Lesson 6 Proof Positive: How Do We Know What We Know in Mathematics? (Building Mathematical Arguments)

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Oct 10, 2013 (3 years and 9 months ago)

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EPPL 612

Unit: Epistemology


How Do We Know What We Know?

Bitto & Goff



1

Lesson
6



Proof Positive:

How Do We Know What We Know in Mathematics?

(
Building Mathematical Arguments
)

PURPOSE & OVERVIEW:

This lesson plumbs the epistemology of mathematics


especially mathematical inductive
and deductive logic


with explicit comparis
ons and contrasts to the natural science
epistemology that we have been studying till now. For example, the previous lesson
(“Salty Seas”) stressed the uncertainty that inhabits all
empirical
data collection, which
means that scientific truth claims are p
robabilistic in nature. This is very different from
the formal, lockstep logic that anchors mathematical claims and conclusions. (Note:
These students will already have studied algebraic and geometric proofs in previous
math courses. The main goal here
is not to master proofs, but rather to grasp
mathematics as a different domain with its own “Rules of Inference.”)

ESSENTIAL QUESTIONS:



How do we know what we know in math, and how is this different from science?



What is a mathematical hypothesis?



What is
a mathematical conclusion?



How do we reach mathematical conclusions?



How do we build a mathematical argument?



How is this
similar
to
and/or
different
from

scientific

hypotheses and conclusions?


KEY CONCEPTS:



Inductive reasoning, deductive reasoning, and p
roof are critical in establishing
general claims.



Deductive reasoning is the method that uses logic to draw conclusions based on
definitions, postulates, and theorems.



Inductive reasoning is the method of drawing conclusions from a limited set of
observati
ons.



Proof is a justification that is logically valid and based on initial assumptions,
definitions, postulates, and theorems.



Logical arguments consist of a set of premises or hypotheses and a conclusion
.

(
VA
Mathematics Standards of Learning Curriculum

F
ramework 2009: Geometry, p.1)



OBJECTIVES:

After this lesson, students will be able to
:



Recognize that some disciplines


such as mathematics and formal logic


derive
claims and conclusions through a combination of inductive and deductive
reasoning, and

ultimately ground them in given definitions, premises, axioms,
and/or theorems (rather than empirical evidence).



A
pply such logical reasoning in such tasks a
s algebraic or geometric proofs



Prove or disprove created conjectures and conditional statements



Distinguish between premises and conclusions, between empirical evidence and
logical judgments, an
d between deductive and inductive
reasoning.

EPPL 612

Unit: Epistemology


How Do We Know What We Know?

Bitto & Goff



2



Identify and apply the elements and principles of algebraic and/or geometric
proofs: givens, claims/statements,
definitions, theorems, properties, postulates,
etc.



Compare/contrast mathematical ways of knowing to other domains, specifically
science


Concept Goals

Process Goals

Content Goals

Goal 1

Goal 2

Goal 1

Goal 2

Goal 1

Goal
2


x




x


V
ocabulary
:



Inductive

reasoning



Deductive reasoning



Conditional statements


Materials:



Article:
Think You’re A Logical Decision
-
Maker? Think Again
,

retrieved from
http://www.bn
et.com/blog/ceo/think
-
youre
-
a
-
logical
-
decision
-
maker
-
think
-
again/5044



Inductive/deductive proof sort



PROCEDURES:

TEACHER NOTES:

Introduction
/Hook
:

Students will read the article,
Think You’re A Logical
Decision
-
Maker? Think Again
,

retrieved from
http://www.bnet.com/blog/ceo/think
-
youre
-
a
-
logical
-
decision
-
maker
-
think
-
again/5044
.
Students will discuss

deductive and inductive reasoning and its applications

as
presented in the article.


Learning Activities:

1.

Discussion: Based on the article read in our
introduction, inductive reasoning is presented as a
lesser form of reasoning. Is this true in all
domains? Why or why not?


2.

