At the end of last year I posted a handful of goals I had for improvement. Here is this year’s version.
- Create clearer expectations for the logarithms unit — or at least make sure the unclarity is purposeful.
- Improve scientific notation materials.
- Revise the vocab lists for my word wall.
- Be careful about continuing instruction while students are still taking notes on prior info.
- Plan for more “discuss/work with a partner.”
- Provide examples of written work by previous students — practice critiquing them as class starter activities or as a way to get the writing juices flowing before a journal assignment.
Graphing Stories was a great addition to my class this year. It started as the Possible or Not activity that I used during our first unit (about functions, operations, input, output, etc.) and grew as I made my own graphs on Desmos to draw out understandings of new function types.
In addition to the learning value of the stories, they were also kind of fun — especially those for graphs that weren’t functions.
At first I required students to write or type their stories; then I provided individual whiteboards so they could just jot them down; then, sometimes, I let students share their stories without writing them anywhere first. Sometimes I meticulously demonstrated an interpretation of the graphed information, while other times it was merely a discussion.
I dictated that we were looking at distance vs. time graphs, but had the students make decisions about what units the time and distance were measured in and what they were measuring distance from. (In the future I might alternate between determining those measurement details myself and asking the kids to do it.) I almost always had to remind students to include observations about speed. (Relatively slow or fast? Constant? Getting faster or slower?)
On tests I included the extending challenge of providing a story and asking students to sketch the matching graph.
During the end of year review, the student who chose this section (Interpreting Graphs) for which to plan his own review activity took the opportunity to bring us all outside. He had prepared 5 graphs using different function types and distributed copies of those graphs to the two teams. On the basketball court the teams took turns acting out a graph’s story while the other team figured out which graph it was.
An extension that I’ve considered but not tried is taking the graph’s action from a book or film students are studying in another class.
In September, while taking that Jo Boaler class I’ve mentioned, I finished a section about asking open questions that allow students to use different approaches and learn from each other. That’s a valuable idea, I thought. Unfortunately it’s just not compatible with my upcoming unit. Simplifying expressions is all about getting a single answer and there’s really just one way to do it.
The idea of asking open questions kept turning in my mind, though, until an application was suddenly obvious. I would keep the goal of having students write expressions in simplest form intact, but would add a dimension by having students also write expressions in non-simplest equivalent forms. This would allow students to follow their own thought processes and provide individualized responses, while also developing the concept of equivalence.
I wrote an expression in simplest form on the board, then asked students to write it in equivalent ways using
- Subtraction (the requirement of like terms for addition and subtraction was a worthwhile challenge)
- Negative Exponents*
We used this activity during the simplifying unit, and returned to it during reviews of previous material. Some students understood the meaning of the activity quickly, while at least one didn’t perceive it until just before the spring final: “Oh! We’re just writing things that will give us that first thing.” Bingo. Sometimes comprehension takes time. That’s why I’m in favor of continual review and intelligently designed cumulative finals rather than the take-a-test-and-forget-about-it model.
*I didn’t think to include negative exponents in the list until late in the year.
I organized this year’s trigonometry final according to theme. The sections of the final were:
- Vocabulary [synonyms, examples, descriptions, comparisons]
- Equivalence [simplifying expressions, factoring expressions]
- Functions [linear, quadratic, exponential, trigonometric] and Graphing [linear, quadratic, sinusoidal; domain and range]
- Interpreting Graphs [distance vs. time]
- Solving Quadratic Equations
- Exponents and Logarithms
- Angles and Trigonometric Values [sine and cosine; deriving the values for angles in quadrant one, providing the values for other standard angles]
I find I’m a big supporter of the cumulative final, even and especially for students who struggle with long-term retention. How else will they train their minds to hold on to things? I’m an equal proponent of intelligently designed cumulative finals. My final this year was not a test designed to congratulate those with natural retention and punish those without it. We spent time throughout the year, plus a good chunk there at the end, building student retention of important skills and information, making my final an opportunity for students to take pride in having actually learned things.
Not that my final was a cake walk. The expectations in the test were high to match the value I intended it to have.
My school has a unique tradition of spending the first two weeks after Christmas break in what’s known as Minimester. Instead of regular academic classes, teachers and a few parents offer other interesting classes, for example, Philosophy and the Twilight Zone and a class about fly fishing. There are some oddities to the Minimester experience, but now that I’ve seen it happen a couple times, I’m thrilled to be teaching two great classes.
The first is called Shakespeare in Performance. Unbeknownst to each other, an SLP and I both proposed classes on Shakespeare, so we decided to join forces and split the class’ focus. My contribution is that we’ll be reading Much Ado About Nothing (my favorite!) and watching clips from three different film versions to compare the performances and broaden students’ views about how Shakespeare can be performed. (When the seniors read Hamlet earlier this year, many of them decided that the 1996 film version was “right” and that other performances were not. I want to emphasize that Shakespeare’s plays are malleable, interpretable.) We’re going to read the modern parallel text in the No Fear Shakespeare edition, which I’m sure makes some people cringe, but it’s a good entry point for these students, especially considering that we’ll be steeping them in the Shakespearean text in other ways.
