polygon 2013
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i
Editorial Note:
We, the editorial committee of Polygon, are pleased to publish the seventh issue of 2013 Spring
Polygon which includes six regular papers. Again we are pleased to present work from a diverse
array of fields written by faculty from across the college. The editorial board of Polygon is
thankful to the administration, staff and faculty of Hialeah Campus and Miami Dade College, in
general, for their continued support and cooperation for the publication of Polygon. The editorial
committee would also like to cordially invite the MDC community to submit their articles for
consideration for the 2014 Spring Issue of Polygon.
Sincerely,
The Editorial Committee of Polygon: Dr. M. Shakil (Mathematics), Dr. Jaime Bestard
(Mathematics), and Professor Victor Calderin (English)
Patrons:
Dr. Ana Maria Bradley-Hess, Dean of Academic and Student Services
Prof. Djuradj Babich, Chair of Computer and Management Sciences
Dr. Caridad Castro, Chair of LAS, EAP and Foreign Languages
Miami Dade College District Board of Trustees:
Helen Aguirre Ferré, Chair
Armando J. Bucelo Jr.
Marielena A. Villamil
Benjamin León III
Marili Cancio
Jose K. Fuentes
Armando J. Olivera
Eduardo J. Padrón, College President
Mission of Miami Dade College
The mission of the College is to provide accessible, affordable, high‐quality education that
keeps the learner’s needs at the center of the decision-making process.
Editorial Notes i
Guidelines for Submission ii-iii
ASSESSING NORMALITY – A CLASSROOM NOTE 1-7 1-7 Dr. Jack Alexander
The Discipline / Program Learning Outcomes Assessment in Mathematics at Miami Dade College, Results from the Academic Year 2011 Application
8-16
Prof. Nicholas Schur, Dr. Jermaine Brown, and Dr. Jaime Bestard
A Model for Secondary-Postsecondary Curriculum Alignment in Mathematics
17-22
Prof. Rene Barrientos, Prof. Nelson de la Rosa, and Prof. Edgar Ginory
A Defense of Epistemological Standpoint Theory 23-31
Dr. Melissa Lammey
A Statistical Analysis of the Effects of Class Attendance and ANGEL Activity Logs on MAC 1105 Students' Final Examination Scores
35-53
Dr. M. Shakil
Dr. Dudley Weldon Woodard, the First African-American Mathematician to Publish a Research Paper in an International Accredited Mathematics Journal – A Historical Introduction
54-77
Dr. M. Shakil
Comments about Polygon
ii
Guidelines for Submission
POLYGON
“Many Corners, Many Faces (POMM)”
A premier professional refereed multi-disciplinary electronic journal of scholarly works,
feature articles and papers on descriptions of Innovations at Work, higher education, and
discipline related knowledge for the campus, college and service community to improve
and increase information dissemination. Published by MDC Hialeah Campus Liberal Arts
and Sciences Department (LAS).
Editorial Committee:
Dr. Mohammad Shakil (Mathematics)
Dr. Jaime Bestard (Mathematics)
Prof. Victor Calderin (English)
Manuscript Submission Guidelines:
Welcome from the POLYGON Editorial Team: The Department of Liberal Arts and
Sciences at the Miami Dade College–Hialeah Campus and the members of editorial
Committee - Dr. Mohammad Shakil, Dr. Jaime Bestard, and Professor Victor Calderin - would
like to welcome you and encourage your rigorous, engaging, and thoughtful submissions of
scholarly works, feature articles and papers on descriptions of Innovations at Work, higher
education, and discipline related knowledge for the campus, college and service community to
improve and increase information dissemination. We are pleased to have the opportunity to
continue the publication of the POLYGON, which will be bi-anually during the Fall & Spring
terms of each academic year. We look forward to hearing from you.
General articles and research manuscripts: Potential authors are invited to submit papers for
the next issues of the POLYGON. All manuscripts must be submitted electronically (via e-mail)
to one of the editors at mshakil@mdc.edu, or jbestard@mdc.edu, or vcalderi@mdc.edu. This
system will permit the new editors to keep the submission and review process as efficient as
possible.
Typing: Acceptable formats for electronic submission are MSWord, and PDF. All text,
including title, headings, references, quotations, figure captions, and tables, must be typed, with
1 1/2 line spacing, and one-inch margins all around. Please employ a minimum font size of 11.
Please see the attached template for the preparation of the manuscripts.
Length: A manuscript, including all references, tables, and figures, should not exceed 7,800
words (or at most 20 pages). Submissions grossly exceeding this limit may not be accepted for
review. Authors should keep tables and figures to a minimum and include them at the end of the
text.
iii
Style: For writing and editorial style, authors must follow guidelines in the Publication Manual
of the American Psychological Association (5th edition, 2001). The editors request that all text
pages be numbered. You may also please refer to the attached template for the preparation of the
manuscripts.
Abstract and keywords: All general and research manuscripts must include an abstract and a
few keywords. Abstracts describing the essence of the manuscript must be 150 words or less.
The keywords will be used by readers to search for your article after it is published.
Book reviews: POLYGON accepts unsolicited reviews of current scholarly books on topics
related to research, policy, or practice in higher education, Innovations at Work, and discipline
related knowledge for the campus, college and service community to improve and increase
information dissemination. Book reviews may be submitted to either themed or open-topic issues
of the journal. Book review essays should not exceed 1,900 words. Please include, at the
beginning of the text, city, state, publisher, and the year of the book’s publication. An abstract of
150 words or less and keywords are required for book review essays.
Notice to Authors of Joint Works (articles with more than one author). This journal uses a
transfer of copyright agreement that requires just one author (the Corresponding Author) to sign
on behalf of all authors. Please identify the Corresponding Author for your work when
submitting your manuscript for review. The Corresponding Author will be responsible for the
following:
ensuring that all authors are identified on the copyright agreement, and notifying the
editorial office of any changes to the authorship.
securing written permission (via email) from each co-author to sign the copyright
agreement on the co-author’s behalf.
warranting and indemnifying the journal owner and publisher on behalf of all coauthors.
Although such instances are very rare, you should be aware that in the event a co-author has
included content in their portion of the article that infringes the copyright of another or is
otherwise in violation of any other warranty listed in the agreement, you will be the sole author
indemnifying the publisher and the editor of the journal against such violation.
Please contact the editorial office if you have any questions or if you prefer to use a copyright
agreement for all coauthors to sign.
Instructions for the Preparation of Manuscripts for the Polygon:
(THE TITLE IS HERE) (12 pt, bold, 32 pt above)
NAME IS HERE (11 pt16 pt above, 32 pt below)
ABSTRACT is here, not exceeding 160 words. It must contain main facts
of the work. (11 pt)
Key words and phrases (11 pt)
iv
Introduction (11 pt, bold, 24 pt above, 12 pt below)
Main Body of the Article
Discussion
Conclusion
Acknowledgements
REFERENCES (11 pt, 30 pt above, 12 pt below)
[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas,
Graphs, and Mathematical Tables . Dover, New York, 1970.
[2] J. Galambos and I. Simonelli, Products of Random Variables – Applications to Problems of
Physics and to Arithmetical Functions , CRC Press, Boca Raton / Atlanta, 2005.
[3] S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers
equations. Chaos, Solitons & Fractals, 28(4) (2006), 930-937.
[4] Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential
equations of fractional order, Int. J. Nonlin. Sci. Numer. Simulat. 1(7) (2006), 15-27. (11 pt)
XXXX YYYYY. Received his Master’s/Ph. D. Degree in Physics from the University of
ZZZZZ (Country) in 1987 under the direction of Dr. M. N. OPQR. Since 1989, he has been
at CCCC College in Hawaii, USA. His research interests focus on the Fractals, Solitons,
Undergraduate Teaching of Physics, and Curriculum Development. (11 pt)
Department of Liberal Arts & Sciences (Physics Program), CCCC College, P. O. Box
7777, Honolulu, Hawaii, USA. e-mail: xxyy@ccc (11 pt)
.
1
ASSESSING NORMALITY – A CLASSROOM NOTE
By
Dr. Jack Alexander
Department of Mathematics
Miami Dade College, North Campus
Miami, Fl. 33167, USA
E-mail: jalexan2@mdc.edu
ABSTRACT
Most courses in beginning statistics do cover the Normal Distribution. However, these courses
tend to ignore or give minimal treatment to Assessing Normality. Having taught statistics for many years,
it is my considered view, that the course would be strengthened if this assessment would be included.
This paper illustrates how the use of Stemplots and Histograms along with the Empirical Rule gives us a
straightforward strategy for determining closeness to normality.
KEYWORDS: Normal Distribution, Stemplots, Histograms, Empirical Rule, Assessment
AMS Subject Classification 2010: 62–07
INTRODUCTION:
Beginning statistics courses typically will include the study of the Normal Distribution. This
endeavor will demand that students learn how to read probability values for both positive and negative
standard deviations above and below the mean from prepared normal tables. What is not usually
discussed is how to assess whether a given set of data is, in fact, Normal. In the view of this writer, the
course would be enhanced if Assessing Normality where included. This paper presents a straightforward
procedure for assessing normality using an arranged Stemplot, a Histogram, and the requirements of the
Empirical Rule.
Narrative:
Many populations from the real world have normal distributions. This is particularly true for
large populations. For example, the heights of adult males in any relatively large community could be
modeled by the normal distribution.
A Normal Distribution is bell-shaped. It turns out that about 68% of values under this bell shape
lie within one standard deviation (positive or negative) of the mean. Further, about 95% of values lie
within two standard deviations (positive or negative) of the mean. And, about 99.7% of values lie within
three standard deviations (positive or negative) of the mean. This phenomenon is called the Empirical
Rule and is illustrated in Figure 1 below.
2
Figure 1 (The Empirical Rule)
. * .
* | *
* | *
* | | | *
* | 34 % | 34 % | *
* | | | *
* | 13.5 % | | | 13.5 % | *
----------------------------------------------------------------------------------------------------------------------
|---------------- 68% ---------------|
|--------------------------------- 95 % ----------------------------------|
|--------------------------------------------------- 99.7 % -------------------------------------------------|
In addition to the Empirical percentages (68%, 95%, 99.7%), a Normal distribution will have the
same mean, median and mode. To demonstrate procedures for assessing normality, we will use the data
set given in Table 1 below. This table gives the ages of 76 actresses at the time they won Oscars.
Table 1
Academy Awards: Ages of 76 Best Actresses
22 37 28 63 32 26 31 27 27 28 30 26
29 24 38 25 29 41 30 35 35 33 29 38
54 24 25 46 41 28 40 39 29 27 31 38
29 25 35 60 43 35 34 34 27 37 42 41
36 32 41 33 31 74 33 50 38 61 21 41
26 80 42 29 33 45 49 39 39 34 26 25
33 35 35 28
3
The first step in the analysis is to create a Stemplot as illustrated below:
Stem (tens) | Leaves (units)
2 | 28677869459945897957169658
3 | 7210805538918554476231383594355
4 | 161032111259
5 | 40
6 | 301
7 | 4
8 | 0
For easier analysis we put each row of the Stemplot in order. This will order the entire data set.
Stem (tens) | Leaves (units)
2 | 124455555666677778888999999
3 | 0011122333334445555555677888899
4 | 011111223569
5 | 04
6 | 013
7 | 4
8 | 0
We can now construct a Histogram as shown below in Figure 2.
4
Figure 2
Histogram for Actress Data
Note that the histogram for the actress data forms a Right Skew. The Mean is 35.7. The Median is
33.5. The Mode is 35. And, the Standard Deviation is 11.06. While the mean, median and mode are
relatively close, the distribution is clearly not Bell-Shaped.
If we round the mean to 36 and the standard deviation to 11, one standard deviation below the mean
(36 -11) and one standard deviation above the mean (36 + 11) gives a range of 25 to 47. From the
arranged stemplot, this yields 64 data. This is about 84% (64/76 = .8421). Two standard deviations
below and above gives a range of 14 to 58. The calculations are: 36 - 2 x 11 = 14 and 36 + 2 x 11 = 58.
There are 71 data in this range, which is about 93% ( 71/76 = .934). Three standard deviations below and
above gives a range of 3 to 69. The calculations are: 36 – 3 x11 = 3 and 36 + 3 x 11 = 69. There are 74
data in this range, which is about 97% (74/76 = .974).
The percentages for two and three standard deviation are within 5% of the Empirical rule standards.
However, the percentage for one standard deviation is much more than 5% different. (84% – 68% =
16%). This along with the skewed histogram would lead us to conclude that the actress data is not close
to normal.
Table 2 below gives the ages of the 76 Actors that received Oscars in the same years that the Actress
received her award. The data will be analyzed in the same manor.
5
Table 2
44 41 62 52 41 34 34 52 41 37 38 34
32 40 43 56 41 39 49 57 41 36 42 52
51 35 30 39 41 44 49 35 47 31 47 37
57 42 45 42 44 62 43 42 48 49 56 38
60 30 40 42 36 76 39 53 45 36 62 43
51 32 42 54 52 37 38 32 45 60 46 40
36 47 29 43
We again construct a Stemplot to begin analysis of this data set.
Stems (tens) | Leaves (units)
2 | 9
3 | 447842965095178069627826
4 | 411103191214977252432890253256073
5 | 212672763142
6 | 22020
7 | 6
Again we construct the associated Arranged Stemplot.
Stems (units) | Leaves (units)
2 | 9
3 | 001222444556666777888999
4 | 000111111222222333344455567778999
5 | 112222346677
6 | 00222
7 | 6
6
Figure 3
Histogram for Actor Data
Note that the histogram for the male data is much more symmetric than the female data. The mean is
43.9. The median is 42. The mode is 41. And, the standard deviation is 9.08. Again, we see that the
mean, median and mode are relatively close.
To make the comparisons with the Empirical Rule, we round the mean to 44 and the standard
deviation to 9. One standard deviation below (44 – 9) and one standard deviation above (44 + 9) gives a
range of 35 to53. From the arranged stemplot, this yields 55 data. This is about 72% (55/76 =.724 ).
Two standard deviations below and above the mean gives a range of 26 to 62. The calculations are:
44 – 2 x9 = 26 and 44 + 2 x 9 = 62.
