ISOMETRIESAND PATTERNS ~ A CREATIVE...

University of Aveiro

Dep. of Education

Lúcia Matos

Isabel Cabrita

ISOMETRIES AND PATTERNS

~ A CREATIVE APPROACH ~


Dep. of Education

Lúcia Matos

Isabel Cabrita

Sructure of the presentation

Problematic of the research

Objectives of study

Theoretical background

Methodology

key Findings


Dep. of Education

Lúcia Matos

Isabel Cabrita

Problematic of the research

AGD

Creative approach, tasks and resolution

Patterns

isometric transformations

Geometry


Dep. of Education

Lúcia Matos

Isabel Cabrita

Geometrical Knowledge

- isometries

Skillsgeneral | transversal | specific

Communication

More affectionate relationship

with the geometry

Autonomy

To Assess the impact of a creative approach of isometries

through the study of patterns and using GSP:

Objectives of the study


Dep. of Education

Lúcia Matos

Isabel Cabrita

Breda et al., 2011; Cabrita et al., 2008, 2009;

Dodge, 2004; ME, 2010; NCTM, 2007; Ponte et al., 2007;)

GEOMETRIC TRANSFORMATIONS

Isometries

PATTERNS(Devlin, 2003; Orton, 1999;

Vale et al., 2006;

Vale & Barbosa, 2009)

DGS(Breda et al., 2011; Cabrita et al., 2009;

Candeias, 2005; Gorgulho, 2005;

Hoylees, Lagrange, 2010;

Kasten, Sinclair, 2009;

NCTM, 2007; Ponte et al., 2007;

Veloso, 1998

Theoretical background


Dep. of Education

Lúcia Matos

Isabel Cabrita

� Essentially qualitative (Bodgan e BilKen, 1994; Huberman & Miles,

2002)

� Multiple case study (Hartley , 2004; Stake, 2007; Yin, 1989, 2005)

� Participants: 21 students of the 9th grade

Half rural

Elementary

school of Aveiro

Class of 9 th

year

•Different school

achievements

•Facility to

communicate

ideas

Methodology


Dep. of Education

Lúcia Matos

Isabel Cabrita

Logbook

Students' productions

Informal talks

Test

PEE/PCE/PCT/PAD/Program

October-November

novembro a dezembro

February

February-March

1st Stage

2nd Stage

3rd Stage

4th Stage

January-March

QuestionnaireCharacterization of the target group

Planification

Pre-test

Didactical approach

inquiry

document analysis

document analysis

Direct observation

Document Analysis

research design


Dep. of Education

Lúcia Matos

Isabel Cabrita

Questionnaire

Test

march

may

january to december

5th stage

6th stage

7th stage

8th stage

Content

Analysisstatistical

analysis

Post-test

Final Questionnaire

Treatment of data

Presentation of data

document analysis

inquiry

Narrative, transcripts,

tables, images, …

research design


Dep. of Education

Lúcia Matos

Isabel Cabrita

� 9 diverse tasks applied during 1 month;

Didactic sequence

� Students solve them in pairs, using GSP;

� Confrontation of the various resolutions;

� A summary of the main mathematician aspects involved to retain

� the test

� a theoretical part;

� a practical one – solved with GSP and another one solved in pairs.


Dep. of Education

Lúcia Matos

Isabel Cabrita

Categories of analisys

Isometries

Patterns

Confidence

Enthusiasm

Motivation

Interest

Geometrical

knowledge

Development

of a new

math vision

Autonomy

Interactions –Teacher and

Students

Math comm.

Communica-

tion

treatment of the data


Dep. of Education

Lúcia Matos

Isabel Cabrita

Key findings

geometrical knowledge

Students improved its knowledge:

to reproduce, continue, complete, create and identify

repeating sets related with friezes, rosaceas and tilings


Dep. of Education

Lúcia Matos

Isabel Cabrita

Key findings

geometrical knowledge

“The work centered in patterns (and the GSP) allowed us to have a better

perception of the geometrical concepts involved” (FQ) - the students

absolutely or partially agreed


Dep. of Education

Lúcia Matos

Isabel Cabrita

Key findings

communication

�reciprocal negotiation

“working in groups was very pleasant and I

learned more”

“the use of this software fosters the

interaction between students”


Dep. of Education

Lúcia Matos

Isabel Cabrita

Key findings

communication

• difficulty in communicating mathematical ideas, mainly in writing,

“For the first shape we used isometries of rotation. And in the

second one, we used isometries of rotation and reflection. The first

one is cyclic and the second one is dihedral”.


Dep. of Education

Lúcia Matos

Isabel Cabrita

Key findings

autonomy

• Gradually, they earned some autonomy,

and recognize their – “it was important

that the students drew their own

conclusions”.

• First, the students would call for help - “Teacher, I don’t

know what to do!”, “What should I answer? We don’t

understand!”

