Oddness

With 18 days off for winter break (16 in the calendar and 2 snow days at the beginning) I have time to take for just me. .. and today is one of those. I’m spending today tweeting, blogging,  thinking, reading, looking through the books I have stacked up, basically learning and reflecting!

I have a blog draft on my personal blog called “Good Questions To Ask Kids” that I WILL get to soon… but conversations on Twitter this morning are making me take one of those questions and tell a story (so for my ardent fans–Kirsten, David and Chad–this is for you! LOL)

A prequel, though:

When we study geometry in my math classes, we always  get to the point where I have to talk about how teachers tell kids a kite-shaped figure is called a diamond, but that is not the accepted mathematical terminology. We then study the exact definition of rhombus, parallelogram, kite and diamond and talk about the need for precise language in mathematics. The kids ALWAYS ask why teachers teach them wrong in Kindergarten, and my response is that I believe teachers don’t deliberately teach wrong information, they simply teach what they know and all of us are human and make mistakes. Kids get that and accept it–and it helps me bring home the fact that they ALWAYS have to think critically about whatever they are told. It helps me reiterate that not all sources are 100% reliable and the more they think about everything they learn, the better they’ll understand the world around them. I tell them teachers–and probably all adults–will tell them something that isn’t quite right at some point, and they have to understand it’s their responsibility to be able to sift and sort right and wrong–in all areas of their lives. The fact that a teacher is talking to them about adults being wrong sticks with them–and they remember this lesson. (See  http://en.wikipedia.org/wiki/Rhombus and http://en.wikipedia.org/wiki/Kite_(geometry) and http://www.mathopenref.com/kite.html if you want to explore the differences between the geometric vocabulary.)

Now to the recent story:

In math class about a month ago–a combined class of 4th and 5th graders–Tyler hypothesized that there was no such thing as an odd number, because if you take any number–say 5–and split it, you can always get 1/2 of it–like  2.5, or 2 and 1/2. That caused quite a stir with these very bright kids, with many agreeing, and others bringing up specific examples where Tyler’s theory didn’t quite play out well, such as when choosing teams or anything involving people.  After all, as several of them said, “You can’t cut a person in half and share them.”

However, I pointed out that you COULD share a person fairly.  I explained that if you had the “leftover kid” always play in the field, or always ride the bench, then the teams would indeed be fair. That sold most of the kids, especially those who were arguing you could not cut  a person in half. I also gave the example of 12 kids going to music while the 11 others were in class reading and then the groups switching making it a fair split as well. Making it a “fair” split met their need for justice.

Not one of them caught that I had deliberately changed the criteria from an “evenly divided” split to a “fair” split. Doing so made them really think about the definition of even and odd, verbalize it more precisely and discuss the difference between evenly divided and fair split.

Some then began arguing and asking why would teachers teach odd numbers if there was no such thing?

That took us back to the prequel, where someone responded with something like, “This is one of those times where teachers only teach what they know–and it might not be completely right. Little kids can’t understand decimals or fractions, so it’s a simple way to teach numbers and fair sharing.” (Fair sharing is a term we’ve used a lot, so they immediately picked it up in this circumstance and ignored the “evenly divided” criteria.)

So they wanted know the “truth.”  I explained to them that there was more information they needed in order to clearly understand this concept, and so I began describing number theory, just a bit.  I began by telling them there were many different kinds of number categories and that they had learned some of them, but not all.  So we began brainstorming kinds of numbers they knew–positive, negative, fractions, decimals, percents, ordinal, counting, and whole numbers. Since it was the end of class, I told them we’d keep the conversation going next class and for them to think about it overnight, but NOT to look it up–they could ask people, but not to consult a book, the internet , or another print resource.

Due to circumstances out of my control, (field trips, a grade level test, no 5th grade on Friday, etc.,) it was literally almost a week before I had both classes together again and I refused to have the conversation without the whole class there. It drove the kids CRAZY–kids were stopping by before school, coming in during lunch and trying to have the conversation–they were arguing it on the busses, in the cafeteria, in the hall. Talk about igniting a fire–I’d hit just the right trigger with that fair sharing example–kids knew enough about even, odd, fractions and decimals to talk parts and pieces, and they had enough experience with school that they had learned a lot about odd numbers–and figured there had be such a thing, but they couldn’t figure out how an odd number could NOT be divided equally.

So I asked them to share their thinking on a wiki, and I have synopsized their comments here. Some of them were written before the class conversation,  some were amended after it. Regardless of when the comment was written, you can clearly see the passion in their comments.  These kids were engaged in trying to figure out this quandary.

While we were waiting for that combined class again, I decided to get the fourth graders to explore more attributes of even and odd numbers, to deepen their understanding.  We delved into adding and subtracting odd and even numbers and looking for patterns.  I asked what happened when you added two odd numbers together and asked them to try some numbers and see.  One kid immediately said the answer would ALWAYS be even. When I asked her why, she said she remembered being in kindergarten and studying the hundreds chart with her teacher and that’s when she learned that.  She could demonstrate with her hands how 2 “leftovers” could always be put together to make another pair. Blew me away she had that vivid memory!

The days I had only the fifth graders, we went into divisibility rules. Both explorations allowed the kids to come to our combined conversation with additional information about even and odd numbers.

When we finally DID get to explore the answer to Tyler’s original question, and I explained categories of numbers, drawing a tree diagram on the board with complex, rational and irrational numbers, we then began placing the  numbers they knew on those tree branches. They were eager, engaged, and involved–asking questions, making connections, and assimilating what they were hearing with what they already knew. Now, did they do this at different levels? Of course–but I guarantee you that each and every one of these kids know that whole numbers can be split into odd and even categories and that definition technically does not apply to numbers like fractions, decimals and percents.

The big question here was “Is there such  thing as an odd number?”  It totally engaged them, kept them thinking and had them questioning what they knew for well over a week, even without teacher direction.  I know this is not a description of a typical math class–but my goal is to have them leave me with deep mathematical understanding AND a reasonable use of arithmetical skills.  I want them to NOT accept the world at face value, but instead to challenge their own thinking. I know that this lesson is an “odd” way of teaching, but I believe it needs to be less odd and more utilized. So my questions to you are:

1. What kind of teaching “oddness” do you see that promotes deep understanding?, and

2 . How do we promote and share and develop expertise in that kind of teaching and learning?

Share your thoughts here, please!