exploring teaching finite mathematics 118 - team-math

EXPLORING TEACHING FINITE MATHEMATICS 118

Running Head: EXPLORING TEACHING FINITE MATHEMATICS

EXPLORING THE EFFECTIVENESS OF DIFFERENT APPROACHES TO TEACHING

FINITE MATHEMATICS

Alabama State University University College Department of Mathematics and Science

Science Building Room 302 915 S. Jackson St.

Montgomery, AL 36101

Dr. Mary Alice Smeal [email protected]

Dr. Sandra Walker [email protected]

Dr. Jamye Carter [email protected]

Dr. Carolyn Simmons-Johnson

[email protected]

Dr. Lisa James [email protected]

Dr. Esenc Balam

[email protected]


Abstract

Traditionally, mathematics has been taught using a very direct approach where the

teacher explains the procedure to solve a problem and the students use pencil and paper to solve

the problem. However, a variety of approaches to mathematics have surfaced from a number of

different directions. The purpose of the study was to examine the effectiveness of three teaching

methods on student achievement in undergraduate Finite Mathematics classes at Alabama State

University. The three teaching methods stressed traditional teaching methods, the incorporation

of graphing calculators, and online learning, respectively. Three hundred sixty-one students

formed the control group and were taught by the traditional lecture method, 202 students formed

one of the experimental groups which used calculator enhanced instruction, and twenty students

formed the other experimental group that utilized online distance learning. The research project

examined each teacher's style and compared achievement outcomes. To investigate the

performance of the students, a pretest was given to each student at the beginning of the semester

before instruction and a posttest was given at the end of each semester after instruction. The

results presented a significant improvement (p < .001) between the pre-tests and post-tests within

each group. The students in the calculator-enhanced group scored significantly higher (p < .001)

on the post-test than the students in the traditional group.

Keywords: Finite, Mathematics, Undergraduate, Technology


Introduction

Traditionally, mathematics has been taught using a very direct approach where the

teacher explains the procedure to solve a problem and the students use pencil and paper to solve

the problem. However, a variety of approaches to mathematics have surfaced from a number of

different directions. The National Council of Teachers of Mathematics (2000) encouraged

teachers to incorporate a more student-centered approach as well as utilize technology. The

American Mathematical Association of Two-Year Colleges (1996) standards also suggested that

technology is an essential part of reform curricula, specifically software and graphing

calculators. In attempts to include more students at the university level, many universities are

offering distance learning courses as an option (Perez & Foshay, 2002). Another consideration of

distance learning is that an estimated 50-75 percent of corporate education uses online

technology (Bourne, Harris, & Mayadas, 2005), so familiarity with online education increases a

students’ job marketability.

The purpose of the study was to examine the effectiveness of three teaching methods on

student achievement in undergraduate finite mathematics classes at Alabama State University.

The three teaching methods stressed traditional teaching methods, the incorporation of graphing

calculators, and distance learning, respectively. The research project examined each teacher's

style and compared achievement outcomes.

Review of Literature

Numerous research projects (McCoy, 1996; McDonald, Vasquez, & Caverly, 2002; Narum,

2008; Perez & Foshay, 2002; Su, 2008; Taylor, 2008) explored a variety of methods for teaching


basic mathematics and science courses at the university level. Alternative approaches to teaching

mathematics included distance learning (Perez & Foshay, 2002; Su, 2008), computer-assisted

curriculum (Taylor, 2008), and graphing calculators (McCoy, 1996).

In a comparison of achievement using graphing calculators and traditional approaches, the

results were mixed. In McCoy’s study (1996) using computer-based mathematics learning, she

reported that technology-related tools improved conceptual learning, but computation skills were

no different. The Mathematics Department at Columbia College in South Carolina revamped the

college algebra curriculum by featuring graphing calculators (Hopkins & Kinard). The new

program of study focused on using graphing calculators to help students to learn conceptually

and incorporating students’ intuitive understanding about mathematics. Hopkins and Kinard

(1998) conducted a study that compared students taught by traditional methods with students

involved in the new program that concentrated on graphing calculators. Students in the graphing

calculator program performed better on the final exam and had better attitudes toward

mathematics at the end of the course. Two studies reported that achievement on final

examinations of students using graphing calculators were higher than students who were not

using graphing calculators (Quesada & Maxwell, 1994; Stiff, McCollum, & Johnson, 1992).

