Independent Study Report
Rong-Ji Chen, 12-14-2000
Abstract:
The focus in this report is active teaching and learning in
undergraduate mathematics classroom. A review of relevant research
on active teaching and learning is first presented, followed by three
research questions. Based on the survey data from the Indicator
Project, an active index is derived as the dependent variable in the
investigation. Finally, the results from the statistical analyses and
discussions on the major findings are presented.
Active Teaching and Learning
I hear, I forget;
I see, I remember;
I do, I understand.
(old proverb)
Feeling that class time is brief and precious, and the material to be communicate to the
students is not only so much but also important, the undergraduate instructor tends to
lecture a lot and tell the students directly what they need to know. Consequently, students
tend to learn mathematics in a passive way. They usually watch the instructor lecture at
blackboard and hurriedly take notes; they seldom speak in class or discuss with the
instructor or classmates; they hardly synthesize major mathematics ideas and they spend
most of their time doing computational problems, usually working individually. They
then develop and suffer from “inert knowledge” (Brown, 1989). This term was
introduced by Whitehead (1929) to describe knowledge which can usually be recalled by
students when explicitly asked to do so, but is not used spontaneously in problem-solving
even thought it is relevant. Inert knowledge is gained through memorization or reading,
but is never used to solve new problems.
One solution to the problem of inert knowledge may be the systematic use of active
teaching and learning (Mitchell, 1997). Active teaching and learning involves the use of
strategies which maximize opportunities for interaction in the classroom with the hope
that students can actively participate in solving mathematics problems, discuss
mathematics with others, reflect on their learning, and synthesize major ideas. The
interaction can be between teachers and students, among the students themselves, as well
as between students and the materials.
The attempt to the paradigm shift from a passive to an active teaching and learning is
demonstrated in the NCTM Principles and Standards for School Mathematics (2000) at
1
the pre-university level and the Calculus reform movement (Hughes-Hallet et al, 1994) at
the university level. Both of them advocate that teaching should make the students more
active participant in their learning. In responding to the call from the Calculus reform
movement, Bookman (1998) summarized three evaluation studies on a reform curriculum
of calculus. The results showed that, comparing to students in traditional calculus classes,
students in the reform classes exhibited a positive attitude that learning mathematics is
useful and involves understanding.
Research on the comparison between junior and senior faculty in their teaching
style
Under construction…(I tried to locate relevant research, but I just could not find
any paper addressing this issue.)
Research on the influence of professional organization on faculty’s teaching
Under construction…
Research on active teaching and technology
While active teaching does not necessarily involve the use of technology, technology
provides a powerful means to engage students in active learning and improve their
learning of college algebra and calculus. Mayes (1995) conducted a quasi-experimental
research to compare a traditional lecture-based college algebra course to an experimental
algebra course. The experimental course stressed active student involvement and the use
of the computer as a tool to explore mathematics. The results revealed that students in the
experimental group scored significantly higher than the control group on a final measure
of inductive reasoning, visualization, and problem solving. This study also showed that
the use of computers to focus instruction on problem solving did not adversely affect the
computation and manipulation skills of students. Cooley (1997) reported an experimental
study in which two sections of calculus were taught using the same materials, except one
section was enhanced with the computer algebra system Mathematica. Results indicated
that the students in the technology group had advantages to understanding certain key
topics in calculus such as limits, derivatives, and curve sketching.
Research questions:
The purpose of this study is to investigate the use of active teaching in college
mathematics classrooms. Specifically, this study addresses the following three questions:
1. With more teaching experience, are senior instructors more active in teaching than
their junior colleagues?
2. Does a connection with professional organizations help an instructor be more
active in teaching?
3. What kinds of technology tools are more associated with active teaching?
2
Deriving Active Index
On the faculty survey part 2, subjects are asked to indicate how frequently they
emphasize each of the following in their instruction:
a) Mathematical reasoning --passive (YOU show your students justification for
mathematical statements; YOU demonstrate deductive proofs of theorems)
b) Mathematical reasoning --active (your STUDENTS construct justifications for
mathematical statements; your STUDENTS construct demonstrative proofs of
theorems)
c) Mathematical procedures (your students develop skill in performing routine
mathematical procedures, especially algorithms)
d) Mathematical facts (your students learn basic definitions and statements of basic
theorems)
e) Applications of mathematics -- passive (YOU show your students how mathematics
can be used to solve a variety of "real world" problems, e.g., in science)
f) Applications of mathematics -- active (your STUDENTS determine how
mathematics can be used to solve a variety of "real world" problems, e.g., in
science)
g) Mathematical communication (you ask your students to speak and/or write clearly
about mathematical ideas)
h) Modeling --passive (YOU show your students how to create and use appropriate
mathematical representations of "real world" situations, e.g., in science)
i) Modeling --active (your STUDENTS create and use their own appropriate
mathematical representations of "real world" situations, e.g., in science)
j) Multiple representations (your students see and work with the same mathematical
concept in a variety of forms; graphical, numerical [tabular], symbolic, and verbal)
Items b, f, g, i, and j are regarded as active teaching strategies and an aggregate score for
each subject is obtained by adding up his/her individual score on each item except for
item f since the subjects’ response to this item is lost. On the other hand, items a, c, d, e, h
are considered as passive teaching strategies and an aggregate score for each subject is
also obtained by summing up his/her score on each of these items. Finally, an active
index is derived by subtracting the passive aggregate score from the active aggregate
score. In the presentation and discussion hereafter, this active index is denoted by AI.
