Low Floor, High Ceiling Math Tasks

Another one of those topics that is hard to write about but important to understand is how we try to teach reading, writing and math with a strong emphasis on deeper thinking. I have been wanting to write this post but have been putting it off because it is so challenging to explain but I am going to try now. I will start with a math example.

The week before last we were going to do a lesson on even and odd numbers. I started it as a regular lesson. I told the crew that I had a game that could only be played with a partner. Then I asked one student to stand up and I asked the crew if that student could play the game. They were confused at first and finally said that she could play if she had a partner. So I had another student stand up and they all agreed that those 2 students could be partners and play the game. We continued having students stand up one at a time and thought about whether that number of people could play the game. We created a chart that looked something like this:

Then I asked the crew what they noticed about the chart. They said things like:

• It is counting by 2s in the yes column.
• You count in a zigzag going back and forth between yes and no.
• The numbers in the yes column are doubles.
• It is a pattern.
• It looks like you are counting by 2s in the no column but those aren't the numbers for counting by 2.
• If you add 2 numbers in the yes column, the answer you get is in the yes column, too.
• I think those are called even and odd numbers.

There was much more but these are the comments that I can remember now. Every student was engaged in trying to put into words what they noticed about the chart. They were talking to each other and building off of each other's ideas.  In education, we call this the social construction of knowledge. Individually we all might notice a few things but when we put our ideas together it leads to much deeper understanding for all.

As they shared their ideas I made notes on the chart to show their ideas. Then we began to delve deeper into some of the ideas shared and this created more ideas.

The next thing we talked about was why does it look like you are counting by 2s in the odd column but those aren't the numbers you say when you count by 2s. After much discussion, we figured out that it was counting by 2s but that they started on 1 instead of 2. (Minds blown!) They were shocked that this was "allowed". Then someone noticed that if you added 2 numbers in the odd that the answer was in the even column. We had much discussion and checking to see if this was actually true. It was. Then someone asked "If you add numbers in the even column and get and even number and add numbers in the odd column and get an odd number, how do you ever get an odd number when you are adding?" Then it was time to go home so we picked up the discussion the next day and spent the whole math time using cubes and adding numbers to try to figure this out. We finally did figure out that when you add a number from each column to each other you will get an odd number. (An odd number plus an even number is an odd number.) They were so excited and feeling like real mathematicians. We continued the lesson in small groups the next day trying to answer the question of what happens when you add 3 even numbers or 3 odd numbers. They weren't ready to stop this discussion at all. This is exactly what you want to have happen in a math classroom.

So this long description is all to explain what a low floor, high ceiling math task is - it is one that all students can work on but take to a wide variety of different places. Some kids stayed at the "floor" level and worked to figure out exactly which numbers are odd and which numbers are even while others carried it through to figuring out patterns of adding even and odd numbers and counting by 2s starting on different numbers. We still have a lingering question out there about does all of this work when you subtract but we haven't answered it yet.

This whole lesson began with a simple activity and then a simple question: What do you notice? that then lasted for 3 days. These types of lessons don't happen every day but they are our goal. We want kids to get curious and want to try to figure things out. The strategy of asking simple questions and then letting the students talk will almost always lead to this type of discussion.

Some questions that we use frequently include:
What do you notice?
What do you wonder?
How do you know?
How did you figure that out?
Are you right?
What will you try next?
Why does that work?
What makes you say that?

We use these same types of questions when teaching reading and writing skills. We may be noticing things about a story, a piece of writing, or a word but we use the same process.

Kids learn more and have deeper understandings when they come up with the "noticings" and ideas that they want to talk more about. This is what Expeditionary Learning is all about.

Note: I should add that the crew was very excited as they were discovering new things about even and odd numbers. I love when they get so excited about content!

Disclaimer: We don't have these types of lessons every single day. There are times when a lesson is very teacher-directed because we need some very beginning skills in order to be able to have discussion like these.

Two resources for you to explore on this topic - one website and a video:

youcubed - website designed by Jo Boaler of Stanford University with tons of articles, resources and math tasks for students, parents and teachers