USE OF GSP WITH
PROOFS AND CONSTRUCTIONS
This topic was chosen because of a question asked by a member of our EMT
- 705 class on Friday July 28 1995. The questions was," Should we continue
to do paper and pencil constructions as well as the GSP construction, or
do we discontinue paper and pencil construction?"
I decided to expand the paper to include proofs, so that I would have more
to talk about and because I think the question is appropriate for proofs
as well as constructions.
The NCTM standards do not seem to address the question posed in the preface
to this document. However, under topics to receive increased attention are:
1. Computer-based explorations of 2-D and 3-D figures.
If a teacher has the computers and software to do the explorations suggested
in #'s 1 and 3 above, then there will be less time for some topic. That
topic would almost obviously be paper and pencil constructions. We presently
present Geometry between Algebra 1 and 2, during the ninth or tenth grade
year. Students selected in middle school for high mathematics ability take
Algebra 1 in the eighth grade and Geometry in the ninth grade. Other students
take Algebra 1 in the ninth grade and Geometry in the tenth grade. They
take the course together with some eleventh and twelfth grade students.
These students do no constructions, as a general rule, in the Algebra 1
course or in any other course.
2. Three-dimensional geometry
3. Real-world applications and modeling.
It is a good idea to start with some simple paper and pencil constructions
such as the perpendicular bisector of a segment, the bisector of an angle,
perpendicular from a point on a line or segment and from a point not on
a line or segment. Then you run up against it so to speak, do you stop here
or go for one more construction? The answer is that you stop here, BUT each
time you do a GSP (or computer based) construction, remind the students
that this can be done by paper and pencil construction. If you are lucky
(or unlucky), depending on the philosophy of the teacher, a student will
ask to see the construction done on the chalk board and the teacher can
demonstrate the skills of compass and straight edge. One of the 'good' constructions
is an angle congruent to an angle, which is difficult for many students.
This can be pictured by edit and copy on GSP or the actual paper and pencil
construction can be done on GSP. If is a woolly bear for many students,
but it does give them facility in handling GSP.
One construction the student may be able to do on the GSP is a dihedral
angle showing the plane angle. An illustration is given below and this involves
extensive use of the construct menu with segment and ray tools. If the student
has trouble with this on the GSP it will be helpful to draw the figure with
pencil and paper. This construction can be printed out and graded for cosmetic
appearance, but be liberal if the student is new to the GSP. A good example
of paper and pencil construction for teacher illustration is the construction
of a line parallel to a given line.
Perpendiculars may be used or congruent angles. This can also be shown using
the features of the construct menu and tools of the GSP. From this point
onward do all the construction on the GSP. The only time to use a paper
and pencil construction after this, is if the lab is not available and you
must use paper and pencil constructions.
A time should be given to let the student draw a design of choice on the
GSP. Year before this past year I watched a student draw a pickup truck
using GSP while employing color combinations etc. Each lab period as he
finished his assignments he would go back to the pickup truck. This brings
up the fact that it is good to let each student have his own personal disc,
teach them to 'save as' to their own disc. Our mathematics department provides
a disc for each student and has written instructions as to save to the disc.
Evaluation is of four types: 1. Print out the construction and grade as
would be done with the dihedral angle. 2. Grade the construction from the
disc.* 3. Grade the construction as it is on the computer. 4. Check the
disc to see that required constructions are completed. *this can take time,
# 3 is a better option.
I have reviewed the content and text of eight textbooks, copyrights 1987
through 1995 and they all have a section devoted to construction by paper
and pencil technique. The only exception is the Serra book from Key Curriculum
The graphing feature of the GSP can be used to construct on the coordinate
plane, increasing coordinate and transformation approach and reducing geometry
from a synthetic perspective.
The number of pencil and paper constructions can be greatly reduced and
the student and teacher will profit by this action. No significant understandings
will be lost! Be brave, technology forever!!
Two column proofs are alive and well. The texts mentioned above that were
reviewed by me are: HOLT, 1991; Houghton Mifflin, 1992; Prentice Hall,1993;
Southwestern Publishing,1992; Merrill,1995; Addison Wesley, 1994 and 1987;
Key Curriculum Press 1989. Excluding Key Curriculum Press the content sections
are almost identical as follows:
Chapter 1 Definitions,
Not only is the content section as described in the Chapter sequence the
same, the theorems are presented in the same order. They could all have
printed one book under separate covers, which is about what they have done.
The publishers are printing a book to sell to teachers, which tells me that
this is the order sequence that most teachers like. Again, excluding Key
Curriculum Press, each book has proofs in chapter 2, after definitions in
chapter 1. The course is then developed by supporting each new topic with
a two column proof. The Southwestern Publishing book does definitions and
proofs in Chapter 1 and seems to rely less on proof than the other books
except, of course, Key Curriculum Press. The above is apparently the teaching
method for geometry. Start with your definitions, then axioms/properties,
then postulates and theorems. This must be the accepted 'American' way!!
