Finding Patterns

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At the end of last class I gave my students the preassessment for the Manipulating Polynomials lesson on the Math Assessment Project website.  I had ONE student even come close to understanding the essence of the problems.  Today I taught the associated mini-lesson.  I was  a little worried, but I trusted the process and jumped in with both feet.

I gave them one extra example, broke them into groups, and had them work on the cut-and-paste-poster activity.  I asked the students who were struggling to take one of the dot patterns and start by just writing down the number of black dots, white dots, and total dots.  Then we discussed how the number of each type of dot could be expressed as a function of n (not using that terminology, but you know what I mean).  There was some struggling.  There was some fussing.  There were even a couple of complaints of, “When am I ever going to use this?”   (More on this below) The satisfying part, though, is that they KEPT WORKING.  They worked through their struggles.  They asked for help when they needed it. (I admit, I broke down and gave them the formula to find the nth perfect square and nth triangular number.)  They discussed the problems with their peers.  They even seemed satisfied when they arrived at a solution.

I have worked very hard this year trying to have my AIS students see beyond the tricks and gimmicks that they may learn in class and to enjoy the rigor and perseverence of math.  This was one of the first times that they showed me that they were willing and able to reason mathematically.  They showed that eventhough the problem was challenging for them, they were willing to think analytically and to persevere.

Looking back, I think this task was successful for many reasons, but mostly because of the relevance of the task. Not the “real world relevance”, but the relavance quoted from Ben Blum-Smith on Dan Meyer’s blog:

The real test of whether a math problem is “relevant” is not “do you use this in ‘real life’,” whatever that means, but “do you want to solve it?” It’s not that you want to solve it because it’s relevant; wanting to solve it is what it means to be relevant.

These students were genuinely interested on some level.  I think that they were intrigued that they could find an expression that could describe what was happening in the picture.  They were excited that this even allowed them to find future values beyond what was on the paper in front of them.  This was empowering to see in a class room of students who dislike and have ven come to fear math.  I am definitely looking forward to finding more and more relevant tasks for my Math AIS students.

Side Note:  Speaking of relevance….I think a surefire way to decide if a task has lost it’s relevance is when a student asks, “When are we ever going to use this?”  This means that a student sees no connection to anything they could possibly ever do in the “real world”.  I have taken two distinctly different paths when being asked this question.  First, I have scrapped the activity and done something new.  When I do this I usually see that there is some underlying skill that they are struggling with.  Once this is remedied, the task at hand usually seems much more
“doable”.  Secondly, I have honest and frank discussions about what the student(s) want to do in life.  What are their goals, dreams, and aspirations.  Once they can remove themselves from the now and look at the bigger picture they realize that the tasks we are looking are appropriate.  Not to mention that it might be beneficial to practice and become efficient in the skills of rigor and perseverence.

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