Students are solving single variable equations with one and two steps.
This is where I can wedge into the process all kinds of reinforcement of the stuff we only pretend you are confident with: operations with variables and integers.
I can say for w/2 = 10 that w is a number so big that when you divide it by 2, the answer is 10. I can make sure you do more than grind out the procedure to get the answer.
When it says -4 + 4y = 16, and you’re not sure whether to add or subtract, when I say “yes, you could subtract the negative four but that would be the same as adding,” and I can see that it connects.
I’m sure some students figuring things out on their own are doing just fine and making a lot of those connections. Most of those are students who actually expect math to make sense.
Rather many students in college learning this honestly don’t think math makes sense, so “figuring out” means getting through that problem set with whatever garbage nonsense it takes. It means making up “rules” to survive that may have nothing to do with mathematical sense (because in your life math hasn’t made sense anyway).
It means asking me for help and informing me ‘confidently’ that -(y/4) is positive because the teacher said it was the same whether the negative was on the top or the bottom, and a negative divided by a negative was a positive. When I suggest that actually, it can be on the top or the bottom, but that it’s not on both, you inform me righteously that well, that’s NOT what you learned in class; my explanation of where the negative can be placed doesn’t quite penetrate. (You end up leaving in tears about the fourth time you insist on telling me why the math can’t work and I insist on asking you questions about what you do know. You go out to the main tutoring lab and I hope you got those last 8 problem sets done…)
Showing students who aren’t confident in connections how to make them makes a big difference… but it takes time and practice.