Taking 20 Minutes for One Problem? A look at a Low Floor High Ceiling Problem

Starting.  Seems so easy.  But for many students this is very difficult.  When all of your students can start on a question, then you as a teacher have accomplished a lot.  So I give a lot of low floor high ceiling questions like this one.  Low floor questions are ones that ANYONE can begin. The high ceiling part is where it is really challenging for EVERYONE to get parts of the question.  This combination is very important.  It's called DIFFERENTIATION.   Now the beauty of low floor high ceiling questions is that you don't tell the students how to do it.  You let them get struggle.  You let them come to you with questions and observations.  You try to HOOK them.  Here is an example.  I got this problem from @davidwees and @rachelfruin.  There are many problems like this at http://www.visualpatterns.org/

This was so fun.  The students really went after the problem.  I had them working in pairs.  I choose 5 or 6 pairs to do all their work on the white board wall.  I have taken pictures of a few examles. 

   

I loved the way this group separated the colors without using colors.  

This group was very methodical and to the point.  I thought it was interesting that they did not have any figure for their 4th step.  

This pair was so proud of their work.  Amazing detail here.  

This group showed the next step correctly.  They did not get the variable expression correct, but we discussed their thinking.  I love the legend on this one.  

My favorite part of this one is the check at the bottom to see if their expression was correct.

Most interesting was the fact that none of the groups used this idea.  We had done some problems previously that were not so colored centered.  See http://tinyurl.com/visualpatternspractice  

So we asked the students to think of this problem in terms of area.  If the step was 1, then the area was 2 times 5.  If the step was 2 the area was 3 times 6 and so on to make....  

However, it was a great teaching moment because we asked the students to show that the variable expression that they found was the same as the variable expression we all found.  It was an ah ha moment.  

Justify that these are the same

It was terrific.  One student did this.

Another student did this...

  The cool thing was that many of the students did this to justify

This took 20 minutes to do this one question.  An it was worth every minute.   I think this is what Micheal Fenton @mjfenton calls Slow Learning Math.  If you are on Twitter you can use the hastag #slowmathchat.    We thoroughly took a look at this problem.  We let the students choose how they wanted to represent the problem.  They didn't have a set way of doing it because we haven't showed them the way to do these.  The were engaged completely because they owned how they did it.  They were open to new ideas because they were confident in how they did it.  

I love being a math teacher.  I can't wait for a new question for the next lesson.

Where do you get your Low Floor and High Ceiling questions?

Do you show how to do things before or after your students have tried to them?  Is there a balance with this approach?

I'm Still Deciding about Flipping the Classroom

The Flipped Classroom Craze is out.  I'm still not completely sold on it.  I need more convincing that it will work for me.   (Flipping the class is having students do traditional classwork at home online through videos or websites and then doing activities and homework in class)  Here are my thoughts.  Please give me yours.  

Pros

• There is value in having students doing preparatory work before class instead of work that is all after the fact (homework)
• I like the idea of having more 1 on 1 time with my students.
• Homework will get a better start because you will have a network of people including the teacher to help you.
• Differentiated learning is much easier to accomplish in this setting.

Cons

• Watching a video is not always the best way of learning something
• Class discussions and engaging activities are vital in the learning process and they might be limited in the completely flipped classroom.
• The classroom environment will be less about the whole team and more about the individual
• Structured learning will decrease
• Not all students have easy access to the internet. 

Moot Point

• Not all students will prepare for the class.  This means that in the flipped classroom they might not do the prep work before class.  In the traditional class they might not do the homework.

I think a combination of flipping and regular might be a possibility for me.  I'm not ready to jump ship completely to flipped, but I like many of the Pros of flipping the classroom.  One thing I do know is that I don't want my students to watch a long "lecture" every night.  I would have to incorporate great online investigations (Geogebra or Desmos Calculator activities)  and some things they could discover on their own and then gather them back the next day to discuss what they did.

STILL DECIDING!!!!!!

Clicker Circuits

My colleague Tony Schlorff made a lesson for our Introduction to Algebra Team. It was a Smart Response 10 question review. I used it the other day and it worked great. He has the questions set up all over the room. They are numbered. The students each take a clicker and log in with their ID. They then can walk around to any problem they choose. They then put their answer at the appropriate number on in the clicker. They then answer all 10 questions. They are constantly MOVING around the room to go to each question. Once they get all 10 questions, they submit their answers. They will then get a chance to see which problems are correct and which are not. Most importantly, I get a can get a printout of the types of questions that were most missed. I also get a idea from tagging my objectives which objective was most missed. I know now which problems I need to work on for the next review.

If you haven't used the CLICKER system, you should try it. It is a great way to get instant feedback on my students learning.

Another thing that I can do is identify my students in thirds for the next review. This little review sets up well for differentiated learning groups for the next day.

If you are interested in looking at the smartboard notebook document click here.

If you would like to see the document as a PDF click here.