Argumentation Day 4

Math Bridging PracticesMonday August 18, 2014

WHAT IS AN ARGUMENT? WHAT IS A MATHEMATICAL ARGUMENT??

We have been wrestling with…

Monte Python – The Flying Circus

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.)

Take-away 1! What is an argument?

Argumentation

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

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 big or small.

Take-away 1! What is an argument?

THINK!

What is 16 x 25?

Take-away 2! There’s a difference between an explanation and an argument.

Explanation of Steps Providing an Argument

I took 16 and split it into 10 and 6. I multiplied 10 by 25, and I multiplied 6 by 25. And then I added those 2 numbers together.

I took 16 and split it into 10 and 6 because I need to find 16 groups of 25, and so I can find 10 groups of 25 and then add to it 6 more groups of 25. So I multiplied 10 by 25 and 6 by 25. I added them together to give me 16 25s total, which is what I need.

Explanation of Steps vs Providing an Argument

Student Work16 = 10 + 610 x 25 = 2506 x 25 = 150250 + 150 = 400

Take-away 2! There’s a difference between an explanation and an argument.

Explanation of Steps Providing an Argument

I took half of 16 to get 8 and doubled the 25 to get 50. I did it again- so half of 8 was 4 and double 50 was 100. 4 times 100 is 400.

I took half of 16 to get 8 and doubled the 25 to get 50. 16 x 25 and 8 x 50 are the same, because if you take ½ of one number (8), and double the other number (25), the product is the same.

Explanation of Steps vs Providing an Argument

Student Work 16 x 25= 8 x 50= 4 x 100= 400

16

2525

8 8

25

I did it again- so half of 8 was 4 and double 50 was 100. 4 times 100 is 400.

8

8

Take-away 2! There’s a difference between an explanation and an argument.

Toulmin’s Model of Argumentation

Claim

Data/Evidence

Warrant

Take-away 3! Argumentation involves claims, warrants and evidence.

Toulmin’s Model of Argumentation

Claim

Data/Evidence

Warrant

THE ARGUMENT

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

Data/Evidence3 examples that fit

the criterion

WarrantBecause if it works

for 3 of them, it will work for all

Take-Away 4! Not all arguments are valid (viable).

Is it a viable argument?

sdf

Reasoning and Proof, NCTM (2000), p 189

I think they have different areas

because the triangle looks a lot bigger.

I think they’re the same because they are both half of the bigger rectangles.

They’re both half, so they have to be the

same.

Take-away 5! (in progress)

What “counts” as an acceptable (complete) argument varies by grade (age-appropriate) and by what is taken-as-shared in a class (established already as true).

Regardless of this variation, it should be mathematically sound.

Example

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.

Claim

Angel’s Response

Data/Evidence2 consec #s are always one odd and one even

WarrantBecause we’ve shown

before odd+even is odd

It is an argument. It is viable.Depending on class, may or may not be complete.

STUDENT WORK – HEXAGON TASK

Purposes To help us get clearer about strengths and

weaknesses of different arguments To help us each develop a vision for what we

value and will count as a strong argument in our classrooms

THE HEXAGON TASK

Each figure in the pattern below is made of hexagons that measure 1 centimeter on each side.

a) Draw Figure 5. Find the perimeter of Figure 5.b) If the pattern of adding one hexagon to each figure is

continued, what will be the perimeter of the 25th figure in the pattern? Justify your answer.

Analyzing Student Work

PTT -Just think. (~3 mins)

Group time! (~15 mins) What is the student’s claim? In your own words,

summarize the student’s argument. Take turns summarizing.

(Column 2) Has this student constructed a viable argument to show the perimeter of the 25th figure? [does it prove the claim?]

(Column 3) Commentary – Explain why or why not. Perhaps address completeness -- what else would you want to see?

Is the response an argument that shows the perimeter of the 25th figure?

The result is justified! Valid, complete argument!

Result is not

justified

[not a valid

argument, or not

complete –too

many gaps]

Place a sticky note on the poster according to the following. Put your Table # on the sticky note.

We’re not sure

Be prepared to share your thinking

Argumentation – solidifying

What will count in your classroom for a valid argument? (What qualities or criteria are important to you?)

How are these criteria communicated to students?

What do you expect at the beginning of the year? Where will growth be?