When the Answer is E: He Falls Off the Roof and Breaks His Neck

Our annual state testing season is almost here. The juniors will partake in the Minnesota Comprehensive Assessments in Mathematics a week from Tuesday. Our department decided issuing a practice test to all of our juniors would help re-familiarize them with long lost skills. After distributing copies during our monthly staff meeting, I’m always curious if any teachers in other disciplines look at the practice materials. Much to my delight, the choir director approached me at lunch on Friday, test in hand.

Mr. Warren: Is this test just like the MCAs?
Me: Most likely similar. Why?
Mr. Warren: Ok, well look at this one.

 

Mr. Warren: I think the answer is E, Xai s going to fall and break his neck.

The conversation went on for another few minutes, with me agreeing  that what’s been called “math education” includes ignoring the context of situations and focusing on a procedure.  In fact, I was curious how many juniors who completed this practice test even noticed that the situation was outrageous.

Since we were running on a 2-hr delay schedule Friday, I thought it would be the perfect opportunity to present the problem to my algebra class. They are mostly juniors who have been continually frustrated with a mathematics curriculum that doesn’t make any sense in the real world.

Me: Read through this problem. Does it make sense?

Student: ok, it looks like 32.

I didn’t expect any of them to apply any trigonometry, so I thought we needed to approach the problem differently.  In fact, I wasn’t even concerned about the angle measure.  I wanted them to look at the scenario itself.

Me: Imagine this scenario. We’ve done a lot of estimating in here. We need to envision a 20-foot ladder, three feet away from a house. Does this seem reasonable?

Unfortunately, it did seem reasonable to most of them. I needed another approach.

Me: ok, how could we simulate this in classroom-scaled size?

Student: Get a ruler.

Me: Perfect. How close does it need to be to the wall?

Students: (a chorus of answers)

After exploring multiple methods of calculating exactly how far, we arrived at 1.8 inches.  With as much drama as possible, I set the ruler against the wall, exactly 1.8 inches away.

Me:  Does this look like a ladder that any of you would want to stand on? (of course, a few did).  Keep in mind, this is a TWENTY foot ladder, not a 12 inch ruler.

Student:  Yea, I don’t think anyone is climbing up that ladder and coming down in one piece.

Another Student:  What if they had a spotter?

A spotter!  Now we’re talking.  To be honest, I have no idea if a spotter could hold a 20-foot ladder so that it could be placed three feet from the wall.  But now I’m interested to find out!

I know Mathalicious investigated a similar scenario using a claim from Governor Janet Napolitano.

In my mind, these are the questions that should be circulating Facebook and aggravating parents.  This is the kind of math that should rile up Glenn Beck and company.  Our state of Minnesota opted not to adopt the Common Core State Standards in Mathematics, but requiring this kind of math instead is what is actually dumbing down the curriculum.  It assumes that the real world doesn’t apply, only rote procedure does.  “Just figure out the answer, don’t question the situation,” is what kids read and do over and over when problems like this are solved without real context.  A richer classroom experience for both teachers and students comes when we ask students to assess the reasonableness of situations, create new scenarios that are more appropriate, and solve the new problems they develop.  The CCSS Standards for Mathematical Practice tell students that it’s vital that they “construct viable arguments and critique the reasoning of others.”  I don’t think “critique the reasoning of others” should be reserved for only reasoning created in the classroom.  I’d like my students to critique the reasoning of the creator of these types of problems and others like it that have been deemed a necessary component of high school math success.

Thank you, Mr. Warren for igniting the exciting conversation in my classroom.

 

Class Commences – an hour I won’t soon forget

Recently, Michael Pershan unearthed a Shell Centre gem straight from the 80’s (literally).  This collection of materials is fantastic, and hopefully demonstrates to both students and teachers that engaging in rich tasks and high-level thinking is timeless.

I decided to give the function unit a shot in my Algebra 2 class today.  Some background on this group of students:  there are 38 juniors and seniors, last hour of the day, in a class geared toward lower-level students.   So far though, the only thing that’s been “lower” in this class is the number of empty desks I have.   I handed out this task, gave minimal directions and let them go for a few minutes on their own:

 

from:  Shell Centre for Mathematical Education, University of Nottingham, 1985

from: Shell Centre for Mathematical Education, University of Nottingham, 1985

It was so interesting to watch the different ways each of them started.  Some began with 7, since that was the first you saw when reading the graph from left to right.  Others insisted to work from 1 to 7, identifying the corresponding people along the way.  A few worked the other way around, from the people to the graph.

I walked around to make sure each student was able to get started and that those who thought they had determined a solution also supported their claims.  Then, I wrote the numbers 1 – 7 on the dry-erase board, stepped back, and let these kids amaze me.
One student volunteered an answer, and then handed the marker off to another.  I intervened only briefly to make sure that every student had an opportunity to contribute if he or she wanted.  Once 7 names were completed, I knew a couple of them were out of place.  I sat and said nothing, and this entire class showed me what they are capable of.  Here was a class full of students labeled mathematical underachievers completely nailing SMP #3.  Their arguments were viable, their critiques constructive, their discussion productive.  It bothered a few of them that I wouldn’t let them know if/when they were correct.   But most of them are starting to understand that my main focus here is not the correct answer, but the incredibly rich and interesting process they used on their journey to finding it.  They came up with multiple ways to support their answers and noticed tiny details about the people that supported their findings.  For example, did you notice that Alice is wearing heels? According to my students, that is perhaps why she appears slightly taller than Errol.

I had a heart-to-heart with this group when we were done about how proud I was at how they conducted themselves throughout this task.  I’m really thoroughly looking forward to a fantastic trimester with this special group of kids.  Their work on this task gives both of us the confidence that they can tackle something more difficult next time, and they are capable of mastering high-level mathematics this trimester.