1. Introduction
We have carried out France and Japan Cross-Cultural
Research on the Role of Figures in Geometry(*) since 1997.
We have interested in students' geometrical proof-problem
solving in secondary schools in each country. We clarified
that geometrical figure is an important mathematical object
that help students in producing their proofs (Harada et
al.,1993).
However, we observed that the geometrical
figures exerted negative influences on the problem
solving,for example,"producing propositions based on visual
judgments" and "depending prototype example of triangle"
(Gallou-Dumiel et al.,1997). We think that the negative
influences depend on their viewpoints of geometrical
figures.
From this viewpoint, we set up the purpose of
this research as follows:
The purpose of this research is to clarify
characteristics of students' apprehensions of figures
underlying their viewpoints of geometrical figures in
proof-problem solving through a case study.
For the purpose of this research, we will consider
"types of apprehension of figure" ("les types
d'apprehension des figures") which are identified by
Mesquita (Mesquita,1989) as a theoretical foundation.
Because we think that it is useful for us to observe
students' cognitions of geometrical figures.
2. Theoretical Foundation of
Research -Types of Apprehension of Figure -
Mesquita insisted the characteristics of four types of
apprehensions of figures as follows:
(1) Perceptual apprehension (l'apprehension
perceptive): it is a type of apprehension based on
perceived properties of figures and relates to the
figural organization laws in Gestalt Psychology.
(2) Operative apprehension (l'apprehension
operatoire): it is a type of apprehension based on
modifications or transformations of a figure. A heuristic
function which gives students an insight to a solution of
problem will be produced by this type of
apprehension.
(3) Sequential apprehension (l'apprehension
sequentielle): it is a type of apprehension based on
construction sequences of figure. The constructions of
figure are necessary for this type of apprehension.
(4) Discursive apprehension (l'apprehension
discursive): it is a type of apprehension based on the
hypothesis in the problem. The hypothetical-deductive
proof will be produced by this type of apprehension.
We especially focused on the "operative apprehension" in
this research because there are various ways of apprehension
of figure and heuristic functions to find solutions of the
problem in the operative apprehensions.
Mesquita describes that there are four kinds of
"modifications of configurations (modifications
configurales)" as the ways of the "operative
apprehensions":
(a) Mereologic modification (modification
mereologique): it is a modification which we can divide a
whole figure into sub-figures and integrate those
sub-figures.
(b) Ensemblic modification (modification ensembliste):
it is a modification which we can divide elements of
figure (line, circle,etc.) into several elements or take
out the elements from the figure.
(c) Optic modification (modification optiqe): it is a
modification which we can modify a figure from a specific
viewpoint, for example, larger or narrower, and
horizontal or slant.
(d) Positional modification (modification
positionnelle): it is a modification which we can modify
the positions or the orientations in the figure.
3.Methods of
Investigation
(1) Problem of
Investigation
Problem: ABC is triangle. Construct a square MNPQ with
Pe(BC), Qe(BC), Me(AB), Ne(AC). Can you explain how to
construct the square ? Also prove that the construction is
valid.
This problem is located in common with Secondary
School Curriculum in each country. The problem is given in
Lycee 1eres in France and Junior High School,the second
grade, in Japan.
The problem have considered by many researchers
based on their viewpoints ofresearches, for example, Polya
(1973) and Schoenfeld (1985). Although they insisted
"heuristics for problem solving" on this problem, they did
not discuss relationships between the geormetrial figures
and proofs. In this investigation, we would like to discuss
such relationships based on their apprehensions of
figures.
(2) Subjects
In France, six students in Lycee 1eres. They are the 11th
grader students and 16-17 years old. In Japan, six students
in Junior High School. They are the 9th grader students and
14-15 years old.
(3) Methods
We explained how to solve the this problem to our
students as follows:
- First, draw an inscribed square in the
triangle based on your expectations and research in this
figure geometrical properties to find how to draw the
square.
- Second, draw an another square in your figure (
e.g.BCDE) and research in this figure geometrical
properties to find how to draw an inscribed square in
triangle.
- Third, explain how to draw the inscribed square in
the triangle.
- Fourth, prove that the construction is valid.
They could spend about 1 hour to solve the problem. After
the investigations, hey were interviewed by investigators
about their thinking processes.
4. Viewpoints of Investigations
and the Results
(I) After students drew an
inscribed square in the triangle based on their
expectations,
what types of apprehensions of figures do they produce
?
