Unit 3: Reasoning and Explaining (MP2 and MP3)

# 3.1.4 Achieving Consensus through a Community of Reasoners

Addressing misconceptions during teaching does actually improve achievement and long-term retention of mathematical skills and concepts."

Askew & William,1995

As the work progressed, the 5th-grade students developed two conjectures the class agreed to:

- If you put cubes into pairs and there are no leftovers, then the number is even.
- If you put the cubes into pairs and there is a leftover, then the number is odd.

The following conjectures were brought forward without class consensus and formed the basis for ongoing arguments.

- If you put cubes into any sized groups and there are two leftovers, then the number is even because the two leftovers make partners.
- Even means the “same amount”, if you divide a number by any divisor, and it doesn’t have a leftover, then it’s even.
- If you divide a number by some divisor and there is no leftover, then the number is even. If you divide the same number by a different divisor, and there is a leftover, then that shows that the number is odd. Therefore, this number is both odd and even. For example, 15 ÷ 3 = 5 (even) and 15 ÷ 2 = 7 R-1 (odd). So 15 is both odd and even. See photo below of student representation.

In your Metacognitive Journal, reflect on how developing a community of reasoners demands a shift in allotting time for students to develop thinking around definitions, content, and arguments.

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