University of Pennsylvania

Journal of Statistics Education v.2, n.2 (1994)

Copyright (c) 1994 by Iddo Gal and Lynda Ginsburg, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

**Key Words**: Affective issues; Assessment
instruments; Anxiety.

While many teachers of statistics are likely to focus on transmitting knowledge, many students are likely to have trouble with statistics due to non-cognitive factors, such as negative attitudes or beliefs towards statistics. Such factors can impede learning of statistics, or hinder the extent to which students will develop useful statistical intuitions and apply what they have learned outside the classroom. This paper reviews the role of affect and attitudes in the learning of statistics, critiques current instruments for assessing attitudes and beliefs of students, and explores assessment methods teachers can use to gauge students' dispositions regarding statistics.

1 In recent years, statistics educators have focused attention on rethinking the process of statistics education at both the college and pre-college levels. Calls for reform of college level statistics education now urge faculty to update their materials and methods and to involve students in more hands-on activities. At the elementary and secondary levels, attempts are being made to integrate statistics education into the mathematics and science curricula.

2 While statistics educators have focused on improving the cognitive side of instruction, i.e., the skills and knowledge that students are expected to develop, little regard has been given to non-cognitive issues such as students' feelings, attitudes, beliefs, interests, expectations, and motivations. This paper aims to examine selected non-cognitive aspects of statistics instruction, especially those related to affective reactions and attitudes towards statistics. We believe that further attention to such factors is warranted, as they may contribute to students' difficulties in learning basic concepts in statistics and probability. These difficulties have been widely documented over the last two decades (e.g., Shaughnessy 1992).

3 Statistics educators routinely mention that many students enter statistics courses with negative views or later develop negative feelings about the domain of statistics. Perney and Ravid (1991, p. 2) describe a familiar scenario:

"Statistics courses are viewed by most college students as an obstacle standing in the way of attaining their desired degree. It is not uncommon to see students who delay taking the statistics courses until just before graduation. . . College professors who teach the research and statistics course are all too familiar with the high level of anxiety exhibited by the students on the first day of the term."

4 Peterson (1991, p. 56) wrote about introductory statistics courses in Science News.

"Few college students escape taking an introductory course in statistics. But for many of these students, the lessons don't seem to stick. They remember the pain but not the substance. `Such initial courses tend to turn students off,' says Barbara A. Bailar, executive director of the American Statistical Association."

5 Simon and Bruce (1991, p. 22), explaining their motivation for developing the resampling method (the process of generating new samples from a given data set or from a data-generating mechanism for the purpose of providing access and meaning to the study of probability and inferential statistics), offer this lament:

"Prob-stats continues to be the bane of students, most of whom consider the statistics course a painful rite of passage---like fraternity paddling---on the way to an academic degree. At the end of the semester, most of those who study it happily put prob-stats out of their minds forever."

6 Some readers may be familiar with students who nickname the statistics course "sadistics." Rosenthal (1992, p. 281) adopted the term in an editorial for the UMAP Journal titled, "No more sadistics, no more sadists, no more victims."

"I come to announce...a future for the elementary statistics course, one utopian possibility for exorcising the malevolent spirits of the standard errors in our statistics teaching...I appropriate the pejorative "sadistics" from student culture, to implore our community to acknowledge and legitimize students' perceptions of the quality of life in the course we create for them...[and] reflect the reality that unintended human suffering takes place under our watch."

7 We take these and other related views (Benson 1989; Harvey, Plake and Wise 1985; Roberts and Bilderback 1980) as strong indicators that students' feelings about statistics education, and the effects of these feelings on resulting learning, knowledge and further interest in statistics, should occupy a more central role in the minds of statistics educators.

8 This paper is organized in three sections. First, we examine the justification for attending to non-cognitive issues within the larger context of the goals of statistics education. Next, we briefly review and critique existing approaches to research on students' beliefs and attitudes towards statistics. Finally, we explore implications for assessment practices in statistics education and directions for further research. (By research we refer both to "academic" research done to increase general knowledge, as well as to local research that individual teachers or statistics departments can, and in our view should, undertake in order to be informed about their students' views and beliefs and to be able to provide a better service to learners.)

9 The above discussion suggests two separate yet related motivations for attending to non-cognitive aspects of statistics education: the outcomes we desire and the educational process itself.

