Changing beliefs about the benefit of statistical knowledge

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Changing beliefs about the benefit of statistical knowledge

Alexandra Sturm, Andreas Eichler

To cite this version:

Alexandra Sturm, Andreas Eichler. Changing beliefs about the benefit of statistical knowledge.

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.761-767.

�hal-01287130�

of statistical knowledge

Alexandra Sturm

and Andreas Eichler

1 University of Education Freiburg, Freiburg, Germany, sturmfr@ph-freiburg.de 2 University of Kassel, Kassel, Germany, eichler@mathematik.uni-kassel.de

In this day and age, statistically-based information is nearly omnipresent in daily media like newspapers or TV. Therefore, people need skills to read and interpret such information adequately. Due to the fact that pre- vious research shows that sometimes statistical knowl- edge was improved without seeing the usefulness of this knowledge, we focus on improving both statistical knowledge and beliefs about the benefit of statistical knowledge. We examine whether a short-term (two hours) intervention has measurable influence on knowl- edge and beliefs of students at school and university, who have different mathematical foci. In this paper we discuss the design of our study and mainly focus on the intervention.

Keywords: Statistics, beliefs, attitudes, intervention study.

INTRODUCTION

The ability to read, to understand and to judge statis- tical information adequately has become increasingly important in our information society for everyone.

The National Council of Teachers of Mathematics [NCTM] (2000, p. 48) highlights this significance in its ‘Principles and Standards for School Mathematics’:

“The amount of data available to help make decisions in business, politics, research, and everyday life is stag- gering.“ Accordingly, the NCTM emphasizes that sta- tistics skills are necessary for students “to becoming informed citizens and intelligent consumers” (ibid.).

There is a wide consensus that the ability to proper- ly interpret quantitative data and, thus, to proper- ly interpret statistical data in daily life is based on knowledge elements, but also on beliefs about the im- portance of statistics for society or the own life. For example, Wallman (1993) describes the term statisti- cal literacy as “the ability to understand and critically

evaluate statistical results that permeate daily life, coupled with the ability to appreciate the contribu- tions that statistical thinking can make in public and private, professional and personal decisions” (p. 1).

Emphasizing knowledge elements as part of statisti- cal literacy seems to be self-explaining. Further, Gal (2004, p. 69) suggests dispositional elements includ- ing “the willingness to invest mental effort” to be a second part of statistical literacy. It can be assumed that beliefs and attitudes have consequences on us- ing knowledge learned before: Schau and Emmioglu (2012, p. 86) suggest “that students who leave their statistics courses with negative attitudes are unlikely ever to use what they have learned. That is, they will not intelligently and literately use statistics in their professional and personal lives or in any educational venture”.

Accordingly, an educational goal all over the world is developing statistical literacy including both knowledge elements and dispositional elements (Shaughnessy, 2007). But research shows that stu- dents being schooled in statistics before could have improved knowledge, but not improved beliefs (Schau

& Emmioglou, 2012; Eichler, 2011). Therefore, a pos- sible assumption is that statistical literacy is only sustainably developed if both parts, i.e. knowledge elements and dispositional elements, are developed appropriately.

As a consequence of the discussion above and taken

into account that research shows that even adults are

predominantly in a struggle with handling statistical

information (Gal, 2004), our research project aims to

investigate the relation of knowledge elements and

dispositional elements when developing statistical

literacy. For this reason, we developed an interven-

tion aiming to improve both knowledge elements

and dispositional elements. Students’ beliefs, their

Changing beliefs about the benefit of statistical knowledge (Alexandra Sturm and Andreas Eichler)

762 perception referring the significance and benefit of

statistics for both society and their own life, are a main focus of our research. We investigate students’

perception referring to different samples. Firstly, we investigate students at university that use mathemat- ics in a different way: Students of mathematics educa- tion, students of health education that have to apply mathematics and students of pedagogy that do not use mathematics in their university studies. Each subsa- mple will comprise at least 60 participants. Further, we will investigate students at school (grade 11, age 17) in a second step of our study.

