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, ora=bresults (whereaandbare 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.

Problems with the “Standard” you have quoted are numerous.

Written by a mathematician I suspect.

1. Linear equation in one variable : x + 2 = x + 3 Why would I think of this, it has two x’s in it.

2. Is the standard an instruction to the teacher or an expectation of meaningful student output ?

3. What makes a = a a linear equation in one variable? Looks like an equation in no variables.

4. Clever student writes 2x=4, x – x = 0 and x – x = 2 . How many marks will he get for being “clever”.

I think that the author of the standard was being too clever for his own and anybody else’s good.

I also think that the quote from barak rosenshine is way too narrow. It views math as set in stone, with a correct pathway ahead. example: you don’t need to know the definition of “median” to pursue an inquiry about the intersection of medians if you start with special cases, equilateral and iscoceles are good choices…….

I’m not really in a position (nor do I feel the need) to defend the standard. It was merely chosen to provide something concrete to an otherwise abstract discussion.

That said, if you’re curious about the “who” behind the writing of that particular standard, you can read about the development process here: http://www.corestandards.org/about-the-standards/development-process/

And I agree that there are examples of situations of where the specific vocabulary isn’t necessarily a prerequisite to exploring a topic, and your example is a pretty good one. However, in the example that you brought up x + 1 = x + 2 with a question over whether that’s two variables or one,a bit of big-group conversation in your class could possibly help everyone understand what it means for an equation to have one variable, even in that variable gets inputted into the equation in multiple places.

It’s so hard to separate cultural practice from should. The teacher and students are amazingly central variables that seem to be left out of these discussions. What I like best about what you’ve got here is the consideration of audience and purpose as the teacher is making choices. What do their students need to be able to do, and how to equip them for it.

The other piece that I think gets left out is the difference between a specific lesson and the arc of the school year. If you’re using the same mix in April as you were in October, I think you could be missing an opportunity to grow agency in your students.

Also think Boaler would be surprised to see her call for fluency as a ratification of direct instructions. In her How to Learn Math course she segues from that to discussing Number Talks.

I’m certainly not advocating one over the other and I also recognize that Jo Boaler is often associated with a very open-ended instructional style.

I also heard (and this may be more evidence of a bias I was listening with than of any actual meaning Dr. Boaler put into it) that more knowledgeable students had more meaningful explorations. That flows well with what other educators and professionals have reported about the need for sufficient amounts of background knowledge for inquiry to bear the kinds of fruit we’d all like it to.

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