Argumentation Day 1

ArgumentationDay 1 June 23, 2014 What is it???

Argumentation Pre-write(~15 mins) When done, please make sure your name is on it and put into the Table Folder.

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?

One way to think about arguments

One way to think about arguments Another way to think about arguments… avoid, avoid, avoid We’ll go for a third, more analytic approach…

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 something is true for reasons that are not necessarily mathematical (e.g., popular consensus; external authority, intuition, 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.

Let’s take a look…

When you add any two consecutive numbers, the answer is always odd. Think • Is this statement (claim) true? • What’s your argument to show that it is or is not true? Pair - Share

Two students, Micah and Angel, explained why the following statement was true. When you add any two consecutive numbers, the answer is always odd. Consider each of their responses: • Is the response mathematically accurate? • Does the response justify the statement? (That is, does the response offer a mathematical argument that demonstrates the claim to be true?)

When you add any two consecutive numbers, the answer is always odd. Micah’s Response Angel’s Response Consecutive numbers go even, odd, even, odd, and so on. So if you take any two consecutive numbers, you will always get one even and one odd number. And we know that when you add any even number with any odd number the answer is always odd. That’s how I know that no matter what two consecutive numbers you add, the answer will always be an odd number. 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.

Students’ views on arguments • What percent of middle school students do you think thought Micah had a valid and acceptable justification? What of Angel’s response? For grades 6 – 8, at each grade, 40% - 50% of students thought each argument was valid. (responded to “Whose response proves the statement is true?”) Slight decrease in thinking Micah had proved the statement true. Slight increase in thinking Angel had proved the statement true.

Student Responses (6th – 8th grade) Whose response proves that the statement is true? Explain your reasoning. Angel’s response, because Micah’s just proves that those 3 number pairs work for the rule. But Angel should have given examples. Angel’s because she provides proof and facts, not just a few different problems by experimenting with different numbers. Micah’s because she proved that it will work with any number, low or high. Both prove that, but they prove it in different ways. Micah’s proves it mathematically and Angel’s proves it by explaining what goes on with the odd and even numbers. Angel’s because God sent her and Micah has several errors.

When you add any two consecutive numbers, the answer is always odd. Roland’s Response The answer is always odd. A number + The next number = An odd number There’s always one left over when you put them together, so it’s odd.

What do you notice about these arguments? How are they similar? How are they different?

Comments on the approaches • Example based (Micah) • Narrative (Angel) • Pictorial (Roland) • Symbolic (algebraic, example shown here) • Consecutive numbers can be represented as n and n+1. • Adding these, we get n+(n+1)=2n+1. • 2n+1 is odd because there’s one left over when you divide by 2.

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 truefor 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.)

Toulmin’sModel of Argumentation Claim Warrant Data/Evidence

Toulmin’sModel of Argumentation Claim Warrant Data/Evidence THE ARGUMENT

Toulmin’sModel of Argumentation Claim 7 is an odd number Warrant Definition of odd/even An even number is a multiple of 2; An odd number is not a multiple of 2. Data/Evidence 2 does not divide 7 evenly

Micah’s Response Example 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 thatno matter what two consecutive numbers you add, the answer will always be an odd number.

Micah’s Response Example 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. Claim

Micah’s Response Example 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 whattwo 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

J: I am a British Citizen B: Prove it J: I was born in Bermuda ?

Toulmin’s Model of Argumentation Claim I am a British citizen Warrant A man born in Bermuda will legally be a British Citizen Data/Evidence I was born in Bermuda

Toulmin’s Model of Argumentation • Claim • Conclusions whose merit must be established. For example, if a person tries to convince a listener that he is a British citizen, the claim would be “I am a British citizen.” (1) • Data/Evidence • The facts we appeal to as a foundation for the claim. For example, the person introduced in (1) can support his claim with the supporting data, “I was born in Bermuda.” (2) • Warrant • The statement authorizing our movement from the data to the claim. In order to move from the data established in 2, “I was born in Bermuda,” to the claim in 1,“I am a British citizen,”the person must supply a warrant to bridge the gap between 1 & 2 with the statement, “A man born in Bermuda will legally be a British Citizen.” (3) Example from Wikipedia, entry on Stephen Toulmin

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.

Applying Toulmin’s: Ex 1 Which is bigger: 73 – 26 or 76 – 26 – 3? • 73 – 26 is the same as 76 – 26 – 3. I add 3 to 73 and then take 3 away at the end. • 73 – 26 is the same as 76 – 26 – 3. If I add 3 to 73 and then take 3 away at the end, I’ve added nothing overall, so the answer is the same. • 73 – 26 is the same as 76 – 26 – 3 because 73 – 26 is 47 and 76 – 26 – 3 is also 47. What’s the claim? What’sthe data/evidence and warrant? (Are they both provided?)

Applying Toulmin’s: Ex 2 Which is bigger? 4 +(x+3)2 or π • Neither because x is a variable • Pi, because you can’t figure out what 4+(x+3)2 is • 4+(x+3)2 because 4 is bigger than pi and (x+3)2 is always positive, so you’re adding a positive value to 4. What’s the claim? What’sthe data/evidence and warrant? (Are they both provided?)

It seems that Calvin’s teachers haven’t helped him see math as something that involves reasoning…

Commentary Argumentation is important for • Teaching 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’) • 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

Argumentation Day 1