assessing a mathematical inquiry course: do students gain an appreciation for mathematics?

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Assessing a MathematicalInquiry Course: Do StudentsGain an Appreciation forMathematics?Barbara B. Ward , Stephen R. Campbell , Mary R.Goodloe , Andrew J. Miller , Kacie M. Kleja , EninkaM. Kombe & Renee E. TorresPublished online: 15 Mar 2010.

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Assessing a Mathematical Inquiry Course: DoStudents Gain an Appreciation for Mathematics?

Barbara B. Ward, Stephen R. Campbell, Mary R. Goodloe,

Andrew J. Miller, Kacie M. Kleja, Eninka M. Kombe,

and Renee E. Torres

Abstract: This study evaluates changes in students’ performance and appreciation

for mathematics as the result of taking a required general education course designed

for calculus-ready students. The purpose of the study is to determine whether the goals

of this rigorous general education course are being met. Changes in students’ perfor-

mance are measured by comparing grades on a pre-course and a post-course test.

Changes in students’ appreciation of mathematics are measured by scores on a pre-

course and post-course survey, including open-ended questions concerning beliefs

about mathematics. The study reveals a significant increase in grades on the post-

course test, indicating gains in students’ performance. It also reveals a significant

increase in scores concerning creativeness, indicating a greater appreciation for the

creativity involved in doing mathematics. The study reveals a significant decrease in

scores on questions concerning attitudes, indicating a decrease in students’ feelings

about mathematics as it relates to them personally. The open-ended responses show an

increase in students’ appreciation for proofs and for the role of mathematics in the

world. This report concludes with a discussion of some of the perceived benefits and

challenges of including a rigorous, non-calculus mathematics course in the general

education requirements for calculus-ready students.

Keywords: Quantitative literacy, general education mathematics course, mathemati-

cal inquiry, student attitudes, student beliefs, assessment.

Portions of the preliminary results of this study were presented at the Joint Meeting of

the Mathematical Association of America and the American Mathematical Society in

January 2006 and at the National Conference on Undergraduate Research in April 2006.

Address correspondence to Barabara B. Ward, Department of Mathematics and

Computer Science, Belmont University, 1900 Belmont Blvd., Nashville, Tennessee

37212, USA. E-mail: [email protected]

PRIMUS, 20(3): 183–203, 2010

Copyright # Taylor & Francis Group, LLC

ISSN: 1051-1970 print / 1935-4053 online

DOI: 10.1080/10511970801992921

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1. INTRODUCTION

The Department of Mathematics and Computer Science at Belmont University

offers Mathematical Inquiry, one of four mathematics courses students can

take to satisfy the university’s general education core mathematics require-

ment. None of these four courses is a traditional algebra or calculus course. As

the name suggests, Mathematical Inquiry is designed to invite students into the

process of thinking deeply about selected mathematical concepts, with the

intention of further developing their abstract and critical thinking skills, expos-

ing them to the breadth of the mathematical enterprise, and hopefully enhan-

cing their respect and appreciation for the subject.

The perceived need for this course grew out of observations made over

a period of many years. We observed a distressingly large number of

students who came to college and graduated, staunchly maintaining atti-

tudes and misconceptions about mathematics that seem harmful to their own

intellectual progress and to the progress of mathematics in our society.

Some of these attitudes and misconceptions about mathematics include:

� The ability to succeed in mathematics is primarily an inherited trait. You

are either gifted for it or not. Most people aren’t. I’m not.

� Mathematics is a dead subject. Everything worth knowing was discovered

hundreds of years ago.

� Mathematics may be useful, but only for a few geeky jobs. My parents

were successful without it. I will never use it.

� Doing mathematics requires one to use the right side of his/her brain. I’m

creative and therefore I only use, and should only be expected to use, the

left side of my brain. (About 40% of our students major in one of the fine

arts or in Music Business.)

� Mathematics is about memorization of formulas and calculating with numbers.

� Mathematics consists entirely of algebra, geometry, and calculus.

In addition to these attitudes and misconceptions, some students were

bored and unmotivated because they already knew the material contained in

the general education mathematics course(s) they were required to take.

Others were intimidated by a required mathematics course that covered the

same material they had failed to master in high school, but which they were

now expected to absorb at a much faster pace.

We wanted to develop a general education mathematics course that

would address at least some of the attitudes and misconceptions listed

above and that would extend its appeal to a larger audience. We felt it should

have some of these characteristics:

� does more than repeat topics in algebra that students have taken in high school

� engages students in a subject that is obviously alive and well

184 Ward et al.

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� explores historical achievements and current topics

� examines a wide spectrum of topics showing something of the breadth of

mathematics

� requires creativity and exploration above memorization of facts/formulas

� encourages critical thinking and develops problem solving skills

� examines various strategies used in mathematics, including proof and

experimentation

� incorporates inquiry as a key feature of the course

� includes some significant applications

� offers a variety of activities and measures for evaluating student performance.

