> \^[@ dbjbjFF "h,,68tda,"$R`9888888@cZ]10a?L?>?d8$
"
TESSELLATIONS, TRANSFORMATIONS AND TECHNOLOGY: TEACHING GEOMETRY IN THE DIGITAL AGE
Dr. Dorothy M. French
Community College of Philadelphia
Philadelphia , PA , U.S.A.
Abstract:
Electronic learning resources are radically altering the landscape of education. For mathematics education in particular, digital technologies can complement hands-on, activity-based methods to create a productive environment for learning geometry. This paper compares the achievement of community college students in a technology-rich geometry course with the achievement of students in a traditional lecture-based algebra course.
Introduction
To investigate whether there is a difference between the achievement of community college students in a technology-rich geometry course and the achievement of students in a traditional lecture-based algebra course, student outcomes for two Community College of Philadelphia mathematics courses, Geometry for Design, (Math 137), and Intermediate Algebra (Math 118) were compared. Math 137 was developed by Community College of Philadelphia faculty with funding from the U. S. National Science Foundation, and was first introduced in 1998. Math 137 is an interdisciplinary, technology - rich activities - based geometry course that must be taken by students who are in the Colleges Construction Technology, Art and Interior Design programs. These students would otherwise take an intermediate algebra course, Math 118, which is a traditional lecture-based course with no designated technology component. Encouragingly, students who study Math 137 Geometry for Design seem to have a better chance of success -- an A, B or C grade -- than students who take a traditional Math 118 Intermediate Algebra course.
Background
Community College of Philadelphia is an urban, open-admission, associate degree-granting institution and is one of the citys top three largest institutions of higher education. Since its inception in 1964, more than 570,000 Philadelphians have enrolled at the college. At present, Community College of Philadelphia serves more than 44,000 students at its main campus, regional centers and neighborhood sites.
The college offers more than 70 associate degree and certificate programs. Students transfer to bachelors degree programs, and others take courses to secure or improve employment or for personal enrichment. Our student body is extremely diverse, especially in age and ethnicity, and many different cultural backgrounds are represented. There are local and international students who come from 65 countries, and they share many different educational objectives.
The majority of students who enter visually oriented programs (Art, Architecture and Interior Design or Construction Technology) do so without a strong knowledge of geometry. This lack of knowledge impedes progress in many ways: students are unable to make simple area calculations or conceptualize three-dimensional physical relationships. Students need well-developed mathematical skills to help them succeed in applied art and graphic design, architecture, construction technology and computer-assisted design. To address this need, faculty members from Community College of Philadelphias Mathematics, Art, Architecture and Construction Technology departments created a math course that would address these deficiencies. Thus, Math 137 Geometry for Design was first offered in 1998, and has run every semester since then, enrolling typically 15-20 students per semester.
Pedagogy of Geometry for Design
Pamela Weiger (2000) quotes Dr. Gloria Gilmer, a proponent of ethnomathematics, in a recent article as having said the following: Math is hard for people because it is not real. Its an abstraction, studying models, not reality.
Fortunately, one characteristic of geometry is its immediacy. In contrast to other branches of mathematical study that rely heavily on synthetic abstraction, the study of geometry can begin with commonly observed and experienced phenomena within the physical world. In Geometry for Design Math 137, the pedagogy of activity-based learning assumes that learning is a hands-on, experience-based endeavor. Math 137 combines a sound preparation in basic geometric concepts and techniques with a thorough exploration of their concrete applications in architecture, art and design. Topics include: traditional straightedge-and-compass and computer-based construction methods; properties of triangles, polygons and circles; plane transformations, symmetry and tessellations of two-dimensional figures; area; 3-dimensional polyhedra ; volume; the Pythagorean Theorem; ratio and proportion and similarity. Throughout the course, students use Geometers Sketchpad computer software to explore conjectures and draw conclusions. Hands-on activities, drawing and three-dimensional modeling are stressed, and the internet is used for examining properties of polyhedra and for doing research for written papers.
