Carnegie Learner Study Clinical Interview

8/14/2019 Carnegie Learner Study Clinical Interview

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Carnegie Learner StudyClinical Interview

Notes: As much as possible, have the interviewee talk through his/her

thinking. If interviewee points to items written on the page, ask him/her to

underline them.

Turn pen on and

TAP RECORDBACKGROUNDBACKGROUND

Well start with some general background questions, before moving on toquestions particular to mathematics. If you feel the first questions are toopersonal, you can simply decline to answer and well move on.

How old are you? Whats your ethnicity? Whats the primary language spoken in your home?

Are your studies at ________ related to a particular career goal?

What role do you think math will play in your future job? If any: Is that kind of math the same kind you learn in math class?

Do you look forward to seeing math on your course schedule? Why?

Do you think of yourself as being good at math? If yes: What makes you good at it?

If no: Was there a time that you did feel good at math?

If someone is good at math, what exactly are they good at? For example,some people say math is about remembering rules and procedures. Otherpeople say its about understanding and reasoning. What do you think?

Do you think math is interesting? If no: If someone likes math, do you think that person thinks its

interesting?

How long has it been since your last math class? What class was it?

What math class are you enrolled in currently?

How were you placed in that class? For instance, did you take a test?

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MATHMATH

Question 1Question 1 (mental math)(mental math)

What number would I need to add to 64 to get 100?

If uses subtraction: Is there a way you could figure this out withoutusing subtraction?

If student struggles: 40 is too much to add, so if I add 30 I get to 94,then I need 6 more to get to 100. That means I need to add 30 and 6more or 36.

Can you try that with this one? What number would I add to 72 to get100?

What number would I add to 457 to get 1000?

If uses subtraction: Is there a way you could figure this out withoutusing subtraction?

If student struggles: You might think about it this way: 457 plus 500 is

957, plus 40 more is 997, plus 3 more is 1000. So, I need to add fivehundred, forty, and three to 457 to get to 1000. (commas added topause after each chunk of 543)

Can you try that with this one? What number would I add to 835 to get1000?

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Now I am going to ask you to do some mental multiplication. As you work oneach problem mentally, I want you to talk about what you are thinking.

(Write these as shown below, one problem at a time, and write Ss answer inthe blank. Go all the way though the list. If S does not use decomposition,be explicit about how 31 x 13 can be thought of as 30 x 13 plus another 1 x13. Ask if S can use that same idea on 22 x 13, then ask about 29 x 13.)

10 x 3 = ___

10 x 13 = ___

20 x 13 = ___

22 x 13 = ___

30 x 13 = ___

31 x 13 = ___

29 x 13 = ___

How would you do 22 x 13 it if you werent forced to do it mentally?

22X13

66220286

What would you get if you did 13 x 22? Ss should say this will be the sameproduct.

Can you show me how you would do 13 x 22?

13x22

2626 *286

Why did you put a * [or 0 or blank] here?

Which of those two (of 22 x 13 or 13 x 22) matches the way you did it in yourhead?

Lets try one more. How would you do 14 x 22 mentally?

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Question 2Question 2Provide a worked, addition problem.

462+253715

How would you check to see if the answer here is correct?

If reworks problem: Is there another way to check?

If no other way: Is there a way you can use subtraction to check?

Of these two numbers, 572 and 86[written horizontally], circle the one thatslarger. How do you know?

If has more digits: Can you always apply that rule? What about 572and 367?

Write 572 > 86

Of these two numbers, 0.572 and 0.86 [written horizontally], which is larger?How do you know?

If has more digits: Can you always apply that rule? What about 0.9and 0.1111?

If incorrect: correct student and ask if s/he can see why 0.9>0.1111.

Write 0.572 < 0.86

Can you show me how you would set up 572 86[written vertically]?

Can you show me how you would set up 0.86 0.572 [written horizontally]?

If incorrectly lined up: Is the placement of the decimals important? How did you decide

where the decimals go?

Can you show me how you would set up 0.572 0.86 [written horizontally]?

If incorrectly lined up: Is the placement of the decimals important? How did you decide

where the decimals go?

Do you think you would get the same answer is you would with the one above[0.86 0.572]? Why or why not?

Here [in whole number subtraction] you lined up the 8 and the 7 and lined upthe 6 and the 2. Here [in subtraction of decimals] you lined up the 5 and the8 and lined up the 7 and the 6. Is there a reason that theyre different, or areyou taught to do it that way just because it looks neater?

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Question 3Question 3

Which of these two numbers is larger [a/5 or a/8]? How do you know?

If struggle: Could you try substituting a number for a? Would that be away to think about it?Will that work no matter what number you choose for a?

Which of these two numbers is larger [5/a or 8/a]? How do you know?

If struggle: Could you try substituting a number for a? Would that be away to think about it?Will that work no matter what number you choose for a?

You may have been told in school that the fraction bar means division. Howare fractions related to division?

Which of these two numbers is larger [4/5 or 5/8]? How do you know?

If 4 and 5 are closer together than 5 and 8: What about these twonumbers [4/5 or 2/3]?

If 4/5 is closer to 1: Tell me a little more about why that strategyworks.

Can you draw a number line and place 4/5 and 5/8 on it? [talk aloud]

If student is not able to draw a number line, draw just this much:

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Can you now add these numbers to it [-3/4 and 5/4]?

