One comment I’ve had from a classroom observer lately is while I may have a clear classroom objective posted, say —
I can isolate a variable in an equation.
— I hadn’t made clear why students would care about such a thing in a first place. Here I was stuck: while I have some lovely activities that interest even 13 year olds, to get to them we needed to tread through some math which motivates some more math which motivates some more math and THAT leads to application projects.*
I don’t know about your students, but no amount of emotional appeal or explanation of how it fits into the larger picture of mathematics would lead mine to care about
Isolate the y: 2x + 3y = 14.
until I did this:
This is essentially project goal oriented sequencing where I make totally explicit what hurdles still need to be leaped before we get to the Stuff My Freshmen Actually Care About (the football lesson is still one of my most popular activities ever).
* It would be lovely to get them interested in the mathematics for its own sake, but some topics are hard to redeem. Even mathematicians don’t get worked up about isolating a variable in simple algebra.