Geometry: Between the Devil
and the Deep Blue Sea
Johnny W. Lott
Thursday, July 17, 2008
Geometry should be taught like
swimming—Freudenthal.
• Agree or disagree
What are your thoughts on axiomatics
in geometry?
• Pro
• Con
• Somewhere in between
What is your goal for the level of rigor
in a geometry class?
• Purely axiomatic?
• Purely utilitarian?
• How do you decide?
What do you want for your students in
their study of geometry?
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Allow manipulatives?
Require axioms?
Do proofs?
Allow technology?
– What are the issues?
How do you grade integrated math
projects?
• How do mathematicians grade application
projects in mathematics?
How much skin can be grafted?
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How much skin do you have?
How do you decide?
Why should I care?
Is this a geometry problem?
The figure below shows the base of a
kaleidoscope that is an equilateral triangle
with sides of length s. A light source is
placed at a point S on segment AB at a
distance d from point A.
s
s
d
A
S
s
B
• If a light ray emerges from S at an angle with
measure of 60˚ as shown, answer the following:
(1) Does the light reflect back to point S? Prove
your answer. Include a Geometer’s Sketchpad
drawing with your proof.
(2) If possible, find the length of the light ray’s path
in terms of s.
• Suppose the light source was reflected at another
angle. Explain whether or not that changes any of
your answers to part (1).
And a problem from other areas!
Can you trust your map?
Can you trust your map?
• Just how much trust do you have?
• Which is bigger? Greenland or Mauritania?
Why are there problems?
• Why do maps contain discrepancies?
• When might one map be better than another?
• What are the assumptions made when maps
are drawn?
• What are some spatial visualization problems
encountered by map-makers?
Questions to consider:
• In which types of maps would you expect
South America to have the greatest area?
• On which maps would you expect to find
the North Pole?
Examples
• What is anamorphic art?
Where does it go?
• Map Making
– Cartography
– Mapping a plane to a cylinder
– Stereographic projections
– Coordinates on different surfaces
AND AN EXAMPLE FROM THE
QUILTING WORLD!
Quilting: what geometry is involved?
RECURSION IS MORE IMPORTANT
THAN EVER!
Would you work for me?
Example: Quilting, where does it go?
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Quadrilaterals and their properties
Parallel lines
Tessellations
Rep-tiles
I pay you $1 to start the first day. I pay you $.50 the second day. After
that you have to figure your own salary. It is 2.5 times today’s salary
minus yesterday’s. I round up to the nearest dime. Would you work for
me or the Cobb County School System?
The School System rounds down to the nearest dime. Would you work
for it?
No. of Day
1
2
3
4
5
6
7
8
9
Salary/Johnny
$1.00
$0.50
Salary/School System
$1.00
$0.50
CAN YOU WRITE A FORMULA FOR
OR DRAW A GRAPH WHAT WE JUST
DID?
You are offered a job with the following conditions. I pay you $1000
to start. You pay me a commission at the end of the first day of $100.
Your net at the end of day 1 is $900. To start day 2, I double your
salary; you double my commission. The process continues. Would
you work for me?
No. of Day
1
2
3
4
5
6
7
8
9
Salary/DayStart
1,000
1,800
Commission
100
200
HOW DO YOU APPROACH THE
PREVIOUS PROBLEM?
Integrated math students typically
are not afraid of such problems.
How do you assess with performance
assessment in integrated
mathematics?
• What are your course goals?
• What are your unit goals?
• What are your student objectives in a lesson?
• Nothing should really be changed in following
good assessment techniques.
Would you consider a modeling
problem as an assessment?
• What is a good problem?
• What tools are allowed?
• What type of student interaction is allowed?
What is important in assessing a model or
a project?
• Depends on the class objectives
• Questions to consider
– Are you building a model to answer a question or
to solve a problem?
– Is the project interdisciplinary?
– Is this a math project or model or being done for
another reason?
Assumptions
• The model or project is for mathematical in nature,
or at least is begin assessed in mathematics.
• The model or project answers some type of question.
• The model or project is to be shared with others.
• Everyone involved takes assessment seriously!
Sample Vermont Rubric
• Problem solving
• PS1
Understanding the task
• PS2
HOW? - approaches/procedures
• PS3
WHY? - decisions along the way
• PS4
SO WHAT? - outcomes of the activity
• Communication
• C1
Language of mathematics
• C2
Mathematical representations
• C3
Presentation
What is important in assessing a model or
a project?
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Identification of the problem
Formulation of assumptions
Construction of the “model”
Solution and its interpretation
Validation of solution
Mathematics
Communication
Identification of the problem
• In your words, what is the problem?
• Can you restate it in a way that it can be
mathematized?
• Can you put it in quantifiable terms?
• Are there subproblems that must be identified
before you can solve the problem given?