Many researchers acknowledge the importance of context. Peter Glidden (2001) writes that: Teachers need a repertoire of mathematics investigations that involve pattern searching, exploration and technology. As means of connecting abstract geometric principles to specific aesthetic- and design-related problems, topics from art and architecture are studied, and include: the development of perspective drawing systems (Brunelleschi, Alberti, Durer); the Golden Mean and its applications in mathematics, classical architecture and the natural world; the development of arches and their relationship to the available technology of their periods; transformations, symmetry and tessellations and their applications to ceramic art and textile designs of ancient and modern cultures (Celtic art, Islamic design, American quilts and African textiles, Native American basket and ceramic designs and Hindu mandalas).
Throughout the semester, as well as being immersed in traditional mathematical formulas and the symbolic manipulation of geometrical problem-solving, students are required to create a portfolio of manual and computer-based drawings and to build models as a means of enhancing their assimilation of course materials. Each week, one 80 minute session meets in a regular classroom location, while the second 80 minute session is held in a computer-equipped classroom, where students use Geometers Sketchpad software to explore ideas, make conjectures and literally draw conclusions. In the computer-equipped classroom they also can access Internet web sites that relate to geometry, architecture, art and design.
Student achievement is measured by traditional computational math-based homework, midterm and final exams, two short written papers and a portfolio of computer-generated drawings.
Pedagogy of Math 118 Intermediate Algebra
In contrast, Intermediate Algebra Math 118 is a course that is taken at some time by the majority of students at the College. However, the popularity of Math 118 is a result of compulsion, and not inclination. This course is a prerequisite for admission to most degree or certificate programs, especially those in the fields of engineering, science and technology. As with many courses at other institutions, our Intermediate Algebra course is really a filter or a gatekeeper course. This phenomenon is described by Orange Coast College Associate Professor Eduardo Jesus Arismendi-Pardi, who is quoted in a recent article by Greene (2000):
At the community college level, math is really a gatekeeperStudents at the community college will take algebra or trigonometry and they cant get out of it. They either dont pass it or are turned off by it.
Although Math 118 may not be the average students favorite course, it has a large enrollment, currently in excess of 1500 students per semester. Topics include the Real number system, systems of linear equations and inequalities, polynomials, rational expressions, radical expressions, and quadratic equations. Most instructors use a customized intermediate algebra text, and some may use a bank of departmental written examinations as part of a students assessment. Most sections are taught in a traditional classroom in a lecture-based format. While there is no general consensus on the use of technology in the classroom, some instructors are progressive about the use of technology, and others discourage and even prohibit the use of calculators. Meanwhile, students have embraced a digital age and many own and would like to use their graphics calculators. Generally, Math 118 is not taught with computer software.
Quantitative Data
To compare the achievement of community college students in a technology-rich geometry course with the achievement of students in a traditional lecture-based algebra course, records of students final grades were analyzed. Data came from 10 sections of Math 137 and 20 sections of Math 118 taught by the author during the last five years from Fall 2002 through Fall 2007.
If success is defined as earning an A, B or C grade in a course then the data are as follows: for Math 137 Geometry for Design: the number of successes = x1 = 110 and the total number of students enrolled = n1 = 226. So p^1 = sample proportion of successes = 110/226 = 0.487
For Math 118: the number of successes = x2 = 257 and the total number of students enrolled = n2 = 681. So p^2 = sample proportion of successes = 257/682 = 0.377.
The null hypothesis is that there is no difference between the populations of sample proportions of success for the Math 137 and Math 118 sections, viz : the means of the two sampling distributions of sample proportions are equal. If combined results for both courses are used, let p = ( x1 + x2) /( n1 + n2), and q = 1 p. Let the standard deviation of the difference in sample proportions be (p^1-p^2) H" " (p q( 1/n1 + 1/n2)) . Thus, the normal score is z = ( p^1 p^2)/ (p^1-p^2) . This gives z H" 2.92 and this would be significant at the 5% level.
