A Study of Affective and Metacognitive Factors for Learning Statistics and Implications for Developing an Active Learning Environment

Carl Lee

Department of Mathematics, Central Michigan University Mt. Pleasant MI 48859 USA e-mail: carl.lee@cmich.edu

Aklilu Zeleke

Department of Mathematics, Alma College Alma, MI 48801 USA

e-mail: zeleke@alma.edu

Maria Meletiou-Mavrotheris Cyprus Ministry of Education

11 Achilleos St., 2370 Ayios Dometios Nicosia, Cyprus

e-mail: meletiu@spidernet.com.cy

Abstract The difficulties in teaching and learning introductory statistics have received considerable attention by educators and professional organizations. Factors associated with learning quantitative contents such as introductory statistics are complicated. Review of the research literature suggests that these may include the cognitive domain on the ability of learning, the affective domain on beliefs and attitudes, and the metacognitive domain on motivation and strategies of learning. The paper describes findings from a study, which investigated the effect of affective and metacognitive factors on student learning of introductory statistics, and discusses the implications for the development of more active learning environments. Keywords: Active Learning, Affective, Learning Styles, Metacognitive, Motivation

Introduction

In most higher education institutions, introductory statistics has become a general

education course or a quantitative literacy course. The difficulties in teaching and learning

introductory statistics have received considerable attention by educators and professional

organizations (e.g. Hogg, 1992). A series of calls for change in statistics education has been

made in recent years by both the American Statistical Association and the Mathematical

Association of America (e.g. Cobb, 1992; Moore, 1997; CUPM Report, 2001). The major

recommendations include: (1) emphasizing statistical thinking, (2) using more data and concepts,

less theory, and fewer recipes; (3) fostering active learning.

The recommendations included in the calls for change in statistics education point to the

need for not only a shift in emphasis from procedures and recipes to data (production,

visualization, analysis and presentation) and from formulae to concepts (in particular, variability

and statistical thinking), but also for the development and fostering of active learning

environments using appropriate technological and other instructional tools in order to motivate

and facilitate student learning. From the learning theory point of view, these recommendations

are closely linked to the four dimensions associated with learning – cognition, affect,

metacognition and environmental limitations (Hartman, 2001). While cognitive abilities are

necessary for performing a task, metacognition is critical to understanding how the task was

performed (Schraw, 2001), and affective factors like motivation and self-efficacy stimulate the

use of metacognitive strategies such as self-regulation (Pictrich & DeGroot, 1990).

Factors associated with learning quantitative contents such as introductory statistics are

complicated. The literature suggests that these may include the cognitive domain on the ability of

learning, the affective domain on beliefs and attitudes and the metacognitive domain on

motivation and strategies of learning (Garfield, 1995; Gal & Ginsburg, 1994; Lee, et al, 2002).

The intent of this article is to present findings from a study investigating the effect of affective

and metacognitive factors on student learning of introductory statistics, and to share our

experiences of using these findings to improve our instructional practices by developing active

learning environments. By contributing to the overall understanding of how the affective and

metacognitive domains are related to statistics teaching and learning, we hope to provide useful

information to statistics educators interested in developing more student-centered classroom

environments.

Review of Literature

Most studies on cognitive issues conducted in the past excluded affective factors from their

considerations. This was the result of the profound influence on educational psychology of

behaviorism, which had little interest in non-cognitive aspects of learning such as beliefs,

emotions, attitudes, and motivation (McLeod, 1992). However, due to the acceptance in recent

years of constructivist learning theories, affect has emerged as an important, inseparable from

cognition, aspect of learning (McLeod, 1992). The literature has shown that affective aspects of

learning (McLeod, 1991; Stage et al., 1998) such as beliefs, attitudes, motivation, expectations,

emotions, and learning styles (Sims & Sims, 1995) greatly influence learning.

The recognition of the central role of affect in learning has had a major impact in

mathematics education. Affective issues have received increased attention in teacher education,

curriculum development, and research on mathematical teaching and learning (McLeod, 1992).