In the first part of this report, we outline the main constructs of our research by describing a model of statistical literacy and by specifying the construct of beliefs. Subsequently, we present a specific statis- tical topic which we address, i.e. theorem of Bayes, and partly the problems of the area of ‘risk commu- nication’ respectively ‘health literacy’ used in our intervention. Afterwards, we outline such a statis- tically-laden situation and its visualization before discussing methods aiming to investigate knowledge and beliefs as a part of statistical literacy. Finally, we present first results of the intervention with 118 stu- dents of health education.

A MODEL OF STATISTICAL LITERACY

We use the construct of statistical literacy to describe students’ ability to cope with statistical-laden situa- tions. For describing statistical literacy, we primar- ily refer to the model of Gal (2004, p. 51; cf. Table 1).

Gal’s model describes both knowledge elements and dispositional elements as constituent parts of statis- tical literacy, similar to Wallman (1993). The left side of the model comprises five components which are briefly presented in the following, starting with the three non-mathematical and non-statistical aspects.

Literacy skills are necessary to perceive information through an oral or written text, whereas context skills are necessary to perceive a certain context in which

data are produced. Critical questions include the ability to be aware of possible manipulations in re- ports that are based on statistics. Further, Gal (2004) distinguishes between mathematical and statistical knowledge. We avoid this distinction in our research approach and subsume mathematical knowledge to statistical knowledge, although it’s possible to differ- entiate these components by defining certain math- ematical procedures as parts of a specific statistical knowledge (Gal, 2004).

Since we focus especially on the dispositional ele- ments of statistical literacy, beliefs and attitudes, we will briefly outline our understanding of these con- structs to discuss the right side of Gal’s model.

BELIEFS AND ATTITUDES AS ELEMENTS OF STATISTICAL LITERACY

Following Hannula (2012), beliefs and attitudes are parts of mathematics-related affect. We understand the term belief as an individual’s personal convic- tion concerning a specific subject, which shapes an individual’s way of both receiving information about a subject and acting in a specific situation (Pajares, 1992). Although sometimes beliefs are understood as stable, we are aware that stability is no inherent and definable characteristic of beliefs (Liljedahl, Oesterle,

& Berèche, 2012). In contrast to rather cognitive beliefs, attitudes embrace the more affective part of mathe- matics-related affect (cf. Hannula, 2012). According to McLeod (1992, p. 581), attitudes could be defined as

“affective responses that involve positive or negative feelings of moderate intensity and reasonable stabili- ty”. For example, beliefs about the benefit of statistical knowledge can be measured by items as ”statistics is necessary to understand decision making in society”, because it’s an indicator of an individual conviction and, thus, a belief. By contrast, agreeing “I like statis- tics” indicates a favor towards an object, statistics, and, thus, an attitude (cf. Eagly & Chaiken, 1998).

Knowledge elements of statistical literacy Dispositional elements of statistical literacy

Literacy skills Beliefs and Attitudes

Statistical knowledge Critical stance

Mathematical knowledge Context knowledge Critical questions

Table 1: Aspects of statistical literacy

Concerning beliefs, which are our main focus in com- parison to attitudes, we further distinguish between beliefs towards the world and beliefs towards the self.

For example, it is possible that a student believes that statistics provides a benefit for the society in a global sense on the one hand, and evaluates further his abil- ity to use statistics in his own life on the other.

Research referring to stability of beliefs in mathe- matics education shows that positive influences on beliefs are partly very rare (Eichler, 2011; Maaß, 2010).

Therefore we discuss in the following ideas which we used in our intervention to address positive beliefs towards statistics.

PROMOTING KNOWLEDGE AND DISPOSITIONAL ELEMENTS Principles of the intervention

Principles for the design of our intervention are based on possible reasons why students’ do not ap- preciate the benefit of statistics for society or their own life. Firstly, it is possible that statistics is not part of the curriculum (cf. Burrill, 2011). A second possible reason is that statistics is part of the curriculum, but teachers do not teach it, because of a self-estimated lack of time or feeling uncomfortable with statistics (Eichler, 2011). Another possible reason is that statis- tics is taught, but in an inappropriate way of teaching:

using not-application-oriented contexts and problems (we point out an example in Figure 2).

