In response to @lmhenry9's post...
I've realized that with my smaller class sizes this year, it seems like I am doing less activities than usual. For one reason, direct instruction is a lot easier with 10 students compared to 28. Last year, I knew 100% that lecturing would not hold the attention of my 28 students and so I desperately searched for anything that I could turn into an activity. I was always looking to free myself up to wander the room and put out fires.
This year, that thought hasn't even crossed my mind. Whatever activities I've used in the past I have used again but I can't really think of one new activity that I've created this year. Sad. :(
But I think there are small things that I do that make me feel like I'm not all direct instruction. I guess it depends on how you define direct instruction. I think of it more like a traditional college setting- a professor writes and talks a lot and students write a lot and don't talk. Some people probably define direct instruction as anytime the students are looking at the board or teacher.
If you peeked in my classroom window, it would probably look boring and just like direct instruction. I think it's more the way you arrange, present, scaffold, and question things.
If the paper my students are working on asks them to analyze graphs, write the data in a table, make comparisons, and find a relationship between two categories, is that direct instruction?
If the paper my students are working on asks them to look at two columns of expressions and asks them to explain why the column of expressions on the right can't go in the left column, is that direct instruction?
If the paper my students are working on asks them to write proofs or cut and paste statements and reasons into coherent proofs, is that direct instruction?
If the paper my students are working on asks them to work out various pattern and look for a common end result, is that direct instruction?
Just because I'm at the board and my students are working at their seats, writing on paper, is that direct instruction?
If I am questioning, asking students to try hard problems on their own, asking students what is different about this problem than the last, asking students how they think we should start, asking students what could we do next, asking students to explain another method, asking students to work problems, compare, and discuss with their group; if I am giving students individual think time, asking more questions than I answer, redirecting questions from me to the whole class, and students are writing and talking more than I am, is that direct instruction?
To me, direct instruction is when I am directly instructing students every step of the way; when I say this is the only way to do this, you must copy exactly what I've written; when there is more writing than thinking and more listening than talking; when students have no input; when there is no puzzle, no pattern finding, no analyzing, no comparing, no questions, no discussion.
And to that effect, math class with less direct instruction does not ONLY mean activities, projects, students out of their seats or on the computer, or peeking in my classroom and seeing chaos.
It means my students ask each other for the correct answer before they ask me. It means that they find different patterns and question whose is correct. It means that they can write without listening to me talk. It means they have to look at things, they have to analyze, compare, observe, sort, organize, write definitions, create examples, explain counterexamples. They have to THINK.
I feel that I can accomplish the majority of these things with the way I design and scaffold my guided notes. I have a lot of good activities. I'm not good at projects. I haven't flipped my classroom. I haven't found anything computer based that I like. But I've found a way to present things, to question my students, to scaffold my concepts, that allows my students to do the skills I think real mathematicians need.
I don't feel like a lecturer. I don't feel like a college professor. I don't feel like a public speaker.
I feel like a highlighter. A pointer outer. An offerer. An observer. A questioner.
And I hope my students feel the same.