Mathematics and Mathematics Education: thetechnology as strategy of a hermeneutical
approach of the History of Mathematics toteach school mathematics content
Tsukuba Global Science Week
University of Tsukuba – September 28-30 2015
Session “Hermeneutics in Mathematics Education: History of Mathematics toImagine the Future and Understand the Perspective of Others
September 30, 2015
Yuriko Yamamoto Baldin (yuriko@dm.ufscar.br)
Senior Professor of Mathematics
Department of Mathematics – Universidade Federal de São Carlos, BRAZILICMI Executive Committee - Member at Large, 2013-2016

Summary Reflections from the perspective of Teacher Educator and some projects for
the Professional Development of Teachers: Mathematics:
 Evolution of mathematics
 The importance of the History of Mathematics to understand the evolution ofmathematical concepts and their essence through the time; Mathematics Education: The importance of the History of Mathematics in the education of teachers: tosupport/improve the teaching and to promote the learning (what, when, how)

 The Hermeneutics as grounding principle in developing teaching materials for theprofessional development of school teachers: Technology of educational software as strategy to improve the interpretation ofmathematics concepts through historical texts, to understand the mathematic activities
for the learning, to perceive the perspective of the others for a better assesment.

Mathematics
 There is strong need to desmistify a common understanding (at least forschool teachers and general people) that Mathematics has beenorganized since ancient times as it appears in the school textbooks, asexact and abstract science with known rules and calculation methods.
 What is the meaning of the historical discoveries and accomplishments that
accompanied the civilizations and societies?

 History of Mathematics is a way to understand the genesis and theevolution of mathematics as a human enterprise through the time; Utmost important questions are raised: Then, what are the meanings of mathematics curriculum at basic schools in theteacher education courses?
 How could we appreciate the deep implication of mathematics discoveries in thecontemporary world of changes?

Mathematics Education
 The challenge of understanding the meaning of mathematics in the school
curriculum of basic education.

 What do the Teacher Education courses need to entail further professional
development to improve the practice?
 The realization of the importance of knowing the historical evolution ofconcepts to understand the difficulties of learning as well as the semiotic
meaning of instruments for learning.

 Objective: Update the teaching methodologies to attend the necessary
paradigm shift in classroom practices, to master the lesson planning todevelop competencies in meaningful Problem Solving (mathematicsactivity).
How can teacher educators prepare and support teachers prepared to this
scenario? Where can the teachers look for the inspiration to get insightful
vision of Mathematics as accumulated scientific knowledge, and at sametime, a school content knowledge suited to the education of citizens?

Hemeneutics in Mathematics Education
 The hermeneutics effort as focused in this Session:

 a conceptualization of hermeneutics as the theory of interpretation, that is, atheory of achieving an understanding of texts; (Forster) Moreover, narrowing into a restricted focus, as a theory of understanding forplanning the problem solving approach; (Isoda)

 Activity of the recursive process of interpretation to explain the role of history foreducation, in order to educate pupils and teachers to think oneself intoanother person who have lived in another time and another culture.... (Jahnke)
 To understand the perspective of others (Isoda)

Technology as strategy of hemeneutical approach tomathematics content, school curriculum topics andmethodologies of teaching that connect the meanings of different
topics, developed with historical perspective.

Two exemplary projects in Mathematics and Mathematics
Education on Hermeneutics of Historical texts, with the aidof technology: M. Isoda et al.
 D-book software to study and to interpret the famous
Schooten´s book (17th century) on Conic Sections inTeacher education courses at U. Tsukuba and Workshops
abroad ( in BRAZIL: Lesson Study at UFRJ and USP)

AWARD WINNING PUBLICATION(ISODA &BUSSI, 2009)
ENCYCLOPEDIA OF CURVES, TOKYO,Kyouritsu sha

Examples from the research in Teacher Education
and Professional Development Projects (Baldin) Technology as strategy of hermeneutical approach to History ofMathematics in: Research activities of preservice teacher
 Didactical material for teacher education courses (preservice and in-service) Design of teaching material for classroom use at basiceducation level

Example1: Research Work of a preservice teacher (HASEGAWA, 2006) on Historyof Mathematics.
Providing a visually concrete model to retrieve/interpret the original 3dimensional definition of conic sections, following Appolonius.

A GEOMETRIA DO LATUS RECTUM NO ESTUDO DECÔNICAS SEGUNDO APOLÔNIO DE PERGAAdemir Cristovão Lucchiari (IC); Yuriko Yamamoto Baldin (O) – DM-UFSCARINTRODUÇÃO: Mesmo conhecido anteriormente aos trabalhos de Apolônio, desde Menaecmus (séc. IV – V A.C.) considerado o descobridor das secções cônicas, o latus rectum (ou parâmetro de uma cônica) foi geometrizado na obra
As Cônicas, por Apolônio de Perga (262 a 190 A.C. aproximadamente).

