Mathing…

Today, while teaching piecewise defined functions, a few students were visibly frustrated. One of the first examples that seemed to work was f(x)=|x| and how it can be split into 2 “pieces” where one piece is y=-x for x<0 and the other piece is y=x for x>=0. As soon as we got to a graph where the boundary was not zero, I could see myself losing students. One graph required them to find the equation of the line y=-x+3 for x<=-1. To many students, I was pulling y=-x+3 out of the blue. I extended the line to show how the line would touch the y-axis at 3 (the y-intercept) and looking at the tick marks from the graph we can calculate slope and see that it’s -1. It addressed the question of some students, but for others it was a tough sell, the frustration was already too high.

I tried different strategies where I isolated parts of the graph by covering up the other parts so that I could draw their attention to the relevant sections, but that obscured the entire picture. I tried redrawing each piece of a function on separate graphs and worked on their understanding of the intervals and choosing the right parts of the graph to put on the final drawing, but I think it was difficult for students who had poor understanding or ability to quickly connect x<-1 and x>=-1 and the sides on the left and right of the line x=-1.

Before this lesson (2 days ago), we practiced predicting the slope of the line by inspection. The slope is positive if the y-values increase as x-values increase (goes up as it goes to the right) and negative if the y-values decrease as x-values increase (goes down as it goes to the right). We also practiced guessing the magnitude of the slope as either 1 (forms a 45 degree angle with the x-axis), less than 1 (forms a “shallow” incline), or more than 1 (forms a “steep” incline) and went over the logic step by step to see that rise and run do match up to what we know about shallow and steep lines.

I expected all students to be able to put the different parts together as we started to work on piecewise defined functions, but no luck. I get this sense that when frustration builds up, it’s hard for kids to drop the frustration and focus on the lesson. So I started thinking about how much and how far I should adjust the instruction to help students in a classroom setting? Should I have spent the additional time in class to help 2 students in class with additional examples and skip covering new material by end of class? I decided to ask them to stay after class to review additional examples. I was glad that they stayed but they had to leave soon after. We went over additional examples and it seemed to answer their questions but I won’t know until I get a chance to assess them either in the opener or on a quiz at the next class meeting.

Over at Questions?, David Cox has a nice little anecdote about Stevie Wonder and how he stopped the band during concert and changed the key of a song to help a crowd volunteer sing the song more easily. David goes on to say that:

As teachers, we need to do the exact same thing every day.  No matter how well we construct a lesson, we need to be ready to adjust to the kid who continues to sing off key.  You can’t plan for that.  The band didn’t practice the song in every possible key just in case they had someone who couldn’t sing with them.  They knew their song, they understood the progression and understood what to do if they started somewhere different than where they had planned.

Beautiful story (read it first if you haven’t) and kudos to Stevie Wonder for recognizing the needs of his guest and adjusting accordingly, you can tell the kind of person that he is. More importantly, I think this illustrates 3 points:

1. A good math teacher should be able to adjust instruction, not just as part of prep in lesson planning, but also during lessons. For it to happen though, he has to have a good grasp of the subject material and be able to quickly understand the problems students are encountering so that he can adjust instruction by pre-empting (band practicing different keys before concert) or dynamically (adjusting on the fly midway through a song). But you don’t have to be Stevie Wonder to do what he did, you just have to have a good grasp of music and a willingness to help. It probably means good instruction that can more closely meet student needs is unlikely to ever be “teacher-proof.”

2. Teachers can adjust instruction when caring for one student (such as during tutoring), but hard to do for an entire class. If the 3 crowd volunteers sang together, who knows what Stevie would’ve done. Maybe Stevie would’ve realized that the 2 better singers could adjust to the new key and changed it like he did in the story. Maybe he would continue with the old key if he knew the 2 other singers would have trouble with the new one. Maybe he would keep the same key to discourage the bad singer to prevent the performance from worsening.

3. There is objectively a right key to sing the song. If the crowd volunteer were a music student/singer in the classroom instead of a guest at a concert environment (where thousands are watching), singing the song at the proper key still needs to be learned. Maybe the song needs to be adjusted to the singer for a better performance. But at some point, we need to tell students that their voice and training did not prepare them for singing properly and they need additional practice or tutoring if they want to continue in the class.

Maybe I’m over-analyzing and the analogy is falling apart.

Next year, I might try creating several entire graphs out of the different parts of the piecewise defined functions that I want the students to work on so that when it comes time to identifying the different parts of the graph, I can point to an opener they did earlier and get them closer and hopefully more focused on the lesson at hand. Things that I thought would have been easy because of the practice we’ve had, became additional burden and frustration while students learned the new topic.

Did I expect too much? Did I not assess properly? Will I be making things too easy by narrowing the question? Is it worth the time to sacrifice covering new material? During instruction, should I adjust instruction to the lowest common denominator or do set the bar high and help the lagging student move up after class? Does it change your answer to the above questions if you knew my school had open enrollment (anyone can take AP and honors if they wish)?