MemberDecember 4, 2019 at 12:37 pm
This is really helpful. I’m trying to use solutions to systems as a specific place in which the lack of conceptual understanding becomes clear, but I do think it goes back to the earlier contexts you describe (and you’re getting exactly what I’m going for with the idea of “solution”). While I think they can say thats the answer, I’m still not sure they even fully associate that with being the solution. I’m wrapping up one variable equations right now and yesterday we looked at no/infinite solutions. I circled back around to the first problem I’d opened with (x=6) and asked if any other numbers would work. They were overall not sure, even my strong students. We opened equations with a double number line, we’ve worked with them conceptually, some of them have worked with manipulatives–but they still seem to be associating that 6 with the result of an algorithm rather than as a value with a relationship to that specific equation (that no other variable has).
I’m wondering where I can start doing the exploration you mentioned before we get to systems–what are the places in other topics, in tasks, in a warm up, that I can continue to look at the idea of a solution. I don’t think it lies in word problems–we’ve talked about answering in a sentence, making sure its reasonable so I don’t know that there is a ton more to get out of there. And I suspect that it is when it is abstract that it falls apart. Something around solutions to one variable equations? Pulling out specific solutions to a linear equation (I semi-tried this last year but I’m not sure I helped much)?
I’d love to hear further thoughts but even just the post above is hugely helpful. My team is really thoughtful and I know they can help design better instruction around this and introduce it earlier–I just can’t seem to get them to get what I mean 100%. We end up going in circles with me going, but that isn’t quite it. I think your post above could help them to see what I’m hoping for the students to understand.