Chapter 2 Proof,
Chapter 3 Parallel lines,
Chapter 4 Congruent triangles,
Chapter 5 Congruent triangles applications,
Chapter 6 Polygons/Quadrilaterals,
Chapter 7 Similar triangles,
Chapter 8 Right triangles,
Chapter 9 Circles,
Chapter 10 Area,
Chapter 11 Volume,
Chapter 12 Coordinate Geometry,
Chapter 13 Loci and transformational geometry.
Why do we do it this way? We seemed to be saying that a student cannot learn
geometry, without first learning proof. The justification seems to be that
when a student reaches the geometry course, that this is the first time
he is required to accept a logical orderly presentation of thought. Further,
we say, the students do not know how to organize their arguments in an orderly
manner and two column proofs help do the job.
Using the internet in our EMT-705 class, I ran across an argument that has
been going on, at least, since January of 1995. John Conway, arguably the
world's leading geometer, makes the point that two column proof is not necessarily
"Euclidian geometry." He says, "Any (caps are his) way
to get the right answer, and be certain you've done so, is OK as far as
he is concerned." [I would bet the tin can full of money in your back
yard that Mr. Conway has never taught geometry to ninth and tenth grade
students in the Southern part of the U.S.] Mr. Conway further states that
he never heard of two column proofs until 11 January 1995. He has negative
comments but sees some virtues in the hands of a good teacher. Mr. Scott
Powell from the University lab School in Honolulu, seems to agree with Mr.
Conway, and gives some arguments such as: "When a student looks at
an Isosceles triangle and notices two of the angles are equal that becomes
something rather trivial and from then on a proof is thought of by the student
as a waste of time." Personally I think it is important to know "Why"
and I am in the habit of showing it in proof form. However, I suspect Mr.
Powell may be right, for it is true that every time a student constructs
an isosceles triangle, (GSP or paper and pencil) the angles opposite the
congruent sides are congruent. Mr. Powell says it is ridiculous for the
student to have to prove this! Maybe. Mr. Powell seems to agree with Mr.
Conway that there are other ways (proofs?) that can convince a student of
a fact in geometry. Some correspondents in this argument are putting forth
the theory that two column proofs are a useful way of documenting the problem
solving process. Two column proofs are a handy way of checking legitimacy
and accuracy of student thinking. Some of the correspondence suggest that
constructions can be used as proofs. Mr. Powell continues by saying that
Geometry is so much more than proof and it should be an outcome from the
experiences and insights gained by discussion and good problem solving.
One correspondent states that; " I believe that many teachers are convinced
that in a geometry class they are teaching geometry. The are not teaching
geometry, but are teaching reasoning in geometry. Another correspondent
answers this : Why not teach geometry in a geometry course and reasoning
in all our math courses? Another correspondent, John Tant, states that,
"The plane geometry course is the first time that students get a systematic
study of the reasoning process. That's the purpose of the course. We have
found this to benefit students in physics, precalculus and calculus."
A lady who teaches math in Wantage, Oxfordshire, England (Jan Hall) states
that she doesn't understand the divisions that we create with our separate
courses. She states that they divide into pure maths, mechanics and statistics
. They sometimes get new students from the American system who are appalled
at the integration. She says these students say that they prefer the English
method after a period of time. One of the students in our EMT-705 class,
from Gwinnett county, showed me her geometry text for next year, which uses
an integrated approach.
After reading all this and talking with some of the students in both my
classes (EMT-705 and EMT-708) it is difficult to know what is best. I have
to do something because I will be teaching geometry next year. Most teachers
probably follow the text which is, as stated above, practically the same
from every publishing company. Then the tests and exercises fall into line.
However I have been trying (and sometimes succeeding) to escape this trap
ever since the Litmus program started in the summer of 1991 and the GSP
appeared on the scene for me.
When you are beginning a course in geometry I think it is important to mention
the idea of proof ( deductive and other types) and gradually work up to
some execises that students can do. This should not be a rigorous development
because it can turn students away from geometry.
Example: Proving the sum of the angles of a triangle can be done inductively
by getting each student to draw (on paper) five triangles, use a protractor
and measure each angle, get a sum for each triangle and an average for the
five. Then we use the TI-81 to get the average of the averages and it is
usually within .1 or 180 . Then we tear off the corners of a triangle and
show that they fit on a straight line. Then we draw it on the sketch pad
and show that the sum is 180 . Later when more geometry is used we can prove
this deductively using the parallel postulate. I do believe that a rigorous
system of proofs, which students try to memorize, can turn the students
away from a logical development of geometry. [One of the students in our
class admitted to me that when she took geometry in HS, that she used her
memory rather than understanding the proofs.]
My conclusion about two-column proofs is that it is something to help the
students understand as a system of logic, but it should not be presented
as a rigorous discipline (within a discipline) throughout the geometry course.
Understanding and appreciating the symmetry and beauty of geometry should
take precedence over the teaching of proof.