Uses of History of Mathematics in School (pupils aged 6 - 13)
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Uses of History of Mathematics in School (pupils aged 6 - 13) Narges Assarzadegan [email protected] Abstract In this paper, I explained some ways we applied in IMH (Isfahan Mathematics House) for teaching math with its history. I am going discuss on the workshops that I have designed as out of school activities for 6-13 aged students. Introduction In Iran formal math curriculum, math history restricted to telling very short story of Iranian mathematician’s life on back of the textbooks. Teachers themselves if they have some interests in history of math could apply methods without any formal education on this area. We in Isfahan Mathematics House (IMH) have designed some workshops for connection between math and history for elementary, middle and high schools students as an out of school activity. Isfahan is a city, full of historical buildings, Islamic arts and tiling therefore, students may be having seen applying the geometry in art and architecture anywhere in the city. So, the link between math, history and art can be valuable area for inspiring and motivating the students. In the other hand, students can understand arts which have been built on the basis of geometric modeling and methods in previous years. Also, regarding to ethnomatematics students become familiar with the cultural backgrounds. Math education and history of math
Transcript of Uses of History of Mathematics in School (pupils aged 6 - 13)
Uses of History of Mathematics in School (pupils aged 6- 13)
Narges Assarzadegan
AbstractIn this paper, I explained some ways we applied in IMH (IsfahanMathematics House) for teaching math with its history. I am goingdiscuss on the workshops that I have designed as out of schoolactivities for 6-13 aged students.
Introduction
In Iran formal math curriculum, math history restricted totelling very short story of Iranian mathematician’s life on backof the textbooks. Teachers themselves if they have some interestsin history of math could apply methods without any formaleducation on this area. We in Isfahan Mathematics House (IMH)have designed some workshops for connection between math andhistory for elementary, middle and high schools students as anout of school activity. Isfahan is a city, full of historicalbuildings, Islamic arts and tiling therefore, students may behaving seen applying the geometry in art and architectureanywhere in the city. So, the link between math, history and artcan be valuable area for inspiring and motivating the students.In the other hand, students can understand arts which have beenbuilt on the basis of geometric modeling and methods in previousyears. Also, regarding to ethnomatematics students becomefamiliar with the cultural backgrounds.
Math education and history of math
The question of the connection between the teaching ofMathematics and the study of that discipline’s history is a vexedone. Everyone assumes that there is a link, but it is quiteanother matter to describe what that link really is, and also toproceed to discuss how that link might usefully be pressed intoservice in the classroom (Michael A. B. Deakin, 2001).
A similar caution is evident in Bruckheimer and Arcavi’s emphasison the history of the idea. But perhaps we can be a little lessdogmatic. If, as several authors urged, a knowledge of thehistory allows us to see the reasons for student “hang-ups” andmisconceptions, then we need to know a little of the messiness ofthe historical course, as well. They even make nice juicyanecdotes that students enjoy.
Many authors have discussed the question of why we should use thehistory of mathematics to mathematics education. For example,Fauvel (For Learn Math, 11(2): 3–6, 1991) mentions at leastfifteen arguments for applying the history of mathematics inteaching and learning mathematics. Knowing how to introducehistory into mathematics lessons is a more difficult step.
Geometry and Islamic arts
Geometry is at the heart of Islamic art. The designs and patternsuse a finite number of geometric shapes that combine in manydifferent ways. Traditionally, Islamic patterns are createdthrough compass and ruler constructions but this method is oftentime- consuming and can be complicated to oversee in theclassroom [1]. Students aged 6-13 frequently do not practice theprecision required to achieve accurate results. An alternativemethod of creating Islamic pattern in the classroom is throughmanipulation of modularity with squares tiles.
Figure1: 13 aged student in Isfahan Mathematics House workshops.
Sarhangi in his article on MAKING MODULES FOR MOSAIC DESIGNSsuggested: “the predominance of geometry constructions ofcompass-straightedge in medieval Persian art exhibits both theartisans’ geometry background and the direct involvement ofmathematicians in the pattern making process (Jazbi, 1997)”. Butthe goal of his article is to study other design-makingapproaches that are older than the domination of sophisticatedgeometry in making ornamental designs. The articles Modularity inMedieval Persian Mosaics (Sarhangi, 2008) and Modularity inMedieval Persian Mosaics: Textual, Empirical, Analytical, andTheoretical Considerations (Sarhangi, Jablan, and Sazdanovic,2004) illustrate one of these methods: modularity.
Sarhangi in other article suggested: A study of early Persianmosaic designs shows that most patterns were created by simplycutting and pasting single-color square shaped mosaics(Maheronnaqsh, 1984). In order to save energy and space formolding, casting, painting, and baking tiles, artisans usedsingle-color tiles. The complicated and sophisticated finalproducts on the walls or domes were from the cutting and pastingof these tiles. The artisans cut patterns using differentformats, and then assembled them so that different colors wereswapped with each other in a new arrangement (Maheronnaqsh,
1984). In this way the artisans would rely on the color contrastof cut-tiles to emphasize a design. In most cases the tiles weresquares. However, other conveniently constructed shapes such asrhombi and triangles were also used.