Research of non-cognitive factors influencing mathematics learning has been extensive, covering

several aspects of the affective domain. A research area of particular interest to mathematics

educators has been the study of students’ attitudes towards mathematics and of their impact on

mathematical achievement (e.g. Fennema, 1989). The effect of students’ perceptions and beliefs

regarding mathematics on their performance is also an area extensively studied (e.g. Schoenfeld,

1989). Studies have also been conducted on factors affecting students’ motivation to learn

mathematics (e.g. Grouws and Lembke, 1996). The main message coming from the body of

research on affect is that affective elements accompanying a student’s thinking and problem

solving can significantly facilitate or hinder mathematics learning. In statistics education, the

importance of students’ attitudes, beliefs, expectations and motivation in learning is well

documented in Gal and Ginsburg (1994). Research studies have shown that students' attitudes

toward statistics are more positive when learning takes place in active technology-rich

environments than when instruction follows the traditional lecture-based approach (e.g. Lee,

1999b).

In addition to recognizing the importance of affect, modern cognitive theory also

acknowledges the importance of metacognition, that is, of students’ ability to monitor, evaluate,

and make plans for their learning (Flavell, 1979). Studies of students’ metacognitive strategies

have indicated that metacognitive capabilities such as self-regulation are keys to the development

of appropriate learning strategies (Rojas-Drummond, Hernández, Vélez & Villagrán, 1998).

The role of metacognition in mathematical problem solving has been investigated by

several researchers (e.g. Schoenfeld, 1987; Lester and Kroll, 1990). Empirical research suggests

that if students' attention is drawn, during instruction and evaluation, to the metacognitive

components of problem solving, then their mathematical performance will improve (Lester and

Kroll, 1990). Attaching attention to the metacognitive components of learning is particularly

crucial in the context of introductory statistics, where the complex nature of statistical knowledge

causes learning difficulties. Since the behavior of stochastic phenomena often runs counter to

people’s intuitions, development of the ability to monitor how decisions are taken under

uncertainty and to mentally step aside to reflect on the process of decision making (Pfannkuch,

1999), can help learners improve their stochastical reasoning. Also, since statistics is a relatively

new and unfamiliar field for most students, statistics instructors ought to facilitate the

development of metacognitive strategies so that learners can be more active, exploratory and

self-regulated during the comprehension-building process (Hollingworth and McLoughlin,

2001).

Research investigating the process of learning (American Psychological Association Board

of Educational Affairs, 1995; Alexander and Murphy, 1998) has suggested fourteen principles to

explain how learning takes place. These fourteen principles can be categorized into four

dimensions as (1) cognitive and metacognitive, (2) motivational and affective, (3) developmental

and social, and (4) individual differences. Success of learning is the result of all dimensions

interacting together. Active learning is a process that attempts to take all four dimensions into

account to facilitate student learning, by actively involving learners in their own learning (e.g.

Moore & Cobb 1995; Rautopuro 1999). The literature has provided some frameworks and

guidelines for creating active learning environments and curricula that promote meaningful

learning (e.g. Glasgow, 1997; McCombs & Whisler, 1997; Stage, Muller, Kinzie & Simmons,

1998, Lambert & McCombs, 1998). Meaningful learning is a process in which new information

is related to an existing relevant aspect of the individual's knowledge structure, under the

condition that the individual is willing to learn, and the material prepared by the instructor is

relevant to the learner’s experience and is composed of adequate new knowledge for the learner

(Novak, 1998).

Methodology

Mathematics educators have developed a variety of instruments for examining non-

cognitive factors affecting learning (e.g. McLeod, 1991). In an attempt to determine factors that

may be influential for developing an active learning environment for introductory statistics, we

adopted the suggestions made by Gal and Ginsburg (1994) by reviewing the literature available

in mathematics education, and identifying several well-validated instruments designed to study

students’ affective and metacognitive domains of learning. These include Fennema-Sherman’s

Mathematics Attitude Scales (1976), Kloosterman & Stage’s Students’ Beliefs about

Mathematical Problem Solving Survey (1995), Bigg’s Study Process Questionnaire (1987), and

Bessant’s Learning Preference Survey (1995). The survey instrument used in our study is a

collection of modified items selected from these validated instruments, with added extra items

for investigating students’ motivation, learning styles and study habits as these relate to

introductory statistics.

The survey instrument developed for the study consists of items related to (1) motivation,

(2) learning styles, (3) learning experience, (4) learning habits, (5) beliefs (6) attitudes, and (7)

anxiety, and also includes items inquiring some background information. Students have to

indicate their degree of agreement or disagreement with each item using a 5-point Likert-type

scale.