• FQ - they affirmed to have enjoyed to use GSP, which contributes to

an active and dynamical learning of Geometry


Dep. of Education

Lúcia Matos

Isabel Cabrita

GSP and patterns motivated the learning and catalyzed their interest

didactic approach contributed “to a more positive view towards

Geometry” as well as “for the development of an affective relation” with

the subject (FQ).

“it made Geometry wonderful”

Key finding

positive view


Dep. of Education

Lúcia Matos

Isabel Cabrita

• Adams, D. & Hamm, M. (2010). Demistify Math, Science, and Technology: Creativity, Innovation, and Problem

Solving. UK: Rowman & Littlefield Education.

• Bogdan, R. & Biklen, S. (1992). Qualitative Research for Education: An Introduction to Theory and Methods.

Boston: Allyn and Bacon.

• Breda, A., Serrazina, L., Menezes, L., Sousa, Hélia & Oliveira, P. (2011). Geometria e medida no ensino

básico.Lisboa: DGIDC.

• Cabrita, I. et al. (coord.) (2007 a 2011). Coleção de livros m@c1/2. Aveiro: Universidade de Aveiro, Comissão

Editorial.

• Dodge, C. (2004). Euclidean Geometry and Transformations. USA: Dover Publications.

• Ferrance, E. (2000). Action Research. USA: LAB - Northeast and Islands Regional Educational Laboratory at Brown

University.

• French, D. (2005). Teaching and Learning Geometry. London: Continuum.

• Greenes, C. & Rubinstein, R. (Eds.) (2008). Algebra and Algebraic Thinking in School Mathematics – Seventieth

Yearbook. Reston: NCTM.

• Hartley, J. Case study research, in: C. Cassell, G. Symon, Essential Guide to Qualitiative methods in organizational

research, Sage, United States, 2004, p. 323- 333.

• Hoyles, C.; Lagrange, J.B. (Eds.), Mathematics Education and Technology—Rethinking the Terrain, The 17th ICMI

Study, Springer, New York, 2010.

• Huberman, M. & Miles, M. (2002). The Qualitative Researcher’s Companion. London: SAGE Publications Ltd.

• Kasten, S. & Sinclair, N. (2009). Using dynamic geometry software in the mathematics classroom: A Study of

Teachers' Choices and Rationales. International Journal for Technology in Mathematics Education, 16 (4), 133-

143.

• Leikin, R. Exploring mathematical creativity using multiple solution tasks, in R. Leikin, A. Berman, B. Koichu, (Eds.),

Creativity in mathematics and the education of gifted students, Sense Publishers, Netherlands, 2009, p. 129-145

• Leikin, R., Berman, A. & Koichu, B. (Eds.) (2009). Creativity in mathematics and the education of gifted students.

Rotterdam, Netherlands: Sense Publishers

References


Dep. of Education

Lúcia Matos

Isabel Cabrita

• McMillan, J. Classroom Assessment. Principles and Practices for Effective Instruction, Pearson, United Kingdom,

2011.

• Robinson, K. (2011). Out of Our Minds: Learning to be Creative. UK: Capstone/Wiley.

• Sanders, C. (2003). Geometric Graphics. Key Curriculum Press: USA

• Schwandt, T. Constructivist, interpretivist approach to human inquiry, in: D. Norman, Y. Lincoln, The landscape of

qualitative research: theories and issues, Sage, United States, 1998, p.221-259.

• Sheffield, L. J. Developing mathematical creativity – Questions may be the answer, in Leikin, R. Berman, A.;

Koichu B. (Eds.), Creativity in mathematics and the education of gifted students, Sense Publishers, Netherlands,

2009, p. 87–100.

• Silver, E. Fostering creativity through instruction rich in mathematical problem solving and problem posing, ZDM

Mathematics Education 3 (1997) 75-80.

• Stam, H. (2001). Social constructionism and its critics. Theory and Psychology, 11 (3), 291-296.

• Usiskin, Z., Andersen, K. & Zotto, N. (Eds.) (2010). Future Curricular Trends in School Algebra and Geometry.

Proceedings of the Second International Curriculum Conference, The University of Chicago and The Field

Museum, Chicago, Illinois, U.S.A.

• Vale e A. Barbosa (Org.) (2009). Patterns – Multiple Perspectives and Contexts in Mathematics Education. ESEVC.

ISBN: 978-989-95980-4-1

• Vale, I. & Pimentel, T. (2010). From figural growing patterns to generalization: a path to algebraic thinking.. In

M.F. Pinto, & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the

Psychology of Mathematics Education, Vol. 4, pp. 241-248. Belo Horizonte, Brasil: PME.

• Vygotsky, L. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard

University Press.

References


Dep. of Education

Lúcia Matos

Isabel Cabrita

Lúcia Matos

Elementary School Castro Matoso

[email protected]

Isabel Cabrita


[email protected]

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