The opportunity to implement a course that deals with some of these issues

arose in fall 2004 when Belmont University decided to implement a new

general education program: BELL (Belmont Experience: Learning for Life)

Core. The mission of the BELL Core is to create lifelong learners—people who

might specialize in a certain career but feel compelled to expand their thinking

in other disciplines. During the process of developing the BELL Core, our

university colleagues agreed to require all students, including mathematics

majors, to take a general education mathematics course that is neither algebra-

nor calculus-based. This was a very significant step. Mathematical Inquiry is

one of four courses we developed and now offer to satisfy this general educa-

tion core mathematics requirement. The others are Introduction to

Mathematical Reasoning, Introduction to Computer Science, and Mathematics

for Elementary School Teachers (for elementary education majors only).

Mathematical Inquiry is designed to provide an appropriate level of

challenge for students who have previously demonstrated mathematical abil-

ity as represented by a Mathematics ACT score greater than or equal to 25, a

Mathematics SAT score greater than or equal to 570, or a comparable

university mathematics placement test score. (An Introduction to

Mathematical Reasoning is the core mathematics course with similar goals

for students who do not have the prerequisites to take Mathematical Inquiry.)

Course experiences range from the study of classical problems to con-

temporary mathematical topics and from theoretical proofs to real world

applications. The specific content of the course varies by instructor, but

usually consists of units selected from the following: problem solving,

number properties, symbolic logic, inductive and deductive reasoning, count-

ing principles, infinity, paradoxes and proofs, Fermat’s Last Theorem, dis-

crete dynamical systems, chaos and fractals, recurrence relations, risk, exotic

geometry, and voting methods. The development of the course is ongoing.

Though the topics may differ from section to section, the common theme

prevails: to develop mathematical thought at a rigorous and challenging

level. Students have described the course as a mathematical counterpart to

First-Year Seminar, Belmont’s required writing-intensive general education

course designed to foster critical thinking.

Assessing a Mathematical Inquiry Course 185

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The primary purpose of the present study is to determine whether or not

students who take the Mathematical Inquiry course show a consequent increase

in appreciation for mathematics as a creative and useful human endeavor. In

addition, the developers of the BELL Core identified improved quantitative

reasoning as a key learning goal of the general education experience and speci-

fied that quantitative reasoning should be measured in terms of problem solving

and critical thinking abilities. Hence, a secondary purpose of the present study is

to determine whether Mathematical Inquiry contributes to the development of

the problem solving and critical thinking abilities of students who take the course.

Our study addresses these two purposes by means of the following

questions about Mathematical Inquiry:

Upon completion of the course, is there a significant difference in the students’:

1. problem solving and critical thinking abilities?

2. attitudes (opinion of mathematics as it relates to them personally) toward

mathematics?

3. ideas about mathematics in general, including the creativity required for

doing mathematics?

4. concepts of the usefulness of mathematics?

5. beliefs about the role of proofs in mathematics?

6. beliefs about the role of critical thinking in mathematics?

7. beliefs about the usefulness of mathematics?

Answers to these questions were obtained by gathering information from

students in three different ways, before and after they took the Mathematical

Inquiry course. Responses to Question 1 are measured by student grades on a

pre/post-course test; responses to Questions 2–4 are measured by scores on a

pre/post-course survey; responses to Questions 5–7 are measured by written

answers on a set of pre/post-course open-ended questions.

2. PREVIOUS RELATED RESEARCH

2.1. Attitudes Toward and Beliefs About Mathematics

Numerous scales have been developed to measure students’ attitudes toward

mathematics. The Fennema-Sherman Attitude Scale [4] which measures

confidence, usefulness, mathematics as a male domain, and teacher percep-

tion, was the hallmark of mathematics attitude scales in its time. Schoenfeld’s

Questionnaire on Attitudes about Mathematics [14] evaluates students’ atti-

tudes as well as attributions of success or failure, perceptions of mathematics,

and views of mathematics as a discipline. When Schoenfeld administered the

questionnaire to students, it revealed a positive relationship between beliefs

about mathematics and mathematical performance.

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As in the present study, Schoenfeld’s survey, which consisted of Likert-

scaled as well as open-ended questions, resulted in contradictory results. He

states, ‘‘. . .students simultaneously claim that ‘‘mathematics is mostly mem-

orizing’’ but that mathematics is a creative and useful discipline in which

they learn to think’’ [14, p. 338]. Schoenfeld goes on to suggest a resolution

to this contradictory pattern: ‘‘There is no contradiction between the two

notions if the former refers to the mathematics that takes place inside class

rooms and the latter refers to the mathematics that (at least hypothetically)

takes place outside them’’ [14, p. 346]. The present study similarly found that

students who completed Mathematical Inquiry gained an enhanced under-

standing of the creativeness and significance of mathematics as it relates to

the world, but not as it relates to them personally in everyday life.

In an overview of the previous 25 years of attitude research reported in

the Journal for Research in Mathematics Education (JRME), Mcleod [9]

wrote, ‘‘In the early years of the JRME, research on affect focused on

attitudes toward mathematics, especially student responses to the subject as

taught in schools.’’ He further stated, ‘‘Studies of attitudes soon broadened to

include research on beliefs about mathematics and more intense emotional

reactions to the subject’’ [9, p. 637]. The move from measuring only attitudes

toward mathematics to measuring beliefs as well as attitudes resulted in

changes in the methodologies of research and teaching.