There are 75 data in this range, which is about 98.7% (75/76 = .987). Three standard deviations below
and above the mean yield a range of 17 to 71. The calculations are:
44 – 3 x9 = 17 and 44 + 3 x9 = 71.
There are 75 data in this range. As before, this is about 98.7%. Note that 72% compares well with 68%
(72% - 68% = 4%) for one standard deviation. Further, 98.7% compares well with 95% (98.7% - 95% =
3.7%) for two standard deviations. And 98.7% compares well with 99.7% for three standard deviations
(99.7% - 98.7% = 1%). We can therefore, conclude that the actor’s data is relatively close to normal.
7
CONCLUSION:
We can see that employing an arranged Stemplot, and a Histogram; we can determine the number and
percentage of data one, two, and three standard deviations from the mean. We then compare these
percentages to those of the Empirical Rule. If all three percentages are within 5% of the Empirical
standards, and the mean median and mode are somewhat close, we can assert that the distribution is
relatively close to Normal.
REFERENCES
Brown, R. and Davis, G. 2005. Ages of Oscar Winning Best Actors and Actresses, Mathematics Teacher
Magazine.
Triola, M. F. 2012. Elementary Statistics, 11
th
Edition, Pages 309 – 3 14; Addison – Wesley, Boston
8
The Discipline / Program Learning Outcomes Assessment in Mathematics at
Miami Dade College, Results from the Academic Year 2011 Application
Prof. Nicholas Schur*. Dr. Jermaine Brown** and Dr. Jaime Bestard***
*Assistant Professor Mathematics, MDC Kendall Campus, nschur@mdc.edu
**Chair, Mathematics, MDC Inter-American Campus/Discipline Coordinator, jbrown@mdc.edu
***Associate Professor, Senior Mathematics, MDC Hialeah Campus, jbestard@mdc.edu
Abstract
In the process to document students learning, Miami Dade College moved forward to produce
discipline based documentation, after have wide experience in results documenting general
education students learning via the authentic assessment of learning outcomes. The discipline
/program process started about three academic years ago with the definition of the discipline
learning outcomes, and the action of a committee created for that task in the mathematics
discipline. The design of authentic instruments of assessment and its respective pilots resulted in
the decision to proceed with the first formal application which results are explained in this paper.
Theme: Educational Research
Key words: Assessment Technique
9
1. Introduction:
Since 2006 Miami Dade College enter in a process to documenting learning that has led to place
this institution in the nation leadership in assessment as per the prestigious national awards
received in the last three years in the matter.
The further development of the mapping process pointed to the need to asses students learning in
a gate keeper discipline as mathematics.
2. Methods:
2.1 Timeline:
The discipline assessment team, based on the experience in the General Education Assessment
process at Miami Dade College organized a timeline that is shown below in Figure 1. The
attempt to organize the discipline process in the frame of an institutional faculty driven process
was successful and resulted ready to apply as agreed
Figure 1. Mathematics Discipline Timeline (2011-12)
2.2 Sample selection
The selection of the sections for the assessment was conducted via institutional research where
the target were 638 students in the following sections corresponding to terminal courses in the
discipline of mathematics as indicated in Figure 2.
10
Figure 2. Structure of the sections selected to conduct the discipline learning outcomes
assessment
Courses Number of sections Number of students
MAC1105 College Algebra 18 289
STA2023 Statistical Methods 14 347
MGF1106 Math for Lib. Arts 1 11
The target discipline learning outcomes were
1) Communication
2) Critical/creative thinking and scientific reasoning
The tasks consisted in the analysis of properties of functions applied to a real life professional
scenario, using an authentic assessment with holistic scoring in a summative and formative
assessment instrument
3. Results:
The classification of the results of the instrument scoring is in communion with the college –
wide General Education Assessment as shown in the Figure 3, 4 with the corresponding rubrics
Figure 3 Results of the Communication Learning Outcome for the course MAC1005
Lrng.
Outcome
Outcome No
Effort/No
Evidence
Emerging Developing Proficient Exemplary
Analysis of
the Profit
Function
Communicate
mathematical
ideas and
relationships
using
appropriate
terminology,
mathematical
symbols,
graphs and
laboratories
Unable to
answer any
of the four
questions
concerning
the graph
and its
properties.
Or no
answers
were
written
Able to
answer
exactly
one about
the graph
Able to
answer
exactly two
questions
about the
graph
Able to
answer
exactly
three
questions
about the
graph
Able to
answer all
the
questions
about the
graph
MAC1105 289 Students 4 24 67 120 74
MAC1105 % of students 1.38 8.3 2.318 4.15 25.6
11
Figure 4 Results of the Critical Thinking Learning outcome for the course MAC1005
Lrng.
Outcome
Outcome No
Effort/No
Evidence
Emerging Developing Proficient Exemplary
Discussion
of the
shortest
path
Use critical
thinking
and
concept
from
quantitative
analysis to
solve
problems
Unable to
answer any
questions
Selecting a
non-
existing
path
Selecting
an existing
but
incorrect
path
Selecting
the correct
path but
incorrect or
no
justification
Selecting
the correct
path and
correct
justification
MAC1105 289
Students
36 26 64 100 63
MAC1105 % of
students
12.5 9 2.1 34.6 21.8
Figure 5 Results of the Communication Learning Outcome for the course STA2023
Lrng.
Outcome
Outcome No
Effort/No
Evidence
Emerging Developing Proficient Exemplary
Analysis
of the
Profit
Function
Communicate
mathematical
ideas and
relationships
using
appropriate
terminology,
mathematical
symbols,
graphs and
laboratories
Unable to
answer any
of the four
questions
concerning
the graph
and its
properties.
Or no
answers
were
written
Able to
answer
exactly one
about the
graph
Able to
answer
exactly two
questions
about the
graph
Able to
answer
exactly
three
questions
about the
graph
Able to
answer all
the
questions
about the
graph
STA2023 347 Students 10 24 65 150 98
STA2023 % of students 2.88 6.92 18.73 43.23 28.24
12
Figure 6 Results of the Critical Thinking Learning Outcome for the course STA2023
Lrng.
Outcome
Outcome No
Effort/No
Evidence
Emerging Developing Proficient Exemplary
Discussion
of the
shortest
path
Use critical
thinking
and concept
from
quantitative
analysis to
solve
problems
Unable to
answer any
questions
Selecting a
non-
existing
path
Selecting
an existing
but
incorrect
path
Selecting
the correct
path but
incorrect or
no
justification
Selecting
the correct
path and
correct
justification
STA2023 347Students 43 35 66 110 95
STA2023 % of
students
12.39 10.08 19.02 31.7 27.38
Figure 7 Results of the Communication Learning Outcome for the course MGF1106
Lrng.
Outcome
Outcome No
Effort/No
Evidence
Emerging Developing Proficient Exemplary
Analysis of
the Profit
Function
Communicate
mathematical
ideas and
relationships
using
appropriate
terminology,
mathematical
symbols,
graphs and
laboratories
Unable to
answer any
of the four
questions
concerning
the graph
and its
properties.
Or no
answers
were
written
Able to
answer
exactly one
about the
graph
Able to
answer
exactly two
questions
about the
graph
Able to
answer
exactly
three
questions
about the
graph
Able to
answer all
the
questions
about the
graph
MGF1106 11 Students 1 1 6 3 0
MGF1106 % of students 9.09 9.09 54.55 27.27 0
13
Figure 8 Results of the Critical Thinking Learning outcome for the course MGF1106
Lrng.
Outcome
Outcome No
Effort/No
Evidence
Emerging Developing Proficient Exemplary
Discussion
of the
shortest
path
Use critical
thinking
and
concept
from
quantitative
analysis to
solve
problems
Unable to
answer any
questions
Selecting a
non-
existing
path
Selecting
an existing
but
incorrect
path
Selecting
the correct
path but
incorrect or
no
justification
Selecting
the correct
path and
correct
justification
MGF1106 11 Students 1 0 1 8 1
MGF1106 % of
students
9.09 0 9.09 72.73 9.09
4. Discussion:
Observe that due to the size of the MGF 1106 sections( Chart 1 and 2) they were not considered
in the statistical analysis.
A Chi-Square analysis was conducted to state association between the results in different courses
and in both cases of the two assessments resulted associated at the 5 % significance level
(Observe Chart 3. Chi-Square display)
14
CHART 3. Chi-Square Tests Display( MINITAB)
Chi-Square Test: MAC1105(1), STA2023(1)
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
MAC1105(1) STA2023(1) Total
1 4 10 14
6.36 7.64
0.877 0.730
2 24 24 48
21.81 26.19
0.220 0.183
15
3 67 65 132
59.98 72.02
0.821 0.684
4 120 150 270
122.69 147.31
0.059 0.049
5 74 98 172
78.16 93.84
0.221 0.184
Total 289 347 636
Chi-Sq = 4.028, DF = 4, P-Value = 0.402
Chi-Square Test: MAC1105(2), STA2023(2)
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
MAC1105(2) STA2023(2) Total
1 36 43 79
35.79 43.21
0.001 0.001
2 26 35 61
27.63 33.37
0.096 0.080
3 64 66 130
58.89 71.11
0.444 0.368
4 100 110 210
95.13 114.87
0.250 0.207
5 63 95 158
71.57 86.43
1.026 0.850
Total 289 349 638
Chi-Sq = 3.323, DF = 4, P-Value = 0.505
ANOVA one way un-stacked was conducted to magnify whether at least one assessment resulted
significantly different from the other applications but resulted not significant as presented in
Chart 4 below.
Chart 4. ANOVA for the different sections
16
One-way ANOVA: MAC1105(1), MAC1105(2), STA2023(1), STA2023(2)
Source DF SS MS F P
Factor 3 697 232 0.13 0.941
Error 16 28704 1794
Total 19 29400
S = 42.36 R-Sq = 2.37% R-Sq(adj) = 0.00%
Individual 95% CIs For Mean Based on
Pooled StDev
Level N Mean StDev ---+---------+---------+---------+------
MAC1105(1) 5 57.80 45.42 (---------------*---------------)
MAC1105(2) 5 57.80 28.87 (---------------*---------------)
STA2023(1) 5 69.40 56.84 (---------------*---------------)
STA2023(2) 5 69.80 32.38 (---------------*---------------)
---+---------+---------+---------+------
25 50 75 100
Pooled StDev = 42.36
Fisher 95% Individual Confidence Intervals
All Pairwise Comparisons
Simultaneous confidence level = 81.11%
MAC1105(1) subtracted from:
Lower Center Upper +---------+---------+---------+---------
MAC1105(2) -56.79 0.00 56.79 (---------------*---------------)
STA2023(1) -45.19 11.60 68.39 (---------------*----------------)
STA2023(2) -44.79 12.00 68.79 (---------------*----------------)
+---------+---------+---------+---------
-70 -35 0 35
MAC1105(2) subtracted from:
Lower Center Upper +---------+---------+---------+---------
STA2023(1) -45.19 11.60 68.39 (---------------*----------------)
STA2023(2) -44.79 12.00 68.79 (---------------*----------------)
+---------+---------+---------+---------
-70 -35 0 35
STA2023(1) subtracted from:
Lower Center Upper +---------+---------+---------+---------
STA2023(2) -56.39 0.40 57.19 (---------------*---------------)
+---------+---------+---------+---------
-70 -35 0 35
Observe that no significant differences among the groups assessed appeared.
17
5. Conclusion:
The information collected showed that in all sections and in both assessment instruments
students performed in a distribution shifted to the Proficient – exemplary levels, which states a
fair terminal level for the discipline.
No significant differences were found neither between the MAC1105 and STA courses nor in the
overall comparison of all the courses and assessment instruments applied during the Spring 2012.
The discipline assessment continues in further academic courses showing paths to improve the
instructional practices in the discipline of mathematics.
References
“Documenting Students Learning in the discipline of Mathematics at Miami Dade College”,
Report presented to the Math Retreat March 2013.
18
A Model for Secondary-Postsecondary Curriculum Alignment in Mathematics
Rene Barrientos
1
,
Nelson de la Rosa
2
and Edgar Ginory
3
1
Chair Person, Miami Dade College, Kendall Campus. 11011 SW 104 ST, Miami, FL 33176-
3393. Room 3427. Email: rbarrien@mdc.edu
2
Associate Professor of Mathematics, Miami Dade College, Kendall Campus. 11011 SW 104 ST
Miami, FL 33176-3393. Room 3245-2. Email: ndelaro1@mdc.edu
3
Teacher of Mathematics, Miami Killian Senior High School. 10655 SW 97
th
Ave Miami, FL
33176. Email: eginory@dadeschools.net
Abstract
The growing number of remedial courses colleges and universities offer to freshmen students has become
the solution to accommodate their poor academic preparedness. National data shows that students lack
basic skills to enroll in college level courses in mathematics. This paper proposes a model to close the gap
of mathematics underperformance in higher education institutions. The model recommends implementing
Algebra I as a regular course in 8
th
grade. According to research studies the sooner students start working
with advanced mathematics tools the more prepared they will be for a more rigorous treatment of the
subject in future courses. Further, the model proposes a college level course in mathematics for high
school students who will not take advance mathematics courses in their senior year. In this way, high
school students may be exposed to the mathematics tools they need to succeed in ulterior educational
engagements.
Introduction
The aim of this paper is to (1) provide an analysis of the authors’ perception of the apparent
divide that exists between secondary and postsecondary curricula in general and in particular as regards to
mathematics, and (2) discuss alternatives that may be explored to provide students a continuous transition
from secondary to postsecondary studies.
According to Lee (2012) one of the reasons for the present debate on the condition of American
education is the poor preparation of students who enter postsecondary institutions. The poor performance
of America’s students in college entrance or placement examinations has led to an increase of remedial
courses offered by higher education institutions as a means to equip students with the necessary tools to
continue with their postsecondary studies. At Miami Dade College (MDC) for example, over 60% of
incoming students need some kind of intervention in mathematics, English or both; and a substantial
number of students need more than one semester of this intervention in order to be ‘college ready’ (Miami
Dade College, 2012). Students who enter at the bottom rung have less than a 5% probability of
graduating with an Associate in Arts (AA). This malaise is not particular to our institution. According to
the National Center for Educational Statistics (NCES), in the year 2010 the number of students that took
remedial courses increased from 34.7 % to 36.2 %. Furthermore, the NCES reports that an estimated 5 out
19
of 10 students who enter colleges require some type of remediation. The conclusions that may be reached
from this evidence are generally along the lines that secondary school curriculum is not aligned with
college curriculum, or that competencies tested on placement exams are not aligned with knowledge
acquired in high school, or a combination of both scenarios.
