# Depth of Knowledge in Mathematics

**Guest Blog Post by Robert Kaplinksy**

Take a look at the problem below. It’s based on a fourth-grade standard on comparing fractions. Spend a minute or two thinking about it and then continue below.

**Directions**: Using the digits 1-9, no more than one time each, fill in the boxes to create a fraction as close to one as possible.

Source: Peter Morris at Open Middle

Not your typical problem, right? So by means of introduction, this is what mathematics feels like when it requires deeper understandings to interact with. The problem above requires much more brain power to solve than figuring out, say, whether 2/5 has a greater value than 3/7. Acknowledging and measuring the differences in cognitive complexity is the basis for Depth of Knowledge (DOK), a term coined by Norman Webb in 1997.

What’s important to realize though is that Depth of Knowledge is not particularly well defined beyond that. Consider this quote from Dr. Webb:

Reality needs to influence the comparison of the cognitive demands of expectations as expressed by standards, goals, and objectives with how students are held accountable to perform on assessments for the purpose of alignment. Ideally cognitive studies would be conducted to delineate in some detail what depth of knowledge is required by an expectation and what mental operations students actually used on the corresponding assessments. Such studies can be costly and time consuming. A more realistic analysis would be to seek some expert help and conduct a content analysis using verbs and their objects to judge the match between expectations and assessments.

He is saying that ideally, we would research what makes some problems more rigorous than others. Unfortunately, that would be costly and time consuming, so he suggests that “using verbs and their objects” may be more realistic. I believe this is the basis for where you see images like the one below which is commonly shared (without the red NO circle).

It would be lovely if using a specific verb was all you needed to do to increase a problem’s cognitive demands. For example, consider the task: “Compare 2/5 and 3/7 to find which one has a greater value.” What DOK level is this? Is it DOK 2 or DOK 3 as the verb “compare” is in both sections? Also, compare that task to the one you did above. I consider the problem I gave you to be DOK 3. So, if that is DOK 3, what does that make “Compare 2/5 and 3/7 to find which one has a great value.”?

If this is something you’re interested in and would like to learn more, I’ll give you some links to check out. First, I’ve made three Depth of Knowledge Matrices to give you more clarity as to how to challenge your students. You can download a high quality, printable PDF of each matrix by clicking on the image. That will take you to my website where you can scroll down and click the “Download PDF” button. You can also check out all my blog posts on Depth of Knowledge.

## Depth of Knowledge Matrix – Elementary Math (Kinder to 5th)

## Depth of Knowledge Matrix – Secondary Math (6 to Calculus)

## Depth of Knowledge Matrix – Elementary and Secondary Math

I hope this has been useful for you. If you have any questions, please reach out to me on Twitter and Facebook. You can also sign up to receive my weekly emails which go out every Tuesday when school is in session.

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