I’ve been on both sides of this conversation. And this discussion can get heated. Do we tell students what we want them to know? Do we let them explore? Do we let the students develop their own understanding? Do we model proper techniques for the students?
And this isn’t a particularly new debate. John Dewey was exploring this question at U of Chicago Laboratory School during the second Grover Cleveland Administration.
This is at the heart of many objections to the flipped classroom and services like Khan Academy.
Opponents of direct instruction models say that students are too passive and reliant on the teachers as the keepers of all the knowledge. It’s not “student-centered.”
Opponents of inquiry-based models say that it is ineffective for students who are behind grade-level and alienates students with disabilities.
What strikes me is how often one side isn’t really objecting as much to the other side’s model, but rather objecting to the other side’s model done really poorly!
The direct instruction advocates are often not objecting to the masterful inquiry teacher who differentiates the instruction and has fantastic formative assessment/feedback loops that keep the lines of communication open non-stop as students self-monitor their progress. They are objecting to the free-lance teacher who gives the students almost no guidance and makes them develop every bit of understanding completely on their own as well as answering each others questions while the teacher sips coffee and reads the paper.
Likewise, the inquiry advocates are rarely objecting to the teacher who provides a variety of worked examples, mixed in among short, focused practice sessions where all students’ progress is monitored and subsequent explorations are based on the each student’s progress through the new material. They are typically objecting to the disconnected boo-boo who stands at the front of the room and scribbles on the white board all hour long every day.
For a teacher to say that he or she teaches with in an “inquiry model” or a “direct instruction model” is not a value judgement unto itself. The teacher still needs to do a good job teaching in whichever model they choose. And teaching done well has some pretty standard qualities regardless of the model you fancy yourself following.
In either model, the effective math teacher has a plan for the experiences his/her students will have that day. There’s some kind of goal. A goal that fits coherently within the unit. This means understanding the goal’s connection to other content both previous and subsequent. Sometimes the goals is about specific math content (Example: the students will be able to solve two-variable linear systems by graphing) or sometimes it’s a learning practice (example: the students will learn how to try several different ways of solving a problem and discern which was the most effective).
In either model, the effective math teacher either A) already knows the current state of each of his/her students in their progress toward that goal or B) has an assessment tool built into the lesson to get that information, perhaps a short practice set in the warm-up or some strategically-chosen discussion questions in a Google Form. Something.
In either model, the effective math teacher stays involved with the students as they engage the teacher’s planned activity. This is essential for formatively assessing the students as they progress through the lesson. This is where the teacher gets to monitor that each student is consistently moving toward the predetermined goal. This allows for the students to try, check, get feedback, and try again.
And finally, in either model, the students are more formally assessed against the goal to see if they made it to where the teacher had hoped they would. This might be the exit ticket, the homework that comes back the next day, the quiz, the group sharing out, etc.
So, it seems clear to me that any teacher that has good, tight, coherent plans, is able to create launch points based on the students’ strengths and weaknesses, stays responsive as the students are exploring the content, and then effectively assesses the progress at the end of the lesson is going to have a major positive impact on their students. Those students are probably going to learn a lot of math.
The area where the differences between direct instruction and inquiry are most evident in when the students are exploring the content in a more self-directed way. When done well, both models have built-in ways for students to explore content in self-directed ways. The main difference is when.
Stay tuned for my (research-based) thoughts on that…
Mr. Shauver - Learner Educator 
Like everything else in education, there is no one “right way” to do things. A good teacher uses both “direct instruction” and “inquiry” every day! And we adapt as the lesson goes on, and as the various children progress through it.
While I would not entirely disagree that there is no “right way,” I would add that there probably are “right times.” There are certain types of content or class situations when one model really does support the students better than the other method. That doesn’t make it “The Right Way”, but it does make it the best choice in that moment.
I agree completely; it is the need to identify the “right way at the time” and to adjust according that is being lost with all of the “programs” out there.
I suppose part of the struggle is finding methods that work tolerably well even when they’re done poorly. Even excellent teachers are going to have off days (or off semesters), and any class is sometimes going to be taught by a worse-than-average teacher. Anything can be done excellently; it’s stuff that can’t be done abysmally that’s interesting.
That’s an interested question that I’ll admit I hadn’t thought of. Which of the two would be safer to expect a novice-level teacher to replicate serviceably on a regular basis?
An important question because, and I eluded to this in the piece, excellent teaching isn’t the problem regardless of the model. But which model will be easier for the non-excellent teacher to adapt effectively? Nice comment.