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QuestionQuestion 44

What does the equal sign mean?

Im going to write a few equations that use an equal sign:7 + 5 = x6 + 9 =x+ 4 = 2/42=2

Are all of these equations legal?

Is the meaning of the equal sign the same in all of these cases, or does itserve different purposes?

Lets say that I have the equation we used up here: 7 + 5=x+ 4 Then Isubtract 2 from the left side [write 5 + 7 2 = x + 4]. Does it change whatxis? If so, how? If not, why not?

Whats the solution to this problem [7 + 5 = __ + 4]?

How did you get that?

How about this problem [7 + 5 = x + 4]?

How did you get that?

Is there a way you can use [your answer] to check to see if its correct?

If answered 12 or 16: Some people answer 8. Would that be O.K.?

Is it possible for it to be a number other than 8?

If the methods student used appear different: Are you allowed to use thesame method to solve the second equation as you used to solve the first?

Can you think of another way to solve either of the equations above?

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QuestionQuestion 55

What happens if you take a number and add 1/3 to it?

We could write it like this: a + 1/3 =x. Do you thinkxis bigger than a,smaller than a, equal to a, or that you cant tell.

What happens if you take a number and multiply it by 1/3?

We could write it like this: a 1/3 =x [Make sure that student understandsthat means to multiply.] Well say that a is a positive, whole number. Doyou thinkxis bigger than a, smaller than a, equal to a, or that you cant tell?

If struggle with notation: If I start with a number and multiply it by 1/3,what can you tell me about what I would get?

If still struggling: Lets consider a similar problem. How would you think about

starting with a number and taking half of it? If you have a number and multiply it by , would it ALWAYS be less

than the number with which you started? What if you had 6 ? If student gets it, return to a 1/3 =x . Otherwise, go on to

number line question

Is that true no matter what number you start with?

What if the number you start with (the a) is less than 1?

Where wouldxbe on the number line? [Leave space for Ss to choose aplacement > a]

0 a

If struggles: Can you show me an example of how you would choose avalue for a in a 1/3 =xand then multiply to findx? Lets say a = 6.Does that match where you put x on the number line?

If we know that a 1/3 =x, then what isx1/3?

Take a look at these [write examples below]. Circle the ones that would giveyou half ofn.

n n 1/2 ofn n n n 2

[Select 2 of the items circled] You said that each of these gives you half ofn.Does this one [that you chose] equal that one [that you chose]? How do youknow?

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QuestionQuestion 66What does a + b = c mean in words?

Lets say we know that a + b =c is true. Can you think of other equationsthat would be true for a, b, and c? Any others?

If necessary: Can you think of other equations that have only a, b, andc?

If I know that a + b = c, is this also true about a, b, and c: c - a = b?

Always? Just sometimes? Never?

Lets pick some numbers that work for the first equation. Will they work forthe other equations you came up with?

If stuck: can you think of values just for [a + b = c]? Do they also workfor [c - a = b]?

Will that work no matter what numbers you plug in? Will it be true if thenumbers are negative? What if theyre fractions?

Sometimes in math it helps to make a drawing that represents the meaningbehind the symbols. If you go to from home to Petco, its like the distance a.If you continue to Noahs Bagels, you go an additional distance b. So thedistance from home to Noahs is a + b. We can give that its own name, sayc.

a b

HOME PETCO NOAHSc

Is there any way you can use this drawing to think about whether any ofthese equations work?

1. b + a = c2. c = a + b3. c b = a4. b c = a5. c a +b = 06. c a b = 0 [If select both 5 and 6, probe how this can be]

If #1 and #2 are true, that means b + a = a + b. Notice how b + a and a + b

are both equal to c? Now lets look at #3 and #4. Ifc b = a and b c = a,then it must be true thatc b = b c. Will that always work? Sometimes? Never?

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QuestionQuestion 77Heres an equation [xy= 1]. Can you tell me what this means in words?

Lets sayygets bigger. What would happen tox? Tell me how you thoughtabout answering that.

If incorrect (i.e., NOT bigger): Can you think of numbers to substituteforxand y?

If cant: Lets just say thatxstarts out as 4.

Is it possible for that NOT to be true? Or is it true no matter what value youchoose fory?

Now lets sayxgets bigger. What would happen toy?

If incorrect (i.e., NOT bigger): Can you think of numbers to substituteforxand y?

If cant, Lets just say thatxstarts out as 4.

Is it possible for that NOT to be true? Or is it true no matter what value youchoose forx?

Can you think of some numbers to substitute for x and y that make theequation true? How many pairs of numbers are there that would work?

Heres a new equation [xy= 0]. In this case, ifyincreases, what happenstox?

Can you write the equation in terms ofx? That is,x= ____

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I have just a few closing questions.

My guess is that youve never been interviewed before about math. Is thatright? But is what we did sort of the same process as you go through in amath class? In what ways is it the same and in what ways different?

If you could give math teachers advice about how to teach in a way thatwould better help you understand math, what would you tell them?

These interviews are incredibly interesting to me, but I chose this as my job.I was wondering if the process was at all interesting to you. I wont be at alloffended if the most interesting part is the Amazon gift card.

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Carnegie Learner Study Clinical Interview

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