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.
Early last fall the speech-language pathologist I collaborate with suggested occasionally starting class with a short pop quiz for extra credit based on the material from the previous night’s homework. I don’t recall what issue it was meant to address, but I think it had something to do with students needing additional incentives/reinforcement to practice solving the problems accurately.
Here’s how they worked. The quizzes were usually 4-6 questions, with each question worth half-a-point added onto their homework score. Since they were worth extra credit, I didn’t guarantee plentiful time to complete them; when I needed to move on, it was time to pass the quizzes in. (Again, whining from the mathematically anxious crowd. And the chronically late crowd.) We didn’t discuss them together, but I passed them back, marked, the next day.
From a class management standpoint I liked that the quizzes helped get class started and reminded the students what kind of information they would be held accountable for. It also succeeded at giving students who completed their homework an extra chance to show what they had learned and boost their grades.
Because I didn’t want to offer extra credit all the time but I still wanted something to help get class started and give prepared students an extra chance to show what they had learned, I started doing “problems on the board” on off days. For these I simply spread problems of varying difficulty levels across the board and told the students to find one they felt comfortable solving, grab a dry-erase marker, and solve it on the board. Unlike the pop quizzes, these we did go over together after everyone was done. A couple additional benefits of this technique were that it started class with a bit of self-assessment as each student determined which problem to volunteer for and a little full-body motion as they went up to the board and solved it.
I had some cool math teaching moments last week. Here are two, one from each of my classes (Trigonometry and Algebra 2).
In Trig I’d been running around like a mad woman answering question after question and feeling like I was getting nowhere. So one day I said, “I’m staying here at the front of the room. If you have a question, come to me. If I’m already occupied with someone else, see if you can find someone in the class to answer your question.” The first person came up and, without thinking it, I put her question on the board while we talked about it. A couple more came up while the remarkable moment developed.
I was working with a single student, but as we progressed I could feel students in certain areas of the classroom growing quiet to listen in. They had apparently been struggling with the same problem and wanted their share of the help I was giving. Students who did not join us on that problem were, at that moment, all engaged in quiet group conversations about how to resolve their own questions. Cool, I thought.
For Algebra 2 I show up as a support teacher, meaning that I don’t set curriculum or plan lessons, but I work closely with students to reinforce the work of the lead teacher. Sometimes I really love that stuff. That day they were graphing inequalities on the number line. I began my work, encouraging students to describe their problems clearly, assess their own work, check their own answers. “Is that a true statement?” I asked 30 or so times. “Let’s test it,” I said probably 50. Cool, I thought.
Since I am teaching my very own trigonometry class this year, I get full access to a projector/whiteboard setup and will be using slides. I didn’t encounter slide-based classes until college, where I learned at first to dread them. Clearly, skilled professors knew how to engage and lead a class without the slide-show crutch, while slide-show users were boring and afraid of lecturing on their own.
I’m trying out slide-based teaching now for two reasons. First, I did later have professors who regularly used slide shows without putting anyone to sleep. Second, it takes a lot of class time to write everything on the board by hand and a lot of paper and ink to print everything out.
I’ve only prepared slides for the first two units, but it has taken a lot of time and several revisions. Here are my current guiding principals:
- Minimize text.
- Use the technology to animate, illustrate, represent whenever possible. That means photos, videos, and graphics — if they’re actually helpful.
- Spreading information over several slides is usually better than putting it all on one slide, even if you can fit it all on one slide.
- Put the starting point on the slide, then plan for live development and exploration on the white board, paper, etc.
- Plan to be mobile and interactive during the lesson, not a human-shaped mouse button-pusher.
- Plan to sometimes turn the projector off.
The trig students gushed unforeseen excitement about seeing me derive the Law of Sines today. They loved it! My plain old plan for it had been this:
- Go through derivation clearly but quickly so students see a connection between SOHCAHTOA and sinA/a=sinB/b=sinC/c
- Demonstrate using the LoS to solve a triangle
- Have students practice solving a triangle using LoS (make sure they do the practice problem)
Little did I know that step one would be so energizing and fascinating to them! Although I told them they would not have to reproduce the derivation, students who normally forget the proper function of notebook paper were furiously copying down the steps, asking questions (“In the area equations, what represents the height?” “What’s the pattern of the letters in the different equations?” “Why is it that way?”), and answering other students’ questions.
My favorite moment was when I asked the class, “Does everyone understand how I went from this to this?” A pedagogically poor question, it has traditionally been answered by silence, a dull or uncertain “Yeah,” or even a perturbed “We get it.” Today in my class it was answered by, “Yeah, you broke off that part and everything cancelled out except the a on the bottom.” The language is not terribly specific or mathematical, but it’s far more so than any of the traditional replies.
I wonder whether all derivations would strike these students as forcefully, or whether this one was especially suited to the task. Either way, I slapped a sticky note down on my lesson plan to remind myself to hit it out of the park again next year.
The experiment was a success! Students walked into class today asking to know what the inverse trig functions do (they phrased it as, “What does the negative-one mean?”), two students had reasonable theories to explain it, and one hit it right on the nose.
Before going over the worksheet, I teamed up the students for an activity where one student would write a sine or cosine value on the board, indicating whether it were a sine or a cosine; the next student would name one or several angles with that sine/cosine value and draw the angle/s on the unit circle. They were performing the inverse sine and cosine functions without knowing it. Later when a student attempted to explain his theory for the meaning of sin−1, he suddenly came out with, “I think it’s what we were doing on the board!”
In a class of nine geometry students, I am assigned to work specifically with three tenth graders. They attend lectures and class activities with the other students, then typically leave to work independently or go over homework with me. This allows students in both groups to get more attention from a teacher, and provides me with a sort of teaching apprenticeship. The students, like their classmates, have learning disabilities that cause me to continually reassess what I think I know about how they think.
My latest realization is that I’ve been overusing the whiteboard. At the beginning of the year we introduced angles (acute, obtuse, right and straight) and I found colored markers a great asset as we practiced identifying angles on the board. The students enjoyed working problems on the board with such a small group and even grew comfortable enough to work on the board in front of the whole class when the groups were together. Based on that experience I continued to use the whiteboard each day, expecting that a good method in one case would be good in all cases.
Only this week I accidentally let the students sit down and get to work on their own without rolling out the whiteboard. They worked at individual paces, and I was soon circulating to answer questions on paper instead of addressing each question with the whole group. I was able to see how each student progressed from question to question and was pleased to find them frequently succeeding. Clearly the whiteboard is an invaluable tool but should be given a rest from time to time.