Speaking of which, my colleague’s contribution is that students will be memorizing and performing monologues (using Shakespeare’s text) from the same play we’re reading and watching. We’ve chosen a speech by Don John (the villain), and two by Benedick (one as a cynic and one as a smitten lover).
My second Minimester class is about creepy stories. Together we’ll read and watch different kinds of creepy stories, rank the creepiness, and identify elements that made it creepy. So the kids can also practice creating creepy stories, applying the creepy elements they’ve observed, they’ll be turning innocent children’s picture books into delightfully creepy stories.
The ignoble motivation for this post is to regain some of the self-respect I lost by not completing 85% of Jo Boaler’s class by the deadline. The due date was clearly stamped on every page, but instead of checking what day of the week that date corresponded to, I said, “I think that’s next Monday,” and moved on as if that were fact. I logged in to complete the necessary tasks today only to discover that everything was due yesterday.
The 79% that I completed before the deadline was not remotely wasted. My teaching has grown and my classes have changed as a result of it. And since the course was free, I can’t lament the sunk cost. But, gosh, I wanted that certificate for passing the course! As superficial as it is, that certificate kept me going in the class when all my other responsibilities and interests threatened to crowd it out.
So to everyone who cares (that’s exactly . . . no one but me), please make note that I did not give up on the course partway through, despite sore temptation. I persevered, and had every intention of reaching the passing mark, if not for a sloppy view I took of the due date.
There. Totally vindicated.
I’ve been loving the new things I’m doing this year. I should be posting more often, but hey.
I’ve started a pattern of spending the day after a test, before the tests have been graded, introducing a journal writing prompt for the kids to do for homework. After the test my kids took this week, I played two short clips from Jo Boaler’s video lesson about making mistakes in math class. The first clip described why mistakes are important to learning, how they can create two rounds of brain activity that don’t exist when problems are solved without mistakes. In the second clip Jo gave guidelines for teachers about how to get students to overcome their fear of mistakes. The writing prompt I gave students suggested that they
- Describe a time when they had made a mistake in my class
- Talk about how they feel and react after making a mistake in math
- Give suggestions for what the class could do to help them view mistakes differently
On average, the journals I received in response to this assignment were less insightful than I’d hoped. But one or two bright spots showed that my work on embracing mistakes as part of the learning process hadn’t been completely lost on the students.
Here are the two clips.
For some reason I can’t get the videos to start at the right point, so navigate to 0:46 on this one to see the bit I showed my class.
Start this one at 1:22.
This is another product of the class I’m taking online from Jo Boaler about learning and teaching math.
My trig students are reviewing functions a bit right now. I used this piece of a chart showing average height by country that I found on Wikipedia to build the idea that functions and relations don’t only exist in the isolated world of equations and numbers. By assigning numerical values to information in a chart, you can turn just about any info into a mathematical function, or at least a relation.
We first assigned numbers to the countries of origin (1-11) so that we could have an input value instead of an input country. Using the male data seemed simpler because of it’s proximity to the list of countries, and had the added benefit of the four N/A entries, so we could talk about whether the input 2 (Argentina) were really part of the domain if no output value exists for it.
We listed the domain and range, illustrated the mapping of input values onto output values, listed ordered pairs, and graphed them.
Viewing a chart as a function was a stretch for some (especially whenever I used the word input–“Input? What are we putting it into?”), but is a step toward breaking math out of isolation and realizing its contented existence beyond the classroom.
Even after my vocab-building epiphany, I was inclined to be overly forgiving of slow vocab development. Fortunately my school has speech-language pathologists on staff to collaborate with teachers, and my SLP was quick to direct me toward more exact standards.
For example, one of our favorite vocab-building activities was Math Catchphrase (later dubbed “Mathphrase”). I laminated little cards with mathematical terms on them (some were review words, others were new for current material). Then we played Catchphrase, but instead of reading the word from the disk you pass around, students grabbed their word from a turned-over stack of cards in the middle of the table. On each corner of the table was a piece of paper with the words “Skip Zone,” where students could put the words they had to skip so we could talk about them at the end of the round. After watching us play it once, my SLP saw that the game needed an obvious external incentive for the students to use mathematical descriptions of the words instead non-mathematical ones. So I spruced up the scoring system to reward mathematical descriptions, which made a big difference. Unlike regular Catchphrase, we would reshuffle the words and play again with the same stack for the sake of repetition and reinforcement.
For her end-of-year review, a student came up with her own vocab game (“Draw, Act, or Describe”) that I’m excited to throw into the mix this coming year to prevent overusing Catchphrase.