I stumbled across this excellent quote in Dylan Wiliam’s Embedding Formative Assessment:
Words do not reflect the world, not because there is no world, but because words are not mirrors.
I’ve found that some topics in math are much easier to explain in words than others.
Here’s some math:
I think I can explain why this is true clearly and in a way that helps students learn. This is also a topic that I’ve found many students can figure out through some structured scaffolding and whole-class discussion. It seems to make sense, and to not be too great a leap from what students already know.
Here’s some more math.
I’m pretty terrible at explaining this. I have some tricks that I think are pretty clever, but they always leave some students unsatisfied, with unanswered questions. This is definitely a bigger jump than the exponent rule question. I taught this topic twice during the spring, and the second time I experimented with an alternate approach. Here is something to give to students:
to help them with this task:
No explanation here. Just some examples and some questions, to see what students can do. To me, this is somewhere between worked examples and discovery. But I find that, in the case of limits, my words often get in the way. There are lots of perspectives with which to look at limits that are mathematically useful, and I think this is an example where students generating their own understanding is particularly valuable, and where sharing those different perspectives with each other can create a useful collective way of looking at limits.
This didn’t work perfectly in class. A number of students still struggled to find tricky limits on their own. But, and I think this is important, the questions they were asking seemed to me to be probing their own understanding, rather than asking me for a perfect explanation that might not exist. It puts the mathematical thinking on them, rather than relying on me to do the hard part for them. And it puts some faith in my students’ ability to make sense of a new piece of math. There are plenty of opportunities for me to clarify and explain as necessary, but this comes later, after giving students a chance to make sense on their own.
Here’s what I’m curious about. Some folks like to argue about different approaches to teaching math, often juxtaposing worked examples and explicit instruction with discovery and minimally guided approaches. Where does this fall?