The means of the distributions of sample proportions seem to be different and so the students who took Math 137 Geometry for Design may have an increased chance of obtaining an A, B or C grade, compared to students who took Math 118 Intermediate Algebra.
Qualitative Considerations
In an effort to better understand the Math 137 students opinions about learning mathematics and geometry, short exit questionnaires were also administered. These contained statements such as, Mathematics is interesting to me, and I enjoy mathematics courses, and students agreement with the statements were measured by a 5-point scale. Interestingly, male and female students had different attitudes about learning mathematics, but most of the students agreed that hands-on activities increase ones ability to learn mathematics, and most liked using drawing tools, math manipulatives and computer-based activities. Written comments included: Wow, a tolerable math class. In a course evaluation written comment, Jennifer, an Art major who took the Math 137 after previously failing Math 118 three times, wrote:
When (the Head of Art Department) told me that I needed to take a math (sic) in order to graduate in May I almost cried. I was really hesitant and nervous, but then I talked to other students who had taken this class last semester, and they gave it really good reviews. I really enjoy this classI can actually say that that I look forward to going to this class. I understand what Im doing because I can see how everything that is taught relates to the world.
After a string of Fs in Math 118, this student got an A in Math 137, graduated from Community College of Philadelphia, and also won a $10,000 scholarship to art school. Another Art major, Chin, received an A for Math 137, graduated, and transferred to Art school. He wrote:
Im thinking more about how technology has made an impact on the art worldI think of the geometry class and how we integrated the computer along with the traditional methods of geometry to come up with a more in depth idea of how the world and geometry work together.
More study is indicated to describe the differences in students attitudes towards learning geometry.
Conclusion
This analysis of outcomes for students who took Math 137 Geometry for Design indicates that they seem to be succeeding at a higher rate than students who took Math 118 Intermediate Algebra. Over the years, a number of students who have taken Geometry for Design volunteer that it was a relevant and even (at times) enjoyable experience. This type of response is rare with students in the Intermediate Algebra course. Of course, more study is required before drawing any broad conclusions. Further studies might, for example, compare the achievement of female students versus male students in each course or compare the achievement of students taught geometry with technology with the achievement of students taught in a traditional geometry course.