10 In general, statistics courses, especially the first course, have one or more of the following goals:

a. Prepare students to take higher-level statistics courses. This is most relevant for college students (who may consider) majoring in statistics and related fields.

b. Prepare students for academic or professional careers. Usually, college students majoring in certain disciplines (e.g., biology, engineering, education, business) are required to take a statistics course to enable them to handle, use, or interpret research or statistical data in their content area. For quite a few students, the first statistics course may be the only statistics course they will ever take.

c. Prepare students to deal effectively with statistical aspects of the worldoutsidethe classroom, as they relate to students' personal lives or to their community or civic activities (e.g., be able to interpret charts, graphs, and statistical claims in a newspaper article or on TV, improve personal decisions, etc.). This may be less of a declared goal at the college level, but more characteristic of the goals of high-school statistics instruction.

11 The above goals of statistics education imply that students' encounters with statistics instruction, particularly their first encounter, should encourage further statistical experiences. It follows that,

a.A statistics course should not block demand for further statistics instruction.Students should emerge from statistics classes without apprehension or negative feelings about learning more statistics. Thus, statistics teachers should aim to engender in students a positive view of statistics and an appreciation for the potential uses of statistics and its role in future personal and professional areas relevant to *each* student.

b.A statistics course should facilitate statistical thinking.Students should emerge from statistics classes with an appreciation for when and how the application of statistics in their professional or personal lives is warranted, and with a willingness to think statistically (or probabilistically) in relevant situations. If, on the other hand, students leave statistics classes with a sense that statistics is not relevant to their everyday lives, that it leads to faulty or biased conclusions, or that they are not capable of thinking statistically (i.e., "I'm not good with statistics"), then one is hard pressed to point to the benefits of taking a statistics class.

12 To ensure they have done their job well, statistics educators should be able to assess students' standing on non-cognitive factors, including: (1) interest or motivation for further learning, (2) self-concept or confidence regarding statistical skills, (3) willingness to think statistically in everyday situations, and (4) appreciation for the relevance of statistics in their personal and vocational lives. (The first of these factors---interest in further learning---may be an obvious and easy one to assess, yet is only one of several factors to which educators must attend, and not necessarily the most important one, given the three goals for statistics courses suggested above.)

13 An altogether different set of reasons for attention to
affective and attitudinal factors concerns the impact such factors
may have on the learning and teaching process in statistics.
Increasingly, one of the stated goals of statistics education is
to develop flexible problem-solving and data-analyzing skills, as
opposed to merely imparting computational and procedural skills (Moore 1990). The creation of a problem-solving environment
for learning statistics requires that teachers build an
emotionally supportive atmosphere where students **feel**
safe to explore, conjecture, hypothesize, and brainstorm; are
motivated to struggle with and keep working on problems which may
not have right or wrong solutions and may require extended
investment of energy; feel comfortable with temporary confusion or
a state of inconclusive results; and are not afraid to experiment
with applying different (statistical) tools or methods.

14 Many students, however, do not come to statistics classes fully ready to embrace and function within a learning environment oriented toward problem solving. Instead, many carry baggage that includes negative or unconstructive beliefs about themselves in relation to the learning of quantitative and mathematical issues, including math anxiety (McLeod 1992); apprehension about taking tests, including math tests (Hunsley 1987); beliefs about the relevance (or lack thereof) of statistics for their future career or job plans (Roberts and Saxe 1982), and more.

15 Beliefs and attitudes related to math may play a powerful role in affective responses to statistics since students often expect that the study of statistics will include a heavy dose of mathematics, including complex algebra and formulae ( Simon and Bruce 1991). The frequent appearance of statistics courses within mathematics departments (or as part of a high school math class) reinforces this perception. Since practically all students have studied some high school level mathematics before starting a formal statistics class, their affective reactions to those math-learning experiences may affect how they relate to statistics learning. Students' predispositions, beliefs, and expectations may interact with aspects of the learning environment created by the teacher in ways that would work against what the teacher is attempting to accomplish.

16 Affective and attitudinal factors are also likely to become
important during a class once students begin to experience
difficulties or frustrations with their statistics studies.
Research on adults' memories of learning math during their school
days (Allan and Lord 1991; Tobias
1994) suggests a sequence of events often triggered when
students experience an initial failure to understand. An initial
confusion is followed by a failure to receive adequate
explanations or assistance from the teacher, leading to loss of
confidence and panic over the sense of lack of control of one's
own comprehension. Eventually, students become bored or
disinterested, "switch off" and disengage from what they perceive
as a futile learning process. This process is often accompanied
by students' development of negative views of their mathematical
skills and their ability to handle quantitative problems (Tobias 1994). We hypothesize that similar processes happen
in statistics classes; therefore statistics educators need to be
sensitive to students' emotional and attitudinal status. It is
thus important that statistics educators have access to assessment
instruments that enable an **initial** diagnosis of
their students' attitudes and beliefs and that also enable
monitoring of the status of such attitudes and beliefs **during**
a course.