The survey was given at the beginning of each semester, to students that took introductory

statistics in two institutions in the Midwest region of the United States. The courses were taught

by Lee and Zeleke. Lee gave this survey to his students over a period of four years, and Zeleke

over a period of three semesters. In total, 402 students took the survey. These students were

mostly sophomores and juniors majoring in business-related areas of study. The prerequisite for

introductory statistics is a College Algebra course. The mathematics background of the large

majority of the students was at the College Algebra level, with a very small percentage having

also taken Pre-calculus.

We used the survey data we collected over the years to adjust our instruction and to

develop active learning environments that are suitable for our classes as part of our student

learning outcomes assessment effort. By examining the survey data each semester and through

our experiences of implementing these active learning environments, we learned some valuable

lessons that may also be of interest to other statistics educators.

We recognize that a Likert-type survey is not the optimal way of investigating the reasons

behind students’ choices. The interpretation of results should therefore – as pointed out in Gal

and Ginsburg (1994) – be treated with caution. Consequently, rather than forming hypotheses for

inferential purposes, we focus in this paper on providing a summary of the data and on

describing how we have used the findings for instructional improvements in our classes.

Findings

According to the research literature, teachers and students often have different perspectives

and expectations (Gordon, 1995). Instructors are experts who learn by taking advantage of their

specific knowledge domain, and who may also teach at a level that requires domain-specific

knowledge and experiences from students. Students, on the other hand, are novice learners. Their

learning strategies tend to be more general and they use their own previous experiences, which

are usually not domain-specific. It is therefore important for instructors to be knowledgeable of

students’ attitudes, beliefs, motivations, expectations, and study strategies, so that they can use

this knowledge to develop learner-centered instructional environments. In discussing the findings

of this study, we identify some important – based on students’ views – factors influencing

statistics learning, in the hope that this can help statistics educators to tailor their instructional

practices to their students’ affective and metacognitive characteristics.

(A) Learning Styles:

Rate how each of the following learning activities best suits you. [Use 1: Very True, 2: Somewhat True, 3: A little True, 4: Not At All]

% of ‘Very True’ or ‘Somewhat True’

In-class exercises 93.3% Hands-on activities 88.9% Listening to lecture and taking notes 84.9% Real-world projects 78.0% Cooperative learning 75.1% Following the textbook 59.8% Doing a lot of homework 54.2% Having a lot of quizzes 41.8%

Table (A) indicates that active learning strategies such as hands-on activities, real-world

projects, and cooperative learning are preferred by the vast majority of students. At the same

time, lectures and in-class problem solving exercises are rated as being equally effective for

learning. This suggests that a radical constructivist learning environment may not be as effective

for introductory statistics as one would think. A social constructivist active-learning environment

that engages students in guided group activities as described in Garfield (1995) might be more

appropriate. The social-cultural activity-based framework presented in Gordon (1995) also

emphasizes learning through a guided process. The PACE model developed by Lee (1999a) for

introductory statistics is designed to integrate hands-on activities, projects, cooperative learning

and both in-class and outside-of-class assessments. Some of the principles that underlie student

learning of statistics, discussed in Garfield (1995) and Lee (1999a) include the following: (1)

Students learn by constructing knowledge themselves through guided processes, (2) Students

learn by actively being involved in instructional activities, (3) Practice and feedback are essential

for understanding new concepts, (4) Students learn to value what they know they will be

assessed on. The responses given by students in our survey seem to be in accordance with these

principles.

Schroeder’s (1993) study, which investigated college-level students’ learning styles using

the Myers-Briggs Type Indicator (MBTI), indicated that over 60 percent of college students

prefer the sensing and extraversion learning mode, while only about 25 percent of faculty prefer

the same learning style. For mathematics faculty, the proportion is down to about 10 percent.

According to the MBTI, people who favor sensing and are extraversion learners are

characterized by preference for concrete experiences, moderate to high degrees of structure, and

linear learning. They often prefer practice-to-theory and are uncomfortable with abstract

reasoning. At the other extreme of the sensing and extraversion learners, are intuition and

introversion learners. These learners prefer theory-to-practice, favor abstract reflective patterns,

and tend to teach the way they learn best.

The difference in learning styles between students and faculty is stunning, and ought to be

an issue of concern for statistics instructors. When most of introductory statistics courses are

taught in mathematics departments by instructors with primary training in mathematics (Moore,

2000) who prefer theory-to-practice and abstract ideas, to students who prefer practice-to-theory

and concrete ideas, one can imagine students´ difficulties in learning statistics. The responses

above provide similar evidence that the majority of the students we surveyed also prefer sensing

and are extraversion learners. Hands-on activities, in-class exercises and cooperative learning fall

into practice-to-theory learning styles, while lecture and note-taking fall into the structured and

linear learning styles.