Wynegar and Fenster (2009) researched achievement of students in college algebra. Each

section used a different instructional methodology. The methodologies included computer-aided

instruction, traditional lecture, and online teaching in a college algebra course. The final course

grade was used to compare the various teaching approaches. Wynegar and Fenster (2009)

reported that students in traditional lecture classes performed better than all of the other courses

using other methodologies. The conflicting implications from the discussed research studies

invite further study and comparison.


Hypothesis

Given the conflicting research found in prior studies between types of instruction in basic

college mathematics courses, the following hypothesis is given:

H1: The performance of students in finite mathematics differs with type of instruction.

Design of the Study

All of the instructors of finite mathematics participated in the study to explore the

effectiveness of three methods of instruction to teaching finite mathematics – traditional lecture,

calculator-enhanced, and online distance learning. Quantitative research was chosen as the

methodology for this study (Hopkins, 2000).

Participants

The participants in this study were 584 students enrolled in a freshman-year course in

finite mathematics during the Fall Semester 2008, Spring Semester 2009, and Fall Semester 2009

at Alabama State University – a regional, comprehensive, historically black state-supported

university. Three hundred sixty-one of these students formed the control group and were taught

by the traditional lecture method, 202 students formed one of the experimental groups which

used calculator enhanced instruction, and twenty students formed the other experimental group

that utilized online distance learning. All classes were capped at twenty-five students, so the

student-to-teacher ratio was comparable in all classrooms. Students registered randomly without

prior knowledge of the method of instruction utilized by the instructor, with the exception of the

online distance learning course. Students enrolled in the distance class had prior knowledge that

all lectures, assignments, quizzes and tests would be administered via the computer. The honors

sections were excluded from the study.


Instruments

The instruments used in this experimental study were a pretest and posttest that had

identical questions (See Appendix). The tests measured the students’ level of comprehension of

the seven Finite Mathematics course objectives. The testing instrument consisted of 28 questions

– 4 on linear functions; 4 on solving systems of linear equations; 4 on the operations of matrices;

4 on sets; 4 on probability; 4 on counting principles; and 4 on statistics. The tests were developed

by the mathematics department faculty using the test bank from MyMathLab software program,

a personalized interactive multimedia resource.

Research Procedure

The Finite Mathematics course consisted of a three-hour credit for one semester. The

pretest was administered on the first day of class and all students were allowed to use calculators.

Students in the control group were instructed using the traditional method of lecture and

classroom discussions in which students develop into a community of mathematicians as

observed by St. Clair, Carter and St. Clair (2009). Students in the calculator-enhanced

experimental group used the TI-83 graphing calculator as an integral part of instruction and

testing. Ellington (2003) found that students’ operational and problem-solving skills improved

when calculators were an integral part of testing and instruction. The online distance

experimental group used the computer to receive course content, assignments, and evaluations.

This method allowed students to learn at their own pace and at any place where there is access to

the computer. In addition, the two experimental groups incorporated the technology from the

MyMathLab software program. The posttest was administered on the last day of class to the

students that remained in the course. Only the students who took both the pretest and the posttest

participated in the study.


Analysis of Data

The analysis on the lecture, calculator-enhanced, or online distance instruction in finite

mathematics was conducted using the Statistical Package for the Social Sciences (SPSS:

Shannon & Davenport, 2001). The results from the pretest and posttest were used in the analysis.

A 3 X 2 mixed between/within ANOVA was conducted to assess the between-subject and

within-subject performance differences. The between-subject factor was the instructional method

(traditional, calculator-enhanced, or online distance), while the within-subject factor was time –

the time between pre instruction and post instruction.

