Okay, so right now, we are working on some skills with decimals. The kids are inherently more comfortable with decimals than they are with fractions, so they are a little freer in their ability to consider how they might think outside of the ‘rules’.
Last week, we spent time building our knowledge of how to multiply decimals using Base 10 blocks. Up to this point in their mathematical careers, their work with decimals has been focused on following the ‘rules’ they were taught in prior grades. Thing such as, multiply up, across, and then add, and then just put in the decimal points. “If there are two decimals in the question, then there are two decimals in the answer.” They are a bit baffled when trying to explain why 3.0 x 0.4 is smaller than the first number, but they do know HOW to do the question — they just follow the rules.
Today, I posed this problem:
2.6 divided by 1.2
And I told them that, just for fun, I looked up what the textbook says to do. “To divide decimals, ignore the decimal points.” I confess, I was, what? Just ignore the decimals? What does that mean? So, I posed it back to them. What does this mean? Why does this work?
Why is 2.6 divided by 1.2 equivalent to 26 divided by 12? What is really happening?
Eventually, we got around to having an interesting discussion about where the decimal is really ‘going’ when we do this. We aren’t really ‘ignoring the decimal’, what we’re doing is multiplying each portion of our statement by 10. We’re creating equivalent statements — we’re not really ignoring the decimals. We’re not really ‘magically making them disappear’. We worked our way around to understanding that we’re multiplying both sides by 10 — creating equivalent statements. The decimal doesn’t disappear – we are multiplying both sides by the same amounts, creating a proportional relationship.
The most important part of the discussion, though, was that the kids literally thought you could just ‘ignore’ decimal points — that you could just make them go away or ignore them. They had never considered that there was a mathematical reason behind what they were doing — they had just followed the rules.
This was and is one of the scariest aspects of being a math teacher — that students are willing to simply follow the rules without questioning WHY. And so I wonder, how do I build this culture in my classroom? How do we make our classrooms places where 1. teaching is more than conveying information and rules and 2. where students question those rules when they are presented? This is the challenge we face.