Absolute Value, Absolutely

Last year was the first year I taught Algebra 2.  I have just begun my 8th year of teaching, and for the first 6 years, I taught mostly Geometry, some Algebra 1, and a host of electives I was coerced selected to develop.  I was thrilled at the opportunity to begin to ascend the ladder of math courses, but, of course, there were some bumps along the way – afterclass moments when I realized how the presentation I thought was crystal clear was only that way to ME – who coined the term – The Tyranny of Knowledge?  I already knew the material, so it was almost impossible for me to understand how my students perceived it the first time it was introduced.  I felt like a completely botched a few topics with muddy introductions.

Among those topics were absolute value equations and inequalities.  I introduced them with a brief discussion of absolute value as distance, and then we worked on the procedures for solving them and finding solution sets.  There was little understanding and some outright confusion as a result of this abstract and mechanical approach.

In discussing this with my #alg2chat tweeps, @lbburke offered to share the Smartboard lessons she had gotten from @k8nowak on Absolute Value Equations and Inequalities (http://function-of-time.blogspot.com/2011/09/algebra-2-solving-absolute-value.html).  As I worked through the problems in both lessons, I was impressed with the simplicity with which the lesson built on the idea of distance, and used perfectly -paced discovery to teach not only the way to find solutions, but also to have genuine conceptual understanding of the Big Ideas behind the absolute value statements.   They are the lessons I wish I had designed myself! (Thanks once again, #MTBoS.)  I was psyched for my classes.

I have 2 sections of ‘gifted’ Algebra 2 (‘gifted’ in quotes because the students are thus tracked because they enter the school through a screening process which has nothing to do with being mathematically gifted) back-to-back – 7th and 8th periods.  Same teacher, same lesson, pretty similar delivery – questioning, prodding, occasionally lauding the beauty of math.  But different groups, and VERY different vibes.  The first class was practically hostile because I wouldn’t ‘just tell them how to do it,’ especially after one student who was transferred into the class 2 days ago announced that her other teacher just told them to set the absolute value term equal to both the negative and positive terms on the opposite side of the equal sign and solve.  Still I persevered, with 34 pairs of eyes glaring at me (well, I might be exaggerating, but it sure felt that way).   And while I appreciate that these students are challenging me to lead them to enlightenment, I can’t say that I wasn’t just a little relieved when the bell rang, and I could regroup and rethink my approach.

Enter 8th period.   A somewhat chattier group (closer to the end of the day?), less afraid to ask questions, seem to be better humored.  As soon as we worked through the warm-up, many students immediately grasped the idea of the absolute value equation as a calculation of distance.  This class also seemed to groove on the progressive nature of the discovery [SUCH a well-designed lesson, @k8nowak!].  They posed questions and answered each other – I felt just a little like the conductor of an orchestra rather than a direct teacher.  While there was still some mystery to the whole process at the end of class (what I wouldn’t give for those periods to be 10 minutes longer), I felt an intrigued energy among the students as they left – and, even better, a basic trust of the process in my classroom.

So I’m contemplating what to do tomorrow – in both classes.  I want to reassure my 7th period doubters, but not give in to procedural memorization; I want their trust and their willingness to contemplate the idea behind the math.  And, truthfully, I can’t wait to dig into the material again with my 8th period class; I am looking forward to seeing what understanding they bring back to class tomorrow.

And, as always, I am open to reflections, observations, and suggestions…