Grade Narratives

At HGS we pride ourselves in truly knowing our students as people and as learners. At the end of both semesters, we do provide traditional grades, like A’s and C’s and the like. For all four quarters we also write narratives. I’m right in the middle of this process right now and not entirely eager to finish, so I though I’d share what types of observations I’m making in these write-ups.

Everyone at school strives to write these narratives using the “Oreo cookie” method of delivering bad news. First you say something positive, then whatever negative thing you need to communicate, then something else positive. It’s not exactly the best metaphor, because the white cream part of the Oreo is delicious, but it’s easy to visualize. I try really hard to make these narratives as objective and descriptive as I can, mainly noting observations about a student’s affect and behavior in class. I feel a little bit icky connecting information about behavior to an academic grade, but often it’s a great way to explain why a student was successful or struggled. Noting behavior doesn’t necessarily have to be about discipline – mainly I note things like asking questions or working smoothly in a group that will naturally help them do well in class.

I also try to be as specific as possible about their struggles and successes. Once I experimented with including test scores on particular objectives, but I didn’t like it laid out like that. If a student didn’t do very well on similar triangle proofs, it will tell them and their parents a lot more if I say “Little Jimmy struggles at times with communicating his ideas, but has a solid grasp of the mathematical processes involved in proving similar triangles,” than Similar Triangles: 44. The kids, however, know how they did on each assessment, and they are always given the opportunity to fix mistakes like that.

Here are some examples of narratives, obviously with names changed, in this case to the names of my cats:

Stanley is still doing very well in the class. He continues to take ownership over his own learning and asking as many questions as he needs. Stanley still prefers to work by himself rather than with his classmates, but he has been willing to try this quarter. In the past, Stanley has seemed intimidated by language-heavy math, but this seems to be improving. He seemed to respond well to the math words task, because although it was based on words instead of numbers, the task was asking him to discern a pattern. He also did very well with our work on fractals, where students were required to explain patterns in both algebraic expressions and words. Stanley can expect continued success in this class.

Olive is still doing well in the class. She has an excellent grasp of not just the mathematical concepts but their connection to the real world. She started the probability unit with lots of solid prior knowledge and was always willing to help her classmates, and to challenge herself to extend her knowledge. Her biggest challenge is still her self-confidence. While she is usually one of the first students to understand new material, she can tend to second-guess her answers. Moving into the end of the year, Olive has every reason to feel confident about herself as a math learner, and can expect continued success if she gains this confidence.

I wonder how I would have reacted to reading such thorough assessments of my learning in high school. It probably would have freaked me out, but maybe ultimately helped me as I moved up through college math.


Math and Jazz

I have been hearing that math and music are strongly connected for quite some time. I’m always interested in integrating art into my classroom, so I have been listening. The only problem is that I have utterly no clue about music. I definitely cannot read music, have never played an instrument (except for the recorder in elementary school, but I accidentally snapped that in half…), and feel like Fall Out Boy is a genuinely good band. Luckily, the internet was there for me.

I looked at these lesson plans for some inspiration:
NCTM illuminations¬† “seeing music,” love you NCTM
PBS Ken Burns JAZZ, “Math and Jazz: The Beat Goes On

We just did the straw graphs, and then all the wrap-up needed for them to be extra familiar with the graphs of sine, cosine, and tangent. They know not just what y=sin(x) looks like, but also why it looks like that. Now they’re working on what happens when the graph is y=4sin(x), or y=sin(220x), or y=sin(x)+1 (which sidenote, I do not believe has a musical meaning).

The first introduction to sine shifts was watching this video. They agreed it was pretty cool. This also helped us to discover that the larger the wave’s amplitude, the louder the sound, and the higher the pitch, the greater the frequency of waves. This was a great class because most of them know more about music than I do, so we were all teaching each other.

Then we followed the NCTM illuminations plan pretty closely. They found the frequencies of the notes in two octaves on a piano. *SPOILERS* In addition to reviewing geometric series, they discovered that the frequency of an A note in the higher octave is twice the frequency of an A note in the lower octave.

Things got exciting when we graphed a lower A, a higher A, and then a B note. I asked them to observe what how these graphs compared. The two A notes are harmonious – they would sound nice played together, and their graphs intersect in even intervals:


An A and a B note are too close to each other. They would clash when played together, and their graphs reflect that, kind of hitting each other randomly, as a student put it “just off”:


For an excellent example of dissonance in jazz, listen to City of Glass by Stan Kenton. It’s on Spotify!

I’m really glad I finally did this. I feel like I understand music more, and the students have something concrete to tie the very abstract concepts of sine graph shifts to.