Teacher Tech blog with Alice Keeler

Paperless Is Not a Pedagogy

Alice Keeler

Is Math Teaching Critical Thinking?

Math class SHOULD be teaching students to critically think, find patterns, analyze, and manage data. Is Math Teaching Critical Thinking? Using traditional math textbooks is this happening? No.
Is Math Teaching Critical Thinking?

“When are we ever going to use this?” No joke, years ago my department policy to respond to students was “We are teaching you critical thinking.” Is math teaching critical thinking? I can’t speak for all math classes, but our curriculum was definitely NOT. Once I learned about Depth of Knowledge (DOK) I realized that the majority of the textbook and all of our worksheets were low critical thinking, DOK 1. I think the purpose of math class should be to teach critical thinking so it is important that as math teachers we are aware of what critical thinking looks like in math.

Depth of Knowledge (DOK) is a framework that categorizes tasks based on their cognitive complexity and the level of thinking required to complete them.

The 4 Levels of DOK

There are 4 levels of DOK to describe the levels of critical thinking.

  • Level 1: Recall and Reproduction – Students are asked to recall information and reproduce previously learned concepts.
  • Level 2: Skills and Concepts – Students are asked to apply skills and concepts they have learned to solve problems or perform tasks.
  • Level 3: Strategic Thinking – Students are asked to reason, plan, and use evidence to solve complex problems or complete complex tasks.
  • Level 4: Extended Thinking – Students are asked to make connections between ideas, synthesize information, and transfer knowledge to new situations.

“Live in DOK 2 and DOK 3. Visit DOK 1 and DOK 4”

-Shelley Burgess

Students should be critically thinking in math class. This means that most of what we ask students to do should be at least DOK 2.

Mathematics DOK Definitions

It is tricky to categorize math tasks to the correct DOK levels. We often mistake hard for being complex. This article from WebbAlign contains some helpful tips on what the four levels of DOK are.

The ultimate goal of using DOK levels in math instruction is to provide a more meaningful and challenging learning experience for students.

Highlights From the Article


Following steps is DOK 1 as is “executing a well-defined multi-step procedure.” A math problem can require multiple steps and still be low critical thinking. “Operating on polynomials or radicals, using the laws of exponents, or simplifying rational expressions are considered rote procedures.”

“Verbs should not be classified as any category…”

It is really important to understand that DOK is not about the verbs. Try to avoid the DOK wheel that is prevalent in Google searches for Depth of Knowledge. It is the level of complex reasoning by the student that defines the DOK level, not the words used in the task.

Show your work is DOK 1.


Students need to do their own thinking for it to be considered DOK 2. My quick and dirty for DOK is “how much figure it out is there?” So it is not the number of steps it would require, but rather “engaging in some mental processing beyond a habitual response as well as decision-making…”

If I were to give you a very large binder with 1 million steps to put together a rocket to the moon, this task would be very difficult. You would probably make a mistake somewhere. However, if you followed all 1 million steps perfectly the rocket would go to the moon. Rocket building is a DOK 1 task if you are only following the directions. To elevate this task to DOK 2 you would need to have some ambiguous parts of the directions, perhaps more like IKEA directions, that would require you figure out some steps.

Asking to student to explain their reasoning is likely a DOK 2 task. This is different from show your work. Listing out the steps in words is still recalling the steps and procedures. Students would need to explain why they made a particular decision that goes beyond recalling the rule.


“DOK 3 requires reasoning and analyzing.” Mathematical reasoning is not recalling math procedures and formulas. Mathematical reasoning is much. more complex. This is what hopefully students gain from taking math classes is the ability to reason through a situation and figure out what math they need to apply and how to apply it. Mathematical reasoning does not need to involve numbers. It is the strategic thinking through of a problem and logically figuring out a solution.

“DOK 3 includes situations that are nonroutine, more demanding, more abstract, and more complex than DOK 2.”