Carter and Norwood [2] used two surveys to study the relationship

between teacher and students’ beliefs about mathematics. Their results suggest

that teachers’ beliefs, and consequently their teaching methods, can influence

students’ beliefs about mathematics. Included in Quantitative Reasoning for

College Graduates: A Complement to the Standards [7] is a recommended

survey entitled ‘‘Beliefs about Mathematics and Problem Solving’’. The ques-

tions on this survey further support the idea that students’ beliefs about mathe-

matics could be instrumental in their learning and appreciation of mathematics.

The majority of initial attitude research studies were conducted on

elementary, middle, or secondary school students. However, the recent

emphasis on quantitative literacy for college students has created a need for

research in post secondary courses. The Committee on the Undergraduate

Program in Mathematics (CUPM) Curriculum Guide [8] includes recom-

mendations not only for courses in the mathematics major, but for college-

level mathematics courses for all students, even those taking just one course.

The single mathematics course a student takes in his or her first year of

college can influence his or hers entire view of the discipline. In a carefully

designed first-year course, the student could develop an appreciation for the

creativeness in mathematics and become a life-long learner, one who realizes

the value of problem-solving, critical thinking, and analytical skills in every-

day life. Such a course should not be a repeat or continuation of secondary

school mathematics courses. In an article in the Mathematical Association of

America (MAA) publication, Quantitative Literacy: Why Numeracy Matters


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for Schools and Colleges, Madison [6, p. 57] states, ‘‘The mismatch in the

articulated curriculum between school and college consists primarily of a

narrowing of the broader school mathematics to a limited set of introductory

college courses dominated by algebra and pre-calculus.’’

In the present study students qualify for Mathematical Inquiry if they

are calculus-ready. All qualified first-year students at the university, in-

cluding those majoring in mathematics, are strongly encouraged to enroll

in Mathematical Inquiry instead of one of the less rigorous options.

Consequently, students in the course are already quantitatively competent

and most likely understand the usefulness of mathematics as it relates to

concrete situations in their previous high school courses. Nevertheless, do

they have a true appreciation for the creativeness and importance of mathe-

matics as a discipline and, if not, can they gain those beliefs in a single

introductory course?

Several research studies similar to the present study have used surveys to

measure student attitudes and beliefs in first-year college courses. At Illinois

State University the philosophy that guided the development and assessment of

the course Dimensions of Mathematical Problem Solving was, ‘‘what and how

our students feel about learning mathematics affects what and how they learn

mathematics’’ [13, p. 192]. Assessment instruments, administered at the begin-

ning of the semester, consisted of the Likert-scaled Learning Context

Questionnaire [5] and a newly developed open response Survey on

Mathematics. The results of the questionnaire led to fewer lectures on solving

problems and more guidance to help students develop their own mathematical

processes. Responses on the survey revealed that students thought that mathe-

matics involves problem solving using numbers, and there is only one correct

solution. Students also thought that the role of the teacher is to show, tell,

explain, and answer questions and the role of the student is to listen, ask

questions, and take notes. The developers of Dimensions of Mathematical

Problem Solving concluded that one focus of their course would be to

allay these misconceptions about the respective roles of student and teacher in

learning mathematics, thus making the students more independent learners [13].

A pre/post study conducted at Mount Mary College evaluated the math-

ematical disposition, skills, and attitudes (using the constructs of confidence,

anxiety, persistence, and usefulness) of students in their introductory courses.

The average attitude scores on a Likert-scaled questionnaire increased for all

constructs. Additional written responses explaining students’ choices on the

attitude questions revealed that the courses positively changed their percep-

tions about their ability to use mathematics [1].

The previous studies seem to indicate that students’ appreciation for and

personal beliefs about mathematics are still transforming when they enter

college. The present study examines, via Likert scaled survey questions,

students’ appreciation by assessing their feelings about:

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1. mathematics as it relates to them personally

2. how useful mathematics is in today’s world

3. creativity in mathematics as a discipline.

Analysis of the results reveals a decline in feelings toward mathematics as it

relates to them personally, yet an improvement in feelings about creativity in

mathematics as a discipline. These contradictory results, similar to the results of

Schoenfeld’s study [14], signify a distinction between students’ personal feelings

about mathematics and their feelings about creativeness in mathematics as

defined in the present study. There was no difference in students’ feelings

about how useful mathematics is in today’s world. A significant improvement

was found in students’ beliefs about mathematics, as assessed by open-ended

question responses. We go into more detail on our results later.

2.2. Studies Assessing Quantitative Reasoning Competency UsingPre- and Post-tests

In addition to the aforementioned studies of student attitudes and beliefs,

other studies similar to ours focused on the assessment of quantitative reason-

ing competency using pre- or post-tests. At King’s College the comparison of

pre-and post-tests in a liberal arts mathematics course revealed that more

progress was made in a section of self-identified weak mathematics students

than the other sections. The results of the pre-and post-tests allowed instruc-

tors to make positive changes in the course to enhance learning [11].

At Northern Illinois University, post-tests to assess the seven courses in

their quantitative literacy program led to curricular changes in the courses, an

improved placement test, and an idea for a required senior project involving a

quantitative component [15]. At Virginia Commonwealth University a pre-

and post-test was used to assess quantitative reasoning skills of their general

education mathematics courses. The increase in post-test scores revealed that

their courses are helping students develop quantitative reasoning skills,

although not at the level they had desired in their introductory course [3].

The present study found a significant increase in post-test scores indicat-

ing that a gain of knowledge and a degree of competency in quantitative

reasoning was achieved. We go into more detail on this result later.

