The Success of Failure

Making mistakes in an important part of life.  I make them daily. I tend to forget how old my students are, and I give them tasks that are way above their level.  I make assumptions about what prior background knowledge they are bringing to a task (Islamic Art is a great topic, but most of the kids don't even know what Islam is).

Today I gave my students (I teach grade 5/6) a very difficult task.  It was a task that puzzled a real Mathematician over 800 years ago.  His name was Fibonacci.  I read about the problem in Jo Boalers amazing The Elephant in the Classroom, and put in on my list to try.  Today, we dove in.

A man puts one pair of Rabbits in a place that is enclosed on all sides by walls. How many pairs of rabbits can be produced from that pair in a year, if it is supposed that every month each pair begets a new pair which from the second month on becomes productive themselves?

The real challenge of this problem, and the reason I choose to do it now, is that there are multiple strands of data that are changing with each iteration (in this case, a new month).  My students knew exactly what they had to do, and they go to work on a table.  I was quite pleased that they knew that a table would be the best way to organize this data.  Next, they experimented with the columns of the table (the rows were the months of the year, another point they noticed that made me smile).  This is where they struggled.  They could not see that there were not 2, but 3 levels of data here.  

A - Rabbits that can have babies

B1 - Rabbits that are just born and can't yet have babies (month 1)

B2 - Rabbits that are a month old and can't yet have babies (Month 2)

They just didn't see it.  I don't usually help with problems like this, I try and get them to do themselves by reviewing their problem solving process, or starting at the beginning, or listing what you know and what you need to know.  This time, I gave cryptic suggestions, hints, clues, and other little nudges.  Still, they didn't see it.  Finally, I sat them down on the carpet for a Math Meeting.  Once again, I tried to nudge them to the answer, but it just wouldn't go.  So, I told them.  I said and wrote on the whiteboard,  "there are three levels of data working here; A -Rabbits that can have babies, B1 - Rabbits that are just born and can't yet have babies (month 1 rabbits), B2 - Rabbits that are a month old and can't yet have babies (Month 2 rabbits).  Does this help?"

A sigh of relief when up from some of the students.  They ran back to their work and started adding third columns to their work.  There was still some confusion in what to do with the numbers after that, put the building blocks were in place, and the problem was staged.  More often than not, this is the most difficult thing about Math, not doing Math, but figuring out what Math to do.

We learned a lot through our mistakes today.