Just-in-Time-Teaching
and other gadgets
Richard Cramer-Benjamin
Niagara University
http://faculty.niagara.edu/richcb
The Class
MAT 443 – Euclidean Geometry
26 Students
12
Secondary Ed (9-12 or 5-12 Certification)
14 Elementary Ed (1-6, B-6, or 1-9
Certification)
The Class
Venema, G., Foundations of Geometry,
Preliminaries/Discrete Geometry
2 weeks
Axioms of Plane Geometry
3 weeks
Neutral Geometry
3 weeks
Euclidean Geometry
3 weeks
Circles
1 week
Transformational Geometry
2 weeks
Other Sources
Requiring Student Questions on the Text
Bonnie
How I (Finally) Got My Calculus I Students
to Read the Text
Tommy
Gold
Ratliff
MAA Inovative Teaching Exchange
JiTT
Just-in-Time-Teaching
Warm-Ups
Physlets
Puzzles
On-line
Homework
Interactive Lessons
JiTTDLWiki
JiTTDLWiki
Goals
Teach Students to read a textbook
Math
classes have taught students not to read
the text.
Get students thinking about the material
Identify potential difficulties
Spend less time lecturing
Example Questions
For February 1
Subject line WarmUp 3 LastName
Due 8:00 pm, Tuesday, January 31.
Read Sections 5.1-5.4
Be sure to understand The different axiomatic systems (Hilbert's,
Birkhoff's, SMSG, and UCSMP), undefined terms, Existence Postulate,
plane, Incidence Postulate, lie on, parallel, the ruler postulate, between,
segment, ray, length, congruent, Theorem 5.4.6*, Corrollary 5.4.7*,
Euclidean Metric, Taxicab Metric, Coordinate functions on Euclidean and
taxicab metrics, the rational plane.
Questions
Compare Hilbert's axioms with the UCSMP axioms in the appendix. What are
some observations you can make?
What is a coordinate function? What does it have to do with the ruler
placement postulate?
What does the rational plane model demonstrate?
List 3 statements about the reading. A statement can be something significant
you learned or a question you have. At least two of the statements should be
questions.
One point geometry contains one point and no
lines. Which Incidence Axioms does one point
geometry satisfy? Justify your answer.
I don't think it satisfies any of the incidence axioms
because there aren't any lines; all of the incidence
axioms involved lines and multiple points.
One point geometry does not satisfy any of the incidence
axioms. Incidence axiom 1 says "for every pair of
distinct points", but one-point geometry only has one
point, not a pair of points. Incidence axiom 2 says "for
every line l", but one point geometry does not have any
lines. Incidence axiom 3 says "there exist three points",
but in one point geometry there do not exist three
points. Therefore, it doesn't satisfy any of the incidence
axioms.
One point geometry satisfies Incidence Axiom 1 (For every pair of
distinct points P and Q there exists exactly one line L such that
both P and Q lie on L) and Incidence Axiom 2 (For every line L
there exist at least two distinct points P and Q such that both P and
Q lie on L). This can be illustrated by using If /Then
statements. For example in Incidence Axiom 1 the If /Then
statement would say: If for every pair of distinct points P and Q
Then there exists exactly one line L such that both P and Q lie on
L. Since both parts of this statement are false for one point
geometry, a false/false statement results in a true statement which
means that one point geometry does in fact satisfy Incidence
Axiom 1. For Incidence Axiom 2, the If /Then statement would
say: If for every line L there exist at least two distinct points P and
Q then both P and Q lie on L. Again, both parts of this statement
are false for one point geometry and again a false/false statement
results in a true statement which means that one point geometry
does satisfy Incidence Axiom 2. However, when it comes to
Incidence Axiom 3 (There exist three points that do not all lie on
any one line) it is not possible in one point geometry because in
this cause there are 3 points.
Compare Hilbert's axioms with the UCSMP
axioms in the appendix. What are some
observations you can make?
There are more undefined terms in Hilbert's axioms than in the UCSMP
axioms.
Distance and angle measure are an integral part of the UCSMP axioms and
are not utilized in Hilbert's axioms. Additionally, UCSMP's axioms deal with
area and volume while Hilbert's do not.
Both axioms are organized by categories. Hilbert's axioms fall under five
categories of axioms: Incidence, Order, Congruence, Parallels, and
Continuity. UCSMP's axioms fall under eight categories of
postulates: Point-Line-Plane, Distance, Triangle Inequality, Angle Measure,
Corresponding Angle, Reflection, Area, and Volume. (Notice Hilbert's
axioms are called "axioms" while UCSMP's are called "postulates.")
