On Friday, I co-presented a session with Lew Douglas: A Deep Dive Intro Transformational Proof. You can see the description in my previous post. Lew and I are gradually building up a logical framework for a transformations-based high school geometry course, and working on this session pushed our project forward. You can see our latest writings on my Transformations page. The session went well, I believe, except that there was not enough time for most participants to do the GeoGebra activities we had planned.
On Saturday, I presented Computing Transformations Using Complex Numbers and Matrices. That session was based on my approach to complex numbers in Algebra 2, and on some of the material I developed for my Space course. It went reasonably well, judging by the evaluations, though I did not dwell sufficiently on the trickier ideas, and moreover I ran out of time at the end. You can find the relevant links on my Talks page.
In both sessions, time was a bit of an issue, as it always is for a new presentation. But time was actually the topic of my Ignite talk: Time Pressure: Bad for Students, Bad for Teachers. Ignite is a format where you have exactly five minutes, and your presentation includes exactly 20 slides. The slides succeed each other every 15 seconds, whether you're ready or not. Preparing for this was pretty stressful. I'll link to my presentation when it goes up on YouTube. Meanwhile, I'll summarize the key ideas here.
If we want to teach for understanding and retention, and if we want to reach a broad range of students, we need to reduce the tyranny of the calendar ("you must learn this by Friday!") and the tyranny of the clock ("you must finish in 5 minutes!") I have written a fair amount about the calendar already. (See for example Lagging Homework (my most popular blog post ever) and following the links therein. Also: Pruning the Curriculum.) What about the clock? Here are some thoughts, at much greater length than was possible in the Ignite format.
Classroom DiscourseEven in a student-centered, collaborative classroom, teacher-led whole-class discussion is helpful to share ideas, to move the conversation to the next level, and to bring students into the institution of mathematics. The goal of classroom discourse is not to get through it as quickly as possible! Most of us tend to ask a question, pick the first student that volunteers an answer and move on to our next question. That approach is absolutely guaranteed to leave most students in the dust, and to limit participation to a handful. Instead, we need to find ways to slow down, so everyone has a chance to think and an opportunity to participate. Here are some techniques I've used to slow things down:
- "I'm glad you're enthusiastic, but you should raise a quiet hand if you have an answer. Blurting it out deprives your classmates of a chance to think." Never call on a student who blurted.
- Before choosing a student, wait until many hands are raised. You may even count. "Five hands... Seven hands..."
- Ask students to give their answers to a neighbor, before choosing someone to speak to the class. This dramatically increases participation, and emboldens some shy students.
- Ask students to rephrase what a classmate has said. This is easier than coming up with an original idea. It makes it possible to call on more than one kid for an important question. And over time, it encourages students to listen to each other, not just to the teacher.
- Write several answers on the board, keeping a poker face, and then discuss them. Or: have the students vote on which one makes sense to them, have the discussion, and then vote again.
- If a student is stuck while speaking to the class, or makes a big mistake, give them the opportunity to choose a classmate to help them (among those who are raising a quiet hand, of course.) This helps take the sting out of being stuck or having made a mistake, as the student's attention turns to their newly acquired power.
- Make mistakes that reflect student misconceptions as observed in this or previous classes, then discuss the mistakes. This models a healthy attitude towards making mistakes, which is a necessary part of learning.
AssessmentAt most schools, math is mostly assessed by way of quizzes and tests. Unfortunately, those are usually conducted under tremendous time pressure. But why does it have to be so? What if your quizzes were shorter? What if you gave students as much extra time as they need? What difference does it make if they answer a question at 10:20 and not at 10:15? What if you used at-home assignments, with no clock pressure, as part of your assessment repertoire? What if those included opportunities to write quiz and test corrections?
This is an equity issue: many students tell us that they understand the math well, but that they freak out when under time pressure. It is only fair to give them a chance to demonstrate that by using a range of untimed assessments.
(Check out my series on assessment, starting here.)
Thanks for reading all the way to the end!
PS: The day before the conference, I did pentomino activities with an after-school math circle for children in grades 2-4. At one point, a girl who had done a great job solving puzzles (perhaps an 8-year-old?) asked her mother: "Is this math?" I interjected: "Yes!" Her response: "I thought math was supposed to be serious." I was happy that my workshop had challenged that misconception! And yes, pentominoes are great with all ages. See my Geometric Puzzles in the Classroom page.