In our intervention, we try to meet requirements for application-oriented contexts that potentially high- light the relevance of statistics for students sustain- ably. Further, we formulated three requirements for an appropriate subject matter that

1. emphasizes the benefit of statistics for both society and, in particular, individual’s life;

2. focuses on an issue that is not common for students;

3. focuses on an issue for that exist elaborated strategies for designing a potentially effec- tive short-term intervention.

In our opinion, the Bayes’ theorem fulfills all these requirements.

Firstly, the Bayes’ theorem is existent in daily media (cf. Figure 3) which is in our opinion an indicator for emphasizing the benefit of statistics for the society and an individual.

Further, concerning the second requirement, the Bayes’ theorem or rather Bayesian thinking is not commonplace (Sedlmeier & Gigerenzer, 2001) and it is taught at school rarely. Thus, this subject potentially gives evidence about the benefit of statistics in a field in which adults without training in Bayesian thinking mostly fail to give correct estimations of probabilities (Sedlmeier & Gigerenzer, 2001).

Finally, as a consequence of findings in educational research, there are different strategies to improve understanding referring Bayesian problems. E.g., Sedlmeier and Gigerenzer (2001) found that repre- senting the statistical information in a problem as natural frequencies increases the rate of correct esti- mations. Further there is evidence for the efficiency of two visualization-forms in short-term interventions:

The tree with natural frequencies (ibid.; Wassner, 2004) and the unit square (Bea, 1995).

The intervention

As mentioned above, we laid emphasis on an authen- tic and application-oriented context. An analysis of textbooks showed a considerable amount of less authentic contexts. For example, the question in the task referring Bayes formula shown in Figure 2 seems

Figure 2: A task with a context that is not authentic

Changing beliefs about the benefit of statistical knowledge (Alexandra Sturm and Andreas Eichler)

764 to be irrelevant, since the car dealer could know the

answer of this question without using Bayes’ formula.

By using actual newspaper articles and leaflets (Figure 3), we tried to present the students an authen- tic situation, i.e. a situation that could potentially be a real problem for the student in their future life.

This material should also contribute to recognize and emphasize the significance of the content. Based on the appropriate examples of Boer (1993), Wassner (2004), Pinkernell (2006) and Beckmann (2013), the in- tervention involves three main-contents: HIV-Testing, breast-cancer-screening and prenatal screenings.

A second characteristic of the intervention should be its comprehensibility. For this, we use natural fre- quencies additional to relative frequencies or rather probabilities for the representation of quantitative information in the tasks. Natural frequencies sim- plify the Bayes’ theorem for students, who have only to relate numbers to each other instead of calculat- ing a complex formula that entails three multiplica- tions. The research of Gigerenzer and Hoffrage (1995) shows that representing statistical information as natural frequencies increases the ability to solve Bayesian problems. A possible reason for this effect is that the nested sets structure becomes more salient.

Nonetheless, in newspaper articles, TV and other ev- eryday situations probabilities are often used, so that we didn’t want to drop probabilities.

We further use the tree diagram (Wassner, 2004) and the unit square (Bea, 1995) with natural and relative frequencies that we illustrate in Figure 4 for a fictive situation of a disease-test.

Both diagrams are helpful to apply and understand the Bayes’ theorem (Sedlmeier & Gigerenzer, 2001).

While the advantage of the tree diagram is visualizing the chronological sequence of the given information, the advantage of the unit square is visualizing the

proportions. This aspect seems to be especially help- ful in cases when parameter were changed (Eichler &

Vogel, 2010). Furthermore, we avoided using formal Bayes’ theorem and difficult terms like sensitivity or specificity to support the comprehensibility.

As mentioned above, the three main-contents of our intervention mainly refer to a diagnosis of a disease (HIV-Testing, breast-cancer-screening and prenatal screenings) and the problems could be solved by us-

Figure 3: Headline in a German online-newspaper (translated)

Figure 4: Tree diagram and unit square with natural frequencies

ing Bayes’ theorem. In each problem, a base rate of a disease or an infection and conditional probabilities representing the right-positive rate (probability of a positive test result given a disease) and representing the false-positive rate (probability of a positive test result given no disease) have to be arranged accord- ing to Bayes’ theorem to compute the probability of disease given a positive test. We think that a further interesting content are doping-tests. They have a high significance in sports and provide a great basis for modeling, because the base rate is unknown (estimat- ed 20–35 %, cf. Pitsch, 2009) and the sensitivity- and specificity-values of used doping-tests are not pub- lished by WADA (World Anti-Doping Agency), so they must also be estimated (Pitsch, 2009). But in respect on this complexity it’s not appropriate for our short- term intervention.