Example of Identification of a Problem
• How much of your skin could be used in a skin graft?
– How much skin do you have?
– How can skin be measured?
– How much skin could be safely grafted in general?
– Do I have subproblems that have to be answered before I
can move on?
Formulation of Assumptions
• What are the critical assumptions to solve the
problem or subproblem?
• Do the assumptions significantly change the
problem?
• Are the assumptions reasonable?
• Why did you make the assumptions?
• Are arguments given to justify the assumptions?
What assumptions do you make?
• According to a newspaper report, the trees in
a certain land area are being cut at a rate of
15% a year. The lumber company claims that
it replaces 2000 trees every year in the area.
Discuss the future tree production of this land
area if the plan continues.
What are reasonable assumptions?
• How many trees in the area to start?
• How long will trees be cut?
• How can you justify your assumptions?
End-of-Year Assessment
Construction of the Model
• Contains a verbal description of the model
• Evidence of effort to use math
• Quantifies important features of the problem
using appropriate math
• Justifies choice of math
• Discusses other approaches
Constructing a Model
• There are about 4,000 students at a university.
Early one morning the student body president
tells the vice president that a famous actor
(whose identity is secret) has agreed to be the
commencement speaker. Develop a model for
the way that information is spread.
Constructing a Model
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P(n) = the total # of students who know after n hrs
Assume that each tells another every hour.
Initially, P(n) doubles every hour.
As # increases, the situation changes.
Once P(n) reaches 4,000, it cannot possibly rise.
Assuming P(n) doubles initially and remains constant when
it reaches 4,000, develop a linear equation for r (the growth
factor) and P(n).
• Formulate a logistic difference equation for this model; use
it to study how long it takes news to spread.
Constructing a Model
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r initially is very close to 100%
As most people know, r approaches 0.
Logistically P(n-1) = rP(n)
And the story continues.
Solution and Interpretation
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Obtains a math solution but omits details
Relates solution to original
Obtains a complete solution
Relates solution to original problem
Relates subproblem to solution of original
Solution and Interpretation
• Solve the following system of equations:
ax  ( a  d ) y  a  2 d
(a  3d ) x  ( a  4 d ) y  a  5d
Validation of Solution
• Compares solution to the original problem
• Reflects on assumptions and conclusions
based on the original problem
Validation of Solution
• Apply the solution in the system of equations
to see if it works.
1x  2 y  3
4x  5y  6
Validation of Solutions
• Are there other ways to solve the problem?
• Where will you find problems like this one?
Mathematics
• Uses appropriate language
• Calculations are correct
• Precise language and notation
Mathematics
2 x  3  5 x  7  3x  4  4 / 3
Mathematics
• Two angle measures are congruent.
• 1/0 = 0
• sin(x) = 4
x = 4/sin
Communication
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Explanations are clear
Explanations are unambiguous
Explanations are elegant
Solid support for all arguments
Well organized
Communication
• How do you deal with grammar?
• How do you deal with incomplete thoughts?
• How do you deal with students whose
communication skills are not up to par?
[email protected]
• Email me for
– Advice (that you don’t have to use)
– Sources (that you may choose not to obtain)
– Good mathematics thoughts.
References
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Cichon, D., and J. Ellis. “The Effects of Math Connections on Student Achievement, Confidence, and
Perception.” In S.L. Senk and D. R. Thompson (Eds.), Standards-based School Mathematics Curricula:
What Are They? What Do Students Learn? (pp. 345-374). Mahwah, NJ: Lawrence Erlbaum Associates.
Harwell, M. R., T. R. Post, Y. Maeda, J. D. Davis, A. L. Cutler, E. Andersen, and J. A. Kahan. “Standardsbased Mathematics Curricula and Secondary Students’ Performance on Standardized Achievement
Tests,” Journal of Research in Mathematics Education 38 (January 2007): 71-99.
Keck, H. L. The Development of an Analytic Scoring Scale to Assess Mathematical Modeling Projects.
Unpublished dissertation at The University of Montana, Missoula, MT 1996
Schoen, H. L., and J. Pritchett. “Students’ Perceptions and Attitudes in a Standards-based High School
Mathematics Curriculum.” Paper presented at the annual meeting of the American Educational Research
Association, San Diego, CA. (ERIC Document Reproduction Service No. ED420518, April, 1998.
Souhrada, T. Secondary School Mathematics in Transition: A Comparative Study of Mathematics
Curricula and Student Results. Unpublished dissertation at The University of Montana, 2001.
Webb, N. L. “The Impact of the Interactive Mathematics Program on Student Learning.” In S.L. Senk and
D. R. Thompson (Eds.), Standards-based School Mathematics Curricula: What Are They? What Do
Students Learn? (pp. 375-398). Mahwah, NJ: Lawrence Erlbaum Associates.
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Geometry: Between the Devil and the Deep Blue Sea