Where DOK gets tricky is that it is not just the description of the task to take into consideration. “Note that the sophistication of a mathematical argument that would be considered DOK 3 depends on the prior knowledge and experiences of the person.” Prior knowledge can change the DOK level of a task for a student.

If you expect a student to get it right the first time, it is not DOK 3.


I cannot emphasize enough that the goal is NOT DOK 4. Requiring daily DOK 4 tasks in math simply demonstrates a lack of understanding of DOK. You would be doing pretty good to do DOK 4 once or twice a school year.

DOK 4 typically has “the need to perform activities over days and weeks (DOK 4) rather than in one sitting…” While it is not necessarily a project or research activity, it would be extremely difficult for students to achieve complex reasoning in a single day. The determining factor is NOT time, however. “The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking.”

Is Coding DOK 4?

Is coding DOK 4? Coding is a verb and it is not the verb that determines the DOK level. When I first learned to code JavaScript and then wanted to write my first Add-on for Google Sheets in Google Apps Script I had to do a LOT of figure it out. I was not following a tutorial with steps and directions.Taking my knowledge of JavaScript and extended it to figure out how it applied in a different context of Google Apps Script.

I had an original project idea and need to adapt what I knew to a new situation. This was DOK 4 reasoning. However, today if I wanted to code in Google Apps Script I am probably at best doing DOK 2 level reasoning and possibly DOK 1. If I can effortlessly produce a script without making an error I am using my recall skills. The same task of creating code to arrange student names into tabs of a spreadsheet was DOK 4 for me, however, now this is a DOK 1 task. I no longer have to figure it out.

Difficult Vs. Complex

One of the reasons it can be so challenging to align a math task to DOK is the perception that if a student struggles with it they must be doing complex reasoning.

Just because it is hard does not make it complex.

Notice in the graphic below that on a matrix of complex and difficult that math tasks are commonly high difficulty but low complexity. Meaning they are DOK 1, but students struggle with them.

Is Math Teaching Critical Thinking? most common are problems that are hard but not complex. this shows a qudarant of difficulty low to high and complexity low to high. Most common are the upper left low complexity and high difficulty.

Surely, if a student is doing Calculus they are doing DOK 4. Actually, most of the Calculus book I had in college was DOK 1. If the book gives you the formula and then you use it to do practice problems then it is doing the thinking for the student and thus DOK 1.

Robert Kaplinsky

Robert Kaplinsky is, in my opinion, the expert in math DOK. If you are looking to increase the DOK of your class take some time to read Robert Kaplinsky’s blog. He is also the founder of Open Middle. There are DOK 2 and DOK 3 level math problems. Notice they are not DOK 4 level math problems because, again, DOK 4 is not the goal!

Building Thinking Classrooms

The book “Building Thinking Classrooms” by Peter Liljedahl took a research based approach to discovering how to get students to do more thinking in a math classroom. The book is an easy read and transformational in how we approach teaching math. Join the Facebook group.

Is Math Teaching Critical Thinking?

Analyze Current DOK Levels

Start by reviewing your existing math problems to identify their current DOK level. This will help you identify areas where you can increase the cognitive complexity of the tasks. Firstly, review…

Develop Learning Objectives

Next, focus on developing learning objectives that require higher-order thinking skills, such as analysis, synthesis, and evaluation. Secondly, develop…

Increase Complexity

Design math problems that align with the learning objectives you have developed. Ensure that these problems challenge students to apply what they have learned in new and complex ways. Moreover, design…

Incorporate Real-World

Incorporate real-world scenarios or data sets into the problems to increase the authenticity and relevance of the tasks. By doing so, students will be able to see the practical application of what they are learning. Furthermore, incorporate…

Design Rubrics

Create rubrics that align with the DOK levels of the problems you have designed. These rubrics will help you assess student performance and provide feedback that will guide their learning. Lastly, create…

By following these steps, you can increase the DOK levels of the math problems you create, providing a more meaningful and challenging learning experience for your students. Is Math Teaching Critical Thinking? It could be!

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