3. METHOD

3.1. Description of Subjects

The study was conducted in the fall semester 2005 at Belmont University by

a team of researchers comprised of three students, one supervising faculty


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member, and three instructors of Mathematical Inquiry. The sample consisted

of 72 first-year students, 45% male and 55% female, who were enrolled in

five sections of Mathematical Inquiry taught by the three instructors on the

research team. As mentioned before, students in Mathematical Inquiry are

typically well-qualified in mathematics. All subjects in the study were con-

ventional, full-time students between 18 and 20 years of age. Students were

not randomly assigned to the sections, but the distribution of demographic

(major, gender, mathematical background) characteristics were similar. All

sections had similar syllabi, schedules, grading rubrics, and covered the

topics that are the focus of the questions on the pre/post-course test.

3.2. Pre-course and Post-course Survey

During the first class period students completed, read and signed the

informed consent and then had 20 minutes to take the pre-course survey.

The survey, adapted from a survey included in Indicators of Quality in

Undergraduate Mathematics, a National Science Foundation-funded project

[16], consisted of a section of multiple choice demographic questions followed

by 20 questions concerning attitudes, usefulness, creativeness, and proofs.

Attitude questions were designed to measure respondents’ feelings about

their personal relationship with mathematics. Usefulness questions concerned

the use of mathematics in everyday life. Creativeness questions were designed to

portray the respondents’ feelings about creativity in mathematics as a discipline.

Proofs questions concerned feelings about the importance of studying proofs.

(Note: In this article, the words attitude, usefulness, and creativeness have

technical meanings that are detailed as constructs in Appendices B and C.)

The survey was measured on a five-point Likert scale from 1 (strongly

disagree) to 5 (strongly agree) and consisted of ten positively stated and ten

negatively stated questions. The negatively stated questions were reverse-

coded (1 to 5, 2 to 4, 3 to 3, 4 to 2, 5 to 1) so a higher Likert score implied a

more positive attitude on the question.

Cronbach’s Alpha [12] was calculated for the complete survey of 20

questions and found to be 0.840,1 indicating that the survey was reliable as a

measure of respondents’ appreciation for mathematics. Cronbach’s Alpha was

0.872 for the five attitude questions, 0.737 for the eight usefulness questions

(including the two proofs questions), and 0.600 for the seven creativeness

questions indicating that the grouped questions that made up the constructs

were reasonably reliable as measures of attitude, usefulness, and creativeness.

The completed survey was placed in an envelope by the student, and the

envelope was sealed by the researcher. After all course surveys had been

1A reliability coefficient of 0.70 or higher generally indicates that the grouped

items are measuring the same underlying construct

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completed, the principal researcher randomly assigned a code number to each

student. The key relating the code number to the student was kept secure by the

principal researcher and the original documents were not seen by the instructors

of the course. Near the end of the semester, students had 20 minutes to complete

the post-course survey which was identical to the pre-course survey. Survey

results were analyzed using factor analysis and paired-difference tests.

3.3. Pre-course and Post-course Test

During the second class period students read and signed the informed con-

sent, then had thirty minutes to take the pre-course test. Identical security

measures as those for the pre-course survey were taken to ensure anonymity.

Near the last day of class students had 30 minutes to complete the post-course

test which was identical to the pre-course test. The test consisted of six

questions, the topics of which were: general reasoning, symbolic logic,

infinite sets, chaos and fractals, irrational numbers, and recursive sequences.

Students were not allowed to use a calculator. The highest possible score on

the test was 18 and a detailed grading rubric was written for each question.

Each test was graded by two researchers and any discrepancies in scores were

resolved. Pre-course and post-course test scores were compared using a

nonparametric Friedman rank test for paired data [12].

3.4. Open-ended Questions

Four open-ended or free-response questions were included with the pre-

course and post-course surveys. Written responses were typed by the

researchers and were not seen by any of the instructors of the course. The

questions, modeled after questions in the Survey on Mathematics [13], were

designed to determine students’ beliefs about mathematics before and after

taking the course:

1. How would you respond to someone who asks, ‘‘What is mathematics? ’’

2. What does it mean to understand a mathematical proof?

3. Explain the role of critical thinking when doing a problem that uses

mathematics.

4. What reasons would you give to someone for studying mathematics?

The written responses to the open-ended questions were analyzed for the-

matic content. Themes were identified inductively as they emerged from the

responses. First, the pre-course question responses were typed and grouped by

question. One faculty and three student researchers, working alone, reviewed and

recorded the main themes or ideas in each typed response for each pre-course


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survey question. Working together, the researchers compiled the common and

most prevalent themes. A code was assigned to each theme and a table was

created. The table contained the codes for each question accompanied by a

description of each code. Each question had a unique set of codes, but if a

common theme prevailed throughout several questions, it was assigned a com-

mon code. The typed responses were then reread and coded by the team of

researchers. Each typed response could contain none, some, or all of the themes

described by the codes for that question.

The same process was followed for the post-course questions, resulting

in a similar coding table. If a new theme emerged in the typed responses for a

post-course question, the pre-course questions were recoded for evidence of

the new theme, and the pre-course coding table was updated. Frequency

analysis of the codes was conducted to determine if there was a difference

in themes for pre-course and post-course open-ended question responses.

