Table of Contents
- Content Home
Welcome
Module Overview- Pre-Assessment
- Unit 1: Teaching and Learning the Standards for Mathematical Practice
- Unit 2: Overarching Habits of Mind: MP1 and MP6
- Unit 3: Reasoning and Explaining (MP2 and MP3)
- 3.0 Unpacking MP2 and MP3
- 3.1 Beginning to Reason: Definitions and Conjectures
- 3.1.1 Student Definitions and Explanations
- 3.1.2 Developing a Community of Reasoners
- 3.1.3 Introduction to Conjectures
- 3.1.4 Achieving Consensus through a Community of Reasoners
- 3.2 Explaining and Justifying
- Taxonomy of Questions in Mathematical Discourse
- 3.3 Identifying Flaws in Reasoning
- 3.4 Making Arguments More Viable
- 3.5 Summary
- Unit 4: Modeling and Using Tools (MP4 and MP5)
- Unit 5: Seeing Structure and Generalizing (MP7 and MP8)
- Unit 6: Summary and Next Steps
- Post-Assessment
Certificate of Completion- Glossary
Resources- Acknowledgements
Module Evaluation
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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Mathematics: Kindergarten through Grade Twelve (K–12) Standards for Mathematical Practice. Brought to you by the California Department of Education, the Regents of the University of California, and the California Mathematics Project.
For questions or feedback, please e-mail commoncoreteam@cde.ca.gov.