Este trabalho busca uma interpretação das proposições de Apolônio que definem as secções cônicas numa linguagem matemática moderna, e demonstra suas construções utilizando régua e compasso. Também traz uma visão bidimensionaldessas construções por meio do rebatimento do espaço para um plano, facilitando o entendimento.
Através da tradução e interpretação da obra de Apolônio a partir da versão inglesa (On Conics Sections, in Great Books of the Western World, Robert Maynard Hutchins (ed), Encyclopaedia Britannica, 22nd ed, 1978, Vol. II), e com autilização do software Cabri Géomètre II, demonstramos como as três secções cônicas eram conhecidas e obtidas, como o latus rectum era construído e como veio a determinar os nomes das secções, permanecendo até os dias atuais.
PALAVRAS-CHAVE: Secções cônicas, latus rectum, uso de tecnologia no ensino.

Visão antes de ApolônioVisão de Apolônio

As secções eram obtidas através do corte de um cone circular reto por um plano perpendicular a uma retageratriz do cone. Cones com um ângulo reto (cone retângulo), menor (cone acutângulo) ou maior (coneobtusângulo) em seu vértice geravam, respectivamente, um orthotome (hoje parábola), um oxytome (hojeelipse) ou um amblytome (hoje hipérbole).

Apolônio passa a utilizar um cone único para obtenção das três secções cônicas, variando a inclinação
do plano que o corta, sem necessidade da perpendicularidade relativa a uma geratriz. Também introduzo cone de duas folhas e o cone obliquo, em substituição ao cone reto.

AS .BC 2

O Latus Rectum (l =AB 2)(1) RO = SD (lados do paralelogramo);(2) OP² = RO.OV (média geométrica);

OV BCBC OV  DO.
DO ABABSD BCBC
 SD  AS .
AS AB
AB(3)eOP 2  SD.OV  AS .
;BCBC AS .BC 2.DO.
.DO
ABAB
AB 2

Então,AS .BC 2
AB 2; AS, BC e AB são constantes
Orthotome

Oxytomepara qualquer ponto P em EDG, portanto
rectum.

Elipse: RO < SD →
é constante e foi chamado de latus OP² < SD.OV.
Então OP² < l.DOHipérbole: RO >SD → OP² > SD.OV.
Então OP² > l.DOAmblytomeA parábola

Parábola: OP² = l.DOA hipérbole

Conforme proposto porApolônio, a parábola é definida a partir do segmento FH (latus2BCFH
 . Na figura, KL² = LF.FH para qualquer K pertencenterectum), onde
BA. AC FA
à secção cônica.

Para a hipérbole, assim como nos casos anteriores, vemos o latus rectum FL, onde

AK 2FHBK .KC FLe MN² = FN.NX > FN.FL, para qualquer M na secção cônica.

Na figura seguinte vemos o rebatimento com a igualdade entre as áreas, MN² = FN.NX.

Na figura seguinte rebatemos a parábola para o plano do triângulo ABC (BC = DE) e vemos a igualdade deáreas entre KL² e LF.FH.

Considerando o mesmo referencial ortogonal, com MN = y e FN = x, temos y² > lx.

Considerando um referencial ortogonal, com a reta suporte do segmento KL = eixo y e a reta suporte dosegmento LF = eixo x, KL² = LF.FH seria y² = lx, onde l é o latus rectum.

A elipse
Para a elipse, com latus rectum EH, temos

Na figura seguinte (rebatimento) vemos a igualdade entre as áreas, LM² = EM.MX.
Considerando o referencial ortogonal acima, com LM = y e EM = x, temos y² < lx.

HERMENEUTICAL EFFORT TO
UNDERSTAND THE
CONCEPTUAL GENESIS AND
TO RELATE TO MODERNPRESENTATION OF CONICS.
FOR THE BETTER KNOWLEDGE
OF TEACHERS!

AK 2DE.
BK .CK EH

Na figura, LM² = EM.MX < EM.EH, para qualquer L na secção cônica.

Example 2: A RESEARCHWORK BYA PRESERVICETEACHER ON THE CONCEPTOF LATUS RECTUM IN THE
ENGLISH TRANSLATION OF
THE ORIGINAL “ON CONICALSECTIONS”, BY APPOLONIUS,
WITH THE USE OFTECHNOLOGY.

Como pudemos observar, a álgebra dos gregos era totalmente baseada na geometria. As demonstrações dasconstruções das secções cônicas saem de comparações e razões entre segmentos, entre áreas e entre segmentos
com áreas, tratando uma figura espacial como algo que pudesse ser geometrizado no plano. Apesar de nãoutilizarem um sistema de coordenadas cartesianas, como entendemos hoje, parece-nos evidente que
trabalhavam com um sistema ortogonal, mesmo que implicitamente.
A comparação de y² com lx, sendo igual, menor ou maior, veio a definir o nome das curvas substituindo os
nomes utilizados até então e permanecendo até os dias atuais. O nome parábola, indicando colocar ao lado ou
comparação, substituiu o nome orthotome; o nome elipse, significando falta de alguma coisa, substituiu
oxytome; e o nome hipérbole, significando um lançamento além, ou um excesso, substituiu o nome amblytome.