(B) Motivation Factors

In general I am motivated to study for a subject: [Use 1: Strongly Agree, 2: Agree, 3: Disagree, 4: Strongly Disagree]

% of ‘Strongly Agree’ or ‘ Agree’

When topics are interesting 97.1% When topics are related to my major 94.1% Because I want to get a better grade 93.5% Because I want to better prepare myself for the future 88.2% When my teacher cares about my learning 88.1% Because my friends know about this subject. 16.6%

Table (B) indicates that factors affecting students’ motivation to study for a subject

generally fall into four categories, as proposed by Lee et al. (2002). These are:

(1) Intrinsically motivating: Ninety seven percent of the students participating in the study

agreed that they are motivated to study when topics are interesting. An important

characteristic of intrinsic motivation is that motivation derives from self-willingness

and curiosity about the subject. When topics are interesting to a learner, this will

usually increase motivation to learn. The question is how instructors can make

introductory statistics more interesting. Hogg et al. (1992) pointed out that statistics is

often taught using meaningless numbers and with an emphasis on formulae and

computation. The consequence is the view shared by many students of statistics as

being boring and uninteresting. More emphasis on data and conceptual understanding

using hands-on activities and connecting the activities with students’ experiences are

ways of improving the motivational climate in statistics classrooms.

(2) Goal/career oriented: Over 93 percent of the students indicated that they are motivated

when topics are related to their major or when they want to get better grades. Since

grade and career are important motivators, teaching statistical concepts and methods in

ways closely related to students’ experience, and giving projects that will be useful for

their future careers motivates student learning.

(3) Value-oriented: Over 88 percent of the students agreed that they want to better prepare

themselves for the future. Better preparation for the future is not necessarily directly

related to students’ major, it however does add value to their credentials. Doing well in

statistics would be a good motivating factor for students.

(4) Socially/environmentally oriented: Over 88 percent of the students indicated that they

are motivated to study if teachers care about their learning. We think this is one of the

motivating factors where an instructor could really make a difference. Providing an

adequate number of office hours, showing enthusiasm and excitement about the subject,

interacting with students, assessing student learning on a regular basis and providing

prompt feedback, are some of the ways in which an instructor can provide a supportive

atmosphere which sends to students the message that he/she cares about their learning.

Introductory statistics is usually not considered by students as being directly related to their

major, nor is it a course to take in order to easily earn a good grade. Therefore, making statistics

interesting and relevant and showing students that we care about their learning of the subject, are

motivators that we, as instructors, should and can easily manage. They again seem to be closely

related with how to develop a learning environment that actively engages students in the learning

process.

(C) Expectations About The Course

What are your expectations about the course? [Use 1: Strongly Agree, 2: Agree, 3: Disagree, 4: Strongly Disagree, 5: Don’t Know]

% of ‘Strongly Agree’ or ‘Agree’

I expect tests to be the same as what was taught in class. 98.9% I expect the course to have a lot of formulae and mathematics. 93.6% I expect that material will be taught at such a level that I will understand everything in class.

91.7%

I expect the course to be difficult. 77.3% I expect to learn and understand every topic taught in the course. 72.3%

I expect this class to require more time than other classes. 59.1%

Students’ expectations regarding introductory statistics seem to gear toward grades and

effort. ‘I expect tests to be the same as what was taught in class’ and ‘I expect that material will

be taught at such a level that I will understand everything in class’ were highly rated by almost

all students. Introductory statistics may be considered useful for most students, it is not however

viewed as a course directly related to their majors or careers. Yet, grade is an important

motivator. However, the fact that less than 60 percent of the students expect to spend more time

studying for statistics than for other classes, even though over 77 percent of them expect the

course to be difficult, is an issue of concern. A set of learning goals and expectations clearly

spelled out by the instructor at the beginning of the course and stressed throughout the semester

would be critical for both instructor and students, since it would establish a system of

accountability that would hold students responsible for their performance and would encourage

them to conduct self-monitoring and evaluation of their learning.