Coding tables for each question, including the theme description for each

code, can be found in Appendix A.

4. RESULTS AND DISCUSSION

4.1. Test Grades: Comparing Pre-course and Post-course Competency

There was a significant increase in students’ performance as measured by scores

on the pre-course test and post-course test. A Friedman rank test for differences in

medians was conducted2 [10]. The test score range was 0 to 18 points. The pre-

course test median was 6.5 and the post-course test median was 9.8, indicating

that there was an increase in median score on the post-course tests. This result

indicates that students gained knowledge and were able to work problems that

involved general reasoning, symbolic logic, infinite sets, chaos and fractals,

irrational numbers, and recursive sequences. More broadly, students improved

their critical thinking and problem solving skills in these subjects.

4.2. Survey Responses: Comparing Pre-course and Post-course

Attitudes

When developing the pre-course survey which measured appreciation for

mathematics, the researchers had subjectively grouped the questions into

three categories: attitudes, usefulness, and creativity. A factor analysis3

2Friedman’s S ¼ 26.73, df ¼ 1, p < 0.001 (adjusted for ties).3A statistical procedure that delineates the factors (or constructs) that underlie a set

of survey questions. A factor is a linear combination of related questions that have

some special affinity among them.

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[10] of this survey confirmed that the questions in each group were related to

one another, each group measuring a unique construct. (For the purposes of

this study, a construct is defined to be the label placed on the broader ideas

measured by the Likert-scaled survey and open-ended questions.) Specific sur-

vey questions which comprise these constructs can be found in Appendix B. An

explanation of the factor analysis and a description of the calculation of the

summated Likert scores for the constructs can be found in Appendix C.

When Likert score responses were compared for each construct, attitude

showed a significant decrease, usefulness showed no significant difference,

and creativeness showed a significant increase. It is possible that the abstract-

ness of the topics taught in Mathematical Inquiry led to the decrease in

attitude (their feelings about mathematics as it relates to them personally),

yet the intrigue of those same topics led to the increase in creativeness (their

feelings about the creativity in mathematics as a discipline). The statistical

results of the tests comparing these Likert score responses can be found in

Table C2 of Appendix C.

The conflicting results provide an interesting challenge to the instructors

of the course. Topics must be chosen that are unique, challenging, and creative

enough to change students’ pre-conceived ideas about mathematics, but not so

much so that students see mathematics as no longer personally applicable. The

result of no significant increase in students’ feelings about usefulness is not

unexpected, because students who place into Mathematical Inquiry are calcu-

lus-ready. They are good quantitative problem solvers, scored well on standar-

dized mathematics tests, and most likely had an existing well-developed

concept of the usefulness of mathematics in everyday life.

4.3. Open-ended Question Responses: Comparing Pre-course

and Post-course Beliefs

Four open-ended questions on beliefs about mathematics allowed researchers

to delve more deeply into the thoughts of the students who completed the

pre-course and identical post-course surveys. Two-sample tests for propor-

tions4 were conducted to compare the frequency of occurrence of individual

themes for pre-course and post-course responses for each question. See

Appendix A for a description of each theme and its corresponding code.

When pre-course and post-course codes (as defined by the emergent

themes) were compared for Question One, there was a significant increase

in the relative frequency of occurrence of the code WORLD5 (helps to

4A more conservative 0.01 level of significance was used for each question in this

multiple-test comparison, resulting in an experiment-wise error rate of 0.12.5z ¼ -3.64, p < 0.001


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understand the world or universe) and LANG6 (involves language, system, or

rules) in the post-course responses. This result seems to indicate that students

gained the belief that mathematics is useful in understanding the systems and

rules of the world or universe (Figure 1).

Question Two revealed a significant increase in the relative frequency of

occurrence of the codes HOWHY7 (tells you how and why a conjecture is true)

and LOGIC8 (gives the logic or reasoning behind why a conjecture is true), and

a significant decrease in the code DKNO 9 (don’t know what a proof is). This

indicates that students gained an understanding of the importance of mathe-

matical proofs in explaining the logic or reasoning behind a conjecture as well

as how and why it is true. The code DKNO was prevalent in the pre-course

responses about proofs, so it was encouraging to see that there was a significant

decrease in DKNO for the post-course responses (Figure 2).

Question Three revealed a significant increase in the relative frequency

of occurrence of the code SOVPR10 (find a solution, or solve a problem),

indicating that significantly more students believed that the role of critical

thinking is important when solving problems, although this result could have

been influenced by the wording of the question (Figure 3).

When pre-course and post-course codes were compared for Question

Four there was a significant increase in the relative frequency of occurrence

of the code WORLD11 (helps to understand the world or universe). This

0

10

20

30

40

50

60

Co

un

t

Before

After

How would you respond to someone who asks, "What ismathematics"?

Figure 1. Total number of times each code occurred in Question 1 open-ended

responses.

6z ¼ -2.58, p ¼ 0.0057z ¼ -2.66, p ¼ 0.0048z ¼ -3.33, p < 0.0019z ¼ 3.16, p < 0.00110z ¼ -2.58, p ¼ 0.00511z ¼ -2.37, p ¼ 0.009

194 Ward et al.

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result, similar to the result for WORLD in Question One, corroborates the

suggestion that students gained the belief that mathematics is useful in

explaining the world and the universe (Figure 4).