Smested on ICMI discussion group5 report (2012) suggested:
As the publication of the 2000 ICMI study raised awareness historyof mathematics in teaching mathematics:
- allows pupils to experience the process of mathematics - problemsolving, proof construction (e.g., Lakatos, 1976; Ernest, 1998);
- provides the landscape of Guided reinvention (Freudenthal,1991);
- expands understanding of nature of mathematics; that is,mathematics is not “finished” and continues to evolve and someideas are subject to change (Ernest, 1998); and
- often relies on not taking the end results of mathematicians’works as starting points (Freudenthal, 1973) aimed at progressivemathematization (Gravemeijer & Doorman, 1999, p. 116).
Regarding these theorical frameworks we have designed workshopsfor students out of school activities in IMH (Isfahan MathematicsHouse).
The aim of these workshops is to ensure that participants becomefamiliar with the methods for making a variety of modules andusing these to create Islamic patterns. As Meenan and Thomassuggested:
This will act as a creative stimulus for designingother patterns and help participants to:
make cross-curricular connections between art andmathematics
make connections between history and geometry
develop two-dimensional spatial awareness
broaden students’ creative outlook and developpractical skills
increase students’ motivation and enjoyment ofmathematics
In this workshop we want to make connection between geometry, art,mathematics history and culture and implement this idea in 6-13 agedclassrooms. This workshop applied in IMH (Isfahan Mathematics House) 5month ago. We had some steps to implement this workshop:
1. Students (13- aged) passed a workshop on tiling.
2. Students give their comments about what they observed and also theinformation on tiling in Isfahan buildings.
3. Presenting a video about some tiling in Isfahan which could applymodularity.
4. Share some blue and white square and triangular tiles between groupswhich make one of the tiling they saw.
5. Providing the participants with some sophisticated patterns thatalready have been prepared on the paper.
6. Finally, the students must make some patterns with their innovation.
Here, there are pictures of tiling on Darbe- Imam Shrineconstruction.
Figure2: Darbe Imam shrine 1453 christian era
Figure3: Darbe Imam shrine 1453 christian era.
And below is a picture from student’s effort for modeling the above tiling without any paper model.
Figure4: Modeling tiling Darbe- Imam by students (with two color squares andtriangular tiles)
Figure5: Geometrical pattern for tiling (designed by Geometer’s Sketchpad).
Figure6: A set of three modules, the generated base design, and the final result for the“maple leaf” tessellation (obtained from Sarhangi’s article).
Figure7: students are modeling patterns.
Figure8: The 3 module set, the generator, a two-coloring of the layout, and an imageconstructed from two proper reflections across the sides of the generator.
(a) The computer generated design using modularity approach and (b) the existing Persianmosaic in the mausoleum of Emamzadeh Soltan Ali, Kashan, Iran (obtained from Sarhangi’s
article).
Figure9: student’s works.
Figur10: Another mosaic design (obtained from Sarhangi’s article).
Figure11: student’s works.
There are some models and pictures which can use with abovemanner in workshops:
Figure 12: The design with the details in an existing mosaic on the wall of Darbe-Imam Shrine, Isfahan, Iran, and a metamorphosis made from the two above
tessellation (obtained from Sarhangi’s article).
Figure 13: the wall of Darbe- Imam Shrine construction, Isfahan, Iran (see figure 11).
Figure 14: the wall of Darbe- Imam Shrine, Isfahan, Iran (see figure 11).
Figure 15: the wall of Darbe- Imam Shrine, Isfahan, Iran (see figure 11).
Figure 16: the wall of Darbe- Imam Shrine, Isfahan, Iran (see figure 11).
Figure 17: the wall of Darbe- Imam Shrine, Isfahan, Iran (see figure 11).
Figure 18: the wall of Darbe- Imam Shrine, Isfahan, Iran (see around the shape).
References:
[1]: E.B. Meenan* and B.G. Thomas. Creating Islamic Patterns fromFolded Shapes. School of Education* and School of Design.University of Leeds. Leeds, LS2 9JT. [email protected]
[2]: Modularity in Medieval Persian Mosaics: Textual, Empirical,Analytical, and Theoretical Considerations. Reza Sarhangi.Mathematics Department. Towson University. Towson, MD 21252, [email protected]. Slavik Jablan and Radmila Sazdanovic. TheMathematical Institute. Kneza Mihaila 35, 11001 Belgrade,Yugoslavia. [email protected].
[3] Using History to Teach Mathematics: The Case of LogarithmsEvangelos N. Panagiotou. Published online: 27 July 2010.
[4] Katz, Victor (Ed.). Using History to teach Mathematics. AnInternational Perspective. MAA Notes (No 51). ISBN: 0-88385-163-6. Paperbound, 2000. Michael A. B. Deakin (Australia)[5] Sarhangi, Reza. MAKING MODULES FOR MOSAIC DESIGNS.Mathematician, Mathematics Educator, Member of Board ofDirectors: Bridges Organization, Mathematical Connections in Art,Music, and Science (www.Bridges MathArt.org), E-mail:[email protected]: Department of Mathematics, TowsonUniversity, 8000 York Road, Towson, Maryland, 21252, USA
[6] Modularity in Medieval Persian Mosaics: Textual, Empirical,Analytical, and Theoretical Considerations. Reza Sarhangi.Mathematics Department. Towson University Towson, MD 21252, [email protected]. Slavik Jablan and Radmila Sazdanovic. TheMathematical Institute. Kneza Mihaila 35, 11001 Belgrade,Yugoslavia. [email protected]