(D) Beliefs about Statistics Rate the following statements. [Use 1: Strongly Agree, 2: Agree, 3: Disagree, 4: Strongly Disagree, 5: Don’t Know]

% of ‘Strongly Agree’ or ‘Agree’

The real world uses statistics to solve problems. 85.6% Statistics is a science comprised of a collection of formulae. 74.5% You must have a strong mathematical background to study statistics. 68.5% Most of the material learned in this class will be useful for my career. 54.7% Statistics is an interesting subject. 39.1% Memorization is the key to passing the course. 31.2% Statistics is only useful to those with science-related careers. 12.4% Statistics is an inquiry-based discipline that only mathematicians can understand.

10.8%

Statistics only deals with abstract mathematics. 8.1% Beliefs can be further divided into beliefs about the subject, about the self, about teaching,

and about the social context in which learning occurs (Mandler, 1984; Mcleod, 1991). The table

above summarizes students’ beliefs about statistics. Their beliefs about the use of mathematics

and formulae in introductory statistics are not surprising; they are nonetheless issues of concern.

Traditional introductory statistics instruction has been emphasizing use of formulae and teaching

of the subject similarly to mathematics. At the same time, however, over 85 percent of the

students believe that the real world uses statistics to solve problems and about 40 percent of them

consider statistics as an interesting subject. These beliefs are encouraging, and suggest that one

should teach introductory statistics using more real-world sets of tasks and fewer formulae. We

were happily surprised to find out that only 31 percent of the students agree on the use of

memorization as being the key to passing the course. This suggests that the instructor can and

should shift from a procedural/algorithmic approach to a problem-based approach that

emphasizes use of real-world data. The fact that just about 12 percent of the students believes

that statistics is only useful to those with science-related careers, is also encouraging. The

experiences students had during high school related to the use of descriptive statistics and

chance, as well as the frequent use of statistics in the media and in sports, may have some

influence on this belief. An instructional implication is that we should take advantage of

students’ beliefs regarding the practical usefulness of statistics by teaching them how the

discipline is applied in daily life, in order for them to become able to understand potential

misuses of statistics and to properly interpret statistical data appearing in the media.

(E) Beliefs about Self – Pride and Anxiety about Statistics and Mathematics

Rate the following statements. [Use 1: Strongly Agree, 2: Agree, 3: Disagree, 4: Strongly Disagree, 5: Don’t Know]

% of ‘Strongly Agree’ or ‘Agree’

I would be proud to be good at mathematics. 96.4% I am good at mathematics and expect to do well in this class. 66.4% Mathematics seems to be unusually hard for me. 37.2% I am afraid of statistics. 34.3% I usually feel confused and uncomfortable solving real-world problems. 30.0% (F) Attitudes towards Statistics and Mathematics

Rate the following statements. [Use 1: Strongly Agree, 2: Agree, 3: Disagree, 4: Strongly Disagree, 5: Don’t Know]

% of ‘Strongly Agree’ or ‘Agree’

I am interested in learning statistics. 70.7% I would do as little as possible in mathematics. 29.1% This course is a waste of my college credit hours. 10.0% All I want is to pass this course. 9.8%

Quantitative anxiety often has a negative effect on learning mathematics and other related

disciplines. Preconceived ideas about the nature statistics as a branch of mathematics made up of

complex algebra and formulae (Simon and Bruce, 1991), can cause anxiety in students and lack

of confidence in their ability to learn statistics (Gal and Ginsburg, 1994). More than one third of

the students expressed their anxiety about both statistics and mathematics. The fact that, in a

class of 30 students over 10 of them would be afraid of statistics, should be of concern to any

instructor. However, it is interesting that over 95 percent of the students agreed that they would

be proud of being good at mathematics and over 70 percent of them that they were interested in

learning statistics. These are very positive signs suggesting that an instructor could be very

influential in developing students’ confidence in relation to statistics and their willingness to

persist. If an instructor is caring and willing to create a supportive classroom environment that

guides students and engages them in active learning, their self-concept would improve, their

interest in the subject would generally increase, and they would be proud of being good at

statistics. The fact that about 30 percent of the students agree that they would do as little as

possible in mathematics is worrying, and it might be related to students’ lack of confidence in

their ability to learn statistics since research findings indicate that lack of self-confidence often

leads to a reluctance to try (Cross and Steadman, 1996). On the other hand, this could also

provide an opportunity for statistics instructors to engender in students more positive attitudes

about statistics and to reduce their anxiety about the subject, by using hands-on activities,

projects, and real-world data that are useful and interesting to learners.

