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Argumentation Day 3. Math Bridging Practices June 25, 2014. A Mathematical A rgument. It is… A sequence of statements and reasons given with the aim of demonstrating that a claim is true or false It is not… ( Solely ) an e xplanation of what you did (steps )
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Argumentation Day 3 Math Bridging Practices June 25, 2014
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… • (Solely) 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.)
Point of Clarification What’s 7 + 11? (a) 7 + 11 is 18 because 7 + 1 is 8 and 8 plus10 is 18. This is more retelling steps. (b) 7 + 11 is 18 because 11 is 10 plus 1. I added the 1 onto the 7, to get 8, and then I did 10 plus 8 instead. This is a mathematical argument. It is given to support the claim that 7 + 11 is 18.
Argumentation Student work : When talking about calculations such as: “I multiplied the cost of one package by 7 because that’s how many packages are needed for 14 days.” • Students offer a mathematical reason for why their method is correct • Students offer a logical argument to show how they know that their result is correct
Point of Clarification Having students generate arguments can happen every day in your class! I would argue it should ha ha ha What can you make an argument for? Any well formulated claim about something in math that could be determined true or false – no matter how bit or small.
Point of Clarification “Arguments in math” – need a claim, need evidence, need to know how the evidence shows the claim true (or false). “Arguments in the courtroom” – need a claim (guilty or not?), need evidence, need to know how the evidence shows the claim true (or false) “Arguments among friends” “Debates”
Language to help us think about and talk about mathematical arguments
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 NOTE: this has the structure of an argument, but thisdoes not show the claim to be true. (not a viable argument) Example Data/Evidence 3 examples that fit the criterion 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. 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
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.
Applying Toulmin’s: Ex 1 Data/evidence included; Missing warrant Which is bigger: 73 – 26 or 76 – 26 – 3? • 73 – 26 is the same as 76 – 26 – 3. I can 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. Warrant included too! Warrant – I did the math. Not “explanatory”
Applying Toulmin’s: Ex 2 Which is bigger? 4 +(x+3)2 or π • 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 • 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.
