How To Create Higher DOK Problems

Now that you’ve read about the three steps I often follow when trying to raise a math problem’s Depth of Knowledge level, I thought it might be useful to take you through the entire thought process of actually implementing those steps.  The experiences I’m describing took place while preparing with some other teachers to teach…

X-Ray Vision Glasses

Have you ever believed something about your students only to find out that you were not correct? This has happened to me many times and it doesn’t feel good when the reality kicks in that you were wrong. I remember a specific instance of this happening to me as I transitioned from teacher to teacher specialist.

Depth of Knowledge Matrix – Elementary Math

I’ve decided to expand upon my previous Depth of Knowledge Matrix that helped make it easier to distinguish between depth of knowledge levels in mathematics. While it is still useful, it didn’t cover every grade level and may be too broad in scope. So, I have made two new Depth of Knowledge Matrices: one for elementary mathematics and one for secondary mathematics. This week I am releasing the elementary mathematics matrix and next week will be secondary mathematics.

3 Steps To Increase A Math Problem’s DOK Level

Here’s my first attempt at articulating my thought process behind increasing a procedural problem’s Depth of Knowledge level. It’s not my intention to say that this is the only way to change the level. This is just one way. Also, I am only addressing the DOK level of the math content, not the conversation that students may have around the problem.

I’m starting out with something really basic: single operation problems. In an effort to make this applicable for a K-12 audience, my examples include addition, subtraction, multiplication, square root, exponents, and trigonometry. Clearly six problems won’t apply to every grade level, but I hope they are close enough to what you teach so that you can imagine a version you could use. Specifically, I want to discuss how to take a problem that is primarily a routine procedure and make it more rigorous.

Open Middle Worksheet

If you are frustrated that your students will try a math problem once or twice and then give up, this blog post is for you.  Let me (re)introduce you to one of my favorite tools for helping students persevere when problem solving: the Open Middle Worksheet.

To explain why many educators find it invaluable, I’ll need to tangent and briefly explain a central theme of one of my favorite books, Freakonomics.  Life is all about incentives.  There are positive incentives (I go to the gym because I want to stay healthy) and negative incentives (I don’t steal because I don’t want to go to jail).  Without these incentives (or if the incentives changed) what you do would change.

Shallowness (not Depth) of Knowledge

Consider the two problems on adding two-digit numbers shown below that are of varying Depth of Knowledge (DOK) levels.
When I gave both of these problems to my then second-grade son (who is fortunately a mini-math geek like his dad), he quickly conquered Problem One with it providing no real challenge. When I showed him Problem Two, he got 98 + 76. This is a great first answer, but it’s wrong.
He knew which four numbers to use but the problem exposed a hole in his understanding of place value and gave me a great opportunity to ask him questions like, “What does the 9 represent? What does the 8 represent? How does having more tens or ones change the sum?”
I’d make the case that the problem on the left is DOK 1 while the problem on the right is DOK 3 (more info here). Recently though, I got some pushback that forced me to think of a new way to articulate the difference between them.