17 The body of research on students' attitudes, beliefs, and
affect related directly to statistics education is very small and
problematic. To date, practically **all** studies
that have been published (either in journals or conference papers
available through ERIC) have been based on data gathered from
paper-and-pencil Likert-type scales. Two measures worthy of
attention, the Statistics Attitude Survey (Roberts and Bilderback 1980) and the Attitudes Toward
Statistics (Wise 1985), are described below.
Educators interested in using the SAS, ATS, or similar instruments
for diagnostic purposes, and researchers interested in
understanding factors affecting statistics learning, should be
aware of several fundamental problems with the SAS and ATS, and
also with the studies in which they were developed or employed.
These problems are identified later in response to four questions
of concern to statistics educators. Finally, a summary of the
state of attitude assessment in science education provides a
larger context for our evaluation of statistics attitude
assessment frameworks.

18 The SAS was developed by Roberts and Bilderback (1980) to improve the prediction of success in statistics courses beyond what was offered by general measures of affective reactions to mathematical domains. The instrument appears to have been designed as an "affective scale couched in statistical jargon" (p. 236) in line with the general item format and item phrasing employed by measures of math anxiety and attitudes towards mathematics (Fennema and Sherman 1976; Richardson and Suinn 1972). Roberts and Saxe (1982) suggest administering the SAS at the beginning and/or end of a statistics course. Students are given 33 statements regarding the perceived usefulness of statistics ("Statistics will be useful to me to test the superiority of one method over another"); personal competence in solving statistical problems ("I normally am able to solve statistics problems without too much difficulty"); beliefs about statistics ("I find statistics to be very logical and clear"); and affective responses to statistics ("The thought of taking another statistics course makes me feel sick"). Students respond using a 1-5 Likert-type scale ranging from Strongly Agree to Strongly Disagree. Roberts and Bilderback (1980) reported that the questionnaire is highly homogeneous (alphas in three samples all exceeded 0.93), and Roberts and Saxe (1982) corroborated that all items in the questionnaire load high on a single general factor.

19 The ATS was developed by Wise (1985) in an attempt to improve on perceived limitations of the SAS, mainly that many of the SAS items seemed to Wise to measure prior achievement in or exposure to statistics, rather than attitudes. (For example, one SAS item states, "I make a lot of errors when I calculate statistics problems.") The ATS scale is composed of two subscales: a 9-item Course subscale, measuring attitudes towards the course in which students are enrolled, and a 20-item Field subscale, measuring attitudes towards the use of statistics in their fields of study. As with the SAS, the students respond to each of the items using a five-point Likert scale, ranging from Strongly Disagree to Strongly Agree. Wise reported a correlation of 0.33 between the two sub-scales. He argued that since the ATS focuses on attitudes rather than students' previous experience with statistics, it is more appropriate than the SAS as a measure of changes in students' attitudes from the beginning to the conclusion of a statistics course.

20 A few other instruments relating to statistics anxiety have been developed but have not been widely tested or evaluated, and their quality is yet to be established. A 24-item instrument, Students' Attitude Toward Statistics (STATS) has been developed by Sutarso (1992), and a small-scale pilot study indicates that this instrument does not particularly differ from the SAS and ATS. Another instrument, the Coping Strategies Inventory for Statistics (CSIS) has been designed by Jarrell and Burry (1989). It appears to have no particular relevance to statistics, as all of its items evaluate general test-taking skills and coping strategies. The word "statistics" is never mentioned in any of the CSIS items and is only used in the title of the instrument.

21 As mentioned earlier, items in both the SAS and ATS deal with respondents' self-evaluation of several different issues: competence in dealing with statistical problems and calculations, competence in and attitudes towards dealing with mathematical tasks, interest in learning statistics, beliefs about the usefulness of statistics, and expectations regarding the relevance of statistics for their careers. While all these topics appear important and are aligned with the goals of statistics education, it is difficult to accept that students' views, feelings, or attitudes towards these diverse issues are all combined into one (SAS) or two (ATS) global scores. The findings reported by Roberts and Reese (1987) regarding the similarity of the SAS and ATS scales (a reported correlation of 0.88 between the two scales), their homogeneity in terms of both high internal consistency (alpha coefficients exceeding 0.90), and their simple factorial structure (a single general factor, though Wise [1985] earlier reported two factors), do not sit well with the diverse content of the items.

