Direct Instruction vs. Inquiry: The What and the When

In my last post, I looked at the characteristics of high-quality classroom instruction and discussed why I felt like those were essential regardless of the model any given teacher used. There were some excellent comments left after I posted that, so I’d encourage you to go join the conversation.

What I didn’t discuss is the role of inquiry and the role of direct instruction. Each tool that gets wielded in a classroom is build to do a certain type of work. To maximize the effect, each tool must be used to do the job for which it was created. Direct instruction does one type of work. Inquiry does a different type of work. In order to highlight this difference, let’s consider a content standard.

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).


Consider how we’d assess this standard. The students need to “give examples of”, which means they need to actively create something and explain why it’s the right kind of something. But, the explanation is predetermined. They can’t explain it anyway they want (according to the standard, at least). They need transform their example to match one of the stated forms.

So, the final assessment of that standard (if we choose to assess it to the letter, so to speak), would include three equations that the student created and then evaluated in a standardized way to support their claim that their equations had one solution, infinitely many solutions and no solutions respectively.

From my perspective, anytime the students are going to be expect to create something on the assessment, they will need some time to freely explore. You can’t assess a student on something they’ve not gotten the chance to practice. So, if you want them to create on the assessment, they need to practice creating. But we aren’t assessing their ability to create just ANYTHING. We want them to create strategically.

There’s also that standardized evaluation process they’ll use on the equations they’ve created. While there may be some value in allowing the students to explore a variety of different, homemade ways to tell what their equations are going to do, in the end, we are going to ask them all to do the same thing. They need to be taught this process.

Also, we need to make sure everyone is on the same page with the words “equation,” “solution”, and “variable.”

Hang on… I need a quote.

“[Highly-effective teachers] provided support by teaching new material in manageable amounts, modeling, guiding student practice, helping students when they made errors and providing sufficient practice and review.”

“Many of these teachers also when on to experimental hands-on activities, but they always did the experimental activities after, not before, the basic material was learned.”

– Barak Rosenshine

Based on his research, Rosenshine is saying that inquiry can work provided students possess the appropriate background knowledge.

He isn’t the only one to say stuff like this.

“[Content and creativity] drive each other. Students need a certain amount of content to be creative. Increased creativity drives deeper understanding of the content.

“Algorithms and problem-solving are related to one another. Algorithms are the product of successful problem solving and to be a successful problem solver one often must have knowledge of algorithms.”

– Dr. Jamin Carson

And also…

“Students need to be flexible problem solvers. We know that one thing that separates high-achieving students from low-achieving students in elementary school, is that the students who are successful can flexibly use numbers.”

– Dr. Jo Boaler

This idea can be found within a variety of researchers in high-quality math instruction. Students need to explore. They absolutely do. They need to freely explore and play with the math.

But in order for that to be effective as a learning tool, it really, really helps to have sufficient background knowledge. Be it the knowledge of algorithms helping to support and drive the problem-solving process, the math facts giving the elementary students flexibility, or in the case of our example 8th grade standard, a solid understanding of “variable”, “equation”, and “solution” to give the sufficient foundation on which to build their exploration.

So, for this standard, I would probably recommend a direct instruction introduction to the standard that ends with making sure that all students are clear on the three essential vocab words as well as the evaluation process.

Then, I’d move to an structured inquiry activity that led them through a chance to practice creating their own equations and evaluating them eventually leading them to make some generalizations about what equations look like when they have one solution, infinitely many solutions, or no solutions. I see the possibility for some small group discussions, reporting out… possibly a Google Sheet or some white boards and a gallery walk, etc.

And from my chair, this exercise through this standard demonstrates the bigger picture. It isn’t whether or not inquiry or direct instruction should be used in eighth grade.

It’s about what we are going to ask the students to do and which of those models supports the students best at which point during the instruction.

It’s not about which. It’s about what… and when.


Quotes taken from:

Rosenshine, Barak (2012) “Principles of Instruction”, published in American Educator, Spring 2012 edition. Quote taken from Pg 12-19, 39. Quote taken from pg 12.

Carson, Jamin (2007). “A Problem With Problem Solving: Teaching Thinking Without Teaching Knowledge.” Published in The Mathematics Educator, Vol. 17, No. 2, Pg 7-14. Quote taken from pg 11.

StanfordSCOPE interview with Professor Jo Boaler. Quotes taken from times 2:40-3:20 in the video.

The Ongoing Debate: Direct instruction vs. Inquiry

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…

Redesigned Wedding Cake Problem with Desmos

It all started with this tweet:

I got a taste that Desmos was capable of conditional images and text and stuff. So, after I did a little of the ol’ one-man PD tinkering around with their link, I decided I wanted to try to model something. So, I figured why not Wedding Cake Problem? It’s gotten a few redesigns already, so what’s one more?

And so I created this on Desmos.

What I like about this one is that, after all this time, I believe that I have finally figured out a way to deal with the issue of the variable “frosting thickness”.

Also, I like that I was able to include the reality checking piece that the top had to be the smallest section, the middle had to be the second smallest, and the base had to be the biggest.

I encourage any feedback.

Also, I have a question for anyone who is able to answer it: Is there a way that I could make the function round up to the next whole value? I’d like the line to only snap to whole number values since the question is “how many jars of frosting will you need”? (That is, if you need 2.3 jars, you need to buy 3 jars.)


The last question received an answer by John Golden (@mathhombre) who taught me how to use the ceil() function. You can now check out the final Desmos worksheet.

