Most people who teach mathematics are aware that it’s useful to have alternative explanations for concepts, and useful to have different ways to approach problems. Given enough time, you are guaranteed to come across students for whom the standard explanation isn’t working today (as long as you give students a chance to tell you about their understanding).
Having worked with thousands of students one-on-one, I have tried quite a few alternative explanations and methods for many things. Sometimes they’re whole different approaches; sometimes they’re just little tweaks. Sometimes they are just a different order of the sentences you might otherwise say; sometimes they use physical manipulatives like the floor graph or play dough. Many teachers, like me, have such a bank of alternatives.
The problem is… Well, you can see it already in the way I’ve talked about these explanations: I have called them “alternative”, as opposed to “standard”. They are different, unusual, other. And the students know they are. A student who always has to have the other explanation can come to feel that they themselves are other.
A prime example of this is when the “dumb class” use physical toys to learn, whereas the “smart class” only uses symbols. (I use “dumb class” and “smart class” because that’s what the kids call them. Don’t fool yourself into believing that they don’t.) If you set up this sort of dichotomy, then any child who ever has to use the physical tool to help them understand knows they are stupid.
Another example is when mathematicians do not provide pictures when showing how to work out problems, and only provide them when someone doesn’t understand the text version. Students come to think that pictures are only for the “dumb kids” who aren’t capable of understanding the text alone, and they try to avoid drawing them, even if they could solve a problem ten times faster with one.
Obviously if the first explanation you try doesn’t help a student, then you do need to try another one – I never want people to stop providing alternatives!
But perhaps the explanation you use as the standard one doesn’t have to be the standard. Perhaps the other one you usually save for second might work as the first explanation for all the people the standard one works for, and also a few more. Each new explanation needs a bit of consideration to decide if maybe it can supplant the one you usually use first. At the very least, when you hear or think of an alternative explanation, don’t say, “I will keep that in mind for my struggling kids.”
Even better, perhaps we should more often just provide more than one explanation to begin with, rather than just one. No explanation can possibly work for all possible students, and even the “smart kids” will benefit from having more than one way to think about something. So maybe we can avoid othering people by simply giving more options from the outset. For example, to stop students feeling like they’re a “dumb kid” when you draw pictures, you can just draw pictures for everyone a lot more of the time.
So please, do seek out and try other explanations, but make sure you are careful for them not to become othering explanations.