However, these conclusions fail to account for the many students who are able to pass placement
and/or entrance examinations and enter the country’s universities and colleges extremely well prepared
for the rigors of college life. At MDC for example, 37% of first-time-in-college students are college
ready (as measured by SAT or placement scores), and those who register for college level courses in their
first semester and who were placed there at the minimum score range complete their course at a rate of
67%. It is therefore clear that those students who place into developmental course did not take advantage
of the preparation afforded in high school and graduate with academic deficiencies that are at the root of
their peril upon entering college.
The model proposed in this paper has at its root to identify the factors that contribute to
successful completion of the high school curriculum and postsecondary placement, and make
recommendations of what actions are needed in order to increase the number of students who place into
college level courses immediately upon graduating from high school.
The Challenges
A comprehensive review of the literature (Conley 20008; Le, 2000; Lee, 2012) reveals that here
is a problem much deeper than academic unpreparedness at the heart of this apparent divide between the
traditional K-12 system and college. The evidence lies in the observation that there are students who
graduate from high school who are fully prepared for postsecondary studies. The challenge is: how do we
replicate the experiences of these successful students at a lager scale that embraces a wider population?
Here are some facts: allowing for students who are returning after many years of not attending a
college or university, lack of preparedness has become a socio-economic issue impacting negatively
higher education institutions, most of which must now offer a substantial number of sections of
developmental mathematics to assist students unable to enroll in postsecondary level courses. At MDC,
the increase in enrollment (as measured by credits) during the period 2009 – 2011 was 3.8% (Miami Dade
College, 2012). At the same time, the increase in developmental mathematics credits went up by 4.3%.
These statistics indicate that the mechanisms that guarantee a smooth transition from secondary to
postsecondary education and ensure college readiness have not been effective. Besides, a pattern of
disconnect in the delivery of the instructional message between secondary and postsecondary programs
has led to the unfortunate state of affairs in which a large number of students are unable to complete their
high school studies with the level required for a smooth transition to college or university. If we admit
that the curriculum itself is not necessarily a factor responsible for such disconnect, then we must seek
other factors that can be modified in order to accomplish our objective.
But, what does it mean for a student to be ready to college? According to David Conley (2008)
college readiness is “the level of preparation a student needs to enroll and succeed—without
remediation—in a credit-bearing general education course at a postsecondary institution that offers a
baccalaureate degree or transfer to a baccalaureate program” (p. 24).
20
A close examination to the learning resources (text books and software programs) used in high
schools and colleges reveals that there is content alignment across levels. However, divergence of
purposes between secondary and postsecondary education is a fact. For example, college entrance exams
and state standardized exams might not be aligned in content. They may differ on solution strategies
(multistep vs. algorithms), level of questions (critical thinking, memorization, abstraction), type of
questions (free response, multiple choice), and topics (mathematics standards) assessed (Le, Hamilton,
and Robyn, 2000). That may be the case of a high school state exit exam which determines student’s
graduation status in high schools. This exam is designed to measure student’s competence in applying
mathematics knowledge to solve real life problems. However, college placement/entrance exam stress on
mastering fundamental mathematics skills and only those students who are highly motivated and
complete foundation courses throughout their high school years are able to develop this mastery (Le,
Hamilton, and Robyn, 2000).
Hence, one of the disconnects seems to be that high school state exams do not respond to college
demands; colleges emphasize mastering problem solving skills while the public school message is
application driven with a teach-to-the-test approach.
In addition, our experience has been that many students who place into developmental courses in
mathematics have not been exposed to the subject for over a year due to the current high school
regulations regarding course requirements. There is a large time gap between the moment the high school
state exit exam is administered and when students take the college entrance exam. Usually, students take
the exit exam in the second year of high school. It is not until few months before graduation, in the last
year of high school, that they take the college’s placement test. Therefore, there are not follow up
strategies to address the student’s lack of skills in mathematics after they pass the high school exit exam
that prevent them from failing the college placement test.
Compounded with the fact that high school students are currently not required to take 4 years of
mathematics, we have a situation wherein the average high school student is at a complete disadvantage
as a potential college candidate.
Finally, school districts have to follow a rigid curriculum. They do not have the liberty to make
adjustments to academic programs so that high school students are ready to begin their journey through
postsecondary education and only those highly motivated students take advantage of dual enrollment and
Advance Placement (AP) courses afforded at the high schools.
Recommendations
Findings of research studies suggest that there may be a relation between taking advanced courses
at an early time in secondary level (9
th
and/or 10
th
grades) and proficiency in college mathematics courses.
Long, Conger, and Iatarola (2012) studied the effect of enrolling rigorous math courses in early high
school years in high school completion and in college going. They found that students who took rigorous
courses at least a rigorous course within the first two years of high school had a greater chance to
graduate high school and move on to college education.
We know that the diversity in the students’ population attending the Miami Dade County Public
educational system and the disparity of schools conditions serving these students are conditions that
hinder from implementing this model at a larger scale. Nonetheless, this model may become a first step to
21
close the disparity of performance in our schools and it may contribute to compensate the academic
difference.
Clark and Lovric (2009) analyzed some of the factors that impede a smooth the transition from
secondary to tertiary education in mathematics. They claimed that “the relationship between high school
teachers and university instructors is far from satisfactory” (p. 762). Additionally, they were concerned on
the quality of instruction universities will provide in the next decades if the pattern of offering remedial
courses persists. Apparently, they believed that remedial support is not a strategy to foster effective
transition. The problem of is more complicated than that. The assumption that eliminating remedial
instruction from the mathematics curriculum in colleges will solve the lack of preparation of colleges and
universities freshmen is weak and it does invalidate the mission of community colleges. In many cases,
community colleges serve students that are not granted university entrance due to their poor preparation.
One of the key factors that lead to student success at any level of his or her academic
development is motivation and discipline. To the extent that high school curricula can be modified to
incorporate study skills and motivation courses, it is our recommendation that MDCPS considers
seriously the possibility of introducing them early in K-12 curriculum. Furthermore, parental
involvement should be fostered in all possible ways. These are very tough challenges, but we feel that
they are key ingredients in the affective domain.
Figures 1 and 2 show the traditional path and the proposed path for 9-12 grades. The former is
taken by average high school students and the latter is often followed by the more math-driven students
who are able to place out of developmental education from the start. As shown in Figure 2, we propose to
move algebra I to 8
th
grade because we feel that students at that age level are ready to begin the study of
the subject and furthermore, the sooner students start working with the elements of algebra tools the more
prepared they will be for a more rigorous treatment of the subject in future courses. The model proposes a
college level course in mathematics for high school students who will not take advance mathematics
courses in their senior year. In this way, high school students in their senior year may be equipped with
the mathematics tools they need to succeed in ulterior educational engagements.
Further, the model includes a battery of assessments which now has become part of Florida’s law.
These are the end-of-year subject tests (EOC) which we feel will generate great gains in the effort to
make the curriculum more uniform for all students, leveling the field to some extend between those
students who have the support at home, who are highly motivated, and who traditionally are the ones who
place into college level course, and those who traditionally fall through the cracks.
MDCPS and MDC have already started something that is essential to the success of the
implementation of this model: direct dialog between mathematics faculty in the school system and college
faculty. This will ensure that school faculty is aware of the expectations college faculty have as far the
level of mathematical maturity of a college-ready student. We realize that the Florida State Department
of Education is much more involved in curriculum issues at the K-12 level than at the college level, but
we believe that these conversations will create a united front among faculty at all levels that can better
lobby for changes at the state level.
We also recommend a systematic way to encourage students to take AP courses or participate in
dual enrollment programs. The latter have the added advantage of bringing the student to campus and
making him/her aware of what college life is all about while he/she is still in an academic formative stage.
Regarding to diagnostic and placement test the proposed path ensures alignment with the New Generation
Sun Shine State Standards and College Readiness Standards. There is consistent with the proposed path
22
since in all academic levels students’ performance in mathematics is assessed and, if necessary, remedial
action is taken via summer school courses.
Final Reflections
We strongly believe that the alternative we offer represents a genuine solution to close the gap
between high schools and higher education for those students who traditionally fail to attain college
readiness upon graduation from High School. The relevance of this proposal strives for public schools
students to be exposed to college level experiences in mathematics before they graduate from secondary
education and assess their progress early and frequently in order to determine what corrective steps must
be taken in order to ensure that they remain on course. Additionally, it will decrease dramatically the
number of resources used to provide remedial instruction in colleges and university.
In order to assess the effectiveness of this project, it is recommended to implement a pilot study
that explores the degree of collaboration between the community college and the high schools in the same
district by doing a careful comparison of course competencies at the two levels, sharing teaching and
assessment methodology, and assessing the viability of the proposed college readiness path outlined in
Figure 2.
23
References
Conley, D. T. (2008). Rethinking College Readiness. New England Journal of Higher Education, 22(5),
24-26.
Le, V. N., Hamilton, L., Robyn, A. (2000). Alignment among secondary and postsecondary assessment
in California Crucial Issues in California
Education. Chapter 9. The RAND Corporation. Retrieved from
http://www.stanford.edu/group/bridgeproject/PACE/Chapter9_a.pdf
Lee, J. (2012). College for All: Gaps Between Desirable and Actual P–12 Math Achievement Trajectories
for College Readiness. Educational Researcher , 41(2), 43-55. Retrieved from
http://web.ebscohost.com.ezproxy.fiu.edu/ehost/detail?vid=3&hid=18&sid=0f5dbfbb-62df-
4fe1-8a33-79f3d2dcf26b%40sessionmgr15&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=
eft&AN=72676254
Long, M., Conger, D., & Iatarola, P. (2012). Effects of High School Course-Taking on Secondary and
Postsecondary Success. American Educational Research Journal , 49(3), 285-322. Retrieved from
http://web.ebscohost.com.ezproxy.fiu.edu/ehost/detail?vid=3&hid=13&sid=4ab840ef-2f83-493a-
b2a7-
2b0b8f22fa3d%40sessionmgr10&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=eft&AN=7
3984716
Miami Dade College. (2012). MDC’s Executive information system. Retrieved from
http://www.mdc.edu/main/
National Center for Education Statistics (2010). Digest of Educational Statistics, Tables and Figures.
Washington, DC: U.S. Department Of Education.
24
Figure 1. 9
th
Grade to College Traditional Path
Figure 2. 9
th
Grade to College Proposed
Path
10
th
Grade:
Geometry
9
th
Grade:
Algebra I
11
th
Grade:
Algebra
II
12
th
Grade:
Analysis of
Functions
Advanced
Topics
Pert Test
Post College
Algebra
courses
Dev.
Math
10
th
Grade:
Algebra II
9
th
Grade:
Geometry
11
th
Grade:
Analysis of
Functions
12
th
Grade:
The
college
experienc
e
EOC
Summer
School math
and English
remediation
Post college
algebra
Courses
8
th
Grade:
Algebra I
PERT
Summer
School math
and English
remediation
EOC
Summer
School math
and English
remediation
EOC
Summer
School math
and English
remediation
PERT
Summer
college
experience
25
A Defense of Epistemological Standpoint Theory
Dr. Melissa Lammey
Associate Professor of Philosophy
Miami Dade College
Hialeah Campus
E-mail: mlammey@mdc.edu
ABSTRACT
Epistemological standpoint theory is a theory of knowledge that takes the
role of the 'knower' as central to knowledge itself. While it is rooted in
Marxism, it is most thoroughly represented in the works of feminist
philosophers such as Nancy Hartsock and Sandra Harding. This view stands in
opposition to epistemological views that champion a particular view of
objectivity; namely, the view that 'objective' knowledge must be abstracted
from the knower in order to eliminate bias. Several feminists thinkers reject
this notion as not only problematic, but also impossible. In this paper, I defend
the idea that the inclusion of perspective is not bias, but is essential to a
complete epistemological theory.
In my view, the most difficult challenge to standpoint theory epistemology is
posed by the problem of ‘conflict’ among standpoints. In other words, when
descriptions of the world conflict, who is right? There seems to be no way to
determine the answer to this question according to epistemological standpoint
theory. In order to address this challenge, I argue that the desire for
epistemological objectivity is always motivated by normative concerns,
namely, by the question, ‘How should we understand the world around us?’ In
my view, the answer to this question must be a matter of what is the best
description of the world. The idea that there can be a ‘best’ description of the
world assumes that some goal is sought and it is not clear that goal is always
‘objectivity.’ In the case of social analysis, for instance, the goal is likely to be
something like justice, fairness, or equality. Here, I argue that within social
analysis, we can speak of one ‘best’ description of the world despite the fact
that epistemological standpoint theory alone does not seem to admit of one.
Keywords: philosophy, epistemology, standpoint theory, social analysis
26
The generic label, standpoint theory, seems to denote two distinct types of
theories. Epistemological standpoint theory aims to characterize the nature of
knowledge. Standpoint theory as social analysis, on the other hand, is the view
that some knowers have privileged access to knowledge about society. The
two are certainly related. Standpoint theory as social analysis assumes, and
often includes arguments for, the truth of epistemological standpoint theory.
For this reason, some arguments that are launched against standpoint theory as
social analysis are misplaced criticisms of epistemological standpoint theory.
My primary purpose in this paper is to discuss the distinction between
epistemological standpoint theory and standpoint theory as social analysis. In
doing so, I consider two types of feminist methodologies for the social sciences
as characterized by Sandra Harding. According to Harding (1991), these
methodologies differ in terms of their epistemic commitments. However, the
role of these commitments in evaluating their corresponding methodologies
needs to be clarified. My purpose in offering this clarification is to analyze the
relation between epistemological commitments and social analysis. I argue
that the merit of social analysis does not depend on its epistemological
commitments. In doing so, I address a species of argument against feminist
standpoint theory as social analysis– namely, those that assume this sort of
analysis is necessarily flawed by bias and ultimately results in relativism. I
argue that these criticisms are better aimed at the epistemological commitments
assumed by standpoint theory as social analysis. As such, they do not address
the merit of including standpoints in social analysis.
