1. Argumentation Tuesday, October 7, 2014 “Argument is the soul of an education” “Argument, in short, is the essence of thought.” Schmoker & Graff (2011)

2. Our focus today Developing 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

3. Guiding Questions What 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?

4. 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

5. A Mathematical Argument It 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.)

6. A journey into ELA

7. 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

8. SLIP OR TRIP

9. 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.

10. Evidence (concrete, observable information; testimony; objects and their condition or appearance) Rule or WarrantClaim (slipped or tripped?) Glass in left hand1) 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

11. Argumentation in ELA backing warrant evidence claim rebuttal qualification Toulmin’s Argumentation Schemata

12. Features/ComponentsELAMATH TypesArguments of Fact, Judgment & Policy ComponentsClaim, evidence, warrant, qualifications, rebuttals EvidenceConcrete, observable information; Testimonies & Quotes; Textual passages; Quantitative data Warrants & BackingRules, laws, agreed upon common sense, scientific findings, and (particularly in arguments of judgment) extended definitions PurposesConvince audience of the rightness of claims using logical reasoning and relevant evidence

13. Features/ComponentsELAMATH TypesArguments of Fact, Judgment & Policy ComponentsClaim, evidence, warrant, qualifications, rebuttals Claim, evidence, warrant (qualifications, rebuttals = refutations) EvidenceConcrete, observable information; Testimonies & Quotes; Textual passages; Quantitative data Graph, table, diagrams (representations); Warrants & BackingRules, laws, agreed upon common sense, scientific findings, and (particularly in arguments of judgment) extended definitions Definition; properties; PurposesConvince audience of the rightness of claims using logical reasoning and relevant evidence Convince audience of the truth of claims using logical reasoning and relevant evidence

14. Features/ComponentsELAMATH TypesArguments of Fact, Judgment & Policy Deductive; proof by exhaustion; induction; proof by contradiction ComponentsClaim, evidence, warrant, qualifications, rebuttals, Claim, evidence, warrant EvidenceConcrete, observable information; Testimonies & Quotes; Textual passages; Quantitative data Representations (graphs, number lines, etc.); calculations; generic examples; details of the objects under inquiry; givens of the problem Warrants & BackingRules, laws, agreed upon common sense, scientific findings, and (particularly in arguments of judgment) extended definitions theorems, properties, definitions, previously established ideas; using models (e.g. area model) PurposesConvince audience of the rightness of claims using logical reasoning and relevant evidence Convince audience of the truth of claims using logical reasoning and relevant evidence

15. We now return you to your regularly scheduled math programming. Problem Solving Time!

17. 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

20. 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?

21. Let’s look at some student work STUDENT A Warrant: (one interpretation) I think it works for numbers 1-10 because if you try it for each number you get the same answer Evidence: none given –alludes to fact that someone may have tested each case Claim: Yes

22. Let’s look at some student work STUDENT A Warrant: Not all types of numbers work Evidence: none Claim: No Conceptual understanding?

23. Let’s look at some student work 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?

24. 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

25. 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

26. One way to frame it Claim: the claim is stated clearly Argument (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

28. Stars & Stairs! Select 2 student work samples Identify 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.

29. Goal – make progress on these questions How 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?

30. 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?

32. WORKING LUNCH! Reminders: * Share your summer preferences If you’re having trouble with the wikispace, find Steve, Maddie or Sharon PLEASE 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?