Unit 3: Reasoning and Explaining (MP2 and MP3)

# Next Phase of Lesson Forming Conjectures

In the next phase of the lesson, Asturias posed a question that empowered students to begin to form conjectures:

Asturias:  Work with a partner and decide whether you think Emily is right or wrong and why. Do you agree with me that she’s a little confused and we need to clarify her thinking? Work together to decide how to help her be clear. You can use words, pictures, diagrams, examples, or non-examples. Be clear enough that you can explain it to the class. I’ll give you 5 minutes.

Below is Daniela’s response as presented to the class. She used multi-link cubes projected on a document camera and followed Emily’s thinking that 50 can be divided by any divisor to determine if it is odd or even.

Daniela:  We figured out that 50 is an even number. And how we found that out is, because we first wanted to prove that (grouping) in any way, meaning in threes, fours, fives, or tens. We wanted to prove that it is even. So we tried (dividing into groups of) three, and it worked. But we had two left over, and we figured out that that’s still even, because the two has a partner. So it was even in any way, even if you have leftovers, the leftovers have even blocks.

Daniela has a beginning conjecture that a number can be divided by any divisor, and if there is a remainder of 2, the number will be even because the remainder is a pair — and pairs are even. Notice her use of multiple definitions of the word “even” in the transcript above.

Evidence shows that class discussion is important in students' development of mathematical conceptions … instances of disagreement arise from diverse ideas generated by children."

Wood, 1999