For many of our students, a college algebra course can be a frustrating, often unsuccessful, and usually terminal mathematical experience. As an alternative, some of our students can study geometry in a course that is immersed in the rich contexts of art, architecture, construction and design. It is possible that, in studying geometry in a course that employs hands - on, activity - based, computer - assisted methods, they may finally succeed in learning some interesting, relevant and powerful mathematics.
Bibliography
Glidden, P. (2001, February). Beyond the golden ratio: a calculator-based investigation. Mathematics Teacher, pp. 138-144.
Greene, E. (2000, October 6). Good-bye Pythagoras? ST " 0 8 9 F J _ g h y z ?
@
x
y
z
ɾɳѣɘɫѳѳxшшшphhR}CJaJh[CJaJhjCJaJha:CJaJh`CJaJhq&-CJaJhq&-h3$CJaJh3$CJaJh!VCJaJhq&-hq&-CJaJhq&-h&nCJaJh&nCJaJhTGCJaJh\CJaJh\h\6CJaJh\5CJ aJ hTG5CJ aJ 'Tjl
y
V#!d@@gd"
!@@gd"$@@a$gd"gdwfOd@@gdTA@@gdt$@@a$gd`@@gd\$@@a$gd5ddSX
g
.7KSrsͽ͵تصآآͽ͚͚ͽآh~SCJaJhX{CJaJh_8h CJaJh
1CJaJh/CJaJh[CJaJh_8h_8CJaJh CJaJh9nCJaJh_8CJaJhR}CJaJhHCJaJ:Ml58PQUV#2UVXQøÝÒڇwoghQ;CJaJhEFzCJaJhTACJaJhSzCJaJhwfOhwfOCJaJhwfOhSzCJaJhwfOh_8CJaJh~SCJaJhwfOCJaJhwfOh#CJaJhwfOh2#CJaJh%ph#CJaJhR}CJaJh_8h2#CJaJh%pCJaJhsh/CJaJ$QRSr@DM_s379V}*.2Xȸ襚wh"h2`CJaJh2`CJaJh"CJaJh3CJaJh"h"CJaJhsh`CJaJhCCJaJh CJaJh%pCJaJhTACJaJh~SCJaJhEFzCJaJhQ;CJaJh_8h2#CJaJh_8h2#5CJaJ.Xnw> ()4GU"$ ` u x j!k!!!!="@"X"\"p"""""""""ŭŭŭ嚏hCJaJhsh CJaJhsh"CJaJh_CJaJh~]kCJaJh+CJaJh%3CJaJh(CJaJh3CJaJhHCJaJh2`CJaJh"h"CJaJh CJaJh9nCJaJ5!""Y%0&))))`+x,-0q225789-:.:9:dgdsgdsgdvD@@gdjd@@gd"@@gd"d@@gd"d@@gd""""""#4#<#K#M#N#W#i#q#r####?$G$$$$U%X%Y%Z%%%%%0&1&:&B&&&&&&'''(Y(`(j(((((()$)ݽݴ͠ݠݐ͐͐h&nCJaJh#CJaJhCJaJhH7gCJaJh"h"CJaJhh2`CJaJhh2`CJaJh+CJaJh_CJaJh%CJaJh"h"CJaJh9nCJaJh~]kCJaJh CJaJ5$)&)()3);)U)i)t)w){))))))))))))))))**** +;+<+@+A+`++++,~vvjh0&h0&6CJaJh0&CJaJhHCJaJh8CJaJhjCJaJhh
CJaJhsh"5CJaJh"h_CJaJh"CJaJh"h"CJaJh\CJaJh CJaJh&nCJaJh+CJaJh#CJaJh_CJaJh%CJaJ&,,4,5,6,A,B,C,D,G,N,,,,,,,,,,,,,,,,--)----..x.z..................b/d/ŽŵŽŽh`?h8CJH*aJh`?h86CJH*aJh`?h86CJaJhfCJaJh8CJaJhp0CJaJhh
CJaJh86CJaJh0&h0&6CJaJh0&CJaJh0&h0&6CJH*aJ6d/f/h/l/n/p/~///////////////////00
0000000 0h00000L1$2%2<2>2n2p2q2٩h;UCJaJhfCJaJhh
CJaJhp0CJaJhp06CJaJh`?6CJaJh`?h`?6CJH*aJh`?CJaJhH6CJaJh8CJaJh`?h86CJaJh`?h86CJH*aJh`?h`?6CJaJ/q22223333:3E3{333O44444585`5m5n5z555566L6M666666666*7+7j7k77777708H8a8譥h9e[h9e[CJaJh9e[h^bCJaJhfCJaJhHCJaJh9e[hvDCJaJh^bCJaJh+CJaJhvDCJaJhu.CJaJhUCJaJhACJaJhvDhvDCJaJhshvD5CJaJ2a888888887989|9}9999999:,:-:.:9:::?:@:K:L:X:ɵxjaXLaLhvDhvDCJaJhh9e[CJaJhh^FCJaJhhvDhvD5CJaJhhshvD5CJaJhhs5CJaJhhvDCJaJhhACJaJhhu.CJaJhhHCJaJhhvDhvDCJaJhh^bCJaJhvDhvDCJaJh9e[h^bCJaJh9e[hvDCJaJh+CJaJhh9e[h9e[CJaJhX:j:s::::::::;;;;;;;;;<<<W<~<.=I=J=j=w=y===========>>>> >(>)>=>X>Z>>>>>>>>>>>>>>>>>>ؽشƴؽؽششhvDh^FCJaJhhCJaJhh+CJaJhhACJaJhhu.CJaJhh^FCJaJhhvDhvDCJaJhh9e[CJaJhh$CJaJhB9:/=2????.dddddddddddddh]hgd*y&`#$gd*y$@@a$gdsd@@gds
@@1$gds
&F@@1$gdsd@@gds@@gds>>>
????%?1?2??????ddd d.dOdrdsdvddddddueUFhAhACJOJQJaJhshs>*CJOJQJaJhshvD>*CJOJQJaJhAh^FCJOJQJaJhAhvDCJOJQJaJhsCJOJQJaJUhvDhvD>*CJOJQJaJhvDhvDCJOJQJaJhshvD5CJaJhvDCJaJhhvDhCJaJhhvDhvDCJaJhhACJaJhhCJaJhChronicle of Higher Education, pp. A16 -A18.
