This summer I was fortunate to hear Virginia Bastable keynote about the work in her book Connecting Arithmetic to Algebra. If you have not read this book, it is a must! It explores the process by which we have students notice regularities, articulate claims, create arguments and representations, and make generalizations.
It is something, as elementary school teachers, we need to really be thinking about more in our math classes. Are we creating environments that encourage students to think about the math behind the strategies and make generalizations based on the properties of operations? I have taken this recent reading and made it a priority in my classroom.
I always have students notice and discuss patterns and regularities but I don’t often have them create generalizations for us to revisit as we move through the year. For example if their claim works for whole numbers, shouldn’t I revisit that as we work with fractions? Does your claim still hold true?
As a class routine, I posted this on the board and asked students to fill in the blanks to make it true:
12 x 4 = ___ x ____
Quickly, students wrote down answers, had their hands up, and one student blurted, “This is easy, you don’t even have to solve it!” Typically blurting out answers before others are done thinking drives me a bit crazy, but this time I was thinking…Yes! I asked who else thought the same thing. I had at least one hand up at each table so I asked them to discuss with their table how that is possible. We came back together and each table said they could double/half to fill in the blanks. I took answers on the board and got the expected: 6 x 8, 24 x 2, 48 x 1, 3 x 16 and then I even had a 96 x 1/2 and 36 x 1 1/3! I asked about the 48 x 1…did you get that by double/halving from the original problem? What is happening there? They noticed that it was x 4, ÷ 4, and then the same with 3 x 16. I asked them to take some individual time to see if they thought their strategy would always work and could prove it with a representation. They then talked at their tables and I asked each table to write a claim, something they think is true about this work.
I got some who kept solving problems to prove it works:
I had a couple try out the representation (exclaiming how hard it was to draw what is happening:)
Here are some of their claims:
This is the class list from my second period class. I especially liked that one of them said it only worked with multiplication. How fun to revisit!
Instead of losing these, I started a Claim Wall to post and have students add to and revisit throughout the year. I am trying to think through how to have students comment on them, possibly agree/disagree post its?
If you would like more information about Virginia’s work, there are courses available here: http://mathleadership.org/programs/online-courses/ Check it out, great stuff!