Opening Question:
How are

indu
ctive reasoning
and deductive reasoning used in mathematical
logic and arguments?

Class will discuss

based on
prior knowledge
, but will

also

revisit this question
at the end of the lesson.




This article

presents the
business side of in
ductive
and deductive reaso
ning and
the flaws of applying
inductive reasoning.




Both inductive and deductive
reason
ing are critical to
proof and logical
mathematical arguments.


In mathematics,
we u
se inductive
reasoning to make conjectures.

Deductive reasoning assumes the
hypothesis (the “if” part) and uses
definitions and theorems to prove
the conclusion (the “then”
part)



Induction is usually described as
EPPL 612

Unit: Epistemology


How Do We Know What We Know?

Bitto & Goff



3


3.

Identify examples of deductive and inductive
reasoning in mat
hematics through a sort
ing

of
examples.

The teacher will give the students
inductive or deductive questions, or even full
proofs. The students are not to solve the proof,
but sort the examples into the two categories to
develop generalizations through si
milarities and
difference of the two types of logic.

Discuss: Are
they both valid forms of proof and logic?

Why or
why not? When do we use which in mathematics?





4.

Students will
work in pairs or small groups and
generate mathematical conjectures

and
c
onditional statements. Then groups/pairs will
swap such conjectures and conditional statements
to prove or disprove one another’s.



5.

Revisit the question: How is inductive reasoning
and deductive reasoning used in mathematical
logic and arguments?

Compar
e and contrast
responses from the beginning of class to now.



Grouping:

For majority of this lesson, groups should be formed to
facilitate the most productive discussion. Whole group is
recommended if all students have an opportunity to
participate. O
therwise,
you may want to form
discussio
n
groups within the whole group
. For many of the
questions posed, think, pair, share is encouraged to
provide ample wait and think time for students to collect
and successfully verbalize their thoughts. For proced
ure
4, pairs or small groups are encouraged.


Debriefing
:

1.

How are hypotheses and conclusions similar and
different in mathematics
vs.

scientific inquiry?

2.

Should we use inductive and/or ded
uctive reasoning
in science? In

other domains? Why or why not?

3.

How would you respond to the last paragraph of the
article we read in our introduction?


moving from the specific to the
general, while deduction begins
with the general and ends with the
specific; arguments based on
experience or observation are best
expressed inductively, while
arguments based on law
s, rules, or
other widely accepted principles
are best expressed deductively.


(Possible examples:

Inductive reasoning: Based on
examples, t
he sum of two odd
numbers is always even
. Prove
this statement is true for any case.


Deductive reasoning:
If n i
s an odd
integer, then 3n is an odd integer.)




























Try to invoke (or get student to
invoke) language that explicitly
connects to our previous lessons


like “Rules of Inference”


and to
draw explicit comparisons to the
Salty Seas lab
, the opening Handful
of Hooks, and other lessons that
stress the uncertainty that inhabits
all empirical data and the
probabilistic nature of scientific
claims.

EPPL 612

Unit: Epistemology


How Do We Know What We Know?

Bitto & Goff



4



Homework:

Find an application (in any domain) of inductive
reasoning AND deductive reasoning
(2 separate
examples). How are these examples of reasoning alike
and/or different from using inductive and deductive
reasoning in mathematics?



Differentiation for:


ELL
:
A tru
th table for conditional statements can be
provided to aid i
n decoding the language into symbols for
applying the deductive reasoning.



Twice
-
Exceptional
: Students can have the option of
completing a graphic organizer to organize and
synthesize differences between inductive and deductive
reason
ing.



Highly Gifted
:

Students are creating their own
conjectures and conditional statements and have the
flexibility to ma
ke them as challenging as they’d like.



Differentiation:


Choice:



Resource
s:

Students can access a variety of resources
while developing their own conjectures and/or
conditional statements


Products:

Students create and solve own conjectures
and/or conditional statements.


Tiered questions/a
ssignments: