argumentation tuesday, october 7, 2014 “argument is the soul of an education” “argument, in...

ArgumentationTuesday, October 7, 2014

“Argument is the soul of an education”

“Argument, in short, is the essence of thought.”

Schmoker & Graff (2011)

Our focus todayDeveloping language and criteria

for discussing argumentation and arguments◦In ways that clarify our thinking◦In ways that help us provide

feedback (and feedforward) to students

Guiding QuestionsWhat are characteristics of **written**

(communicated) mathematical arguments?

What counts as quality?What can we do to promote high

quality arguments? To use argumentation to promote conceptual understanding? ◦What are we already doing?◦What can we tweak?◦What else can we do?

One model: A pedagogy of mathematical reasoning

New question(s)

Generate ideas

Elicit and Publicize ideas

Press on and develop ideas collaboratively

Solidify and/or refine new meanings

A Mathematical ArgumentIt is…

◦A sequence of statements and reasons given with the aim of demonstrating that a claim is true or false

◦“an argument is a collective series of statements to establish a definite proposition” (Monte Python)

It is not… ◦ (Solely) 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 Adrianne said it, and she’s always, always right.)

A journey into ELA

Argumentation in ELA

Persuasion• Claim Based on Opinion

• Claim not always substantiated

• Pathos – appeals to emotions, desires, needs

• Ethos – appeals to writer’s/speaker’s trustworthiness or character

• May not consider opposing view

Argumentation• Claim (Position, Hypothesis, Thesis,

Opinion)• Claim substantiated with relevant &

sufficient evidence• Logos – appeals to logical reasoning &

evidence (e.g. facts, examples, historical & legal precedents, extended definitions)

• Ethos – appeals to writer’s/speaker’s credibility – established through knowledge & merits of evidence & reasoning

• Considers opposing view accurately and uses evidence and reason to refute it

SLIP OR TRIP

Slip or Trip?

With your tablemates, review the picture and text. Determine whether you think that Queenie Valentine is telling the truth or lying. Identify the evidence that supports your decision. Be ready to defend how your evidence supports your decision.

From Hillocks, G. (2011). Teaching Argument Writing, Grades 6-12: Supporting Claims with Relevant Evidence and Clear Reasoning. Portsmouth, NH: Heinemann.

Evidence (concrete, observable information;testimony; objects and their condition or appearance)

Rule or Warrant Claim(slipped or tripped?)

Glass in left hand 1) When people walk downstairs, they hold onto a railing, and that should have been his left hand for downstairs. 2) When people fall, they try to stop themselves, so he would have dropped his glass, or dropped it when he hit the ground

Queenie is lying

His body position is legs up and facing up

When people fall coming downstairs, they don’t generally land face up

Queenie is lying

She’s only 5’ 6” and he’s 5’ 10”

Generally, when someone is significantly shorter, it’s hard to kill them

Queenie is not lying

Argumentation in ELA

backing

warrant

evidence claim

rebuttal

qualificationToulmin’s Argumentation Schemata

Features/Components ELA MATHTypes Arguments of Fact, Judgment &

PolicyComponents Claim, evidence, warrant,

qualifications, rebuttals

Evidence Concrete, observable information; Testimonies & Quotes; Textual passages; Quantitative data

Warrants & Backing Rules, laws, agreed upon common sense, scientific findings, and (particularly in arguments of judgment) extended definitions

Purposes Convince audience of the rightness of claims using logical reasoning and relevant evidence


PolicyComponents Claim, evidence, warrant,

qualifications, rebuttals Claim, evidence, warrant(qualifications, rebuttals = refutations)


Graph, table, diagrams (representations);


Definition; properties;


Convince audience of the truth of claims using logical reasoning and relevant evidence


PolicyDeductive; proof by exhaustion; induction; proof by contradiction

Components Claim, evidence, warrant, qualifications, rebuttals,

Claim, evidence, warrant


Representations (graphs, number lines, etc.); calculations; generic examples; details of the objects under inquiry; givens of the problem


theorems, properties, definitions, previously established ideas; using models (e.g. area model)


Convince audience of the truth of claims using logical reasoning and relevant evidence

We now return you to your regularly scheduled math programming.

Problem Solving Time!

A NUMBER TRICK!

Think of a number between 0 and 10 (inclusive)….

Opportunity to use our language of claims, warrants and evidence to analyze student work

Example of a task targeting the develop-ment of a core conceptual understanding

Your task… What is the claim?Identify the argument

◦ What’s the evidence the student offers?◦ What’s the warrant(s) that links the evidence to

the claim?Critique the argument

◦ Is the approach (chain of reasoning) mathematically sound?

◦ Are there logical gaps? Must the reader fill in connections or pieces of evidence?

Conceptual understanding◦ What can you infer about the student’s

(developing) understanding of the distributive property?

Let’s look at some student work

STUDENT A

Warrant: (one interpretation) I think it works for numbers 1-10 because if you try itfor each number you get the same answer

Evidence: none given –alludes to fact that someone may have tested each case

Claim: Yes


STUDENT A

Warrant:Not all types of numbers work

Evidence:none

Claim:No

Conceptual understanding?


STUDENT B

Claim:Yes

Warrant: if you multiply a number before adding it vs just adding a number, the second number has to be more [for the end result to be the same]Evidence: appeals to shared common understanding of a products [of positive integers]Conceptual understanding?

Your turn… What is the claim?Identify the argument

◦ What’s the evidence the student offers?◦ What’s the warrant(s) that links the evidence to

the claim?Critique the argument

◦ Is the approach (chain of reasoning) mathematically sound?

◦ Are there logical gaps? Must the reader fill in connections or pieces of evidence?

Conceptual understanding◦ What can you infer about the student’s

(developing) understanding of the distributive property?

Please do C – H first

I & J “knew” the distributive property

Stars

Stars: comments that highlight aspects of the response that are part

of a competent performance

Stairs: comments that indicate “next steps” that would help

improve the quality of the work

and Stairs

One way to frame itClaim: the claim is stated clearlyArgument (warrants & evidence)• presents a chain of reasoning that links

evidence to build to the claim• uses previously established ideas and

facts, including definitions, and/or ideas that are established as true within the argument

• uses representations (words, symbols, graphs, tables, pictures, etc.) that help convey ideas; the representations used are those the class knows or is able to understand

• addresses all cases covered by the claim Other: Contains no errors in calculations that detract from the argument

Stars & Stairs!

Select 2 student work samplesIdentify what the student is doing

well with respect to argumentation. Write a comment that conveys to the student what s/he is doing well.

Identify an area of improvement for the student with respect to argumentation. Write a learning promoting comment that conveys to the student how s/he might grow.

Goal – make progress on these questionsHow does argumentation help us

promote conceptual understanding of important ideas (e.g., the distributive property)? How does argumentation reveal students’ understanding of important ideas (here, the distributive property)?

How does our work getting better at identifying the claim, evidence and warrant help us give feedback and feedforward to students’ arguments?

Argumentation – solidifying

• What currently counts in your classroom for a valid argument? What qualities or criteria are important to you?

• How are these criteria communicated to students?

• What kinds of proficiencies did you find at the beginning of the year? Where will growth be?

WORKING LUNCH!

Reminders: * Share your summer preferences• If you’re having trouble with the wikispace,

find Steve, Maddie or SharonPLEASE RETURN AT 12:20

• What currently counts in your classroom for a valid argument? What qualities or criteria are important to you?

• How are these criteria communicated to students? • What kinds of proficiencies did you find at the

beginning of the year? Where will growth be?

argumentation tuesday, october 7, 2014 “argument is the soul of an education” “argument, in...

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