Reasoning Activities Using 0.999... = 1 in

Developmental Mathematics

Chris L. Yuen, CLYUEN@buffalo.eduSUNY University at Buffalo

November 13, 2014

Cross ExaminationO Inherent the Wind (1960)

https://www.youtube.com/watch?v=l5Kdc0LLSW8

O Witness for the Prosecution (1957) https://www.youtube.com/watch?v=Fq3UK04pNrY

O Legally Blonde (2001) Start the clip at 3:14https://www.youtube.com/watch?v=ytWGiOuzpe4

.999… as an Object of Interest

O Why use .999…? Yopp et al. (2011) studied in-service teachers’ perception toward .999…

Another Finding…

More findings from Yopp

O Some teachers believe that .O Some teachers believe “small”

quantities are irrelevant.O Teachers’ (mis)understanding of real

numbers can be tied to their teaching as well as perceptions of their students learning.

And More…O Some teachers modify “truth” to fit

their subjective beliefs.O Some teachers dismiss conflict as

unimportant or uninteresting as it relates to their teaching.

O Some teachers teach that approximations are good enough and “small” does not matter.

Developmental MathO Much instruction directed to

development math is pedagogically delivered differently from higher math.

O Emphasis of proceduresO Emphasis of formulasO Emphasis of performing well on

traditional tests and exams

Proving .999… = 1

Proofs requiring addition:

A Multiplication Proof

An algebraic proof

Let

Proof by LimitDefine Find the limit:

How to Engage Students?

O The proofs alone are not going to motivate students.

O Developmental students often do not know much about the concepts of “limits.”

O One can show an application of proof in legal arguments – Reductio ad absurdum

O So, involving a trial for the students to craft arguments may be a productive alternative.

What is Reductio ad absurdum?

O Big Bang Theory https://www.youtube.com/watch?v=ytWGiOuzpe4

Reductio ad absurdumO Reduction to absurdityO To demonstrate a statement is true

by showing that a false, untenable, or absurd result follows from its denial

O This sounds very much like “Proof by Contradiction

The Trial

Is ?

Mathematics Zealot vs. Point Nine Repeat

Analysis of the ActivityO The trial encompasses all the proofs

that a developmental mathematics student could comprehend.

O The trial co-create mathematical discourse with students in a social manner, as opposed to proving a mathematical fact in a solitary setting.

O Mathematics can be system/postulate specific, and the court room setting is analogous to a specific system.

ApplicationO This activity allows students to see how

mathematical argumentation and logic be applied in an unexpected setting.

O Other form of argumentation can be used in this activity, such as “Proof by Contradiction” (e.g. is irrational).

O The overall effect of this activity is to show that argumentation is indeed applied, and in this case, there is a link between math and the legal field.

Questions/Comments

Contact Information:

Chris L. Yuen, Ed.D., CLYUEN@buffalo.eduEOC Assistant Professor of MathematicsUniversity at Buffalo Educational Opportunity Center555 Ellicott StreetBuffalo, NY 14203