3 Weiger, P. (2000, August 17). Re-calculating math instruction. Black Issues in Higher Education, pp. 59- 62.
PAGE
PAGE 6
dddddddddddddddddhHhH0JmHnHuh/
h/0Jjh/0JUhvDh`CJaJhvDhvDCJaJ 1h/ =!"#$%@@@NormalCJ_HaJmH sH tH >@>" Heading 1$dh@&aJH@H" Heading 2$$dh@&a$aJhJ@J" Heading 3$$dh@&a$
5aJhDA@DDefault Paragraph FontRiRTable Normal4
l4a(k(No ListHHZ@HBalloon TextCJOJQJ^JaJH+@H2#Endnote Text1$OJQJaJh<B@<2# Body TextdhaJh< @"<"Footer
!CJaJ4@24*yHeader
!.)@A.*yPage NumberB^@RBwfONormal (Web)dd[$\$6hTjlyV
#Y0!!!!`#x$%p'q((+-./-0.090/325?55!6666666666660000000000000000000000000(00 0!0!0 0!p(00o(0o(p0(0o(0((0(00900900(0 0=5 0=50<500<500<500x@0@0@0@0@0@0h0ly6O900H&O900O900^>00^>00^~00^>00RQX"$),d/q2a8X:>dd #$%&()*+,-.03!9:d!'/d"!!.t,t/t,0t,1t,l2t,}3t,U4t,d"5t,\6t,]7t,4N8t,L9t,\:t,;t,@t,M?t,tA@t,AAt,Bt,TCt,4"Dt,t"Et,Ft,<jjll)--.6
{}%55##...6
B*urn:schemas-microsoft-com:office:smarttagscountry-region9*urn:schemas-microsoft-com:office:smarttagsState8*urn:schemas-microsoft-com:office:smarttagsCity=*urn:schemas-microsoft-com:office:smarttags PlaceName=*urn:schemas-microsoft-com:office:smarttags PlaceType9*urn:schemas-microsoft-com:office:smarttagsplace
)9U^6BDKMR **#6)6666
s;#E#%%&&&&--/-066B6b666666333333333333lxss!!@#A#>$x$&&*8+--//+0,0.0:025^555f66666666e?u>_&0^`0OJPJQJ^Je?u
q-6
,B-6qa`H3*
u\h
u[J-bSEc[aw!2#6m#3$%8%(q&-u.
18_8a:-=`?TAvD^FZ@HwfO~S;U9e[2`^bZIcH7g i~]k9nJr*yEyEFz~A%p" s0&6\p0R}+!VX{t/j$+m8BDbL#%35Uef#C*pQ;&nSzA\xsTG_/FG`/!H.6)@-0-0tY-0-04&(56P@P.Pd@P@UnknownGz Times New Roman5Symbol3&z Arial5&zaTahoma?5 z Courier New"qhFFk&t.c&t.c24d663QH)?ZIcMathematics E-Learning: CCPOh+'0
8DP
\hpxMathematics E-Learning:ath thththNormal.dot CCP2PMicrosoft Word 10.0@Ik@@!V@~>]@~>]&t.՜.+,0hp|
mc6
Mathematics E-Learning:Title
!"#$%&'()*+,-./012346789:;<>?@ABCDEFGHIJLMNOPQRTUVWXYZ]Root Entry F JZ]_Data
51Table=OWordDocument"hSummaryInformation(KDocumentSummaryInformation8SCompObjj
FMicrosoft Word Document
MSWordDocWord.Document.89q