Standpoint theory in feminist epistemology was developed primarily as a
response to the positivist view of knowledge (Duran, 1998). Positivism is a
philosophical movement that was inspired by Auguste Comte and is the root of
contemporary logical positivism. As an empiricist, Comte believes that
knowledge can only arise from experience. He hoped to create a methodology
for solving social problems that relies upon positive knowledge. Positive
knowledge is free from bias, perspective – and as a result, speculation. For
Comte, the only candidate for positive knowledge is scientific knowledge.
Scientific knowledge is purported to be free of bias to the extent that controls
are able to eliminate the influence of perspective. For this reason, positivists
are concerned with determining the conditions under which reliable knowledge
is produced. As such, positivists propose the verifiability principle as a theory
of meaning. According to this principle, the only assertions that have meaning
are those that can be verified through observation. The required standard of
observation here is normal observation. What constitutes normal observation
is unclear, but it is clear that knowledge which relies upon one’s perspective or
standpoint is not verifiable and is interpreted as bias on this view.
Feminists largely reject the positivist criterion for knowledge. In fact, the
very concept feminism seems to depend on a feminine standpoint or
perspective. This is the basis of feminist standpoint theory, a cluster of views
that rely upon Marxist historical materialism (Harding 1991). Historical
materialism is the view that the economic structure of society determines the
nature of its citizens. In other words, one’s mode of production determines
one’s standpoint. The idea here is that the proletariat can offer a more
27
complete analysis of social structures than the bourgeois. This is because the
proletariat experience, and so have knowledge of, oppressive social conditions
that result from class difference. Similarly, feminist standpoint theorists argue
that women have privileged access to knowledge about sexist oppression. This
theory is articulated in the work of Nancy Hartsock insofar as she focuses on
the unique nature of women’s relation to the means of production within
capitalism in developing a feminist standpoint. On her view, standpoint theory
reveals a dual reality – men’s reality and women’s reality.
According to Hartsock (1997), women’s work within every society is
different from men’s. She takes this to be the primary division of labor and
suggests that it forms the basis of all other class divisions in Western capitalist
societies. Like men, women produce goods for consumption, yet the goods
they produce differ from those produced by men. Men produce commodities to
exchange while women produce the means of subsistence within the home. In
addition, women produce human beings. This involves biological functions
such as menstruation, sex, pregnancy, and childbirth; but also includes a
particular form of socialization that prepares women for childbirth and
childrearing. Hartsock argues that women’s role as mothers grounds a feminist
standpoint. As mothers, women’s work involves change and growth. A
mother produces a child within her own body and then, through her own labor,
gives birth to a child who eventually separates completely from her. This
situation produces a unique conception of boundaries for women. At least at
certain times, a woman and the product of her labor are difficult to
differentiate. For this reason, Hartsock contends that women are not rigidly
distinct from the natural world as men are.
Hartsock relies on a psychoanalytic account of gender differentiation in
arguing that women are socialized in a manner that connects them to nature. A
psychoanalytic account defines woman as the primary caregiver of children.
As such, an infant’s first and closest contact is with a woman. As the infant
grows to a child, self-identification takes place. Importantly, the child must
separate from the mother as she is the primary caregiver. When mother is all
that is known, masculinity exists as an abstract that boys come to embody. The
situation is different for girls. Eventually they will become the mother, so
there is no abstract ideal that defines their identity. This results in the further
identification of women with nature and concrete, as opposed to abstract,
existence (Tong, 1998).
As Hartsock explains, man’s interaction with the world is guided by
boundaries which produce a set of abstract social relations. This produces a
series of dualisms in his understanding of the world. The abstract is distinct
from the concrete, the mind is distinct from the body, and society is distinct
from nature. Woman, by contrast, does not experience boundaries as such
because her role as mother produces a different set of relations. She is tied to
nature as she is the producer of human beings. For this reason, Hartsock thinks
that women’s understanding rejects the dualisms that man creates. As she
explains, woman values the concrete over the abstract and connectedness over
separateness.
Given woman’s unique understanding of the world and her relation to it,
Hartsock contends that the female standpoint produces the feminist standpoint.
28
From this standpoint, women are uniquely positioned to diagnose the social
problems that arise as a result of class distinction. Because classes are defined
in opposition to one another, the standpoints of men and women vary in an
important way. Namely, because man experiences the benefits of class
distinction, his standpoint is less able to reveal its injustice. Woman, as man’s
opposite on her view, experiences and so can reveal social injustice. As a
result, a feminist standpoint can provide justification for social change.
Although Hartsock’s idea of a feminist standpoint is useful to feminist
inquiry, it seems that she links it to women as a matter of biological necessity.
She contends that the feminist standpoint is grounded in women’s experiences
as mothers and this is a matter of biological fact. While Hartsock provides the
groundwork for much work in feminist epistemology, notions of biological
essentialism are a source of dispute amongst feminists. For instance, Sandra
Harding (1997) rejects essentialism regarding women’s experiences. As
Harding explains, women have very different experiences from one another.
For this reason, she contends that there are no experiences that are particular to
women in general. Thus, there is no woman’s standpoint. However, she does
agree with Hartsock that there is a feminist standpoint.
The bulk of Harding’s work in epistemology is aimed at offering an analysis
of feminist methodologies in the social sciences. Harding (1997) characterizes
a methodology as “a theory and analysis of how research does or should
proceed.” On her view, feminist methodology suggests a technique for
gathering evidence and an account of how theory should be applied in specific
disciplines. Namely, feminists advocate evidence gathering techniques that do
not render the researcher invisible. The researcher’s role, in addition to the
phenomena she investigates, is subject to analysis. Feminist accounts of how
theory should be applied differ according to methodological goals. Harding
distinguishes between two types of feminist methodologies, feminist
empiricism and the feminist standpoint (1991).
According to Harding, feminist empiricists champion the traditional
methodological norm of achieving epistemological objectivity – knowledge
free from bias. They are feminist insofar as they believe that adhering to this
norm requires controlling for sexism and androcentrism. Harding points out
that the individual biases of researchers as well as the historically biased nature
of the work of science can be exposed by feminist analysis. However, she
argues that a tension exists between “the feminist uses of justificatory
strategies and the parental empiricist epistemology” (Harding, 1997). In other
words, the epistemological commitments of the feminist empiricists are in line
with traditional empiricist methodology – the best sort of knowledge adheres
with an unbiased or value-neutral interpretation of social phenomena.
However, the justificatory strategy they offer for feminist claims is that they
are somehow able to expose bias, or reveal objectivity. The tension Harding
points out is that the feminist empiricist wants to include perspective as a
means of eliminating perspective.
This tension does seem problematic on the face of it. However, it might be
alleviated via an analysis of the connections between epistemological theory
and social analysis. It seems that the feminist empiricist is in fact committed to
utilizing standpoints in social analysis, but undermines the significance of
29
standpoints in articulating the goal of her methodology. She justifies their use
insofar as she views them as advancing a commitment to epistemological
objectivity. Her reasoning is this: if standpoints can expose bias, then they
should be used as a means of achieving knowledge free from bias. Articulated
as such, the tension Harding points out is indeed problematic. However, it
seems that the desire for objectivity as a goal could be articulated in a less
problematic way. For the purposes of social analysis, we can aim at achieving
knowledge in abstraction from perspective even if this task can never be
fulfilled. The underlying epistemological commitment here is problematic, but
the methodology it sets out might be useful in its own right. The feminist
empiricist can advocate the incorporation of standpoints in gathering
information about what is in fact biased. She does not have to be committed to
the notion that a standpoint itself directly reveals bias. Rather, she could
articulate her project as an evaluation of input from different perspectives in
order to abstract commonalities and locate challenges to address. Understood
in this way, her project of incorporating perspectives to eliminate perspective
appears to be less problematic.
In contrast to feminist empiricism, Harding defends the feminist standpoint,
but does so in such a way that rejects the biological commitments of
Hartsock’s theory. On Harding’s view, the experiences women have in
struggling against oppression produce a feminist standpoint when guided by
feminist theory. In other words, women can come to see, or earn the feminist
standpoint by realizing and resisting male domination. By focusing on
women’s experience of social structures rather than childbirth and mothering,
Harding rejects the biological essentialism of Hartsock’s view. However, she
accepts the idea that women are in a position to gain a feminist standpoint in
virtue of their experiences as women in a male dominated societies.
To the extent that standpoint theorists are committed to epistemological
standpoint theory, they hold the view that epistemological objectivity is an
incoherent concept. On this view, abstraction from perspective neglects the
gendered nature of experience and so renders women’s experiences invisible
while disguising men’s experiences as objective. However, standpoint theory
as social analysis is the view that certain perspectives offer privileged access to
particular sorts of knowledge. Just as with feminist empiricism, there is a
disconnect between the method of social analysis advocated by feminist
standpoint theorists and the epistemological commitment from which it stems.
The idea of a privileged perspective cannot be derived from epistemological
standpoint theory alone. The notion that knowledge is had from a standpoint
does not on its own imply that any one standpoint is better than any other. This
claim depends upon a goal of social analysis. Namely, standpoint theory as
social analysis aims to reveal the nature and implications of power disparity.
While feminist standpoint theorists demonstrate congruity between their
epistemological commitments and their method of social analysis, the merit of
the social analysis they offer does not depend on this congruity. Although
epistemological standpoint theory seems to more accurately characterize the
nature of knowledge, it does not in itself support the claim that a privileged
perspective exists.
30
Epistemological standpoint theory seems to accurately characterize the nature
of knowledge because it rests on the seemingly obvious assumption that
knowledge can only be had from the perspective of a knower. In what follows,
I will attempt to explain that perspective is not necessarily biased while being
mindful of Harding’s view that biological determinism should be rejected.
Suppose the traditional view of knowledge as justified true belief is
unproblematic. An agent has a bit of knowledge, x, if and only if x is a true
belief she acquires in the appropriate way. In other words, it is no accident that
she has the true belief in question. She knows x because she bears witness to
the truth of the relevant belief. On this account of knowledge, the role of the
agent is significant for two reasons. First, agents acquire many of their beliefs
through experience. Second, exactly what beliefs an agent has depends on her
actual experiences. One might attempt to circumvent the necessity of the
agent’s actual experiences by suggesting that if the agent had the right sort of
experiences, she would be able to have the beliefs that correspond to them.
However, the sorts of experiences an agent is capable of having must be
determined.
Brute sensory perceptions certainly provide us with crucial information about
our environment. However, it seems that we depend upon social structures to
organize our sensory perceptions so that we can make sense of them in terms of
the goals we set out to achieve. Social structures are not materially extended
like the objects that prompt our sensory experiences, but they play a role in
characterizing our experiences. For instance, my experience of a book is
characterized by the fact that I know what to do with it at least as much as it is
by the qualitative features that the materially extended object I call ‘book’
impresses upon my consciousness. A book is not simply a bundle of blunt
sensory experience. It is something that I read, cite, or put on a shelf. These
facts are significant to my conscious understanding of the book. Likewise, my
experience of a particular person is not simply comprised of a materially
extended body – it is gendered, among other things. Gender is the social
structure that, in part, characterizes a body as a particular sort of person. As
such, it dictates how I understand and interact with that body. I understand it
as a person and social norms regulate how I interact with that person.
The manner in which I interact with a person is certainly part of her
experience. My interaction with her plays a role in shaping her understanding
of herself. In this way, she internalizes the social consciousness that
determines my understanding of her. Of course, to the extent that she is a self-
determining agent, I am not in fact determining her social consciousness.
Regardless of this, her social consciousness relies in part upon her
interpretations of my behavior. For instance, she can accept, modify, or reject
the characterization of her that I have in mind when treating her as I do.
Further, I am but one person with whom she interacts. My role in shaping her
social consciousness is limited and depends upon our relation to one another.
If she is my child, my role would likely be greater than if she were a passing
acquaintance. Thus far, I have discussed the limited effects that I might have
in shaping a person’s social consciousness. However, it is clear that the same
phenomena exists on the macro level as well – a person’s experiences taken as
31
a whole determines her social consciousness insofar as they provide her with
the information by which she understands herself and her society.
On the account I have just described, each person has a unique form of social
consciousness. This is because each person has unique experiences – the sort
of experiences that influence her understanding of herself and her relation to
the world. It follows from this that each person has a unique form of
consciousness. If each individual’s consciousness is framed by her social
consciousness, then there must be an irreducible subjective element of
consciousness, the sort that Thomas Nagel (1986) argues for, and that element
is crucial in determining social identity. How one views oneself, behaves, and
is treated depends on the nature of her consciousness. Now, a distinction
between consciousness and social consciousness makes sense when discussed
in abstraction from social identity. Indeed, such a distinction is needed to
understand the acquisition of social identity. However, when the goal is to
characterize the nature of an individual’s knowledge, it doesn’t make sense to
say that a person can have knowledge apart from her social identity. This
identity determines what sorts of beliefs she has in the first place. Further,
what she is capable of experiencing determines what she may bear witness to.
Social identity – how one views oneself, behaves, and is treated – is a factor in
determining the sorts of things a person might claim to know, especially
concerning what it is like to have a particular social identity.
Of course, social identity is particular to the individual, but this does not
mean that social identities cannot overlap one another – and indeed it seems
they do. As gender is concerned, it seems that persons of a particular gender
have experiences that are certainly unique, but are shaped by common social
structures. To the extent that one thinks of herself as a woman, behaves as a
woman, and is treated as a woman, her social identity is female regardless of
how she embodies this identity. There is a female bourgeois and a female
proletariat, a female who has ovaries and a female who doesn’t, a female who
conforms to feminine beauty standards and a female who doesn’t. Some
aspects of gender as a social identity are more fundamental than others, but the
point is that regardless of what positions a woman may occupy in society, there
is a space carved out for her to occupy that position as a woman. Exactly what
experiences she has and how she chooses to interpret herself in the context of
her surroundings will, of course, be unique to the individual. However, the
experiences she has to work with are the experiences of a woman. This is the
basis of my own formulation of epistemological standpoint theory. Knowledge
depends on a knower and the particular nature of a knower’s interaction with
society determines what sorts of things she can know.