22 If students appear to answer such diverse items in more or less the same way, there may be underlying sources that cause responses to converge. As mentioned earlier, many students expect a statistics course to have a heavy mathematical bent (and such courses usually do; see Simon and Bruce 1991), or they may expect that "doing" statistics later on in their career will involve lots of math and computations. Thus, students' responses to the ATS or SAS may reflect mainly their attitudes towards mathematics, or beliefs about their own ability or knowledge in mathematics, which may then pass for attitudes towards statistics.

23 Not only do instruments such as the ATS and the SAS ignore the possibility that students who report negative attitudes towards statistics are actually reacting to the perceived mathematical component of statistics, they also do not enable their users to distinguish the effects of (a) generalized anxiety, (b) test anxiety, and (c) mathematics anxiety on students' responses. The nature of mathematics anxiety and its relationship to generalized anxiety and test anxiety have been the focus of much research in the mathematics education and educational counseling communities (Hembree 1990; McLeod 1992), yet are not reflected in the development of the ATS or the SAS. It is entirely possible that students' scores on the SAS and ATS, and the homogeneity of these scales, reflect influences of other types of anxieties or attitudes which are not unique to statistics.

24 An altogether different problem lies with the fact that the
SAS, ATS and similar questionnaires never ask students to
**explain** their answers to Likert-type items. This
limits interpretability of scores as it is unclear what motivates
students to answer as they do, whether answers represent a
negative attitude towards the domain in general or reflect a local
(classroom) reality to which the student reacts. Assume, for
example, that a student circled "Strongly Disagree" for the SAS
item "I would like to study advanced statistics." What does this
indicate? Should this response be taken as indicating a
**negative** attitude towards statistics? If so, is
the student afraid of the topic of statistics in general, or only
of "advanced" statistics? In this case, what does "advanced"
mean? Is it possible that the student is uninterested because she
believes (whether rightfully or not) that advanced statistics are
not required to graduate from college, or perhaps the student does
not think statistics would be required in her professional
career **after** graduating from college? Alternatively,
could the student be uninterested because she has heard
rumors about the difficult exams given by a professor whose class
is the only one available at present? We argue that very little
can be learned from responses to Likert-type scales without the
use of an open-ended procedure enabling respondents to elaborate
on their initial answers; further suggestions in this area are
made below.

25 We also have concerns about the meaning of the construct of
"usefulness of statistics" being measured by SAS items such as
"Statistics will be useful to me when I describe my professional
activities to other people" or by the ATS Field subscale. (Items
about the usefulness of **mathematics** in
out-of-school contexts similarly appear in many inventories of
attitudes towards mathematics.) Consider that students come to
statistics classes with different career goals and expectations
which may be at different levels of crystallization when the
assessment is administered. Some students have little or no
information about the actual content or requirements of their
future occupations. (See Dick and Rallis 1991
for a discussion of career development issues in making course
selections in mathematics.) Depending on their stage of career
development and the amount of factual information they have,
respondents might not think (rightfully or not) that statistics
will be useful in their future profession. Thus, a student's
responses to items assessing usefulness-of-statistics issues might
have little to do with
**feelings** or attitudes towards statistics as a
subject; instead, they may only reflect the student's vocational
development (Osipow 1973) or knowledge about
requirements or content of certain jobs. In sum, we argue that
it is difficult to establish the construct validity of Likert-type
items or scales measuring usefulness issues without further
information about the process or factors that account for a
response.

26 As mentioned earlier, one reason for developing the SAS and
the ATS was to aid in **initial** diagnosis of
students' attitudes towards statistics. In this context any
examination of the meaning of the scores generated by the ATS and
SAS (or other instruments using similar items) needs to consider
that **all** the items on these instruments include
statements using the word
**statistics**. (Examples from the SAS include "It
takes me a long time to understand a statistical concept" and
"There are so many statistical concepts to learn that I get
confused.")

27 How the word "statistics" is interpreted by respondents to
surveys of attitudes towards statistics is a major point for
concern. While the word "statistics" should be imbued with some
meaning for students who are **finishing** a statistics
course, many students who are just **beginning** their
first statistics course have little sense of what "statistics"
includes; to the extent that students attach some meaning to the
term "statistics," our concern is that students' generalized
expectations may reflect stereotypical, distorted, or partial
perceptions which may differ from student to student.