The tasks potentially promote a critical stance as a crucial part of statistical literacy. For example, pos- sible critical questions could refer to the following issues:

― What happens, if the sensitivity of the test would be better (the probability to get a positive test result given the HIV infection)?

― What happens, if the test is used in another coun- try like Germany (the base rate for HIV is 0.4% in US, but 0.1% in Germany)?

― What happens if the specificity of the test is not as good as the producer of the test indicates (the pro- ducer indicates that a person that is not infected gets in 99.98% of the cases a negative test result).

Other questions that integrate context in a broader sense could be: “Why could a confirmatory test be necessary?” or “Should the German administration permit the approval of HIV-rapid-tests for home us- age? Which reasons are arguments against it, which do support the approval?”

As the context like disease could be potentially signif- icant for students, we hypothesise that particularly critical questions could improve the students’ beliefs referring the relevance of statistics for both society and the students’ own life.

First results of the ongoing study

In November 2014, 118 students of health education (age: mean=21.72, sd=2.9) were assessed pre- and post-intervention and after two week follow-up.

Their beliefs and attitudes were measured by a ques- tionnaire which comprised e.g. two components of the Survey of Attitudes Towards Statistics (SATS- 36©): value and interest of statistics (Schau, 2005).

The range of coefficient alpha values for the inter- est-component varies from .868 to .879 for the dif- ferent measurement times. The scale shows thereby a high internal consistency. The mathematical and statistical knowledge elements of statistical literacy were assessed by Bayesian-situation-tasks. Students were asked to estimate the right percentages of those people who are infected given a positive test result. As expected, students showed a low performance before the intervention and a high performance after the intervention for these tasks.

DISCUSSION AND CONCLUSION

There seems to be a discrepancy between the enor- mous relevance of statistics in our society and the poor relevance of statistics often assigned by students and adults. It is on the one side possible that these students (or adults) have little statistical knowledge and, hence, do not appreciate the benefit of statistics for both, society and own life. However, research shows that even students with considerable statis- tical knowledge assign statistics little or rather no relevance outside school mathematics (Eichler, 2008).

As a consequence to these findings the main aim of our

research is to investigate the relation of developing

knowledge elements and developing dispositional

elements of statistical literacy. For this reason a main

challenge of our research is to develop an intervention

that potentially improves both elements of statistical

literacy, i.e. knowledge elements and dispositional el-

ements. We provided the example of the HIV-test that

potentially fulfills three requirements of the interven-

tion that we defined. Firstly the HIV-test represents

an authentic context that is controversially discussed

in daily newspapers and official statements. Further

the context of Bayesian problems is not common for

most of the students or adults and, thus, could serve

as a new example showing the significance of statis-

tics. Finally, there exist elaborated and empirically

proved ways to teach Bayesian thinking that facilitate

an intervention that is not linked to a regular statistics

Changing beliefs about the benefit of statistical knowledge (Alexandra Sturm and Andreas Eichler)

766 course at school. In this paper we provided the tree di-

agram and the unit square connected with natural and relative frequencies as possible strategy to represent the information in a Bayesian problem appropriately.

To investigate the mentioned connection of the de- velopment of knowledge elements and dispositional elements we vary the population in different samples.

Since we regard students at school as possible future audience for our intervention, we regard also stu- dents at university that show different characteristics that are, from a theoretical perspective, important referring the status of statistical literacy: The first sample consists of prospective mathematics teach- ers who potentially have a considerable amount of mathematical knowledge as part of statistical litera- cy. The second sample consists of students of health education who potentially hold context knowledge referring to the HIV-test and who potentially need this knowledge in their professional careers. Students of both samples could either show positive beliefs or negative beliefs about the relevance of mathematics or statistics. However, first results show that it seems to be possible to influence the statistical literacy re- ferring to his dispositional elements in a short term intervention as our investigation referring the stu- dents of health education imply. We expect to pres- ent further results of the impact of our intervention described in this paper at the conference.

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