Returning to standpoint theory as social analysis, a particularly stubborn
challenge arises. Harding (1991) argues that labeling a particular standpoint as
privileged allows feminist standpoint theorists a response to the charge that
standpoint theory is a form of relativism. As she explains, feminist standpoint
theorists do not suggest that women’s and men’s experiences ground equally
reliable knowledge. Rather, they argue that the feminist perspective produces
more complete knowledge because knowledge that results from men’s
experiences is unable to characterize the nature of women’s oppression.
However, articulating privileged experience that reveals the nature of women’s
32
oppression is best understood as a goal of social analysis. As such, the idea
that women’s experiences are able to produce privileged knowledge does not
rely on epistemological standpoint theory alone. Epistemological standpoint
theory only requires the existence of unique forms of knowledge – not
privileged forms. It does not, in itself, suggest that power disparity is
problematic in such a way that the feminist standpoint reveals a more complete
understanding of social structures. Yet, it does suggest that power disparity
produces unique forms of knowledge. For this reason, it seems that the charge
of relativism is better aimed at epistemological standpoint theory. However, if
I am correct that the nature of one’s knowledge depends on her social
consciousness, it seems that the relativism in question is not so problematic.
While knowledge is unique to the individual, overlapping experiences of social
structures which themselves can be understood as objective lessen the impact
of the charge of relativism. While there seems to be an irreducible subjective
element involved in experience, this does not mean that knowledge is
problematically relative to the individual. It is true that we cannot directly
access other people’s consciousness, but this fact places no special burden on
the standpoint theorist to answer the charge of relativism. On epistemological
standpoint theory, our unique experiences of social structures understood in
abstraction from social consciousness can overlap.
Feminist standpoint theory as social analysis, by contrast, just is the view that
women’s epistemological standpoints ground privileged access to knowledge
about social structures. This type of theory is necessarily goal oriented. To say
that women might possess a privileged perspective assumes that their
knowledge is desirable for social analysis. On this view, women’s experiences
do not simply produce knowledge that differs from men’s; they produce a more
complete form of knowledge. The fact that women’s experiences ground
unique research methodologies offers standpoint theorists a legitimate goal.
Uncovering these methodologies is important to social analysis because they
reveal information that is not captured by traditional research methodologies.
This information, in turn, allows the possibility of a unique form of social
analysis – namely, feminist social analysis. On this view, feminist social
analysis is necessarily a type of standpoint theory as social analysis. As such,
its merits should be judged independently of its assumption of epistemological
standpoint theory. Despite its epistemological commitments, a form of social
analysis is useful insofar as it achieves the goal of the analysis.
To summarize, I have argued that the merit of a particular social analysis
does not stand or fall on the success of its epistemological commitments. A
methodology for social analysis must rest on independent justification insofar
as it aims to uncover the best sort of knowledge. For this reason, it might be
the case that standpoint theory as social analysis is valuable regardless of its
epistemological commitments. Likewise, the merit of social analysis that rests
on a commitment to epistemological objectivity must be determined in its own
right. In my view, feminist standpoint theorists often conflate the claims of
epistemological standpoint theory and standpoint theory as social analysis.
This seems to explain why they are found to offer epistemological arguments
in support of their social analyses. In turn, criticisms of these sorts of theories
buy into and so argue against these sorts of justifications. I have argued in this
33
paper that both sorts of projects are problematic. The problem I have
characterized here seems to result from the reluctance of feminist researchers
to offer some independent justification for their methodologies. This
inclination is understandable because to offer such a justification would likely
involve an ethical commitment. The best sort of epistemological theory,
according to feminists, is the sort that allows a characterization of women’s
experiences – particularly those that reveal the nature of women’s oppression.
However, an analysis of social structures needs to offer a reason why we
should accept its particular characterization of society in organizing our
research methodologies – and this reason is likely to appeal to an ethical view.
Ultimately, the main problem I see facing standpoint theorists is this: When
descriptions of the world conflict, who is right? Epistemological standpoint
theory seems to offer no resolution to this problem because it incorporates
subjective experience as a necessary condition of knowledge. Settling a dispute
between or among conflicting perspectives requires that some goal of analysis
is sought and epistemology alone seems to admit of none. It is true that
epistemological standpoint theory suggests that the world is such that it admits
of various and sometimes contradictory descriptions, but I cannot see why this
is a problem in itself. However, this might be a very real problem for
standpoint theory as social analysis. Yet, when differing perspectives, and so
differing descriptions of the world, are considered in order to best understand
social structures, an independent goal is clearly sought. For feminists, that goal
is usually social justice, fairness, or equality. When these are the goals of
analysis, it is important to include various perspectives so that they may
receive equal, fair, or just consideration.
34
BIBLIOGRAPHY
Alcoff, Linda and Elizabeth Potter, eds. (1993). Feminist Epistemologies. New
York: Routledge.
Antony, Louise M. and Charlotte E. Witt, eds. (2002). A Mind of One’s Own:
Feminist Essays on Reason and Objectivity, 2
nd
ed. Cambridge: Westview
Press.
Duran, Jane. (1998). Philosophies of Science/ Feminist Theories. Boulder:
Westview Press.
Harding, Sandra and Jean F. O’Barr, eds. (1975). Sex and Scientific Inquiry.
Chicago: University of Chicago Press.
Harding, Sandra. (1998). Is Science Multicultural? Postcolonialisms,
Feminisms, and Epistemologies. Bloomington: Indiana University Press.
Harding, Sandra (1991). Whose Science? Whose Knowledge? Thinking From
Women’s Lives. Ithaca, Cornell University Press.
Harding, Sandra, ed. (1987). Feminism and Methodology: Social Science
Issues. Bloomington: Indiana University Press, 1987.
Harding, Sandra, ed. (1983). Discovering Reality: Feminist Perspectives on
Epistemology, Metaphysics, Methodology, and Philosophy of Science. Boston:
D. Reidel Publishing Company.
Harding, Sandra (1997). ‘Is there a Feminist Method?’ In: Kemp, Sandra and
Judith Squires, eds. Feminisms. New York: Oxford University Press.
Hartsock, Nancy (1997). ‘The Feminist Standpoint: Developing the Ground for
a Specifically Feminist Historical Materialism.’ In: Kemp, Sandra and Judith
Squires, eds. Feminisms. New York: Oxford University Press.
Nagel, Thomas. (1986). The View From Nowhere. New York: Oxford
University Press.
Tong, Rosemarie Putnam. (1998). Feminist Thought: A More Comprehensive
Introduction, 2
nd
ed. Boulder: Westview Press.
35
A Statistical Analysis of the Effects of Class Attendance and ANGEL Activity Logs on
MAC 1105 Students’ Final Examination Scores
Dr. M. Shakil
Associate Professor, Senior
Department of Mathematics
Miami-Dade College
Hialeah, Fl. 33012, USA
E-mail: mshakil@mdc.edu
ABSTRACT
The purpose of this paper is to examine whether the performance of MAC 1105 students in their final examinations are
affected by their Class Room Attendance and ANGEL Activity Logs in these classes.
Keywords: ANGEL Activity Logs, ANOVA, Chi Square Independence Test, Class Room Attendance, Hypothesis
Tests, Regression Analysis.
Mathematics Subject Classification 2010: 97B10; 97D40; 97D60
1. Background and Introduction: It has been found through empirical investigation and research by some researchers
that student performance is inversely correlated with absenteeism. That is, the grades of students with higher number of
absences are poorer than the grades of students with less number of absences. However, it appears from the literature that
not much significant work has been conducted to study the effects of attendance on the performance of students in math
classes at the college level. In this research project, the effects of attendance on students’ performance in their final
examinations in some mathematics classes have been investigated. We intend to examine whether the performance of
MAC 1105 students in their final examinations are affected by their Class Room Attendance and ANGEL Activity Logs
in these classes.
2. Materials and Methods: The studies are conducted on the students enrolled in two different sections of MAC 1105
classes during the Fall Term 2012 by analyzing the performance of these students in their final examinations based on
their attendance and ANGEL Activity Logs in these classes. According to the ANGEL Student Guide , “ANGEL is a web-
based course management and collaboration portal that enables educators to manage course materials and to communicate
with students. ANGEL can function both as a complement to traditional courses and as a site for distance learning.” It is
being used at Miami Dade College for both web-enhanced and online teaching.
In order to study the effects of attendance on MAC 1105 students’ performance in their final examinations, students of
two different sections of MAC 1105 classes are divided into two different groups as follows:
(i) Group 1 with 100 % class attendance, and Group 2 with class attendance less than 100 %.
(ii) Group 1 with
60 hours Angel Activity Logs, and Group 2 with
60 hours Angel Activity Logs.
A statistical data analysis is conducted using statistical methods such as Chi Square Test, Analysis of Variance,
Hypotheses Tests, Regression Analysis, among others. Some statistical and mathematical software such as MINITAB,
EXCEL, STATDISK, among others, have been used for these analyses.
36
3. Literature Review: As stated above, it appears from the literature that not much significant work has been conducted
to study the effects of attendance on the performance of students in math classes at the college level. However, some
studies have been reported in the literature on the relationship between class attendance and performance of students at the
university and school level as investigated by different researchers. For example, the studies conducted by Gottfried
(2010) on the relationship between student attendance and achievement in elementary and middle schools have
demonstrated the positive effect of the students’ presence in school on their learning outcomes. Similar studies have been
conducted by other researchers. For example, Oghuvbu (2010) has studied correlationship between attendance and
academic performance of students in secondary schools. For additional and relevant references on the relationship
between student attendance and achievement in elementary and middle schools, the interested readers may also visit
Gottfried (2010). Many authors have also studied the relationship between class attendance and performance of students at
the university level in various disciplines including medical sciences, among them Jaykaran et al (2011), Fjortoft (2005),
and Khan et al (2003) are notable.
4. Discussion and Analysis: In this section, data analysis of MAC 1105 Students’ Final Examination Scores Based on
their Class Attendance and ANGEL Activity Logs using statistical methods such as Chi Square Independence Test,
Analysis of Variance, Hypotheses Tests, Regression Analysis, among others, are presented and discussed.
4.1. Chi Square Independence Test: Below are presented the effects of Class Room Attendance and ANGEL Activity
Logs on MAC 1105 students’ final examination scores using Chi Square Independence Test.
(A) Effects of Class Room Attendance on MAC 1105 Students’ Final Examination Scores:
(AI) Chi-Square Test: RESULTS-A, Score < 70 (MAC1105-A), Score>= 70 (MAC1105-A)
(Group 1 with 100 % class attendance, and Group 2 with class attendance less than 100 %)
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
Score < 70 Score>= 70
RESULTS-A (MAC1105-A) (MAC1105-A) Total
1 14 3 11 28
14.00 4.50 9.50
0.000 0.500 0.237
2 14 6 8 28
14.00 4.50 9.50
0.000 0.500 0.237
Total 28 9 19 56
Chi-Sq = 1.474, DF = 2, P-Value = 0.479
2 cells with expected counts less than 5.
Based on the above results, it appears that there is not enough evidence to support the claim that MAC 1105
Students’ Final Examination Scores are dependent on their class room attendance , for example, at α = 0.10 level of
significance.
37
(AII) Chi-Square Test: RESULTS-B, Score < 70 (MAC1105-B), Score>= 70 (MAC1105-B)
(Group 1 with 100 % class attendance, and Group 2 with class attendance less than 100 %)
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
Score < 70 Score>= 70
RESULTS-B (MAC1105-B) (MAC1105-B) Total
1 13 3 10 26
13.00 4.88 8.13
0.000 0.721 0.433
2 19 9 10 38
19.00 7.13 11.88
0.000 0.493 0.296
Total 32 12 20 64
Chi-Sq = 1.943, DF = 2, P-Value = 0.378
1 cells with expected counts less than 5.
Based on the above results, it appears that there is not enough evidence to support the claim that MAC 1105
Students’ Final Examination Scores are dependent on their class room attendance, for example, at α = 0.10 level of
significance.
(AIII) Chi-Square Test: RESULTS-A&B-Comb, Score < 70 (MAC1105--A&B-Comb),
Score>= 70 (MAC1105--A&B-Comb)
(Group 1 with 100 % class attendance, and Group 2 with class attendance less than 100 %)
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
Score < 70 Score>= 70
RESULTS-A&B-Comb (MAC1105--A&B-Comb) (MAC1105--A&B-Comb) Total
1 27 6 21 54
27.00 9.45 17.55
0.000 1.260 0.678
2 33 15 18 66
33.00 11.55 21.45
0.000 1.031 0.555
Total 60 21 39 120
Chi-Sq = 3.523, DF = 2, P-Value = 0.172
Based on the above results, it appears that there is not enough evidence to support the claim that MAC 1105
Students’ Final Examination Scores are dependent on their class room attendance, for example, at α = 0.10 level of
significance.
38
(B) Effects of ANGEL Activity Logs on MAC 1105 Students’ Final Examination Scores:
(BI) Chi-Square Test: RESULTS-A, Score < 70 (MAC1105-A), Score>= 70 (MAC1105-A)
(Group 1 with
60 hours Angel Activity Logs, and Group 2 with
60 hours Angel Activity Logs)
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
Score < 70 Score>= 70
RESULTS-A (MAC1105-A) (MAC1105-A) Total
1 13 2 11 26
13.00 4.18 8.82
0.000 1.136 0.538
2 15 7 8 30
15.00 4.82 10.18
0.000 0.984 0.466
Total 28 9 19 56
Chi-Sq = 3.125, DF = 2, P-Value = 0.210
2 cells with expected counts less than 5.
Based on the above results, it appears that there is not enough evidence to support the claim that MAC 1105
Students’ Final Examination Scores are dependent on their Angel Activity Logs.