28 The potential fuzziness of the term "statistics" for many beginning students is illustrated by results from a preliminary study conducted as part of Project STARC, an NSF-funded project on the acquisition of statistical skills conducted at the University of Pennsylvania (Gal 1993; Gal and Wagner 1992). A group of twelfth graders from a prestigious private school in the Philadelphia area, all of whom were college bound and in the process of applying to high-level universities or colleges, were asked, "What do people study when they take a statistics course? What comes to your mind when you hear the term statistics?" The following quotes illustrate the range of responses obtained:

"When I hear the term statistics, I usually think of basketball statistics (% of shooting, number of rebounds, etc.) or survey statistics (as in 40% of teenagers hate peanut butter). I'm not exactly sure what people study when they take a statistics course, but I think it's along the lines of percentage and graphing, etc."

"I really am not sure what you learn when you take statistics. I guess it has to do with taking averages of things."

"Numbers and figures of surveys come into my head. I think of people having a boring life if they make a profession of it, because I know it's a lot more complicated than what I said."

"Although I have never taken a statistics course, I hear they are very difficult. Being a huge sports fan, when I think of statistics I think of how many goals or touchdowns someone has."

"I imagine a statistics course as boring and factual as math. Statistics are gathered data (1000 people live in PA), information good for newspapers, writers, and lawyers."

"Statistics is when someone takes the scores of many things, such as baseball statistics. Math is used a great deal in finding statistics."

29 Some of the above statements contain elements that reasonably portray some of what actually happens in statistics classes, but others give tenuous or incorrect information. Most convey some fuzziness regarding what "statistics" might be about, or a narrow view regarding life domains where statistics may be used. It is difficult to see how these students, who are positioned to begin a high-school or college-level statistics class, would effectively relate to attitude items which repeatedly use the word statistics, thus raising concerns regarding the diagnostic value of attitude surveys if used before the onset of statistics instruction. In fact, the attitude questionnaire itself may unnecessarily cause some students to begin to believe that statistics is very computationally-oriented and difficult.

30 As suggested above, one goal of statistics education is to
engender in students a positive outlook about statistics and its
uses, and confidence in themselves as (intuitive, fledgling)
statisticians. It is thus important to examine the ability of
current assessment instruments to detect **changes**
in students' standing on such issues. Considering the
"slipperiness," or subjectivity of some of the above target
constructs, or the multiplicity of factors affecting achievement
in a statistics class, it is surprising that researchers have so
far paid little attention to instrument sensitivity in this
regard.

31 We have argued earlier that students' standing on some of the constructs being measured by attitude instruments, such as beliefs about the usefulness of statistics, may change over time as a result of vocational maturation or increase in career-related knowledge. There is reason to suspect that such changes, if they occur, will develop gradually (Osipow 1973). In contrast, students' standing on some of the other facets being measured, such as affective responses or self-confidence regarding statistics, may be more labile and likely to fluctuate depending on changing circumstances and classroom events. Thus, interpretation of score changes needs to take into account the expected stability over time of the constructs being measured. Unfortunately, the developers of statistics attitude measures currently in use have not provided data about the instruments' test-retest reliability.

32 Even when score changes may be noticed from the beginning to
the end of a course, there is little information on how to
interpret such changes. A key question relates to the magnitude
and nature of a score change that would be considered meaningful
from an educational perspective. For example,
Roberts and Saxe (1982) report that SAS scores rose from
105.29 to 109.94 with a standard deviation of approximately 15 and
a maximum scale score of 165. Presently, no large-scale data exist
which describe score patterns within the general population of
students, and there are no norms comparable to those available for
instruments measuring **math** anxiety, such as the
MARS (Richardson and Suinn 1972) or the
Fennema-Sherman scales (Fennema and Sherman
1976). In the absence of such information, it is difficult
for teachers or researchers using statistics attitude surveys to
know if score changes (e.g., due to an intervention designed to
affect students' attitudes) are educationally, rather than
statistically, significant.

33 Assume, for example, that the **pre**-instruction
class mean on SAS items such as "I would like to study advanced
statistics," or "I find it easy to explain a statistics topic
to someone else" was 3.5 (given a 1-5 scale, where 1 is "strongly
disagree" and 5 is "strongly agree") and that the
**post**-instruction mean was 3.9. Assume further that
this pre-post difference was calculated for a large
introductory statistics class and is statistically significant.
What does the finding mean? Is it good news, bad news, or no news
for the instructor? (Even after the change, scores may still be
in a range indicative of negative attitudes or affect towards
statistics.)