For all questions, there were no significant differences in the occurrence

of other themes such as those pertaining to everyday life, career, job, and

improving the mind (see Appendix A for a complete list of emergent themes)

when pre-course and post-course written responses were compared.

However, for each question, there was at least a 20 percent increase in the

occurrence of themes in the post-course written answers (for example,

Question 1 had a total count of 133 instances of the mention of any of the

identified themes for that question in the pre-course responses and a total

count of 177 instances of the mention of any identified theme in the post-

0

5

10

15

20

25

30

35

40

DKNO HOWHY LOGIC RULES SOVPR STEPS

Co

un

t

What does it mean to understand a mathematical proof?

Before

After


responses.

0

10

20

30

40

50

60

Co

un

t

Explain the role of critical thinking when doing a problemthat uses mathematics.

Before

After

CHECK

CREAT

DIFBA

DKNOEVAN

FORM

GUESS

LOGIC

NEGTV

SOVPR

STEPS


responses.


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course responses). This increase in the total occurrence of any of the identi-

fied themes, although not statistically significant, could indicate that a more

developed thought process went into the written answers after the students

had completed the course.

The results of the qualitative analysis for thematic content corroborated

the results of the Likert-scaled opinion survey: students seemed to gain an

enhanced understanding of the creativeness and significance of mathematics

as it relates to the world, but not as it relates to them personally. Also, their

opinion of the usefulness of mathematics in everyday life did not change,

perhaps because they had previously developed a feel for the usefulness of

mathematics from previous mathematics courses.

These assessment results suggest that Mathematical Inquiry is meeting

many of the goals of the designers of the course. Students increase their

knowledge of interesting mathematical topics, and the survey results indicate

that they obtain a better understanding of the role of proof in mathematics, a

broader awareness of the field of mathematics, and a greater appreciation of

the creative aspect of mathematical practice. However, we are troubled by

the fact that despite all these positive changes in their view of mathematics,

students seem to overall have a poorer opinion of mathematics as relevant

and useful to their individual lives.

In retrospect, this result is not surprising. Mathematical Inquiry was

designed to focus on more conceptual and abstract ideas in mathematics—

symbolic logic, different sizes of infinity, fractals, and chaos theory, for

example—specifically to combat the prevailing notion that mathematics is

primarily computational and algorithmic. In our attempts to portray mathe-

matics as a rich and interesting discipline, we deemphasized more applica-

tion-oriented topics. One lesson from our study is that efforts such as ours to

interest students in abstract mathematics, even if successful, can produce

05

101520253035

Co

un

tWhat reasons would you give to someone for studying

mathematics?

Before

After


responses.

196 Ward et al.

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other undesired negative opinions in students. Clearly, this is a challenge for

future Mathematical Inquiry instructors, especially since Mathematical

Inquiry is a key piece of the general education curriculum at our university.

Since this study was completed, instructors of the course have made

changes in an attempt to enhance student perceptions of the relevance of

mathematics. Some instructors have emphasized the practical aspects of the

abstract topics in the course. While iterative systems can be used to generate

fractals, they can also used to model population dynamics and compound

interest. A Mobius band may appear to be an interesting but useless geometric

oddity, but a Mobius-twisted conveyor belt will wear more evenly than a

cylindrical belt. Investigations of crossing numbers of graphs have connections

to circuit design.

Another instructor has added a two-week unit that looks at how graphical

analysis and expected value computations can be used to measure and model

income inequality in the United States. This unit shows that sophisticated

mathematical ideas can help students make decisions about economic matters

and confront social issues. Informal surveys indicate that this unit has a

positive impact on student opinions: This unit is generally well-liked by the

students, and a majority of students who study this unit agree that the

mathematics they learned in the course will benefit them in the future. At

the same time, students in these sections report many of the same positive

changes in attitudes seen in the present study.

Ultimately, what our design and assessment of Mathematical Inquiry has to

offer instructors at other schools may be this: By teaching students abstract,

unconventional ideas in mathematics it is possible to broaden students’ views of

mathematics. However, we must be careful to balance such esoteric topics with

more practical ones as well, or, in promoting one aspect of mathematics, we may

find ourselves damaging students’ perceptions of another.

REFERENCES

1. Al-Hasan, A. D., and Jaberg, P. 2006. Assessing the general education

mathematics courses at a liberal arts college for women. In L. Steen,

B. Gold, L. Hopkins, D. Jardine, & W. Marion (Eds.), Supporting

Assessment in Undergraduate Mathematics. Washington, DC:

Mathematical Association of America.

2. Carter, G., and Norwood, K. S. 1997. The relationship between teacher

and student beliefs about mathematics. School Science and Mathematic.

97(2): 62–67.

3. Ellington, A. J. 2006. An assessment of general education mathematics

courses’ contribution to quantitative literacy. In L. Steen, B. Gold,

L. Hopkins, D. Jardine, & W. Marion eds., Supporting Assessment


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in Undergraduate Mathematics. Washington, DC: Mathematical

Association of America.

4. Fennema, E., and Sherman, J. A. 1976. Fennema-Sherman Mathematics

Attitudes Scales: Instruments designed to measure attitudes toward the

learning of mathematics by females and males. Journal for Research in

Mathematics Education, 7(5): 324–326.

