December 5, 2023

Low Floor High Ceiling High Cognitive Demand Activities

I won’t speak for other schools or classes but my students are far from homogeneous, in any way, but especially with regard to their mathematical journeys. This is why every activity I use is designed to have a  low enough entry point so that all of my students can engage but be able to be scaled up on the fly enough to challenge even my most advanced students. While this might seem like a heavy lift it’s not as challenging as one might think. This isn’t a result of great publishers producing awesome curriculum designed with this in mind (in fact the opposite is true). Mostly it’s a result of me reframing my own mindset.

A few years ago I was lucky enough to be able to attend CPM’s Academy of Best Practice where I was introduced to the Cognitive Demand scale (See Below). Now every time I moderate a student math activity I am constantly thinking “How can I raise the ceiling by increasing the Cognitive Demand?” For instance: I created a series of Desmo (Click here) activities based on Youcubed’s Week of Inspirational Math. At this point lots of math teachers at our site and across our district are using it to start off the school year. This year I have a new math teacher across the hall so I had the opportunity to hear about his experience using it in real time. He was going through the activities twice as fast as I was which got me thinking about how I engaged. All in all the activity was fairly simple but I engaged students on multiple levels with multiple objectives appropriate for my program. While the students have this activity already assigned to them and have it up on their computers I also have it up on the screen in front. This is only day 4 of school and day 2 of actually doing work in class. After we watch Jo Boaler’s short video about the power of having a growth mindset and I talked a little bit about it I flipped to the next screen and simply said, “Answer the questions’ and began to circulate. Because this is only day 2 the kids just sat there and worked quietly by themselves. A lot of kids actually typed all of the answers in the box provided during the time I allowed them to struggle on their own. 

After a couple of minutes I stated that we do not do math like this and had everyone stand up and copy the cube and questions on the board. This had multiple purposes. 

  1. It gets them in the habit of always doing their work standing up on their boards. (research says we think better on our feet)
  2. There is value in representing three dimensional objections in two dimensions (MP1, 3 and 8)
  3. When I circulate and see their answers I can instantly ask the clarifying questions that raise the cognitive demand. What do those numbers mean? How do you know? Where are they? Explain in words and graphically. Have you explained your thinking well enough so that when I give you a problem that may use a similar strategy it will help? 

Depending where each team is in their progress I direct them to make a cube out of blocks (I have a supply out on the table) for multiple reasons.

  1. Some teams get stuck and have trouble “seeing” all of the blocks in the drawing so by building the 3D representation it helps them to make the connection between the drawing, words and real life and keeps their struggle productive.
  2. Every team benefits from being able to see the problem with multiple representations
  3. I never tell a team if they are right but make them check their own answers using the cube they made.
  4. I encourage them to add drawings to their board that portrays their cube in a deconstructed fashion that allows them to see the missing middle cube.

In my class we don’t take notes but they are able to use all of the pictures they have taken (they embed directly into the Desmos activities which allows me to check it as well) of their work on the boards on their tests. One of the other multiple objects I hope to help students meet with this activity is to really create a great picture that explains the reasoning and solutions with multiple representations. Hence a simple activity that can be a 5 minute, CD1/2 answer getting exercise becomes a low floor – high ceiling CD3/4 activity that helps every student move forward.

Cognitive Demand Level 1 (CD1) – (memorization):

• reproducing previously learned facts, rules,  formulas, definitions or committing them to memory

 • Cannot be solved with a procedure

 • Have no connection to concepts or meaning that underlie the facts rules,  formulas, or definitions

Cognitive Demand Level 2 (CD2) – (procedures without connections):

• are algorithmic

• require limited cognitive demand 

• have no connection to the concepts or  meaning that underlie the procedure 

• focus on producing correct answers instead of understanding 

• require no explanations

Cognitive Demand Level 3 (CD3) – (procedures with connections):

• use procedure for deeper understanding of concepts

• broad procedures connected to ideas instead narrow algorithms

• usually represented in different ways

• require some degree of cognitive effort; procedures may be used but not mindlessly

Cognitive Demand Level 4 (CD4) – (doing mathematics):

• require complex non-algorithmic thinking

• require students to explore and understand the mathematics

• demand self-monitoring of one’s cognitive process

• require considerable cognitive effort and may involve some level of anxiety b/c solution path isn’t clear

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