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

en

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

qu

en

cy

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

qu

en

cy

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

IINNSSTTIITTUUTTEE OOFF MMAATTHHEEMMAATTIICCSS ·· PPOOLLIISSHH AACCAADDEEMMYY OOFF SSCCIIEENNCCEESS

FFUUNNDDAAMMEENNTTAA MMAATTHHEEMMAATTIICCAAEE

ffoouunnddeedd iinn 11992200 bbyy ZZyyggmmuunntt JJaanniisszzeewwsskkii,, SStteeffaann MMaazzuurrkkiieewwiicczz aanndd WWaaccłłaaww SSiieerrppiińńsskkii

ccoonnttiinnuueedd bbyy KKaarrooll BBoorrssuukk aanndd KKaazziimmiieerrzz KKuurraattoowwsskkii

IISSSSNN:: 00001166--22773366((pp)) 11773300--66332299((ee))

Polska Biblioteka Wirtualna Nauki

Kolekcja Matematyczna

Fundamenta Mathematicae

Tom 13

Warszawa 1929

Spis treści

D. Woodard: On two-dimensional analysis situs with special reference to the Jordan curve-theorem

63

64

65

66

67

68

69

70

71

72

73

74

75

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

COMMENTS ABOUT POLYGON

***********************************************************************

Dr. Norma Martin Goonen

President, Hialeah Campus

Miami Dade College

Thank you, Dr. Shakil, for providing scholars a vehicle for sharing their research and

scholarly work. Without opportunities for sharing, so many advances in professional

endeavors may have been lost.

N

N

M

M

G

G

Dr. Norma Martin Goonen

President, Hialeah Campus

Miami Dade College

***********************************************************************

Dr. Ana María Bradley-Hess

Academic and Student Dean, Hialeah Campus

Miami Dade College

Welcome to the third edition of Polygon, a multi disciplinary peer-reviewed journal of

the Arts & Sciences! In support of the Miami Dade College Learning Outcomes, one of

the core values of Hialeah Campus is to provide “learning experiences to facilitate the

acquisition of fundamental knowledge.” Polygon aims to share the knowledge and

attitudes of the complete “scholar" in hopes of better understanding the culturally

complex world in which we live. Professors Shakil, Bestard and Calderin are to be

commended for their leadership, hard work and collegiality in producing such a valuable

resource for the MDC community.

Ana María Bradley-Hess, Ph.D.

Academic and Student Dean

Miami Dade College – Hialeah Campus

1800 West 49 Street, Hialeah, Florida 33012

Telephone: 305-237-8712

Fax: 305-237-8717

Comments About …

82

***********************************************************************

Dr. Caridad Castro, Chairperson

English & Communications, Humanities, Mathematics, Philosophy,

Social & Natural Sciences

Hialeah Campus

Miami Dade College

POLYGON continues to grow and to feature our local MDC scholars.

Thanks to you and your staff for providing them this opportunity.

Cary

Caridad Castro, J.D., Chairperson

English & Communications, Humanities, Mathematics, Philosophy,

Social & Natural Sciences

Miami Dade College – Hialeah Campus

1776 W. 49 Street, Hialeah, FL 33012

Phone: 305-237-8804

Fax: 305-237-8820

E-mail: ccastro@mdc.edu

***********************************************************************

Dr. Arturo Rodriguez

Associate Professor

Chemistry/Physics/Earth Sciences/Department

North Campus

Miami Dade College

I want to congratulate you and the rest of the colleagues who created the POLYGON that

is occupying an increasingly important place in the scholarly life of our College. Now,

the faculties from MDC have a place to publish their modest contributions.

arturo

Dr. Arturo Rodriguez

Associate Professor

Chemistry/Physics/Earth Sciences/Department

North Campus

Miami Dade College

11380 NW 27th Avenue

Miami, Florida 33167-3418

phone: 305 237 8095

fax: 305 237 1445

e-mail: arodri10@mdc.edu