First off, I have to apologize (I guess) for the infrequency of posts this term so far. I’ve been REALLY BUSY! But I am keen to keep at this, and hope to have the kids up and running as contributors in the next month. (There, I’ve now said that so I have to try and make it true).
I wanted to share a couple of bits of math work we’ve been up to. We do a bunch of arithmetic type of math around here–times tables, and multiplication methods and so on. Stuff that you would probably find familiar. As much as possible, we are trying to remain open to the possibility of different methodology and even different answers. (Where the question is clearly 2 x 67, we’re happier with different methodologies than different answers, but it sure is interesting to see kids argue through their differing answers and learn something about how they got there). So, in several “open” problems recently, we had some pretty fascinating discussions.
The first, you may have seen: a caterpillar crawling out of a jar, up three centimetres and sliding down two each day. You can hopefully make out from the work below, that there was some serious disagreement about the answer. The message from some: we need more information! Like, how long is the caterpillar? There were at least three defendable answers!
This was also a good entry into the quality of our mathematical communication. If you (or I, or anyone else) don’t understand what the math shown below means that could be a place where we talk about communication. (It is also fairly often a place where I discover a child can see a problem–correctly–in a way that I am unable to see it. I love that).
Later, a party problem: ten people at a party, everyone shakes hands just once with everyone else. One answer, but many routes to that answer. This problem had some sweet traps, like having to realize that this means nine handshakes for each person; and that one handshake means a handshake for each of those people. Thus, we were into understanding the problem, and having to use logic. The variety of strategies was rich. It was particularly lovely to see that one of our most capable, super-advanced math citizens who often leaves us all with our jaws hanging open solved this one literally using a little pile of stones. Sometimes the simplest way is the best, and making models is more than OK.
The bonus question took us a whole awesome class: Would doubling the number of people double the answer? Is there a way to predict? Digging into patterns together is so important to seeing the beauty and potential of mathematics.
This week we had this problem:
Not the most elegant problem I’ve written, to be honest. But once we got past my over-wordiness, this was a good example of the openness of even fundamental multiplication. All the students could see that this was a pair of multiplication questions. Our talk about which we predicted would be largest was very interesting. Giving the students a chance to share their thinking is so important. (I’ll do another comparison one like this with fewer words–it was very interesting). And then we got to see the different ways that students were able to show the multiplication. For some, this involved a lot of counting. A lot! This opens questions about efficiency (and also about learning their times tables). The strategies the students choose must first be rooted in understanding, rather than just mimicking some method I’ve shown them. As their understanding grows, they can then see the logic of choosing the most efficient strategies. (Which will, in the end, involve a calculator–a tool that is useless without understanding).
So, openess. This is not meant to replace accuracy or the vital role of automaticity in fundamental arithmetic (so yes, please practice times tables–your child should know what they are working on; if they don’t, tell me!). And it is not all the math we do. But by being open we allow everyone a door into the math we are doing, and hopefully each child also begins to see a direction for their growth and learning.
Upcoming: It is high time I did a post about Orthography. The stuff your children already know about words that I didn’t know by this point in Grade 5 (or, frankly, at age 40) is pretty encouraging. Stay tuned, there may be one or two things for them to teach you as well!