(BII) Chi-Square Test: RESULTS-B, Score < 70 (MAC1105-B), Score>= 70 (MAC1105-B)
(Group 1 with 60 hours Angel Activity Logs, and Group 2 with 60 hours Angel Activity Logs)
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
Score < 70 Score>= 70
RESULTS-B (MAC1105-B) (MAC1105-B) Total
1 11 3 8 22
11.00 4.13 6.88
0.000 0.307 0.184
2 21 9 12 42
21.00 7.88 13.13
0.000 0.161 0.096
Total 32 12 20 64
Chi-Sq = 0.748, DF = 2, P-Value = 0.688
1 cells with expected counts less than 5.
Based on the above results, it appears that there is not enough evidence to support the claim that MAC 1105
Students’ Final Examination Scores are dependent on their Angel Activity Logs.
39
(BIII) Chi-Square Test: RESULTS-A&B-Comb, Score < 70 (MAC1105--A&B-Comb),
Score>= 70 (MAC1105--A&B-Comb)
(Group 1 with 60 hours Angel Activity Logs, and Group 2 with 60 hours Angel Activity Logs)
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
Score < 70 Score>= 70
RESULTS-A&B-Comb (MAC1105--A&B-Comb) (MAC1105--A&B-Comb) Total
1 24 5 19 48
24.00 8.40 15.60
0.000 1.376 0.741
2 36 16 20 72
36.00 12.60 23.40
0.000 0.917 0.494
Total 60 21 39 120
Chi-Sq = 3.529, DF = 2, P-Value = 0.171
Based on the above results, it appears that there is not enough evidence to support the claim that MAC 1105
Students’ Final Examination Scores are dependent on their Angel Activity Logs.
4.2. Regression Analysis: Below are presented Regression Analysis to examine the effects of Class Room Attendance
and ANGEL Activity Logs on the performance of MAC 1105 students in their final examinations.
(A) Regression Analysis for MAC1105(a) Class: FinalExam versus Attendance, AngelActivity:
The regression equation is
FinalExam = 37.4 + 0.510 Attendance + 0.255 AngelActivity
Predictor Coef SE Coef T P
Constant 37.44 21.12 1.77 0.088
Attendance 0.5102 0.5195 0.98 0.336
AngelActivity 0.25487 0.07936 3.21 0.004
S = 13.3073 R-Sq = 32.9% R-Sq(adj) = 27.6%
Analysis of Variance
Source DF SS MS F P
Regression 2 2175.6 1087.8 6.14 0.007
Residual Error 25 4427.1 177.1
Total 27 6602.7
Source DF Seq SS
Attendance 1 349.0
AngelActivity 1 1826.5
40
Unusual Observations
Obs Attendance FinalExam Fit SE Fit Residual St Resid
4 29.0 80.00 78.74 7.72 1.26 0.12 X
7 44.0 100.00 70.59 3.51 29.41 2.29R
17 44.0 40.00 64.98 4.60 -24.98 -2.00R
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
We notice that there are three unusual (or influential) observations which possibly might affect the equation of the
regression line. Therefore, we feel it necessary that we should exclude these observations so that they do not influence the
results of our study, see, for example, Bluman (2010, page 548). For this reason, we have reanalyzed our data and
determined the second equation of the regression line, including the Residual Plots for FinalExam (Figure 1) and
Scatterplot of FinalExam vs Attendance, AngelActivity (Figure 2), excluding the above three observations, as provided
below.
2nd Regression Analysis for MAC1105(a) Class: FinalExam versus Attendance, AngelActivity
The regression equation is
FinalExam = 35.3 + 0.511 Attendance + 0.269 AngelActivity
Predictor Coef SE Coef T P
Constant 35.25 19.06 1.85 0.078
Attendance 0.5109 0.4932 1.04 0.312
AngelActivity 0.26898 0.07241 3.71 0.001
S = 10.3890 R-Sq = 49.1% R-Sq(adj) = 44.2%
Analysis of Variance
Source DF SS MS F P
Regression 2 2182.4 1091.2 10.11 0.001
Residual Error 21 2266.5 107.9
Total 23 4449.0
Source DF Seq SS
Attendance 1 693.2
AngelActivity 1 1489.3
41
Residual
Pe
rce
nt
20100-10-20
99
90
50
10
1
Fitted Value
Residual
90807060
10
0
-10
-20
Residual
Fre
qu
en
cy
151050-5-10-15
4
3
2
1
0
Observation Order
Resid
ual
24222018161412108642
10
0
-10
-20
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Histogram of the Residuals Residuals Versus the Order of the Data
Residual Plots for FinalExam
Figure 1: Residual Plots for FinalExam
Fina
lExa
m
45403530
100
90
80
70
60
50
40
120906030
Attendance AngelActivity
Scatterplot of FinalExam vs Attendance, AngelActivity
Figure 2: Scatterplot of FinalExam vs Attendance, AngelActivity
42
(B) Regression Analysis for MAC1105(b) Class: FinalExam versus Attendance, AngelActivity:
The regression equation is
FinalExam = - 25.4 + 2.87 Attendance + 0.109 AngelActivity
Predictor Coef SE Coef T P
Constant -25.44 78.70 -0.32 0.749
Attendance 2.869 2.585 1.11 0.276
AngelActivity 0.1088 0.1001 1.09 0.286
S = 21.7258 R-Sq = 8.7% R-Sq(adj) = 2.4%
Analysis of Variance
Source DF SS MS F P
Regression 2 1299.2 649.6 1.38 0.269
Residual Error 29 13688.3 472.0
Total 31 14987.5
Source DF Seq SS
Attendance 1 741.6
AngelActivity 1 557.5
Unusual Observations
Obs Attendance FinalExam Fit SE Fit Residual St Resid
6 31.0 65.00 83.07 13.17 -18.07 -1.05 X
27 32.0 10.00 67.23 7.25 -57.23 -2.79R
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
We notice that there are two unusual (or influential) observations which possibly might affect the equation of the
regression line. Therefore, we feel it necessary that we should exclude these observations so that they do not influence the
results of our study, see, for example, Bluman (2010, page 548). For this reason, we have reanalyzed our data and
determined the second equation of the regression line, including the Residual Plots for FinalExam (Figure 3) and
Scatterplot of FinalExam vs Attendance, AngelActivity (Figure 4), excluding the above two observations, as provided
below.
2nd Regression Analysis for MAC1105(b) Class: FinalExam versus Attendance, AngelActivity
The regression equation is
FinalExam = - 10.7 + 2.70 Attendance + 0.101 AngelActivity
Predictor Coef SE Coef T P
Constant -10.73 43.56 -0.25 0.808
Attendance 2.700 1.429 1.89 0.072
AngelActivity 0.10058 0.08472 1.19 0.248
S = 11.1933 R-Sq = 20.0% R-Sq(adj) = 12.7%
43
Analysis of Variance
Source DF SS MS F P
Regression 2 687.6 343.8 2.74 0.086
Residual Error 22 2756.4 125.3
Total 24 3444.0
Source DF Seq SS
Attendance 1 511.0
AngelActivity 1 176.6
Residual
Pe
rce
nt
20100-10-20
99
90
50
10
1
Fitted Value
Residual
8580757065
10
0
-10
-20
Residual
Fre
qu
en
cy
151050-5-10-15
8
6
4
2
0
Observation Order
Resid
ual
24222018161412108642
10
0
-10
-20
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Histogram of the Residuals Residuals Versus the Order of the Data
Residual Plots for FinalExam
Figure 3: Residual Plots for FinalExam
44
Fina
lExa
m
323028
100
90
80
70
60
50
10080604020
Attendance AngelActivity
Scatterplot of FinalExam vs Attendance, AngelActivity
Figure 4: Scatterplot of FinalExam vs Attendance, AngelActivity
(C) Regression Analysis for MAC1105 (a and b Combined) Classes: FinalExam versus Attendance, AngelActivity:
Below are presented Regression Analysis for MAC1105 (a and b Combined) Classes to examine the effects of Class
Room Attendance and ANGEL Activity Logs on the performance of MAC 1105 students in their final examinations. In
our Regression Analysis, we have excluded the unusual (or influential) observations so that they do not influence the
results of our study, and determined the corresponding equation of the regression line, including the Residual Plots for
FinalExam (Figure 5) and Scatterplot of FinalExam vs Attendance, AngelActivity (Figure 6), as provided below.
Regression Analysis for MAC1105 (a and b Combined) Classes: FinalExam versus Attendance, AngelActivity
The regression equation is
FinalExam = 70.2 - 0.192 Attendance + 0.206 AngelActivity
Predictor Coef SE Coef T P
Constant 70.226 9.983 7.03 0.000
Attendance -0.1919 0.2937 -0.65 0.517
AngelActivity 0.20641 0.06020 3.43 0.001
S = 11.8127 R-Sq = 20.7% R-Sq(adj) = 17.3%
Analysis of Variance
Source DF SS MS F P
Regression 2 1679.2 839.6 6.02 0.005
Residual Error 46 6418.8 139.5
Total 48 8098.0
45
Residual
Pe
rce
nt
30150-15-30
99
90
50
10
1
Fitted Value
Residual
8580757065
20
10
0
-10
-20
Residual
Fre
qu
en
cy
20100-10-20
10.0
7.5
5.0
2.5
0.0
Observation Order
Resid
ual
454035302520151051
20
10
0
-10
-20
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Histogram of the Residuals Residuals Versus the Order of the Data
Residual Plots for FinalExam
Figure 5: Residual Plots for FinalExam
Fina
lExa
m
4540353025
100
90
80
70
60
50
40
120906030
Attendance AngelActivity
Scatterplot of FinalExam vs Attendance, AngelActivity
Figure 6: Scatterplot of FinalExam vs Attendance, AngelActivity
46
4.3. Hypothesis Tests for Student’s Class Room Attendance and ANGEL Activity Logs: Below are presented the
hypothesis tests for Student’s Class Room Attendance and ANGEL Activity Logs for the two sections of MAC 1105
classes.
4.3.1. Hypothesis Test for the Two Independent Means of Student’s Class Room Attendance using t
Distribution: For the two sections of MAC 1105 classes, below are presented the descriptive statistics, respective
histograms (Figure 7), and hypothesis tests for Student’s Class Room Attendance, including the respective Figure 8.
Explore Data: MAC1105 (a)
Sample Size, n: 28; Mean: 40.5; Median: 43.5; Midrange: 36.5; RMS: 40.79303; Variance, s^2: 24.7037;
St Dev, s: 4.970282; Mean Abs Dev: 4.071429; Range: 15; Coeff. Of Var. 12.27%; Minimum: 29; 1st Quartile: 38;
2nd Quartile: 43.5; 3rd Quartile: 44; Maximum: 44
Explore Data: MAC1105 (b)
Sample Size, n: 32; Mean: 30.59375; Median: 31; Midrange: 29.5; RMS: 30.63036; Variance, s^2: 2.313508;
St Dev, s: 1.521022; Mean Abs Dev: 1.294922; Range: 5; Coeff. Of Var. 4.97%; Minimum: 27; 1st Quartile: 29.5;
2nd Quartile: 31; 3rd Quartile: 32; Maximum: 32
Fre
qu
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cy
4442403836343230
18
16
14
12
10
8
6
4
2
0
323130292827
14
12
10
8
6
4
2
0
Attd_MAC1105(a) Attd_MAC1105(b)
MAC1105 (a) & (b) Student's Attendance
Fall 2012 - Starting 8/27/2012 Ending before 12/21/2012
MAC1105(a) - Total # of Lectures: 44 (3 Days Per Week)
MAC1105(b) - Total # of Lectures: 32 (2 Days Per Week)
Figure 7: Histograms of Attendance of MAC1105(a), Attendance of MAC1105(b)
47
MAC 1105 Student’s Attendance
(Hypothesis Test for the Two Independent Means using t Distribution)
Not eq. vars: No Pool
Claim: µ1 not equal µ2
Test Statistic, t: 10.1441
Critical t: ±2.038415
P-Value: 0.0000
Degrees of freedom: 31.4171
95% Confidence interval:
7.91863 < µ1-µ2 < 11.90137
Reject the Null Hypothesis. Sample provides evidence to support the claim: µ1 not equal µ2.
Figure 8: Hypothesis Test for the Two Independent Means using t Distribution
4.3.2. Hypothesis Test for the Standard Deviations of Two Samples using F Distribution: For the two
sections of MAC 1105 classes, below are presented the hypothesis Test for the standard deviations of two samples using F
Distribution, including the respective Figure 9.
48
MAC 1105 Student’s Attendance
(Hypothesis Test for the Standard Deviations of Two Samples using F Distribution)
Claim: SD not equal SD(hyp)
Test Statistic, F: 10.6912
Lower Critical F: 0.4705152; Upper Critical F: 2.084818; P-Value: 0.0000
95% Confidence interval: 2.264533 < SD1/SD2 < 4.76679; 5.12811 < Var1/Var2 < 22.72228
Reject the Null Hypothesis. Sample provides evidence to support the claim.
Figure 9: Hypothesis Test for the Standard Deviations of Two Samples using F Distribution
4.3.3. Hypothesis Tests for the Two Independent Means of Student’s ANGEL Activity Logs using t
Distribution: For the two sections of MAC 1105 classes, below are presented the descriptive statistics, respective
histograms (Figure 10), and hypothesis tests for Student’s ANGEL Activity Logs, including the respective Figure 11.
Descriptive Statistics: ActivityMAC1105(a), ActivityMAC1105(b)
Total
Variable Count N N* CumN Percent CumPct Mean StDev
ActivityMAC1105(a) 116 116 0 116 100 100 17.22 14.17
ActivityMAC1105(b) 116 116 0 116 100 100 16.32 14.77
Variable Minimum Q1 Median Q3 Maximum
ActivityMAC1105(a) 1.00 8.00 13.00 23.00 86.00
ActivityMAC1105(b) 0.000000000 6.00 11.50 20.75 78.00
49
Fre
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75604530150
30
25
20
15
10
5
0
75604530150
ActivityMAC1105(a) ActivityMAC1105(b)
MAC1105 (a) & (b) Angel Activity Logs
Fall 2012
Angel Activity Logs: Starting 8/27/2012 Ending before 12/21/2012
(Total # of Angel Activity Days: 116)
Figure 10: Histograms of Student’s ANGEL Activity Logs of MAC1105(a) and MAC1105(b)
MAC 1105 Students’ Angel Activity Logs
(Hypothesis Test for the Two Independent Means using Student t Distribution)
Not eq. vars: No Pool
Claim µ1 = µ2
Test Statistic, t: 0.4736
Critical t: ±1.970348
P-Value: 0.6362
Degrees of freedom: 229.6056
95% Confidence interval: -2.844473 < µ1-µ2 < 4.644473
Fail to Reject the Null Hypothesis. Sample does not provide enough evidence to reject the claim.