CONCLUSÃO: O presente trabalho é uma contribuição para o ensino e aprendizagem da geometria, com um material que serve como apoio a professores e estudantes de ensino médio e das licenciaturas,sobre um tópico que está relegado
a um plano secundário, a despeito da sua importância na matemática.

For the knowledge of teachers: didacticalmaterials developed for teacher education
 Hermeneutical approach to the Conics following the conceptions ofAppolonius of Perga , 3-dimensional setting explored with CAS-technology
dynamical features to enhance the interactivity. Visualization to interpretthe concepts.
 Technology aided interpretation of the eccentricity of a conic section.

Baldin YY& Furuya, YKS (2001)

CAS graphic representation

Profiles of conical sections
Interpreting the 3-dim setting
From 2-dim perspective.
sin(∝)
Eccentricity =sin(𝛽)Dandelin Spheres and Conic sections (1822)

https://en.wikipedia.org/wiki/Dandelin_spheres

Dandelin Spheres: construction method for the profile of a conical section with DGS

Actual visualization of a conical section on its plane through rotating view in the space.

CAS interpretation of Dandelin Spheres in Space
Case of a hyperbola

The case of a parabola

The case of a ellipse

FOR CLASSROOM USE
Simulating ORIGAMI
Exploring Focal Properties
Understanding ageless applications
in real world (Historicalaccomplishments in sciences andtechnology development formodern facilities and progress for thesociety)Paper folding (origami) to understand the geometric property of a parabola as loci.
Focus, diretrix, symmetry, concavity, envelope of perpendicular bisectors.

Technology that permits to explore, to conjecture, to discover, to prove (focal property)

Focal Property of a Parabola

Name: FOCUSInspiration to burning mirrors??
HISTORICAL IMPORTANCE TO IMPROVE THE QUALITY OF TELESCOPES (FROM EARLY AS 11TH CENTURY) TOGALILEO´S TELESCOPE (16TH CENTURY) TO GREGORY´S PARABOLIC REFLECTOR TELESCOPE AND SUBSEQUENTPROGRESS (NEWTON,HOOKE, CASSEGRAIN). SCIENTIFIC ACCOMPLISHMENTS FROM MATHEMATICS.

Parabolic Satellite Dish (signal receiver)

Fonte:http://www.floy.com.br/img/class/122011/antenas_parabolicas_22815.gif

More applications for classroom: headlamps of cars
Projects on modern astronomical telescopes.
Trajectory of a thrown object (mechanics principle - gravitation)
Some profiles of suspended bridgeconstruction (and if the extreme points are close?)

CONNECTING THE KNOWLEDGE: GEOMETRY AND ALGEBRA (AFTER DESCARTES)
What has the school content about graphic of a second degree function
to do with parabola and FOCUS? Hermeneutics effort to connect the concepts.

𝑦 = 𝑥2𝑥2
𝑦=6Conclusion Hermeneutic effort at the service of a better education enables theunderstanding of mathematics lessons through problem solving
activities as a human enterprise and construction of the knowledge asa human heritage through the ages.
 The mission of teacher educators and researchers of teachingmathematics should be focused in providing opportunity toeverybody to achieve mathematics literacy through problem solving.
 The effective use of technology is not only a communication device
for information, but it should permit to each student trace the paths
already walked by others, so he/she can construct own knowledge
understanding the perspective of others.
 Lesson Study is grounding methodology to stimulate teachers tobecome better teachers and researchers of own practice, in ahermeneutic effort for mathematics education.

References Baldin, YY, Furuya, YKS, (2001), A study of conics with Maple V and Cabri-Geometry II, M.
Borovcnik et al (eds) Proceedings of ICTMT 5, International Conference on Technology inMathematics Teaching, Klagenfurt.
 Baldin,YY, (2001 – 2015), Teaching materials on Conics developed for teacher education
courses and for classroom use, UFSCar, São Carlos
 Catesby Taliaferro,R. (1939), On Conic Sections by Appolonius of Perga, Great Books of theWestern World, 595-804, Encyclopaedia Britannica, Univ. Chicago.
 Forster, MN (2002), HERMENEUTICS,//philosophy.uchicago.edu/faculty/files/forster/HERM.pdf (consulted September 2015)
 Isoda, M (2008), Getting Others´Perspectives through the Hermeneutic Effort: A Theory ofUnderstanding for Planning the Problem Solving Teaching Approach, Proceedings ofICME11, TSG 26, Mexico.
 Jahnke, HN (1994) (apud( Isoda, M, 2008)),The historical dimension of mathematics
understanding: Objectifying te subjective. Ponte et al (eds), Proceedings of theInternational Conference for the Psychology of Mathematics Education, 139-156.
 Lucchiari, AC, (2008) Um estudo de cônicas sob abordagens diversas: história, geometria
espacial e plana, construções geométricas e aplicações (A study of conics underdifferent approaches: history, spacial and plane geometry, geometric constructions andapplications), UFSCar, São Carlos, (in Portuguese)Thank you very much for your attention!