(G) Reasons of Getting a Good Grade in a Quantitative Course

I do well in a quantitative course because: [Use 1: Strongly Agree, 2: Agree, 3: Disagree, 4: Strongly Disagree, 5: Don’t Know]

% of ‘Strongly Agree’ or ‘Agree’

I work hard 88.0% The teacher teaches well 87.4% I am good at mathematics 65.7% Just a matter of luck 21.3% I don’t know what happens 17.2%

(H) Reasons for Getting a Bad Grade in a Quantitative Course

I did poorly in a quantitative course because: [Use 1: Strongly Agree, 2: Agree, 3: Disagree, 4: Strongly Disagree, 5: Don’t Know]

% of ‘Strongly Agree’ or ‘Agree’

I didn’t study hard 51.8% I was careless 48.2% The teacher didn’t teach well 47.6% I am not good at mathematics 28.5% Just a matter of bad luck 9.5%

We did not specifically ask students to rate the statements above in the context of statistics,

since introductory statistics is the first statistics course for most students. Instead, we focused on

students´ experience with quantitative courses they had taken in the past. Nevertheless, students’

responses seem to agree with our observations over the years in statistics classes. It is not

surprising that most students are quick to claim that they are either good at mathematics or work

hard for their good grades, but less willing to admit that they did not study hard or are not good

at mathematics. However, we also see that students give credit to their instructors for good

teaching. This is another indication that instructors could exert big influence on student learning.

(I) Beliefs about Teachers and Teaching -Students’ View of a Good Teacher and Good In-

class Learning Strategies

A good teacher would: [Use 1: Strongly Agree, 2: Agree, 3: Disagree, 4: Strongly Disagree, 5: Don’t Know]

% of ‘Strongly Agree’ or ‘Agree’

Guide student learning 97.5% Use different strategies to help students 97.2% Use procedures and algorithms 63.7% Give answers right away 17.3% I will learn better if: [Use 1: Strongly Agree, 2: Agree, 3: Disagree, 4: Strongly Disagree, 5: Don’t Know]

% of ‘Strongly Agree’ or ‘Agree’

I am an active participant 89.5% Information is presented in a variety of formats 80.5%

Students’ views of a good teacher are overwhelmingly associated with how well an

instructor can facilitate learning by using multiple strategies, guiding students, and creating

active learning environments that help them to discover and construct knowledge. However, the

fact that over 60 percent of the students agreed that the use of procedures and algorithms is a

characteristic of an effective teacher may be an issue of concern when teaching statistical

concepts. It seems that traditional views about the nature of mathematics are imported into

statistics instruction, affecting students´ perceptions about the subject and the way it should be

taught. At the same time, students indicated that they learn better if they actively participate in

the learning process and if information is presented in a variety of formats. This suggests that

when leading students to discovering and constructing statistical knowledge, the instructor

should use various strategies and provide guidance to students in order for them not to get lost in

the process of acquiring new knowledge. He/she should facilitate the learning process by helping

students to get the large picture of the new concepts and how they are associated with the

concepts previously learned. This is in agreement with our previous observation about the

importance of a ‘guided’ constructivist learning environment for introductory statistics.

(J) Study Strategies

Rate the following statements that describe your study habits. [Use 1: Most of the Time, 2: Sometimes, 3: Occasionally, 4: Not At All]

% of Most of Time

I prepare for tests by reviewing problems that we did in class and trying to remember the different steps for solving the problems.

82.7%

When I do homework, I try to solve the problem first. If I cannot do it, I go back to the notes, handouts, and/or book.

77.9%

I study a subject by reading notes, handouts and/or the book to make sure I understand the material.

57.2%

I do the homework one or two days before the due day. 45.2% I attend class, try to understand the material, and I do not need to study much after class.

34.2%

I study only when the examination is approaching. 31.8% I keep up with the topics taught to make sure I don’t fall behind. 29.3% When I do homework, I study the notes and material first and then try to do the problems.

18.7%

Cognitive skills are critical for knowledge acquisition, but equally important for managing

the learning process are metacognitive skills such as planning, monitoring and evaluation.

Student’s metacognitive learning strategies seem to be limited. We often hear from students that

they thought they understood the material in class, but when coming to work on homework

problems or tests they find that they do not have good understanding. The results in Table J seem

to provide an explanation regarding students’ lack of an effective study strategy. About 80

percent of the students reported that they prepare for tests by trying to remember the different

steps for solving problems. Over 75 percent of the students agreed that, instead of first trying to

thoroughly understand the material, they first try to do homework problems and study only if

they cannot solve them. Only 29 percent agreed they try to keep up with the topics taught in class

to ensure that they do not fall behind.

