This is in agreement with what many statistics educators have previously reported on this matter: that decision-making heavily depends on skills related to statistical literacy such as understanding, explaining and quantifying the variability in the data (e.g., Gal, 2004; Pfannkuch & Ben-Zvi, 2011).

Only 3 out of 12 participants expressed that decision-making requires the enactment of personal or societal values, such as social fairness. In this respect, it is important to highlight that two of these teachers – i.e., T5 and T9 – selected tasks requesting either a personal or a civic decision. This is in agreement with previous studies on decision-making – e.g., Arvai et al., 2004; Edelson et al., 2006 – , which point out that creating a set of appealing and purposeful alterna-

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670 tives, as well as making thoughtful and high-quality

decisions, needs the application of decision-maker’s values and technical information.

Just 3 out of 12 of the respondents highlighted that decision-making involves an opportunity for students to build their own criteria or rules for decision, which is emphasized in the literature as one of the main char- acteristics of the decision-making process (cf. Brown, 2005; Edelson et al., 2006; Garfield & Ben-Zvi, 2008, p. 277).

The fact that decision-making involves engagement with a familiar real problem was pointed out by only two teachers. This could be a likely reason for having five teachers selecting tasks requesting object-related questions to students. According to many authors (e.g., Brown, 2005, p. 155; Edelson et al., 2006), students relate more easily to realistic decision-making sce- narios, particularly those in which they are asked for personal or civic decisions.

Nine out of the 12 participants pointed out that decision-making involves engagement with different steps of the open-ended approach. These teachers mentioned aspects such as “coming up with a diver- sity of ways of thinking”, “generating alternative de- signs”, “dealing with problems related to uncertain events”, “opportunity to decide about parameters and methodologies”, and “opportunity to determine what

properties would be good to examine”. The number of teachers falling into this category is not surprising at all, because in Japan the lessons usually involve mathematical activities using ill-defined questions, which are often solved by engaging with the open-ended approach method. In fact, “dealing with the openness”

during the decision-making process is a main feature of it (cf. Edelson et al., 2006; Edwards & Chelst, 2007;

Makar & Fielding-Wells, 2011).

Finally, only 2 out of the 12 respondents explicitly related the decision-making

process to social and inter-personal processes such as discussion, communication, argumentation, per- suasion and negotiation to build consensus and common understanding. This result was quite unexpected, since a typical Japanese mathematics lesson has the

“takuto” and “neriage” phases, in which students present their solutions or ideas to the whole class, and discuss the validity and pertinence of the proposed ideas, respectively. Many researchers (e.g., Arvai et al., 2004; Edwards & Chelst, 2007) are of the idea that the decision-making process is much stronger with discussion and feedback, and therefore recommend that students should at first brainstorm on their own a list of objectives related to the decision, and then convene to discuss them in groups and with the rest of their class, in order to overcome particular biases related to individual and group thinking, and then to

TEACHER

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12

ASPECTS

Decision-making involves opportunity to

build students’ own decision criteria ✔ ✔ ✔ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘

Decision-making involves personal or soci-

etal values ✔ ✘ ✘ ✘ ✔ ✘ ✘ ✘ ✔ ✘ ✘ ✘

Decision-making demands from students to make use of their own mathematical and

statistical literacy skills ✔ ✔ ✔ ✔ ✘ ✔ ✔ ✔ ✔ ✔ ✔ ✘

Decision-making involves engagement with different steps of the open-ended ap-

proach ✔ ✔ ✘ ✔ ✘ ✔ ✔ ✔ ✘ ✔ ✔ ✔

Decision-making involves engagement

with a familiar real problem ✘ ✔ ✔ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘

Decision-making requires inter- personal processes such as discussion, communication, argumentation, negotiation, and collaboration

✘ ✘ ✔ ✘ ✔ ✘ ✘ ✘ ✘ ✘ ✘ ✘

Table 2: Competence aspects associated with decision-making, according to the information provided by the participants on the questionnaire

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effectively undertake the decision-making process in the classroom.

CONCLUSIONS

In general, this study found how the surveyed Japanese mathematics teachers conceptualize decision-making and the tasks intended to promote it. A majority of the respondents selected tasks embedded in a real-life context, having multiple ways of solving, requiring the connection of different statistical and mathematical ideas, meant to be carried out entirely inside the classroom, involving a partial or complete engagement with the statistical investigation cycle, and requiring to communicate and justify procedures and solutions. All these features are, indeed, char- acteristics of tasks able to promote decision-making skills indicated by previous researches.

The information about how the participants conceptualize decision-making was obtained from the different cognitive aspects identified in teachers’ explana- tions of why they chose their tasks. The majority of participants related to decision-making aspects such as engagement with steps of the open-ended approach, and practical use of mathematical and statistical literacy skills. These features of the decision-making process are in line with those identified in the spe- cialized literature. However, other main character- istics of decision-making – e.g., engagement with a familiar real problem, formulation of personal decision criteria, and engagement with particular social processes – were not explicitly noted by most of the surveyed teachers. Also, just two teachers explicitly considered affective aspects of decision-making, such as the enactment of personal and societal values. Due to the relevant place promoting decision-making in students has in the Japanese mathematics curriculum, it seems that secondary school mathematics teachers need to improve their professional knowledge about what decision-making is and how to promote it, in order to make them appreciate decision-making as a statistical rather than a mathematical work.

Realization of the potential of any task to promote decision-making heavily depends on teachers’ statistical knowledge for teaching (González, 2014). Thus, exam- ining how teachers may implement their chosen tasks seems a natural next step for this study.

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