Solving of reverse constructions Solving of reverse constructions with GeoGebra in mathematics with GeoGebra in mathematics

teacher trainingteacher training

RNDr. Kitti Vidermanová, PhD. PaedDr. Janka Melušová, PhD.

Department of Mathematics Constantine the Philosopher University in Nitra

Idea while checking homework Students sent their solutions in GeoGebra and Cabri II

by e-mail Mistake of one student – she used wrong given data in

construction task – time consuming and demanding

The Constructional TasksThe Constructional Tasks Belong to most difficult tasks in school geometry, because

they cannot be solved/taught by any algorithm, they have to be solved by deduction

…should be taught because they offer space for independent creative activity of

students, they help to development of creativity and visualization, Logical thinking, Sence for aesthetics.

The Reverse ConstructionsThe Reverse Constructions Teach how to read from figure, While the solving process, students do not construct

but inquire and explain separate steps of the construction,

Transformation of wording to symbolic language, Deeper understanding.

Study – brief descriptionStudy – brief description 3 tasks (2 planar, 1 spatial) – constructions done in

GeoGebra,

8 student working independently with the computer and wrote their reasoning on the paper,

Tasks: „Based on given construction, try to find out, what shape should be constructed and which objects were given.“

TaskTask 1 1Given• length of base AB,• size of angle by A,• height of the trapezoid.

TaskDraw the isosceles trapezoid ABCD.

Task 1Task 1 - - resultsresultsNumber ofstudents

Given data Drawn object Note

3 students AB, v, Isoscelestrapezoid

Correct solution

2 students AB, v, + BC

Trapezoid Unawareness of isosceles

2 students Not written Isoscelestrapezoid

Instead of formulating the task, studentswrote how to draw the object

1 student Not written Trapezoid Instead of formulating the task, student wrote how to draw the object,Student used the equality of legs BC and AD, but he did not write that thefinal object is isosceles trapezoid

TaskTask 2 2Given• lengths of medians.

TaskDraw the triangle ABC.

TaskTask 2 - 2 -resultsresultsNumber ofStudents

Given data Drawn object Note

2 students Isosceles triangle ABC

Rectangular triangle ACG´

They did not use the dynamicity of GeoGebra

2 students triangle ABC Isosceles triangle ACG´

Wrong interpretation of symmetry according to point

Dynamicity not used

2 students triangle ABC Isosceles triangle ACG´

2 students triangle ABC Triangle witharea 1/3 of given Triangle

We consider it as the correct solution

TaskTask 3 3

Given data• length of side of the base• altitude.

TaskDraw the regular pyramid with triangle base in sloping parallel projection

TaskTask 3 - 3 - resultsresultsNumber ofStudents

Given data Drawn object Note

5 students Not writen Regular 3-faced pyramid in sloping parallelProjection

According to the reasoning of students the task was solvedcorrectly, but given data ismissing

1 student Side of the base and the altitude of the pyramid

Regular 3-faced pyramid in sloping parallelProjection

Only complete solution

1 student Not writen Pyramid The object specification is missing

1 student Not writen Regular tetrahedral The given object was not regular

Students’ opinions (future teachers)ICT in geometry It is interesting and motivating, but it should be used more for revising, it

is important to get the skills in drawing on the paper, Students spend too much time sitting by a computer, I prefer the standard

(pen and paper) way to teach.

Reverse constructions Interesting, new and demanding, Like a puzzle, so they can be motivating I would use this kind of task because students usually do not like classic

drawing with pencil, it may be useful to alternate this task with classic ones

I would implement them at upper secondary and tertiary education, students should develop the drawing skills.

Survey of in-service teachersSurvey in 2007 (129 respondents, average length of practice

17, 8 years) 66% of teacher have the possibility to use the

computers in mathematics lessons Only 40 % of these really use it.

Reasons: Lack of time for innovative methods (teach students

to work with program), Lot of teachers is not able to use the software.

Additional informationSince 2008/09 – curricular reform Less content, less time.

Project MVP - design of teachers book aimed to use of ICT in school mathematics (including the training of teachers)

Thank you for your attentionThank you for your attention