Bowling… and supporting struggling students…

We recently went bowling. It was my nephews’ birthdays. We brought the whole family. I have a three-year-old who doesn’t take well to being left out of the fun, and he shouldn’t have to. Bowling is for everyone.

But he’s little. Really little. A 6-pound ball is about 16% of his body weight. That would be like a 200-lb man throwing a 32-lb bowling ball. No one bowls like that.

So, we provided a support to make sure that he could meaningfully engage.

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It’s a ramp. He put the ball on the ramp. The ball rolls down the ramp and now he’s bowling.

My son just plain ol’ isn’t big or strong enough to roll the ball down the lane. His body doesn’t have the capacity to provide enough energy to the ball. That said, there are still things he can do. He can’t supply kinetic energy, but he can provide gravitational potential energy by lifting the ball up to the ramp. The modification is the ramp facilitating the conversion to kinetic energy that ends with the ball rolling toward the pins.

In this case, we haven’t removed the responsibility to pick the ball up, carry it toward the lane, and add energy to the ball. We’ve also not stolen from him the experience of seeing his ball head toward the pins, to compete with the other children, and to have fun.

All we’ve modified is the conversion to kinetic energy that he is unable to do. And if you think that he was feeling unfulfilled because we modified the activity, then think again. He was hopping around like he had springs in his shoes every time he knocked down some pins.

And the progression toward full participation was visible. My six-year-old daughter was bowling, too. No ramp. She would get a running start and heave the ball with two hands.

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Not exactly a textbook technique, but effective as long as we include a different modification: bumpers. (Full disclosure, the three-year-old had bumpers, too.) We were setting the standard that the the ramp isn’t HOW you bowl. It’s a modified way to bowl until you grow strong enough to no longer need the support.

I thought these things as we watched him bowl. And I considered the implications for our students who need their classroom experiences modified to be full participants. Like a student who is still developing fine motor skills needing someone to dictate his/her spoken words. Or a student who doesn’t sit still well being able to work standing up and helping that student develop the ability to produce quality work while standing.

Take the effort that the student is able to give and provide support in the areas that… well… they just aren’t there yet. The expectation is that the student will be given support in their areas of weakness at another time to facilitate their growth, but at the moment they are in your class, it is important for them to be able to participate as fully as possible with the classroom activities. There are learning outcomes that don’t depend on their weaknesses.

For example, there will be a time when that student works on developing the ability to work sitting down, but the teacher is currently leading an activity on the civil war, and “learning while sitting down” isn’t one of the stated learning goals for that activity.

Or consider a student who is in Algebra I, but has a weakness in basic computation. This is a fairly common weakness among our Algebra I students, wouldn’t you say?

If we could create a structure whereby those students are receiving the specific supports in the area of computation (perhaps a 15-min per day intervention right before school… or something), would it be possible to allow strategic use of a calculator to support students who are weak in computation. They’d still have to know what numbers need to be computed. They still need to be able to check their answers and apply them back to a situation, but the calculator acts as the bowling ramp. It supports their effort until they grow strong enough to heave the ball down the lane themselves.

I think it’s important that we consider ways to help our students find success in our classes despite their weakness. The alternative is watching those weaknesses become bigger and bigger barriers until our students are sitting in the back of the bowling alley watching everyone else have fun.

Excellent Classroom Action – Art and Geometry in the Elementary Classroom

I’ve written about the connections in math and art before. The visual nature of Geometry lends itself quite nicely to this. I found that the right pairing could bring in the engagement of the visual arts while maintaining fidelity to the content.

Exhibit A: Sarah Laurens, 5th grade teacher at North Elementary in Lansing. Mrs. Laurens reached out to me excitedly while I was in her building to see a math activity that she was leading her students through. They involved quilts hand sewn by Sarah’s grandma.

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The activity went like this: Students were in groups of threes and fours gathered around one of grandma’s quilts. Each quilt was made of a series of geometric shapes. Sketch the primary “unit” shape of each quilt and identify each of the polygons that are contained within it. On the surface, it is a fairly simple activity, but listening to the students talk to each other.

[Students looking at the black and blue quilt above left]

Student 1: “Those are just a bunch of hexagons.”

Student 2: “Hexagon’s, no… no… those are octagons”

Student 1: “Yeah, yeah… same thing.”

Student 2: “They’re not the same, one’s got six sides and one’s got eight.”

Student 1: “Well… wait… one… two… three… four… five… six… six sides! See I told you!”

[Students looking at the purple and while quilt on the above left]

Student 1: “That’s an octagon with kites around the outside.”

Me: “Are they kites?”

Student 2: “They look like kites.”

Me: “They sure do. How many sides do they have?”

Student 1: “One… t, th, f… oh! five…. They’re pentagons!

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Ms. Laurens and her students were comfortably saying and hearing words like “regular”, “tessellation”, and using definitions to make sense of what these shapes are, and using the definitions to settle disagreements (the foundations of proof…)

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Interesting images like these were leading to some interesting conversations as well. I’m thinking of  a conversation I heard between two students who were trying to make sense of the shapes they were in the picture directly above to the right. They were trying to to determine if the purple section in the middle was one big shape or two smaller shapes put back-to-back.

Then after some discussion, they realized that their answer would be the same either way. (Quadrilateral was their choice for the shape name. The word “trapezoid” was getting thrown around, but the students were having to be prompted for it).

I very much enjoyed getting to see these fifth graders exploring. Ms. Laurens was excited, the students were engaged (and this was clearly not the first time they were expected to be a self-directed and collaborative).

I’m just bummed that my schedule forced me out the door before I got to see Ms. Laurens’ closure of the activity. The students were wrapping up their discussions as I had to head out the door.