The Limits of Words

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.

-Roger Shattuck

I’ve found that some topics in math are much easier to explain in words than others.

Here’s some math:

Screenshot 2016-06-24 at 10.19.59 AM

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.


Screenshot 2016-06-24 at 10.26.25 AM.png

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:

Screenshot 2016-06-24 at 10.27.41 AM
Screenshot 2016-06-24 at 10.36.23 AM.png

to help them with this task:

Screenshot 2016-06-24 at 10.29.53 AM.png

I realize my notation is a bit sloppy

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?

2 thoughts on “The Limits of Words

  1. Brett Gilland

    I would say this is worked examples as investigation (which I prefer to ‘discovery’ for esoteric reasons).

    Also, one technique I might encourage you to try out is Socratic questioning to help students model that probing of their own understanding. In my experience, students (especially those coming from traditional classrooms) struggle in particular with crafting and testing theories based on prior mental schema. Struggling through, even with expert assistance, often gets overwhelming. So if I can help with well chosen ‘what happens when…?’, that is a big deal. Often I use the questions my more reflective students ask me as prompts for the others (and just reflect them back at the former- That’s a good question. What do you think happens? With some examples that blow holes in incomplete models.)

    1. dkane47 Post author

      I like this. I think one of the essential pieces of mathematical expertise students should gain is to get better at recognizing what the important features of a problem or situation are, and to notice those and not get distracted by irrelevant surface features. That type of questioning would be really useful in moving kids along that path.


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