Wagner and Sternberg (1984) argue that teaching needs to emphasize metacognitive skills

because: (1) teaching contents will not transfer the domain knowledge in the long term until

students learn general principles and how to apply them over a wide variety of tasks and

domains; (2) the majority of students have a history of blindly following instructions and have

not acquired the habit of questioning and going beyond what is taught in class; (3) students have

problems in determining the difficulty of a task, effectively monitoring their level of

comprehension, planning ahead, monitoring their effort, and using all relevant information.

These characteristics are more evident in novice learners. Our findings seem to agree with the

argument made by Wagner and Sternberg. How one could assist students in developing

metacognitive skills is an issue of concern for statistics instructors. The PACE model proposed

in Lee (1999a) suggests closely monitoring and assessing students, and informing them about

their performance throughout the semester. A more recent development are online assessment

instruments that allow students to conduct self-monitoring and assessment of their learning

progress. Such tools could be useful in helping students develop more effective metacognitive

learning strategies.

Discussion

Findings of this study point to the diversity of learning styles, the importance of active

learning, and the critical role of the instructor in introductory statistics. Through the years of

collecting this survey data, we have made a variety of changes to our own instructional practices.

The major change has been a shift from traditional lecture-based instruction to an activity-based

active learning approach, called the PACE approach (Lee, 1999a). PACE stands for Projects,

hands-on Activities, Cooperative learning in a computer-equipped classroom, and Exercises and

assessment. Online technology has also allowed us to incorporate into our instruction tools for

students to conduct self-assessment whenever they choose to.

Based on findings from the survey study, our experience, and the learning principles set by

Garfield (1995) and Alexander and Murphy (1998), we identify five major components that we

think are important for developing an active learning environment when teaching introductory

statistics. These are:

(1) The learners: Students have different backgrounds, beliefs, attitudes, learning strategies,

and cognitive abilities. The comparison of learning styles reported in Schroeder (1993)

ought to be alarming for statistics educators. A shift from a traditional theory-to-practice

towards a practice-to-theory instructional approach would be an important step for statistics

instructors to take. The literature suggests that learning will be more effective when

students fully engage in the process of learning and are properly assessed. An active

learning environment that gives different students ample opportunities to participate in

learning and to construct and evaluate their own knowledge, under the guidance of their

instructor, seems to be a natural choice. Such an environment contributes towards

improved self-confidence and attitudes about statistics by making the subject less fearful,

less frustrating, and more interesting for learners (Gal, Ginsburg, and Schau 1997).

(2) The instructional tasks: People have several misconceptions about statistical and

probabilistic concepts that are often extremely difficult to change (Shaughnessy, 1992).

Since for most students the introductory statistics course is most likely the only statistics

course they will ever take, development and implementation of instructional tasks that

facilitate meaningful learning and help students overcome their misconceptions about the

stochastic is of utmost importance. Constructive use of technology has an important role to

play in this regard. Statistical software traditionally used in introductory statistics courses

afford computing power and ease the carrying out of complicated computations which

would otherwise take a lot of time or would simply not be possible to do in class. However,

learning about uncertainty or overcoming misconceptions about the stochastic often

requires more than calculation and interpretation of results; understanding the process

through which uncertainty occurs and the role that chance plays is also essential. A variety

of new technologies have been recently developed, with emphasis not on outcomes but on

helping students understand the process of statistical investigation. These new technologies

support content and pedagogy by affording learners with tools they can use to construct

their own conceptual understanding – tools for not only data display and visualizations, but

also thinking and problem solving. Use of such software in the statistics classroom,

supports active knowledge construction by "doing" and "seeing" statistics in a powerful and

flexible learning environment (Ben-Zvi, 2000), and provides opportunities for students to

reflect upon observed phenomena, and develop their metacognitive capabilities. Examples

include the dynamic learning environment Fathom (Finzer, 1999), online JAVA applets,

online assessment tools (Garfield, delMas and Chance, 2002), etc.