Results

In this section, the results from the quantitative analyses will be described. First, the

descriptive statistics will be given for each teaching approach. Following that will be a

comparison of the means of the three groups. The last two analyses will include a discussion of

the results of the tests of within subject effects and multiple comparisons.

Descriptive statistics include the means and standard deviations of the pretest and the posttest

for each approach to teaching (see Table 1 on p.8). The descriptive statistical table shows that

583 students participated in the study. There were 361 students who were taught using the

traditional method; 202 students who were taught using the calculator-enhanced method; and 20

students who were taught in an online distance format.


Table 1

Means and, Standard Deviations of the Pretest and Posttest (n = 583)

Method Mean Std. Dev. N

Pretest Traditional 10.01 3.240 361

Calculator-enhanced 9.04 3.042 202

Online Distance 9.15 3.558 20 Total 9.64 3.212 583

Posttest Traditional 15.40 3.865 361

Calculator-enhanced 20.68 3.964 202

Online Distance 17.40 4.695 20 Total 17.30 4.646 583

Figure 1 (p. 9) compares the means of the pretest and the posttest for the traditional,

calculator-enhanced, and online distance groups. The graph shows the growth in achievement

within each group. The calculator-enhanced group, with the lowest mean score of 9.04 on the

pretest, has the highest mean score of 20.68 on the posttest. The traditional group, with a mean

score of 10.01 on the pretest, has the lowest mean score of 15.40 on the posttest. The online

distance class, with a mean score of 9.15 on the pretest, has a mean score of 17.40 on the

posttest.


Figure 1 Pretest and Posttest Means for Traditional, Calculator-enhanced, and Online Distance Groups


A statistical analysis was run within subjects to investigate the significance of the

achievement gain within subjects from the pretest to the posttest. Table 2 (below) shows the

within-subject effect. At F (2, 580) = 139.04, p < .001, η2= .32, a statistically significant within-

in subject interaction occurred.

Table 2 Results of the Tests of Within-Subject Effects

Tests of Within-Subjects Effects

Measure: MEASURE_1

5538.041 1 5538.041 609.342 .000 .5125538.041 1.000 5538.041 609.342 .000 .5125538.041 1.000 5538.041 609.342 .000 .5125538.041 1.000 5538.041 609.342 .000 .5122527.333 2 1263.666 139.039 .000 .3242527.333 2.000 1263.666 139.039 .000 .3242527.333 2.000 1263.666 139.039 .000 .3242527.333 2.000 1263.666 139.039 .000 .3245271.362 580 9.0895271.362 580.000 9.0895271.362 580.000 9.0895271.362 580.000 9.089

Sphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-bound

SourceTIME

TIME * METHOD

Error(TIME)

Type III Sumof Squares df Mean Square F Sig.

Partial EtaSquared

Follow-up univariate analyses from Table 3 (see p. 11) reported that at α =.05, p <.01,

participants in the traditional group (M=10.01, SD=3.240) achieved statistically significantly

higher scores than participants in the calculator group (M= 9.04, SD=3.042) in the pretest.

Despite the statistical significance, no practical significance was indicated due to the small effect

size (η2 = .021). At α =.05, p = .653, no statistically significant difference between the online

group (M= 9.15, SD=3.558) and the traditional group was reported. At α =.05, p = .999, no


statistically significant difference between the online distance group and the calculator-enhanced

group was reported.

In the posttest, on the other hand, follow-up univariate analyses indicated that at α =.05,

p <.001, the traditional group (M= 15.02, SD=3.521) achieved statistically significantly lower

scores than the calculator group (M= 20.38, SD=4.244). At α=.05, p < .05, the online

group (M = 17.40, SD = 4.695) achieved statistically significantly lower scores than the

calculator group. At α=.05, p = .205 no statistical significant difference between the online

distance group and the traditional group was reported when using multiple comparisons.

However, when a t-test was run between the traditional and online distance group, the traditional

scores were statistically significantly lower than the online distance group (α=.05, p < .05). At F

(2, 580) = 116.81, p < .001, η2= .287, a statistically significant between-subjects interaction

occurred.