50
Figure 11: Hypothesis Test for the Two Independent Means using Student t Distribution
4.3.4. Hypothesis Test for the Standard Deviation of Two Samples of Student’s ANGEL Activity Logs
using F Distribution: For the two sections of MAC 1105 classes, below are presented the hypothesis Test for the
standard deviations of two samples using F Distribution, including the respective Figure 12.
MAC 1105 Students’ Angel Activity Logs
(Hypothesis Test for the Standard Deviation of Two Samples using F Distribution)
Claim: SD = SD(hyp)
Test Statistic, F: 0.9204
Lower Critical F: 0.6925655; Upper Critical F: 1.443907
P-Value: 0.6573
95% Confidence interval: 0.7983986 < SD1/SD2 < 1.152813; 0.6374403 < Var1/Var2 < 1.328978
Fail to Reject the Null Hypothesis. Sample does not provide enough evidence to reject the claim.
51
Figure 12: Hypothesis Test for the Standard Deviation of Two Samples using F Distribution
4.4. Hypothesis Test for Student’s Final Exams Performance: For the two sections of MAC 1105 classes, below are
presented the descriptive statistics, respective histograms (Figure 13), and hypothesis tests for Final Exams Performance,
including the respective Figure 14.
Descriptive Statistics: MAC1105(a)_FinalExam, MAC1105(b)_FinalExam
Variable N N* Mean StDev Variance CoefVar Minimum Q1
MAC1105(a)_Final 28 0 73.39 15.64 244.54 21.31 40.00 61.25
MAC1105(b)_Final 32 0 68.13 21.99 483.47 32.28 10.00 61.25
Variable Median Q3 Maximum
MAC1105(a)_Final 72.50 85.00 100.00
MAC1105(b)_Final 70.00 85.00 95.00
52
Fre
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96806448
9
8
7
6
5
4
3
2
1
0
80604020
10
8
6
4
2
0
MAC1105(a)_FinalExam MAC1105(b)_FinalExam
Fall 2012
MAC1105 (a) and (b): Final Exam Scores
Figure 13: Histograms of Student’s Final Exam Scores of MAC1105(a) and MAC1105(b)
MAC 1105 Student’s Final Exams Performance
(Hypothesis Test for the Two Independent Means using t Distribution)
Not eq. vars: No Pool ; Claim µ1 = µ2
Test Statistic, t: 1.0771; Critical t: ±2.003402; P-Value: 0.2861
Degrees of freedom: 55.7940; 95% Confidence interval: -4.523355 < µ1-µ2 < 15.04335
Fail to Reject the Null Hypothesis. Sample does not provide enough evidence to reject the claim.
Figure 14: Hypothesis Test for the Two Independent Means using Student t Distribution
53
5. Concluding Remarks: In this paper, we have examined the effects of class attendance and ANGEL Activity Logs
on MAC 1105 students’ final examination scores. Based on our studies, it appears that there is not enough evidence to
support the claim that MAC 1105 Students’ Final Examination Scores are dependent on their class attendance and Angel
Activity Logs. We have also conducted Regression Analysis to examine the effects of Class Room Attendance and
ANGEL Activity Logs on the performance of MAC 1105 students in their final examinations. Finally, we have presented
the hypothesis tests for Student’s Class Room Attendance and ANGEL Activity Logs for the two sections of MAC 1105
classes. It is hoped that this study will be useful for further research on similar topics.
REFERENCES
Bluman, A. G. (2010). Elementary Statistics: A Step by Step Approach, 5th Edition, New York: McGraw-Hill.
Fjortoft, N. (2005). Students’ motivations for class attendance. American Journal of Pharmaceutical Education , 69(1),
107-112.
Gottfried, M. A. (2010). Evaluating the Relationship Between Student Attendance and Achievement in Urban Elementary
and Middle Schools: An Instrumental Variables Approach. American Educational Research Journal , 47(2), 434-465.
Jaykaran, P. Y., Chavda, N., and Kantharia, N. D. (2011). Factors associated with performance of second year student in
pharmacology examinations. Journal of pharmacology & pharmacotherapeutics , 2(2), 123.
Khan, H. U., Khattak, A. M., Mahsud, I. U., Munir, A., Ali, S., Khan, M. H., Saleem, M., and Shah, S. H. (2003). Impact
of class attendance upon examination results of students in basic medical sciences. J. Ayub Med. Col., 15(2), 56–58.
Oghuvbu, E. P. (2010). Attendance and Academic Performance of Students in Secondary Schools: A Correlational
Approach. Studies on Home and Community Science , 4(1), 21-25.
54
Dr. Dudley Weldon Woodard, the First African-American Mathematician to
Publish a Research Paper in an International Accredited Mathematics Journal
– A Historical Introduction
Dr. M. Shakil
Associate Professor, Senior
Department of Mathematics
Miami-Dade College
Hialeah, Fl. 33012, USA
E-mail: mshakil@mdc.edu
ABSTRACT
The purpose of this paper is to present in historical perspective the life and achievement of an African-
American Mathematician, Dr. Dudley Weldon Woodard, in the field of mathematical sciences, and highlight
one of his research papers published by him in Fundamenta Mathematicae in the year 1929 which appears to be
the first research paper published in an international accredited mathematics journal by any African-American
Mathematician.
Key Words: African-American, Analysis Situs, Closed Curve, Jordan Curve Theorem, Topology,
Mathematician.
Mathematics Subject Classification 2010: 01A05; 01A07; 01A70; 01A85
1. Introduction: The accomplishments of the past and present mathematicians can serve as pathfinders to their
contemporary and future colleagues. The achievements of many mathematicians, and their contributions, both small and
large, have been overlooked when chronicling the history of mathematics. By describing the academic history of these
personalities within mathematical sciences, we can see how the efforts of individuals have advanced human understanding
in the world around us. History bears testimony to their achievements, abilities and accomplishments. It should be the
responsibility of the present mathematical world to highlight the achievements of the past mathematicians. The purpose of
this paper is to present a biographical sketch and contributions of an American Mathematician of African Diaspora, Dr.
Dudley Weldon Woodard, in the field of mathematical sciences, with special reference to one of his research papers
published by him in the year 1929 which appears to be the first research paper published in an international accredited
mathematics journal by any African American-Mathematician.
2. Some Prominent African-American Mathematicians (19th Century to 1950):
2.1. African-American mathematicians have contributed in both large and small ways that is overlooked when
chronicling the history of science and mathematics. By describing the scientific history of African-American
55
men and women within mathematical sciences, we can see how the efforts of individuals have advanced human
understanding in the world around us. The contributions of the African-American scholars and their abilities in
the fields of science and mathematics are enormous. Their accomplishments in the field of mathematical
sciences are remarkable and noteworthy. The achievements of African-Americans in mathematics can be
divided into four different periods beginning from 18th century to the present. These periods can be further
classified and indexed by the year as provided in the following Tree Diagram, (see, for example, the
“Mathematicians of the African Diaspora” website created and maintained by Professor Dr. Scott W. Williams,
Professor of Mathematics University at Buffalo, SUNY, among others).
A TREE DIAGRAM
2.2 AFRICAN-AMERICAN MATHEMATICIANS OF 19TH CENTURY: Below is the list of three
African-American mathematicians of 19th century, who are prominent for their contribution to the knowledge
and advancement of mathematical sciences.
(i) Charles Reason (1814 - 1893) is considered to be the first African-American to receive a faculty position in
mathematics, in the year 1849, at a predominantly white institution - Central College in Cortland County, New
York.
(ii) Edward Alexander Bouchet was the first African-American to earn a Ph.D. in Physics (Science), in the
year 1878, from Yale University, and only the sixth American to possess a Ph.D. in Physics. It should be noted
that Yale University became the first United States of America institution, in the year 1862, to award a Ph.D. in
mathematics.
(iii) Kelly Miller was the first African American to study graduate mathematics, in the year 1886, at Johns
Hopkins University. It will be interesting to note that Johns Hopkins University was the first American
University to offer a program in graduate mathematics.
2.3 AFRICAN-AMERICAN MATHEMATICIANS OF 20TH CENTURY: The list of African-American
mathematicians of 20th century is very exhaustive. The statistics on the numbers of African-Americans
African-American Mathematicians
18th Century 19th Century
20th Century
21st Century
2000 - 2004 1925 - 1999 1700 - 1799 1800 - 1899
56
receiving Ph.D.’s in the field of mathematical sciences during the period 1925-2004 have been presented in the
following graph.
STATISTICS OF AFRICAN-AMERICAN Ph.D.’s IN MATHEMATICS (1925 – 2004)
The statistics on the numbers of African-Americans receiving Ph.D.’s in the field of mathematical sciences
during the period 1925-2004 have been presented in the following graph.
2.4 AFRICAN-AMERICAN MATHEMATICIANS (1925 – 1950):
It appears from the above graph that from1925 to 1950 not many African-American had accomplished much in
mathematics except a few. Among them, the names of the following African-American Mathematicians are
notable because of their excellence and contribution to the knowledge and advancement of mathematical
sciences during this period.
(1) 1925: Elbert Frank Cox was the first African-American to earn a Ph.D. in Mathematics in 1925 from
Cornell University. There were 28 Ph.D.'s awarded in the United States that year.
(2) 1928: Dudley Weldon Woodard was the second African-American to earn a Ph.D. in Mathematics in 1928
from the University of Pennsylvania.
57
(4) 1933: William Schieffelin Claytor was the third African-American to earn a Ph.D. in Mathematics
(University of Pennsylvania). Dr. Claytor had an extraordinary promise as a mathematician.
(5) 1934: Walter R. Talbot was the fourth African-American to earn a Ph.D. in Mathematics (University of
Pittsburgh).
(6) 1938: Ruben R. McDaniel (Cornell University), and Joesph Pierce (University of Michigan) were the fifth
and sixth African-Americans to earn a Ph.D. in Mathematics in the year 1938.
(7) 1941: David Blackwell was the seventh African-American to earn a Ph.D. in Mathematics, in the year
1941, from the University of Illinois. Dr. Blackwell earned his Ph.D. at the age of 22. He is regarded as one of
the greatest African-American mathematician of the 20th century. Dr. Blackwell is famous and well-known in
the world of mathematics for his seminal “Rao-Blackwell Theorem” which gives a technique for obtaining
unbiased estimators with minimum variance with the help of sufficient statistics (see, for example, Dudewicz
and Mishra (1988), Kapur (1999), and Rohatgi and Saleh (2001), among others). In 1954, Dr. David Blackwell
became the first African-American to hold a permanent position at one the major universities, University of
California at Berkley.
(8) 1942: J. Ernest Wilkins became the eighth African-American to earn a Ph.D. in Mathematics, in the year
1942, from the University of Chicago. Dr. Wilkins earned his Ph.D. at the age of 19. He is also regarded as one
of the greatest and rarest African-American mathematician of the 20th century
(9) 1943: Euphemia Lofton Haynes (Catholic University), the first African -American woman, and Clarence
F. Stephens (University of Michigan) were the ninth and tenth African-Americans, respectively, to earn a Ph.D.
in Mathematics (see, for example, the websites “Black Women in Mathematics” and “Timeline of African
American Ph.D.'s in Mathematics,” among others). The Morgan-Potsdam Model is the name given to a method
of the teaching of mathematics developed by Dr. Clarence F. Stephens at Morgan State University and refined
at the State University of New York College at Potsdam. Dr. Clarence F. Stephens also received the
Mathematical Association of America Gung-Hu Award for the Pottsdam Miracle. Under the direction of Dr.
Clarence Stephens, Morgan State University became the first institution to have three African-Americans of the
same graduating class (1964), who obtained a Ph.D. in Mathematics. These people were Dr. Earl Barnes
(University of Maryland, 1968), Dr. Arthur Grainger (University of Maryland, 1972), and Dr. Scott Williams
(Lehigh University, 1969). This is still a record that stands among all U.S. universities and colleges.
(10) 1944: This is the year when the eleventh, twelfth and thirteenth African- Americans, Joseph J. Dennis
(from Northwestern University), Wade Ellis, Sr. and Warren Hill Brothers (both from University of
Michigan), respectively, earned a Ph.D. in Mathematics.
(11) 1945: Jeremiah Certaine was the fourteenth African-American to earn a Ph.D. in Mathematics, in the
year 1945, from the University of Michigan.
(12) 1949: Evelyn Boyd Granville was the fifteenth African-American and the second African-American
Woman to earn a Ph.D. in Mathematics, in the year 1949, from Yale University.
(13) 1950: Marjorie Lee Browne (University of Michigan), the third African-American Woman, and George
H. Butcher (University of Pennsylvania) were the sixteenth and seventeenth African-Americans, respectively,
to earn a Ph.D. in Mathematics, in the year 1950.
58
For a chronology of African-Americans, who have excelled and contributed to the knowledge and advancement
of mathematical sciences, during the period 1925 – 2004, the interested readers are referred to Shakil (2010)
and the “Mathematicians of the African Diaspora” website created and maintained by Professor Dr. Scott W.
Williams, Professor of Mathematics University at Buffalo, SUNY, among others).
3. Dr. Dudley Weldon Woodard – A Biographical Sketch: The materials presented here are based
on and adapted from Professor Scott W. Williams’ website “Mathematicians of the African Diaspora,”
www.math.buffalo.edu/mad/index.html.
3.1. A Biographical Sketch: In what follows, a biographical sketch and contributions of Dr. Dudley
Weldon Woodard in the field of mathematical sciences will be presented, with special reference to one of his
research papers published by him in the year 1929 which is the first research paper published in an international
accredited mathematics journal by any African American-Mathematician.
