Just-in-Time-Teaching and other gadgets Richard Cramer-Benjamin - - PowerPoint PPT Presentation

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,

2 weeks Transformational Geometry 1 week Circles 3 weeks Euclidean Geometry 3 weeks Neutral Geometry 3 weeks Axioms of Plane Geometry 2 weeks Preliminaries/Discrete Geometry

Other Sources

Requiring Student Questions on the Text

Bonnie Gold

How I (Finally) Got My Calculus I Students

to Read the Text

Tommy 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

• bservations 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

• rder 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

• ther 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