Listening for and Listening to NCETM Assessment Project Session 3 Nov 2010

Marbles 1 If Barbara gives one of her marbles to John, they will have the same number of marbles. What can you say about the number of marbles they started with?

Marbles 2 If Barbara gives one of her marbles to John, they will have the same number of marbles; if John now gives two of his marbles to Tandi, they will have the same number. What can you say about the relation between Barbara’s and Tandi’s marbles to start with.

Marbles 3 If Barbara gives John one of her marbles, she will then have one more than twice as many marbles as John then has. If John started with 12 marbles, how many did Barbara start with?

Marbles 5 If Barbara gives John one of her marbles, she will then have one more than twice as many marbles as John then has. However, if instead, John gives Barbara one of his marbles, he will have one more than a third as many marbles as Barbara then has. How many marbles have they each currently?

What is being attended to? Semantic Babble 5 yr old: Asked a sequence of questions like “what is 3 + 5; what is 5 + 3” He announces: “anything plus anything is anything plus anything” Arithmetic is about relationships, not about calculations! What is being attended to? What is being ignored?

Syntactic Babbling “she done it”; “the book was took by …” 3 + 4 = 7 x 2 = 14

Speculating on Choices 07:43 Student: Um, I was thinking along the lines of why. [Is it because] 07:45 Teacher: [Good, good] 07:46 Student: you’ve given three and four and the lowest common factor of three and four is twelve, so you’ve divided into twelve. And I was thinking if you’d given us three and six you’d have to divide the box into six because the lowest common factor of three and six is six. 07:59 Teacher: Fantastic. And, um, great use of language there as well, I really liked that. Um, lowest common factor, really key idea. And as the lowest common factor of 3 and 4 is 12 it’s no coincidence that I’ve got twelve boxes. The students are asked to think about why the fractions 1/3 and 1/4 might have been chosen by the teacher for shading on a rectangle containing 12 squares. Teacher reflects learner’s language back rather than correcting immediately!

Listening … … for what you expect or hope for … to what is actually being said … sympathetically to what is or might be being expressed

Semantic & Syntactic-babble (1073) Syria: Nikol divided her square into seven equal parts and took the three. Then she coloured with a dark blue marker the five sixths of the area she coloured first. She divided into six parts and took the five. (1074) T: But which five did Nikol take? (1075) Syria: She took the five. (1076) Xenios: Please sir now I got it… (1077) Jim: This is what I wanted to tell you sir. (1078) Syria: Ah, she did wrong. She should have taken five pieces. (1079) T: What do you mean by five pieces? (1080) Syria: Five rows. What is being attended to? Students are working on finding 5/6 of 3/7. (1072) T: Let’s see what Nikol did. Nikol would you like to explain us what you did? But no, let’s hear Syria explain to us what Nikol did.

(1081) T: So class do you all see here [He points to Figure] (1081) T: So class do you all see here [He points to Figure]? Nikol took five small squares. And what is her answer? (1082) Syria: 5/42. (1083) T: Correct, it is 5/42 because she took five small squares from a total of 42. Do you agree? (1084) Class: No. The case of Syria is a representative example of a pupil whose awareness of the meaning underlying multiplication is both “in action” and “in articulation”

Diamond Multiplication