I learned more about math yesterday than I think I might have in all of Elementary. It was fascinating. I was particularly intrigued by a series of "Rightbrainmath" videos by Mr. Numbers which visually demonstrated the patterns which result from multiples. I felt pretty excited about Math Conference today. We started by handing out number sheets and asking the kids to color in multiples of one. It became apparent immediately that none knew where to start, so rather than instruct we introduced the idea of "mathematical conjectures", ideas about what something might mean which might prove to be incorrect, partially true or completely correct. The kids volunteered conjectures on definitions for multiples of one. Initially, we had suggestions which included "any number with a one in it" and "every second number from one to nine".
Eventually, we visually demonstrated that a multiple of one is any original number that can be split into groups of one and then asked the kids to support or refute the conjectures on the board based on our explanation. Partial agreements about some conjectures lead to conversation about whole numbers. Could one student and one half of a second student be split into groups of one? Why not?
The kids, on their own, debated whether zero and negative numbers should be included in a definition of multiples of one. We looked at whether two non-identical conjectures could mean the same thing and both be completely correct. We modified conjectures to make them true.
Eventually, we visually demonstrated that a multiple of one is any original number that can be split into groups of one and then asked the kids to support or refute the conjectures on the board based on our explanation. Partial agreements about some conjectures lead to conversation about whole numbers. Could one student and one half of a second student be split into groups of one? Why not?
The kids, on their own, debated whether zero and negative numbers should be included in a definition of multiples of one. We looked at whether two non-identical conjectures could mean the same thing and both be completely correct. We modified conjectures to make them true.
How will we know what they understand, how will they know that they do, unless we allow them to debate, to defend and to support their ideas?
We can't. They won't.
I have three things to think about for next time:
We can't. They won't.
I have three things to think about for next time:
- How can I guide a discussion without directing it?
- How do I end a conversation without eliminating further possibilities for exploration within the topic?
- How do I know whether everyone "gets" the conversation? How do I continue to provide opportunity for everyone to contribute?
1 comment:
Deirdre,
I enjoyed reading your blog and I appreciate your rich descriptions of the involvement of your students in making meaning. I like the emphasis on helping students become comfortable in making mathematical conjectures and exploring them with others. Rather than take the time for discussion and exploration, it is tempting to simply provide a definition that can be memorized and applied but, as you so effectively highlight in your blog, you would miss the magic of abstract mathematical discussion. You raise some significant questions for further exploration. I am looking forward to reading more of your blogs.
Garry McKinnon
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