Argumentation
Day 1
June 23, 2014
Standards of Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Overarching Guiding Questions:
What is a mathematical argument? What
“counts” as an argument?
 What is the purpose(s) of a mathematical
argument in mathematics? In the classroom?
 What does student argumentation look like
at different grade levels/levels of proficiency?
What are appropriate learning goals for
students with respect to constructing viable
arguments?
 What will Smarter Balanced “count” as a
quality response to prompts that target
Claim 3?

A mathematical argument

It is…
◦ A sequence of statements and reasons given
with the aim of demonstrating that a claim is
true or false

It is not…
◦ An explanation of what you did (steps)
◦ A recounting of your problem solving process
◦ Explaining why you personally think it’s true
for reasons that are not necessarily
mathematical (e.g., popular consensus;
external authority, etc. It’s true because my John
said it, and he’s always always right.)
Argumentation
Mathematical argumentation involves a host
of different activities: generating
conjectures, testing examples, representing
ideas, changing representation, trying to find
a counterexample, looking for patterns, etc.
When you add any two consecutive
numbers, the answer is always odd.
Think
1) Is this statement (claim) true?
2) What’s your argument to show that it is or is
not true?
Pair - Share
Toulmin’s Model of Argumentation
Claim
Warrant
Data/Evidence
Toulmin’s Model of Argumentation
Claim
Warrant
Data/Evidence
THE
ARGUMENT
Toulmin’s Model of Argumentation
Claim
7 is an odd number
Data/Evidence
2 does not divide 7 evenly
Warrant
Definition of odd/even
If it is even,
2 will divide it evenly;
if it is odd,
2 will not divide it evenly
Example
Micah’s Response
5 and 6 are consecutive
numbers, and 5 + 6 = 11 and
11 is an odd number.
12 and 13 are consecutive
numbers, and 12 + 13 = 25
and 25 is an odd number.
1240 and 1241 are
consecutive numbers, and
1240 +1241 = 2481 and
2481 is an odd number.
That’s how I know that no
matter what two
consecutive numbers you
add, the answer will
always be an odd number
Data/Evidence
3 examples that fit
the criterion
Warrant
Because if it works
for 3 of them,
it will work for all
Claim
Note: What “counts” as a complete or
convincing argument varies by grade (ageappropriateness) and by what is “taken-asshared” in the class (what is understood
without stating it and what needs to be
explicitly stated). Regardless of this
variation, it should be mathematically
sound.
Commentary
Argumentation is important for

Teaching By eliciting reasoning, you gain insight into students’

Learning
thinking – can better address misconceptions and scaffold their
learning
◦ By reasoning, students learn and develop knowledge (conceptual,
linked knowledge, not memorized facts)
◦ Equity issue – provide students access
◦ In the end, it’s more efficient (retention; it’s not ‘you know it or you
don’t’)


Assessing
Positive classroom culture
◦ Reasoning is empowering; merely restating or memorizing
information is disempowering and not engaging; reasoning is
mathematics
◦ Many students can reason very well, even when they have weaker
computational skills