34 Studies using statistics attitude surveys usually report correlational data or mean score changes, but not absolute score levels. Such data, while perhaps showing the existence of relationships between attitude scores and variables of interest, provide insufficient information about the nature of that which is changing over time. A basic requirement from studies of attitudes towards statistics is that they report absolute scores in addition to any other statistical data, so that the educational implications of any score change can be evaluated. Correlational data, as well as reported mean score changes, may be of limited value if they are not accompanied by data about trends of educational significance, such as the proportion of students whose scores stayed the same, improved (i.e., attitudes became more positive), or worsened (i.e., attitudes becomes more negative); such figures are masked when a single statistic is calculated on a whole sample.

35 There has been a long tradition of using Likert-type items
in questionnaires measuring attitudes towards mathematics and
science (Helgeson 1993). Instruments such as
the SAS or ATS, which build on this tradition, yield scores that
are easily reportable and are convenient to use for a broad
description of the outcomes of statistics education. Such
instruments have not been designed to provide diagnostic
information that can point to particular issues of concern to
**individual** students; therefore, we argue that present
measures of attitudes towards statistics have very limited
capability to inform teachers interested in improving the **process**
or **content** of (statistics) education through
either remedial interventions or preventive measures.

36 A key deficiency discussed earlier is that responses to Likert-type scales reveal little about the causes for answers. Especially when it comes to mathematics or statistics anxiety, which may negatively influence students' interest, motivation, and comprehension, it appears that Likert-type scales have very limited usefulness for identifying what individual students are anxious about, their beliefs about learning statistics that might be counter-productive, and what types of support or educational experiences might be useful for students.

37 Mathematics educators are beginning to recognize this fundamental deficiency in the design of measures of students' beliefs and attitudes towards mathematics; some (see McLeod 1992 for a recent review) are beginning to conduct interviews, lead focus group discussions, or ask students to write journals or histories of their present or past mathematical experiences to gain a closer look at the factors underlying students' outlook on their educational experiences in quantitative fields.

38 Above we expressed reservations about the quality of the instruments presently used to assess beliefs, attitudes, and affect towards statistics, and raised questions about the interpretability of results of such instruments. To put these comments in perspective, it might be useful to examine the state of the art in science education, a related, and presumably more mature, field of attitude assessment.

39 In contrast to about 12 studies and four instruments so far published in the area of assessment of statistics education, in a recent review, Helgeson (1993) notes that more than 700 studies have been published so far on students' attitudes towards science (including both attitudes towards science and scientific beliefs) and that more than 50 different instruments have been developed over the years. Major reviews of these studies and instruments repeatedly point to two problems: lack of conceptual clarity in defining attitudes towards science and other constructs used by researchers, and "technical" limitations of instruments used to assess attitudes. Helgeson (1993, p. 6) cites Germann (1988):

"First, the construct of attitude has been vague, inconsistent, and ambiguous. Second, research has been often conducted without a theoretical model of the relationship of attitude with other variables. Third, the attitude instruments themselves are judged to be immature and inadequate."

40 Helgeson (1993) further reviews results of an evaluation by Munby (1983) of the Scientific Attitude Inventory (SAI), then the most popular of the attitude instruments used in science education research. The SAI uses 60 items with a four-point, Likert-type response scale. An examination of the instrument itself and of 30 studies which used it by 1983 revealed that the instrument's validity was highly questionable. Munby (1983) further commented that the field of measuring attitudes was replete with instruments, but that these were being used in a rather "cavalier fashion," without attention to the various aspects of their validity and reliability.

41 We believe that much of what has been said about assessment of attitudes in science education is directly applicable to the emerging field of assessment of attitudes in statistics education. Work on assessing statistics attitudes so far has proceeded with little attention to the meaning of the complex constructs being measured (e.g., what do "attitudes towards statistics," or "statistics anxiety" mean?). Few visible attempts have been made to benefit from accumulating experience and from methodologies developed in other disciplines with similar assessment needs, such as in mathematics and science education. Research designs used by researchers studying attitudes towards statistics show little sophistication, and statistical analyses and presentation of results are often ill-suited to the analytic task. Studies so far do not appear to have produced information of much educational value about students' affective reactions to statistics learning, in part due to exclusive reliance on Likert-type scales. In fact, studies conducted so far serve as rather unimaginative and disheartening models for the skills we would like our statistics students to develop.

43 We find that current approaches towards assessing attitudes towards statistics are ill-suited for the tasks identified above, due to: (1) exclusive use of Likert-type scales, (2) the inclusion of items that are not appropriate for students who have not had extended experience with statistics, or who are not at a very advanced stage of their career development, (3) the tendency not to seek explanations from subjects for their answers, (4) the practice of using total scores which aggregate responses to different item types, and (5) inattention to the links between attitudes towards statistics and other constructs, such as attitudes towards mathematics, when interpreting results. These problems severely limit interpretability of obtained scores at both the personal and group level.

44 The important progress made by mathematics educators in clarifying the differences among emotions, attitudes, and beliefs, and between these and other related constructs such as anxiety, confidence, self-doubt, self-concept, or self-efficacy (see McLeod 1992), could provide an excellent starting point for statistics educators seeking to understand factors affecting their students' performance and learning. Further, the emerging literature on the role of non-cognitive factors in cognitive performance, including metacognitions, affect, and beliefs (Dweck 1986; Schoenfeld 1992), highlights the practical importance of attending to non-cognitive factors.

45 Progress in understanding the role of non-cognitive issues
and their effects on statistics learning must be supported from
two different but related directions. First, we must improve our
ability to assess students' attitudes toward statistics so that we
can understand the meaning as well as the existence of anxious or
uncomfortable feelings. Second, we must explore the interaction
between these negative attitudes and students' **beliefs**
about statistics as a field of study and about themselves as
learners and users of statistics. These issues are discussed
below.

46 Given the above criticisms, a necessary first step is to create a better instrument which would include fine-tuned subscales with an acceptable factorial structure. The instrument should enable separation of students' attitudes towards mathematics, test-taking, statistics, and perceived usefulness of statistics. Reliability and validity data for this instrument should be presented separately for students at different stages of academic careers, and for those majoring in different fields, where the role and relative importance of statistics can be estimated. Our analysis of deficiencies in current practices suggests, however, that little would be gained from additional development of "improved" assessment instruments, if these instruments continue to rely exclusively on Likert-type items.

47 We believe that as a minimal requirement, an assessment
instrument for initial diagnosis of students' attitudes and
beliefs towards statistics should combine the use of Likert-type
items with **open-ended questions**. This design
should enable students to **explain** what feelings,
attitudes, expectations, or beliefs underlie their responses to a
Likert scale; describe the intensity and frequency of specific
emotional responses; and elaborate on their source(s) or causes.
So that the validity of the scale will not be compromised by
interspersed requests for response explanations, students could
first respond to a series of statements using the Likert scale and
then, on a separate page, be directed to explain each of their
responses. Examples of common five-point Likert-type items
(Strongly Disagree to Strongly Agree) and suggestions to guide
students' explanations of their ratings are:

- I think statistics will include a lot of mathematics. (How do you know? What mathematical skills, if any, do you expect to use in this course? Would this create a problem for you?)
- Solving math problems in math class makes me anxious. (Why? What types of problems, if any, make you anxious? How anxious?)
- I was successful in the last math course I took. (What percentage of the material did you feel you understood? What did "success" mean for you?)
- I am nervous about taking statistics. (Why? What aspect(s), if any, of the course make you feel this way?)
- I am looking forward to taking statistics. (Why? What aspect(s), if any, of the course make you feel this way?)
- I think that statistics will be useful in my profession. (How do you know? Why do you assume statistics will or will not be useful? What profession are you considering?)

48 Alternatively, open-ended items in an initial assessment could ask students to express concerns they have about taking a statistics course and about the extent to which their prior academic background may assist or impede their learning of statistics; relate factors that may cause poor performance in this course; describe feelings about learning mathematical topics in general; outline expectations for the extent to which the course will involve mathematical work; explain motivation(s) for taking the course; or describe how the course may fit into future career plans, if any (after all, the course may be a requirement forced upon the student...). Possible open-ended "sentence completion" items might include:

- I think statistics is ... (e.g., useful, boring, frightening) ... because...
- I think statistics is about... (what topics? What skills?)
- I am taking this statistics course because ...
- I expect that for me, personally, statistics may be useful later for... (write "not at all" if you so feel)
- When I think about this course, I'm concerned that ... (write "not at all" if you so feel)