(3) The context in which learning occurs: Among the major differences between mathematics

and statistics is the fact that statistics usually involves real-world contexts while

mathematics is often taught as strictly involving numbers, as well as the fact that statistics

deals with stochastic uncertainty while mathematics deals with deterministic exactness. If

statistics is learned similarly to learning mathematics, it would be difficult to relate it to

real contexts, which always involve uncertainty. The expectation that students will transfer

the understanding obtained through coins, dice, and games of chance to everyday contexts

seems to be a naïve assumption, as research studies (e.g. Pfannkuch and Brown, 1996;

Meletiou-Mavrotheris and Lee, 2002) have indicated. Students bring to each situation a

great variety of prior beliefs, conceptions, and interpretations that instruction ought to take

into account. Therefore, when developing an active learning environment for introductory

statistics, one should always keep in mind the social and cultural context behind statistical

concepts. This component echoes the recommendation regarding use of more real-world

data and problems included in Cobb (1992).

(4) The learning environment: In order to help students learn about practical uses of statistics,

instructors should make use of real-world data and activities. Statistics instruction should

be based on contexts directly connected to students’ experience, since adequate statistical

reasoning requires more than understanding of different ideas in isolation. It demands

“integration between students’ skills, knowledge and dispositions and ability to manage

meaningful, realistic questions, problems, or situations” (Gal and Garfield, 1997, p. 7).

When real-world tasks are used, students are more motivated to engage in the process of

discovering the solutions. Hence, it is only natural that statistics be taught in an active

learning environment. A variety of exemplary activity-based curricula are available (e.g.

Rossman and Chance, 2001; Scheaffer, et al., 1996).

(5) Assessment: Regardless of how well we think we are teaching our students, if they do not

acquire the necessary content knowledge, instruction cannot be effectively carried out.

Therefore, it is essential for students to be assessed throughout the learning process, using a

variety of assessment tasks and multiple-forms of assessment that complement each other,

in order to determine the degree to which the intended learning objectives and outcomes

have been accomplished. Both formative and summative assessment, are important.

Formative assessment allows the instructor to monitor student learning throughout the

duration of the course, whereas summative assessment allows evaluation of student

performance at the completion of the course. Online assessment tools, for example, can be

used to provide real-time assessment and immediate feedback to students (Garfield, delMas

and Chance, 2002) so that they can continuously evaluate and monitor their learning

progress. Garfield and Gal (1999) offer a detailed discussion of the challenges and

directions in assessing student learning of statistics.

The survey data we have collected provide other additional information that is also worth

mentioning. The following list summarizes some of the study findings:

a. Students’ learning experiences are greatly influenced by the instructor.

b. Students’ attitudes and motivation can be improved by the instructor.

c. The most important motivation factors are major, fun, grade, instructor’s caring for

students, and career.

d. Factors affecting students’ motivation to study for a subject fall into four types:

intrinsically motivating, goal/career oriented, value-oriented, and

socially/environmentally oriented.

e. The five most effective learning tools according to students are: (1) in-class exercises, (2)

hands-on activities, (3) lecture, (4) real-world projects, and (5) cooperative learning.

f. Most students study by memorizing steps and by attempting to solve problems without

first trying to understand the material.

g. Most students indicate that they learn better when actively participating in the learning

process, and when learning through a variety of instructional approaches.

Implications for Instruction

Too often, instructors focus solely on cognitive strategies – how to build student

knowledge or their ability perform a task – and ignore the affective and metacognitive domains

of learning. However, in addition to helping students acquire content knowledge, instructors

ought to also help them develop the skills for self-regulation, self-monitoring and evaluation of

their learning progress. They also ought to work towards improving students’ affective reactions

and attitudes towards statistics since, as findings of this study indicate, non-cognitive factors

might often contribute to students' difficulties in learning statistics (Gal and Ginsburg, 1994).

Clear and concise learning objectives and expectations for the course are the first step

towards helping students to conduct self-monitoring and evaluation. Moreover, as already

pointed out, most students are sensing or extraversion learners, while most faculty are intuition

or introversion learners (Schroeder, 1993) who tend to teach the way they learn, not the way

most students learn. There is a need for statistics educators to incorporate a variety of

instructional approaches in order to motivate students with different learning styles.

In an active learning environment, teaching is a process of guiding and facilitating the

development of meaningful learning. All students are encouraged to actively participate in

instruction. This improves students’ attitudes and maximizes learning outcomes. Effective

instruction usually integrates several teaching pedagogies in order to motivate and facilitate

learning at the individual level. The pedagogical tools employed include the use of real-world

data and hands-on activities, use of teamwork, emphasis on conceptual understanding, and

provision of adequate reinforcement. Advanced learning technologies also provide a means for

making problem-based active learning possible.

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