Hilbert's axioms seem to use spatial relationships of angles, lines, and
points. UCSMP's seem to branch off into many perspectives other than
spatial.
What does the rational plane model
demonstrate?
The rational plane model demonstrates that every rational number is
also a real number.
The rational plane model demonstrates the existence of rational
numbers through ordered pairs. Not only does the rational plane
demonstrate rational numbers but it also demonstrates real numbers
since all rational numbers are part of the set of real numbers.
Although, the rational plane model demonstrates that the rational
plane satisfies all five of Euclid's postulates, it also shows how the
proof of Euclid's very first proposition breaks down in the rational
plane. This means that the Euclid was using unstated hypotheses in
his proofs, which also means that there is a gap in Euclid's proofs.
Questions
Every Warm Up assignment will have one
area for “Give three questions or important
things you learned from the reading.”
Questions must be stated as questions.
Stating “I didn’t understand…” is not a
question. Also, your question should not
be something that can easily be looked up
in the index.
What indications or hints were Euclid given in
order to know the importance of his postulates
and the remarkable role they would play in
geometry? How did he pick and choose the
axioms as well?
I was under the impression that a model satisfies
a postulate or a proof. The book states that
there are two models used to show Euclid?s fifth
postulate yet one of them satisfies it and the
other model does not sitisfy it. How is then a
model for Euclid?s fifth postulate?
In the section involving indirect proof it seems as
if they tell you the exact manner in which to do a
RAA and then show cases in which it is
considered sloppy when it seems as if they are
following the steps listed above in the
description of the method of an indirect proof,
why is this considered sloppy? Are certain
elements of geometry more worthy or do they
lend themselves better to a RAA? How do we
know when to use an RAA, when is it most
effective?
When we are writing proofs in class, should we
try and use all of the six kinds of reason? Would
they all apply at once or would that rarely be the
case?
Why is it not enough to provide experimental
evidence that a theorm is true?
When writing a proof, how would one know to
write a proof using the logic formula, induction,
proof by contradiction, etc.? what are some hints
to tell when to use each proof process?
‘Loud Thinking
H. A. Peelle, “Alternative Modes for
Teaching Mathematical Problem Solving:
An Overview”, The Journal of Mathematics
and Science: Collaborative Explorations 4
(Spring 2001) 119-142
‘Loud Thinking
Students work in pairs.
Solver works on the problem
Should
verbalize thoughts while trying to
solve problem.
Recorder records solving process
Can
give hints and read back previous
attempts.
Should not be paired problem solving
‘Loud Thinking
Recorders found it difficult not to take part in the
problem solving
Students liked that it forced them to be aware of
the problem solving process.
Many students are afraid to ‘write down or say
something incorrect.’ This is counter to what we
think of as good problem solving.
Weak students got to observe strong problem
solvers.
Collaborative Oral Take Home
Exam
MAA Notes #49 - Annalisa Crannell
Students work in groups of 3 or 4
4-5 challenging problems
1 week to solve the problems
Each group schedules a half-hour exam time
Each member selects a problem at random and
solves the problem at the board with no notes.
Group is graded on written solutions, individuals
are graded on oral solutions
Collaborative Oral Take Home
Exam
Originally made the grade 100% oral.
Too
much pressure for students
Most groups did very well.
Some difficulty with student schedules.
Nice way to make collaborative learning
part of a higher stakes assessment.
Reading Out Loud
Students pair up
Take turns reading a sentence from the
text.
The listener then asks a question about
the passage just read.
Students read very quickly and have
trouble asking probing questions.
Circle Time
Groups of three
Each person in the group gets a unique
problem to work outside of class.
Students then exchange solutions and
evaluate groups members solutions
Circle Time
Students have very little ability to evaluate
the correctness of a proof.
Students can not follow the logical
arguments of another student
Students can comment on notation
Web pages
My Home Page
http://facutly.niagara.edu/richcb
JiTT
http://134.68.135.1/JiTTDLwiki/index.php/Mai
n_Page
MAA Inovative Teaching Exchange
http://www.maa.org/t_and_l/exchange/exchan
ge.html
Requirements
Secondary
Calc I, II, and III
Foundations
Linear Algebra
Abstract Algebra
History of Math
Prob and Stat I
Math Modeling
Euclidean Geometry
Senior Seminar
Elementary
Intro to Statistics
Elem. Ed Math I and II
Calc I and II
Foundations
Linear Algebra
History of Math
Math Modeling
Euclidean Geometry
Senior Seminar