Not much is known about Dr. Dudley Weldon Woodard's early childhood. He was born in 1881. His father was
employed with the U.S. Postal Service. After completing his early education in Texas, he attended Wilberforce
College in Ohio, where he obtained a bachelor degree (A.B.) in mathematics in 1903. He then attended the
University of Chicago where he received a B.S. degree and an M.S. degree in mathematics in the years 1906
and 1907 respectively.
Dr. Dudley Weldon Woodard (1881-1965) B.S., Wilberforce University, 1903;
B.S. and M.S. University of Chicago, 1906 and 1907;
Ph.D., University of Pennsylvania, 1928.
Ph.D. Thesis: “On Two-Dimensional Analysis Situs with Special Reference to the Jordan Curve Theorem”;
Advisor Professor John R. Kline, a renowned topologist.
This portrait taken from the 1927 issue of the Bison, the Howard University yearbook, when Dr. Woodard
59
was Dean of the College of Arts and Sciences. Photograph courtesy of Moorland-Spingarn Research Center,
Howard University Archives, Washington, D.C.
After receiving his M.S., Dr. Dudley taught mathematics at Tuskegee Institute (now University) from 1907 to
1914 and at Wilberforce faculty from 1914 to 1920. He then joined the mathematics faculty at Howard
University in 1920, where he also served as the Dean of the College of Arts and Sciences from 1920 to 1929.
Dr. Dudley devoted his entire professional life to the promotion of excellence in mathematics through the
advancement of his students, teaching and research. In the early 1920s, he began taking advanced Mathematics
courses during the summer sessions at Columbia University. It was during this period that he became
recognized as one of the gifted Mathematicians in the nation. In 1927, Dr. Dudley took scholarly leave from
Howard University and spent a year at the University of Pennsylvania, working under the direction of Professor
John R. Kline, a famous topologist. He received the Ph.D. degree in mathematics from the University of
Pennsylvania on June 28, 1928, becoming the thirty-eighth person to receive a Ph.D. degree from the University
of Pennsylvania, and the second African American to earn a Ph.D. degree in mathematics. The first African-
American to earn a Ph.D. degree in mathematics was Dr. Elbert Cox in 1925. Dr. Dudley and his wife had a
son who joined the faculty at Howard. Dr. Dudley retired in 1947 and died July 1, 1965 in his home in
Cleveland Ohio.
3.2. Research and Other Accomplishments of Dr. Dudley: In the area of research, Dr. Woodard
published the following three papers:
(i) Woodard, D. W., Loci Connected with the Problem of Two Bodies. His Master’s Thesis.
(ii) Woodard, D. W., On two dimensional analysis situs with special reference to the Jordan Curve
Theorem, Fundamenta Mathematicae 13 (1929), 121-145.
(iii) Woodard, D. W., The characterization of the closed N-cell, Transactions of the American
Mathematics Society 42 (1937), no. 3, 396--415.
As stated above, Dr. Dudley devoted his entire professional life to the promotion of excellence in mathematics
through the advancement of his students, teaching and research. In 1929, Dr. Dudley established the M.S.
degree program in mathematics at Howard University, one of the prestigious institutions among the Historically
Black Universities and Colleges. At Howard University, he also directed many students for their M.S. degree
theses in mathematics. A mathematics library was also established by him at Howard. He also organized and
sponsored many scholarly seminars in mathematics at Howard University. A number of prominent African-
American Mathematicians were Dr. Dudley's students, among them Dr. W.W.S. Claytor, Dr. George Butcher,
Dr. Marjorie Lee Browne, Dr. Eleanor Green Jones, Dr. Jesse P. Clay, and Dr. Orville Keane are notable.
At Howard University, Dr. Dudley was highly respected professor and administrator. Outside Howard
community, Dr. Dudley was also well-known, and revered by the mathematical sciences community. As noted
by Professor Scott W. Williams in his “Mathematicians of the African Diaspora”, Dr. Deane Montgomery,
former president of the American Mathematical Society and the International Mathematical Union described
Dr. Dudley as, "an extremely nice man, well-balanced personally." Dr. Leo Zippin, who was an internationally
known specialist in Dr. Dudley's field, said that he was "one of the noblest men I've ever known."
4. The Research Paper Published by First African-American Mathematician in an
International Accredited Mathematics Journal: One of the most important landmarks and special
60
achievements in the history of contributions of African-Americans in the field of mathematical sciences is the
following paper of Dr. Dudley Weldon Woodard:
Woodard, D. W., On two dimensional analysis situs with special reference to the Jordan Curve Theorem,
Fundamenta Mathematicae 13 (1929), 121-145,
which appears to be the first research paper published in an international accredited mathematics journal by any
African-American Mathematician. For details, see the “Mathematicians of the African Diaspora” website
created and maintained by Professor Dr. Scott W. Williams, Professor of Mathematics University at Buffalo,
SUNY. In the above referred paper, Dr. Dudley Weldon Woodard has discussed the two dimensional analysis
situs (that is, plane topology) with special reference to the Jordan Curve Theorem.
4.1. Some Prerequisites: For the sake of completeness of the present paper, we first describe below some
preliminary and useful definitions and results of mathematical analysis that are relevant to Dr. Dudley’s above
referred paper of 1929 and will be helpful in understanding the purpose of his paper, see, for, example, Jordan
(1887), Brouwer (1912), Moore (1916), Woodard (1929), Kline (1942), and Tverberg (1980), and references
therein.
Definition 1: Simple Curve: A plane curve is called simple if it does not cross itself; otherwise it is called non-
simple, as illustrated in the figures below.
Definition 2: Closed Curve: A plane curve is called closed if it has no endpoints and completely encloses an
area; otherwise it is called an open curve, as illustrated in the figures below.
61
Definition 3: Jordan Curve: A Jordan curve is a plane curve which is topologically equivalent to the unit
circle, that is, a homeomorphic image of the unit circle. A Jordan curve is simple and closed.
As pointed out by Professor J. R. Kline, in his paper "What is the Jordan Curve Theorem?," The American
Mathematical Monthly 49, 5 (1942): 281-286, we may define a simple closed curve or Jordan curve, in non-
technical terms, as the most general set which can be obtained from a circle by bending and stretching without
breaking or crossing. More precisely, a simple closed curve is the image of a circle under a homeomorphism,
i.e., under a (1-1) continuous transformation with a continuous inverse.
Jordan Curve Theorem: In 1887, C. Jordan gave the first proof of the following theorem, that is, any
continuous simple closed curve in the plane separates the plane into two disjoint regions, the inside and the
outside, which in the literature is now well-known as the Jordan Curve Theorem. In 1905, Veblen gave the first
complete proof of the Jordan Curve Theorem which was published in the Transactions of the American
Mathematical Society. After Veblen, many other mathematicians gave the proofs of the Jordan Curve Theorem,
among them Schönflies (1906), Brouwer (1912), and Alexander (1922) are notable.
MOORE’S AXIOM 8: In one of his papers, that is, “ON THE FOUNDATIONS OF PLANE ANALYSIS
SITUS,” Transactions of the American Mathematical Society, Vol. 17, No.2 (1916): 131-164, Professor
ROBERT L. MOORE presented three Systems of Postulates (Axioms), namely 1 , 2 , and 3 , for the
development of plane topology (that is, plane analysis situs). In his own words, Professor Moore states that each
of these systems is a sufficient basis for a considerable body of theorems in the domain of plane analysis situs or
what may be roughly termed the non-metrical part of plane point-set theory, including the theory of plane
curves. As pointed out by Professor Moore further, the axioms of each system are stated in terms of a class of
elements called points and a class of point-sets called regions.
One of the important axioms proposed by Moore is his Axiom 8, which belongs to all of Moore’s three systems
of axioms. The Axiom 8 is as stated below:
Moore’s Axiom 8: “Every simple closed curve is the boundary of at least one region.”
4.2. Dr. Dudley Weldon Woodard’s Contribution to Moore’s Axiom 8: Dr. Dudley in his paper “On two
dimensional analysis situs with special reference to the Jordan Curve Theorem”, Fundamenta Mathematicae 13 (1929),
121-145, considered and investigated Moore’s Axiom 8. According to Moore’s Axiom 8, since every simple
closed curve is the boundary of at least one region, it implies that every simple closed curve defines a bounded
connected set of connected exterior having further properties implied by certain other axioms of Moore’s three
systems of axioms, see Dudley (1929). As pointed out by Dr. Dudley in his said 1929 paper, the chief purpose
of his investigation was to replace Moore’s Axiom 8 by another axiom of such nature that no property of the
simple closed curve is assumed. For details, the interested readers are referred to Dudley (1929).
4.3. Link to Dr. Dudley’s 1929 Cited Paper: After a thorough search and investigation, I was able to find the
original paper of Dr. Woodard, the original copy is freely available at the following link of the European Digital
Mathematics Library (EuDML): https://eudml.org/doc/211923, which, for the interest of readers, is being
produced here.
62
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Spis treści
D. Woodard: On two-dimensional analysis situs with special reference to the Jordan curve-theorem
63
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66
67
68
69
70
71
72
73
74
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5. CONCLUDING REMARKS: The purpose of this paper was to highlight the achievements of Dr.
Dudley Weldon Woodard in the field of mathematical sciences, and highlight one of his research papers
published by him in the year 1929 which appears to be the first research paper published in an international
accredited mathematics journal by any African-American Mathematician. It is evident that Dr. Dudley remains
as a source of inspiration to us to excel in mathematics and other fields of knowledge, and achieve our goals.
The achievements of Dr. Dudley, despite the difficulties he had to overcome, stand as a beacon for us. In fact,
the accomplishments of the past and present mathematicians can serve as pathfinders to their contemporary and
future colleagues. The achievements of many mathematicians, and their contributions, both small and large,
have been overlooked when chronicling the history of mathematics. By describing the academic history of these
personalities within mathematical sciences, we can see how the efforts of individuals have advanced human
understanding in the world around us. History bears testimony to their achievements, abilities and
accomplishments.
ACKNOWLEDGMENTS
I would like to acknowledge my sincere indebtedness and thanks to the works of various authors and resources
on the subject which I have consulted during the preparation of this research project. Special mention must be
made of Dr. Scott W. Williams, Professor of Mathematics, The State University of New York at Buffalo, whose
works I have liberally consulted, particularly his website “Mathematicians of the African Diaspora,”
76
www.math.buffalo.edu/mad/index.html. Also, the author acknowledges the Institute of Mathematics Polish
Academy of Sciences, the owner of Fundamenta Mathematicae, for granting permission (please see Appendix)
to reproduce the photocopy of research article of Dr. Dudley Weldon Woodard, published in Fundamenta
Mathematicae (1929), and also the European Digital Mathematics Library (EuDML) for making the paper of
Dr. Dudley Weldon Woodard freely available.
REFERENCES
Alexander, J. W. (1922). “A Proof and Extension of the Jordan-Brouwer Separation Theorem”, Transactions of
the American Mathematical Society, Vol. 23, No. 4, 333-349.
Brouwer, L. E. J. (1912). “Beweis des Jordanschen Satzes fur den n-dimensionalen Raum”, Mathematische
Annalen, 71, 314-327.
Dean, Nathaniel, editor (1996), “African Americans in Mathematics,” DIMACS 34, American Mathematical
Society.
Dean, N., McZeal, C., Williams, P., Editors (1999), “African Americans in Mathematics II,” Contemporary
Math., 252, American Mathematical Society.
Donaldson, James A. (1989), “Black Americans in Mathematics,” in A Century of Mathematics in America, Part
III, HISTORY OF MATHEMATICS, Volume 3, American Mathematical Society, 449-469.
Jordan, M. C. (1887). Cours D'analyse de L'ecole Polytechnique, Gauthier-Villars, Paris.
Kenshaft, Patricia C. (1981), “Black Women in Mathematics in the United States,” American Mathematical
Monthly, 592-604.
Kenshaft, Patricia C. (1987), “Black Men and Women in Mathematical Research,” Journal of Black
Studies, 19, 2, 170-190.
Kline, J. R. (1942). "What is the Jordan Curve Theorem?," The American Mathematical Monthly, 49, 5 281-
286.
Moore, R. L. (1916). “ON THE FOUNDATIONS OF PLANE ANALYSIS SITUS,” Transactions of the
American Mathematical Society, Vol. 17, No.2, 131-164.
Shakil, M. (2010), “AFRICAN-AMERICANS IN MATHEMATICAL SCIENCES - A CHRONOLOGICAL
INTRODUCTION,” Polygon, Vol. 4, 27-42.
Schönflies, A. (1906). “Beiträge zur Theorie der Punktmengen”, Mathematische Annalen, 62, 286–328.
Tverberg, H. (1980). “A PROOF OF THE JORDAN CURVE THEOREM”, BULL. LONDON MATH. SOC.,
12, 34-38
Veblen, O. (1905). “Theory on Plane Curves in Non-Metrical Analysis Situs,” Transactions of the
American Mathematical Society, 6, 1, 83-98.
77
Williams, Scott W. (1999), Black Research Mathematicians, African Americans in Mathematics II,
Contemporary Math. 252, AMS, 165-168.
Williams, Scott W. “A Modern History of Blacks in Mathematics,” www.math.buffalo.edu/mad/madhist.html
Williams, Scott W. “Mathematicians of the African Diaspora,” www.math.buffalo.edu/mad/index.html.
Woodard, D. W.(1929). “On two dimensional analysis situs with special reference to the Jordan curve
theorem,” Fundamenta Mathematicae, 13, 121-145. Available at: https://eudml.org/doc/211923.
APPENDIX
From: lukasz Stettner [mailto:L.Stettner@impan.pl]
Sent: Saturday, March 02, 2013 12:39 PM
To: Shakil, Mohammad
Cc: torunczy@mimuw.edu.pl
Subject: permission
Dear Prof. Shakil,
In the name of the Institute of Mathematics Polish Acad. Sci. the owner of Fundamenta Mathematicae I
agree for reproducing the article
Woodard, D. W., On two dimensional analysis situs with special reference to the Jordan Curve Theorem,
Fundamenta Mathematicae 13 (1929), 121-145.
in the paper you are writing on Dr. Dudley Weldon and His Research Paper on Two-Dimensional
Topology in Historical Perspective.
Yours sincerely,
Lukasz Stettner
Scientific Director of IMPAN
81
Polygon
Spring 2010 Vol. 4, 81-82
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