5. Griffith, J. V., and Chapman, D. W. 1982. LCQ: Learning Context

Questionnaire. Davidson, NC: Davidson College.

6. Madison, B. L. 2003. Articulation and quantitative literacy: A view from

inside mathematics. In L. Steen (Ed.). Quantitative Literacy: Why

Numeracy Matters for Schools and Colleges. Washington, DC: The


7. The Mathematical Association of America. 1998. Quantitative

Reasoning for College Graduates: A Complement to the Standards.

http://www.maa.org/past/ql/ql_toc.html. 15 February 2010.

8. The Mathematical Association of America. 2004. Committee on the

Undergraduate Program in Mathematics (CUPM) Curriculum Guide.

http://www.maa.org/cupm/curr_guide.html. 15 February 2010.

9. McLeod, D. B. 1994. Research on affect and mathematics learning in the

JRME: 1970 to the present. Journal for Research in Mathematics

Education, 25(6), 637–647.

10. Mertler, C. A., and Vannatta, R. A. 2005. Advanced and Multivariate

Statistical Methods, 3rd ed. Glendale, CA: Pyrczak Publishing.

11. Michael, M. 1999. Using pre- and post-testing in a liberal arts mathe-

matics course to improve teaching and learning. In B. Gold, S. Keith,

W. Marion (Eds.), Assessment Practices in Undergraduate Mathematics.

Washington, DC: The Mathematical Association of America.

12. Munro, B. H. 2005. Statistical Methods for Health Care Research, 5th ed.

Chestnut Hill, MA: Lippincott,Williams & Wilkins.

13. Otto, A. D., Lubinski, C. A., and Benson, C. T. 1999. Creating a general

education course: A cognitive approach. In Assessment Practices in

Undergraduate Mathematics (pp. 191–194). Washington, DC: The


14. Schoenfeld, A. H. 1989. Exploration of students’ mathematical beliefs

and behavior. Journal for Research in Mathematics Education. 20(4):

338–355.

15. Sons, L. R. 1999. A quantitative literacy program. In Assessment

Practices in Undergraduate Mathematics (pp. 187–190). Washington,

DC: The Mathematical Association of America.

16. Travers, K. J. et al. 2003. Indicators of Quality in Undergraduate

Mathematics. Retrieved July 20, 2006, from http://www.mste.uiuc.edu/

indicators/.

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APPENDIX A

Codes for Open-ended Questions (Beliefs About Mathematics)

Table A1. Codes for Question 1: How would you respond to someone who asks,

‘‘What is mathematics’’?

Code Description/theme

CREER career, jobCRITH critical thinking, logicDKNO I don’t knowELIFE everyday lifeFORM formulas, equations, relationshipsGRAPH figures, graphsLANG language, system, rulesNEGTV negative answerNUMR numbers, calculations, manipulationPATRN patterns, sequenceQUANTR quantitative reasoningSOVPR solution, solve problemWORLD explain world, universe

Table A3. Codes for Question 3: Explain the role of critical thinking when doing a

problem that uses mathematics.


CHECK check answer, validateCREAT creative, think outside the boxDIFBA different, best approach(s)DKNO I don’t knowEVAN evaluate, analyzeFORM formula, equations, relationshipsGUESS guesswork, trial and errorLOGIC logic, reasoningNEGTV negative answerSOVPR solution, solve problemSTEPS steps, beginning to end

Table A2. Codes for Question 2: What does it mean to understand a mathematical

proof?


DKNO I don’t knowHOWHY how and whyLOGIC logic, reasoningRULES rulesSOVPR solution, solve problemSTEPS steps, beginning to end


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APPENDIX B

Appreciation for Mathematics Survey

Likert-Scaled Questions Sorted by Construct

Attitude

Question 5: I am looking forward to taking more mathematics.

Question 7: No matter how hard I try, I still do not do well in mathematics.

Question 8: Mathematics is harder for me than for most people.

Question 13: If I had my choice, this would be my last mathematics course.

Question 18: I am good at solving mathematical problems.

Usefulness (Includes Proofs)

Question 1: I expect to use the mathematics that I learn in this course in my

future career.

Question 2: It is not important to know mathematics in order to get a good

job.

Question 9: Mathematics is useful in solving everyday problems.

Question 10: Very little of mathematics has practical use on the job.

Question 15: The content of this course will be useful in my future.

Question 16: There is no value in studying proofs in mathematics.

Question 17: I already know enough mathematics to get a good job.

Question 20: Proofs are essential to the understanding of mathematics.

Table A4. Codes for Question 4: What reasons would you give to someone for

studying mathematics?


CREAT creative, think outside box

CREER career, job

DKNO I don’t know

ELIFE everyday life

FUN fun, patterns

IMPMD improve logical, critical thinking, develop mind

KNOW expand knowledge

NEGTV negative answer

SOVPR solution, solve problem

WORLD explain world, universe

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Creativeness

Question 3: Trial and error can often be used to solve a mathematics problem.

Question 4: Learning mathematics involves mostly memorizing.

Question 6: Mathematics is a good field for creative people.

Question 11: There is little place for originality in solving mathematics

problems.

Question 12: There are many different ways to solve most mathematics

problems.

Question 14: New discoveries in mathematics are constantly being made.

Question 19: Mathematics is mostly learning about numbers.

APPENDIX C

Analysis Results: Comparing Pre-course and Post-course SurveyResponses

Factor loadings choosing three components resulted in the grouping of

questions as shown in Table C1. Component 1 consisted of questions that

the researchers’ had previously subjectively interpreted as measuring atti-

tude. Component 2 consisted of questions previously interpreted as useful-

ness, but interestingly, the two proofs questions (questions 16 and 20) were

loaded onto Component 2 as well. Component 3 consisted of questions, with

the exception of Question 9, that the researchers’ had subjectively interpreted

as measuring creativeness. Question 9 was previously considered by the

researchers to be a usefulness question, and was retained in that category

for the subsequent data analysis, since its factor loadings for usefulness and

creativeness were comparable.

The survey sample consisted of n ¼ 72 students who completed both the

pre-course and the post-course survey. The factor analysis of the pre-course

survey resulted in three distinct factors or constructs: attitude with five

questions, usefulness/proofs with eight questions, and creativeness with

seven questions. Summated Likert scores were created by adding the indivi-

dual question scores (ranging from 1 to 5) by construct for each student. For

example, if a student chose responses 3, 4, 3, 2, 5 on questions 7, 18, 8, 13,

and 5 respectively, her summated Likert score for Attitude would be 17.

Paired t-tests were conducted to compare the average summated Likert

scores for each construct for pre-course and post-course surveys (Table C2).

Summated Likert scores are considered to be continuous and approximately

normal, thus a paired t-test is a valid measure of differences in average

summated scores. Histograms of differences in the summated Likert scores

between pre-course and post-course surveys for each of the three constructs


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were convincingly mound shaped, thus supporting the assumption of

approximate normality.

Table C1. Rotated Component Matrix of Factor Loadings of Survey Questions

Factor (Construct)

1 2 (Usefulness) 3

Question 7 .816 .051 .185

Question 18 .798 .000 .086

Question 8 .788 .070 .288

Question 13 .741 .405 .058

Question 5 .736 .454 -.017

Question 15 .194 .812 .048

Question 1 .108 .739 -.049

Question 10 .121 .614 .383

Question 16 .288 .515 .246

Question 17 -.115 .511 -.142

Question 20 .234 .472 .259

Question 2 .046 .329 -.308

Question 11 .102 .226 .708

Question 12 .132 -.080 .630

Question 6 .230 .392 .491

Question 19 .049 -.109 .466

Question 9 .165 .450 .466

Question 14 .274 .258 .465

Question 4 .189 -.039 .346

Question 3 -.146 .159 .297

Note: Extraction Method: Principal Component Analysis. Rotation Method:

Varimax with Kaiser Normalization. Rotation converged in 4 iterations.

Table C2. Results of Paired t-tests Comparing Summated Likert Scores

Construct

(range of scores)

Pre Course

Average Score

Post Course

Average Score t-score

Degrees of

Freedom

P-

value

Attitude (0–25) 17.39 16.60 2.14 71 0.018*

Usefulness/Proofs (0–40) 28.04 27.39 1.32 71 0.190

Creativeness (0–35) 24.25 25.51 -3.59 71 0.001*

*Significant difference at a ¼ 0.02

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BIOGRAPHICAL SKETCHES

Barbara Ward began her career as a systems engineer for IBM and now teaches

mathematics and statistics at Belmont University. She is currently the coordi-

nator of the statistics program for the Department of Mathematics and

Computer Science at Belmont and has published research in statistics education.

She received her M.S. in mathematics at the University of Memphis. Her

interests include helping undergraduate and graduate students use statistics in

their research, directing a summer sailing camp, and racing sailboats.

Stephen Campbell is a faculty member at Belmont University where he enjoys

teaching a variety of math classes in general education, for majors, for students

in the Honors Program, and for students in other disciplines. Currently, he is

having fun with a course entitled, ‘‘Modeling for the Environmental

Sciences.’’ He received his Ph.D. from Vanderbilt University. In addition to

mathematics, his interests include family, music, and woodworking.

Mary Goodloe received her Ph.D. from the University of Kentucky. She is

currently head of the mathematics department at Belmont University. Her

interests include teaching Calculus, Real Analysis, and preparing students for

careers in mathematics and computer science. Her interests include collecting

Newberry Award books and reading children’s literature.

Andrew Miller has taught mathematics at Belmont University since receiving

his Ph.D. from the University of California, Berkeley in 2003. He is one of

the co-developers of Mathematical Inquiry. His interests include dynamical

systems, applications of mathematics to social justice issues, and marathon

running.

Kacie Kleja received her B.S. degree in mathematics from Belmont

University and her M.S. in professional science with a concentration in

biostatistics from Middle Tennessee State University. She is currently pursu-

ing a career in biostatistics.

Eninka Kombe received her B.S. degree in Computer Science from Belmont

University. She is currently working as a Data Analyst for Healthways, Inc.

Renee Torres is a current graduate student at the University of Nevada, Reno,

where she is working toward a Master of Science in statistics. She also works

for UNR as a teaching and research assistant. Her current research involves

building statistical models for predicting subjective financial decisions. She

received her B.S. in mathematics from Belmont University in Nashville,

Tennessee. Outside of school her interests include traveling, running, and

playing tennis.


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assessing a mathematical inquiry course: do students gain an appreciation for mathematics?

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