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).

CCSS 8.EE.C.7.A

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.

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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.

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