.


Table 3 Multiple Comparisons

Multiple Comparisons

.97* .274 .001 .31 1.62

.86 .814 .653 -1.25 2.96-.97* .274 .001 -1.62 -.31-.11 .824 .999 -2.23 2.01-.86 .814 .653 -2.96 1.25.11 .824 .999 -2.01 2.23.86 .731 .303 -.68 2.39

-.11 .746 .964 -1.67 1.45-5.28* .345 .000 -6.10 -4.45-2.00 1.069 .205 -4.77 .775.28* .345 .000 4.45 6.103.28* 1.086 .019 .48 6.082.00 1.069 .205 -.77 4.77

-3.28* 1.086 .019 -6.08 -.48-2.00* .903 .037 -3.89 -.113.28* .921 .001 1.35 5.21

(J) METHODCalculatorOnlineTraditionalOnlineTraditionalCalculatorOnlineOnlineCalculatorOnlineTraditionalOnlineTraditionalCalculatorOnlineOnline

(I) METHODTraditional

Calculator

Online

TraditionalCalculatorTraditional

Calculator

Online

TraditionalCalculator

Dunnett T3

Dunnett t (2-sided)a

Dunnett T3

Dunnett t (2-sided)a

Dependent VariablePRETEST

POSTTEST

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound95% Confidence Interval

Based on observed means.The mean difference is significant at the .05 level.*.

Dunnett t-tests treat one group as a control, and compare all other groups against it.a.

Results and Conclusions

The use of instructional technology within mathematics classrooms has increased

significantly over the past two decades. With the improvement and expansion of graphing

calculators and the increased demand for online distance education, educational institutions

around the world have incorporated these instructional tools within their mathematics

classrooms.

This research conducted a comparison of the effectiveness of three methods of instruction

within finite mathematics at Alabama State University--the traditional lecture, calculator-

enhanced instruction, and online distance learning instruction. The quasi-experimental study


analyzed the results of identical pretests and posttests for 583 finite mathematics students during

the fall 2008, spring 2009, and fall 2009 semesters.

The results of the analysis showed five major points. First, a multiple comparisons test

showed only one statistically significant (p = .001) difference in pre-test scores. This difference

was between the calculator-enhanced group and the traditional group. However, due to a small

effect size (η2= .021), the difference has no practical significance. Second, there was a

significant improvement (p < .001) between the pretests and posttests within each group—

traditional group, 54%; calculator-enhanced group, 129%; and online distance learning group,

90%. Third, the students in the calculator-enhanced group scored significantly higher (p < .001)

on the post-test than the students in the traditional group. Fourth, the posttest mean scores for

the calculator-enhanced group were higher than that of the online distance learning group, the

difference was statistically significant (p < .05). Finally, while the posttest mean scores for

students in the online distance learning group were higher than that of the traditional group, the

difference was only significant on a t-test (p < .05).

Implications

Since there were either no practical or no significant differences in mean scores between

groups on the pretest, the students in each group started off at relatively the same level.

Significant improvement between tests within each group suggests that regardless of the method

of instruction, learning has taken place. Thus, all three methods of instruction are meaningful.

However, significantly higher mean scores on the posttest in the computer-enhanced group over

the traditional group and online distance suggest that usage of graphing calculators within

instruction provides enhanced learning over that of other methods of instruction. One reason for

this may be that students who use graphing calculators tend to attempt more problems than


students that are not taught using graphing calculators (Harskamp, et. al, 2000). Also, students in

the calculator-enhanced group are likely to make fewer errors than students in the traditional

group, who complete many of their calculations by hand.

Future Research

Further investigation of the effects of the three methods of instruction discussed in this

study will be conducted and will include the following: 1) an increase in the sample size of the

online distance learning group; 2) a survey of students’ attitudes on taking online classes, using

the graphing calculators, and using other technology in the classroom; and 3) a survey of

students’ opinions on what helps them to succeed in finite mathematics.


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APPENDIX

FINITE MATHEMATICS POSTTEST

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