ArgumentationDay 1

June 23, 2014What is

it???

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 argumentIt 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.

Think1) 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

Data/Evidence

Warrant

Toulmin’s Model of Argumentation

Claim

Data/Evidence

Warrant

THE ARGUMENT

Toulmin’s Model of Argumentation

Claim7 is an odd number

Data/Evidence2 does not divide 7 evenly

WarrantDefinition of odd/even

If it is even, 2 will divide it evenly;

if it is odd, 2 will not divide it evenly

Example5 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

Claim

Micah’s Response

Data/Evidence3 examples that fit

the criterion

WarrantBecause if it works

for 3 of them, it will work for all

Note: What “counts” as a complete or convincing argument varies by grade (age-appropriateness) and by what is “taken-as-shared” 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.

CommentaryArgumentation is important forTeaching By eliciting reasoning, you gain insight into students’

thinking – can better address misconceptions and scaffold their learning

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’)

AssessingPositive 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