Results
Question 1: With more teaching experience, are senior instructors more active
in teaching than their junior colleagues?
3
Question one explores the relationship between an instructor's department rank and
his/her active index.
Department rank is coded as the following:
1-Tenured
2-Tenured-Eligible
3-Other full-time
4-Part-time (not TA)
5-Graduate TA
6-Adjunct (Full time)
7-Other (Under TA)
Subjects are categorized into 2 groups: Group 1(1-4 and 6, ie. Faculty), Group 2(5 &7, ie.
TAs)
A T-test is conducted to test the difference in the means of AI between these two groups.
Results show that P= .028, indicating that the faculty members are not as active as TAs.
*n
AI Mean
StdDev
T-Test
Group 1 (Rank 1-4)
18
-6.056
3.523
t = -2.329
Group 1 (Rank 5&7)
49
-3.898
2.874
P= 0.028
* Original sample size was 70, but the number of valid cases is 67.
Difference Between Means = -2.158, t-Statistic = -2.329 w/25 df, p = 0.028
Reject Ho at Alpha = 0.05
4
Question 2: Does a connection with professional organizations help an instructor
be more active in teaching?
Question two explores the relationship between active index and the instructor's
connection to professional organizations.
If a certain instructor is a member of one or more professional organizations, he/she is
coded 1. Otherwise, he/she is coded 0, meaning that he/she doesn't subscribe to any
organization. So, subjects are made into two groups: member and non-member.
A T-test is conducted to test the significance of the difference in the means of AI between
the two groups.
Results show that P= .045, indicating that members of professional organizations are
more active in their teaching than non-members.
Group 1 (non-member) Group 2 (member)
n
13
55
AI Mean
-6.154
-4.255
StdDev
2.764
3.411
T-Test
t = -2.124
P = 0.045
* The original sample size was 70, but 2 observations were duplicated.
Difference Between Means = -1.899, t-Statistic = w/21 df, p = 0.045
Reject Ho at Alpha = 0.05
5
Question 3: What kinds of technology tools are more associated with active
teaching?
Question three explores the relationship between active index and the use of technology.
The use of each technology tool is indicated by:
1- Not used at all
2345- Almost always
For each technology tool, if the response is 3 or higher, it's coded 1. Otherwise, it's coded
0. In other words, subjects are break down into two groups: frequent use and rare use of
the technology tools.
For each technology tool, a T-test is conducted to test the significance of the difference in
the means of AI between the two groups. Here, the active index (AI) is the dependent
variable (Y).
Summary of T and P values from the T-test Results
Technology Tool t value
p value
Calculator
-1.711
0.093
e-mail
-0.545
0.589
Internet
-2.239
0.030
Probe
NA**
NA**
CAS*
-4.206
0.0002
* CAS stands for Computer Algebra Systems: Mathematica, Derive, Maple, etc.
** Only one observation is coded 1
(1) Use of calculator
Group 1 (Rare Use)
n
42
AI Mean
-5.143
StdDev
3.544
T-Test
-1.711
Group 2 (Freq. Use)
25
-3.76
2.976
P = 0.093
Difference Between Means = -1.383, t-Statistic = -1.711 w/57 df , P = 0.093
Fail to reject Ho at Alpha = 0.05
6
(2) Use of e-mail
Group 1 (Rare Use)
n
22
AI Mean
-5
StdDev
3.450
T-Test
-0.545
Group 2 (Freq. Use)
45
-4.511
3.442
p = 0.589
Difference Between Means = -0.489, t-Statistic = -0.545 w/41 df, p = 0.589
Fail to reject Ho at Alpha = 0.05
(3) Use of Internet
Group 1 (Rare Use)
n
41
AI Mean
-5.512
StdDev
3.115
T-Test
-2.239
Group 2 (Freq. Use)
25
-3.56
3.618
p = 0.030
Difference Between Means = -1.952, t-Statistic = -2.239 w/45 df, p = 0.030
Reject Ho at Alpha = 0.05
(4) Use of CAS package
Group 1 (Rare Use)
n
52
AI Mean
-5.384
StdDev
3.225
T-Test
-4.206
Group 2 (Freq. Use)
15
-2
2.591
p = 0.0002
Difference Between Means = -3.384, t-Statistic = -4.206 w/27 df, p = 0.0002
Reject Ho at Alpha = 0.05
Overall, the results show that the use of calculator and e-mail does not related to the
teaching style as active or passive. However, active instructors tend to incorporate the
Internet and CAS into their teaching.
