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

1. If you put cubes into pairs and there are no leftovers, then the number is even.
2. 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.

1. If you put cubes into any sized groups and there are two leftovers, then the number is even because the two leftovers make partners.
2. Even means the “same amount”, if you divide a number by any divisor, and it doesn’t have a leftover, then it’s even.
3. 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. 