49 A different format which has been used to explore students'
feelings about learning mathematics and which could also be
used as part of an assessment of attitudes towards statistics,
includes a series of 12 to 15 pictures depicting faces with
various emotions (or just a list of emotion words), such as
"anxious," "puzzled," "fearful," "frozen," "interested,"
"indifferent," "confused," etc. Individuals can mark these to
express their feelings about mathematical or statistical
situations. (See Allan and Lord 1991for an
application with adult learners of math in basic skills programs.)
This type of assessment provides limited information about
students' feelings, but is useful to break out of the mold of
perceiving students' attitudes as lying across **linear**
paths, and of "attitude change" as moving students "higher" or
"lower" along such paths, as is the case when five-point Likert
scales are used. This form of assessment can be easily adapted to
assess students' feelings at various points during the class and
to detect students who develop high degrees of anxiety or
frustration. Such an assessment is especially useful if students
are asked to explain their answers and be specific whenever they
circle faces or labels that indicate some distress or
difficulty. (These responses can also serve as a useful source of
in-class data on which students can practice various data analysis
techniques.)

50 Statistics educators interested in a deeper understanding of how their students perceive statistics and statistics courses could opt for the use of structured interviews (Tobias 1994) or discussions in focus groups. A closer look at clinics for treating math anxiety among college students or adults engaged in continuing education, which are now emerging on various campuses around the United States (Hembree 1990 cites recently published reports), can point to additional formats for assessing students' attitudes and affective reactions to statistics learning. The information garnered may help a statistics department form policies regarding interventions or support for students with strong negative feelings.

51 The research on students' beliefs about statistics is much more sparse than that concerning attitudes towards statistics. Other than the commonly held belief that statistics is heavily mathematical, students' beliefs about statistics remain unexplored. This gap in our knowledge is disturbing since recent work on the relationship between beliefs and attitudes in mathematics learning (McLeod 1992) suggests that beliefs may be filters through which experiences and events are interpreted by learners.

52 Since students' beliefs towards statistics have generally not yet been explored, and given that many students identify statistics with mathematics, it is informative to first examine related research on beliefs about mathematics and mathematics problem solving. Schoenfeld (1992, p. 359) lists some of the typical student beliefs about the nature of mathematics and mathematical activity:

- Mathematics problems have one and only one right answer.
- There is only one correct way to solve any mathematics problem---usually the rule the teacher has most recently demonstrated to the class.
- Ordinary students cannot expect to understand mathematics; they expect simply to memorize it and apply what they have learned mechanically and without understanding.
- Mathematics is a solitary activity, done by individuals in isolation.
- Students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less.

These beliefs could be similar to students' beliefs about statistics, but there may also be beliefs unique to statistics and statistics education that have not been recognized so far.

53 The relationship between beliefs and attitudes such as anxiety is also being explored in the field of mathematics education. Carter and Yackel (1989) argue that mathematics anxiety is an appropriate response when certain beliefs are present. For example, if an individual believes that mathematics is a collection of rules and procedures, then success in mathematics is determined by one's ability to memorize the rules and procedures and produce them at appropriate moments in the problem-solving process. For routine exercises and practice problems, this belief system allows success and comfort. If an appropriate rule or solution path is not apparent during a problem-solving situation, however, then the learner is at a standstill since there is no mechanism in place for modifying and/or developing rules or procedures. This situation causes feelings of panic, inadequacy, and anxiety. On the other hand, some individuals believe that mathematics is "relational," that is, mathematical knowledge is an interconnected, meaningful network. These individuals are not afraid to enter their "mathematical network" and try to derive or develop an appropriate solution when the solution to a mathematical problem is not immediately apparent. Relational mathematics learners realize that there are many points from which to enter their networks of mathematical knowledge and will thus feel comfortable using their experiences and emerging knowledge as problem-solving tools.

54 To make the learning of statistics less frustrating, less fearful, and more effective, further attention by both statistics educators and researchers should be focused on beliefs, attitudes, and expectations students bring into statistics classrooms or develop during their educational experiences. Instructors can use assessments of attitudes and beliefs to understand students' presuppositions and, with continuous monitoring, to identify areas of frustration for individual learners, guiding the provision of supportive interventions. The discussion above suggests that the development of assessment instruments capable of providing relevant information of value for instructional and research purposes involves many challenges, both conceptual and methodological. Unless significant progress on such fronts is achieved, however, the vision of statistical literacy for all may remain out of reach for too many learners.

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Iddo Gal

Graduate School of Education

University of Pennsylvania

3910 Chestnut Street

Philadelphia, PA 19104

gal@literacy.upenn.edu

Lynda Ginsburg

Graduate School of Education

University of Pennsylvania

3910 Chestnut Street

Philadelphia, PA 19104

ginsburg@literacy.upenn.edu

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