7
Discussion
With more teaching experience, are senior instructors more active in teaching than
their junior colleagues?
Surprisingly, the junior instructors (specifically, the teaching assistants) demonstrate a
more active teaching style than their senior colleagues. One possible reason is that the
younger generation has experienced the changes proposed by NCTM and/or the Calculus
reform movement when they were undergraduate students and they tend to teach the way
they have been taught.
The tenure and promotion system in research universities is another issue. In his
discussion on what is wrong with university mathematics education, Hersh (1992)
expressed his disapproval of the practice of many universities to neglect a faculty
member’s teaching quality in deciding his/her tenure and promotion. While it is not
feasible to impose a sharp separation between research and teaching in mathematics, if
we could account for more credits by his/her accountability on teaching, it would
alleviate the problem.
Does a connection with professional organizations help an instructor be more active
in teaching?
As expected, members of professional organizations tend to be more active in teaching
than non-members. Among the subjects in the study, 53 are members of AMS; 15 are
members of MAA; 5 are members of SIAM; and 9 are members of AMATYC, NCTM
and other organizations. As stated on its Website, the AMS
“fulfills its mission through programs that promote mathematical
research, increase the awareness of the value of mathematics to society,
and foster excellence in mathematics education.” (AMS, 2000)
Members of such professional organizations would be more likely to update their
knowledge about the current trends and issues in mathematics education and apply them
into their own teaching.
What kinds of technology tools are more associated with active teaching?
The results show the use of calculator and e-mail doesn’t associate with active teaching
(p = .093 and .589, respectively). However, differences are seen in the use of the Internet
and Computer Algebra Systems (CAS). These software tools allow opportunities for
students to explore multiple representations, visualize abstract concepts, manipulate
certain components and observe the change on other components, and facilitate
collaboration and communication. The embedded editing tool in Mathematica, for
example, provides a means for the students to write about mathematics and thus enhance
active learning. The writing activity is a good way to encourage students to think about
what they are learning and to see course material in a larger context (Rosenthal, 1995).
However, one should note that, while technology has the great potential to enhance
student’s active learning of mathematics, the use of software tools would not have much
8
of an effect without a proper pedagogy. Take the use of Mathematica for example, the
instructor can either use it to do the demonstration (passive) or have the students conduct
projects on Mathematics (active). Depending on the strategy used, the effects of such
software tool will be different. This leads to a suggestion for future research to obtain
qualitative data on how the instructors and students utilize software tools and how their
use affects their teaching or learning of mathematics.
Finally, in the investigation reported above, unlike the controlled laboratory environment,
one cannot tightly control variables and study the causal relationship among variables. In
this project, subjects were studied in an intact, complex environment. This fact gives rise
to an alternative interpretation for the results. While the results show a positive
relationship between an instructor’s active teaching index with his/her connection to
professional organizations, it is not adequate to state that, for example, the membership of
professional organizations causes an instructor to be more active in his/her teaching.
Interpretation of non-experimental research should be made with great cautions.
Future Plan:
1. Continue doing literature review.
2. Study Mike’s reports and extend the investigation to the student’s side. I can do a
comparison between faculty’s self-report and student’s perception on the same
issue.
3. Since the instructors are categorized into two groups, I can go ahead and break the
students into two groups and do some comparisons.
References
American Mathematical Society (AMS). (2000). American Mathematical Society
Overview. [Online] http://www.ams.org/ams/ams-info.html. Access date: December 14,
2000.
Bookman, J. (1998). Student Attitudes and Calculus Reform. School Science and
Mathematics, 98 (3), pp 117-122.
Cooley L. A. (1997). Evaluating student understanding in a calculus course enhanced by
a computer algebra system. Primus, 7 (4), pp308-16.
Hersh R. (1992). A university mathematician’s view of what’s wrong with university
mathematics education. Humanistic Mathematics Network Journal, v12, pp 24-27.
Hughes-Hallet, D., Gleason, A. M., Gordoa, S. P., Lomen, D. O., Lovelock, D., and
McGallum, W. G. (1994). Calculus. New York : Wiley & Sons.
9
Mayes, R. L. (1995). The application of a computer algebra system as a tool in college
algebra. School Science and Mathematics, 95(2), pp 61-67.
Mitchell, M. (1997) The use of Spreadsheets for constructing statistical understanding.
Journal of Computers in Mathematics & Science Education, 16 (2/3), p201-22.
National Council of Teachers of Mathematics. (2000). NCTM Principles and Standards
for School Mathematics. [Online] http://www.nctm.org/standards/. Access date:
December 1, 2000.
Rosenthal J. S. (1995). Active learning strategies in advanced mathematics classes.
Studies in Higher Education, 20 (2) pp 223 – 228.
Whitehead, A. N. (1929) The Aims of Education. New York: Macmillan.
10