In France, Student BK applied Midpoint Connector Theorem
to the triangle ABC. Student GN and Student AT found that
triangle AMN is drawn by reductions of triangle ABC. Student
LI applied Thales' Theorem to the triangle ABC.
In Japan, Student KY tried to find length
segment BP in the case of QC=x, but she could not a solution
by using equations. Student YA drew segment NP and tried to
find relationships between triangles AMN, MNP and NPQ.
Student HK drew another square which adjoined to the square
MNPQ and found a similarity transformation of two
squares.
French students produced "operative
apprehension" connecting with "discursive apprehension" and
used "optic modifications". On the other hand, Japanese
students produced "operative apprehension" connecting with
"perceptual apprehension" and used "mereologic
modifications".
(II) After they drew an another
square in the figure,
what types of apprehensions of figures do they
produce?
In France, Student LI could find that there is an
homothetie of center A and produced several expressions:
AM=kAB,AN=kAC, AQ=kAB', AP=kAC'.
In Japan, all of students (6) could find that
there is a similarity transformation between two
squares.
In this step of problem solving, both of French
and Japanese students produced "operative apprehension"
connecting with "discursive apprehension" by using "optic
modifications".
(III) In their explanations for
how to draw an inscribed square in triangle,
what kinds of inferences do they produce?
In France, Student LG, AT and LI explained that they
could draw two lines AD and AE based on an homothetie
between MNPQ and BCDE and decide positions of two points P
and Q on BC as intersections of those lines and BC.
In Japan, Student (HK) could explain the reason
based on ratios of length of sides of similar triangles
based on measurements of segments. Students (KM and YA )
checked the fact that two squares are similar by drawing the
lines through vertexes of each squares using ruler. Student
KM explained the fact that two quadrilaterals are squares,
then MNPQ is similar. She made mistake in how to infer in
this problem solving.
French students produced deductive inferences
based on "operative apprehensions" and "discursive
apprehensions" of figure. The "discursive apprehension"
depended on a theorem (i.e.homothetie). On the other hand,
Japanese students produced inferences based on "operative
apprehensions" and "perceptual apprehensions" of figure. The
"perceptual apprehensions" depended on their measurements of
segments using ruler. One student made mistake in how to
infer because she observed two squares based on "perceptual
apprehensions" and not "sequential apprehensions" of
figures.
(IV) In their proofs for validity
of construction,
what kinds of inferences do they produce?
French students LG, AT and LI as well as Japanese
students (except KM) could explain how to draw an inscribed
square in the triangle but they did not write a proof of the
fact.
Because, French students implicitly could
understand the inference that the quadrilateral BCDE is a
square and then it is transformed to MNPQ. On the other
hand, Japanese students could not understand the inferences
that if "the quadrilateral M'N'P'Q' is a square" then "MNPQ
is a square".
5. Conclusions
For the purpose of this research, we could clarify the
characteristics of apprehensions of figures underlying their
viewpoints of geometrical figures.
(i) French students produced "operative
apprehension" connecting with "discursive apprehension"
and used "optic modifications". On the other hand,
Japanese students produced "operative apprehension"
connecting with "perceptual apprehension and used
"mereologic modifications".
(ii) French students produced deductive inferences
based on "discursive apprehensions" of figure. The
apprehensions depended on the theorems. On the other
hand, Japanese students produced inferences based on
"perceptual apprehensions" of figure. The apprehensions
depended on concrete figures or numerical values.
Especially one student who produced "perceptual
apprehension" and not "sequential apprehension" of figure
made mistake in her inference.
References
Gallou-Dumiel E., Harada K., Nohoda N.
(1997) France-Japan cross-cultural research on the role
of the figures in geometry. Report of Geometry Working
Group in the 21st PME Conference, Finland.pp.1-11.
Harada. K., Nohda N., Gallou-Dumiel E.
(1993) The role of conjectures in geometrical proof-problem
solving. France and Japan collaborative research
-.Proceedings of the 17th PME Conference, Vol.3,
Japan,pp.113-120.
Mesquita A. (1989) L'influence des aspects
figuratifs dans l'argumentations des élèves en
geometrie. These Universite Louis Pasteur,
Strasbourg.pp.9-23.
Polya G. (1973) How To Solve It. Princeton
University Press.(2nd Edition)
Shoenfeld A. (1985) Mathematical Problem
Solving. Academic Press, Inc.
Note:
(*) France and Japan Collaborative
Research of an agreement between Universite de Grenoble and
University of Tsukuba.
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