algebra in 100 minutes per week

We are counting down the number of lectures in my algebra classes. Only two more left, and then my math students have an end-of-year, cumulative final exam. One question for every concept we’ve covered this year, 70-90 questions.

Naturally, students are feeling a little stress. My email inbox is full, with requests for assistance on various types of problems. Rather than tediously sketch out explanations, I collect them all.

Then, on the day of their live lecture, I log in 10-15 minutes early, and allow them to chat with each other while I set up the class. I load my prepared slides, but in other tabs, I also load all the problems that students have sent me, as well as commonly missed questions from their weekly quiz.

When the clock ticks over to the top of the hours, I turn on my mic and greet all my students. Then I hit the record button and take attendance, acknowledging each missing student, (“Mary, where are you?!”), as well as the delayed students (students who never attend the synchronous lecture.)

At this point, I take questions, observations, and anything else students want to share with the class. In the last year, we’ve sung Happy Birthday many times, as well as offered condolences on the loss of grandparents, congratulations on winning awards, and reassurance about moves and dance competitions. Sometimes, students even bring jokes (“I would tell a joke about a line, but there’s no end point”) and memes to share with the class.

When we’re ready to get started, I ask if anyone brought me questions, and sometimes they do. I always start with those, opening a blank whiteboard and working through these, step-by-step, with students. These are re-teaching opportunities, making sure that all students understand the materials. I often end these problems with a survey, asking everyone about their confidence level. The third option is frequently crying with my 13-year-old self.

Then I move on to questions I received in my email. I always approach these anonymously, respecting a student’s right to privacy in their confusion. Sometimes, they’re easy concepts that were missed, but often students have difficulty with a slightly different approach to their problem.

Students can work to 100% mastery on their homework with the built-in examples, so by the time they send the problem to me, they’re frustrated. I work through these straightforwardly, modeling my decision-making process aloud, so they can see how each problem works.

I don’t mind running through a given problem, because they’ll see these types again when they’re working independently, later. My class runs on an 80/20 old/new, so they’re rarely “done” with any give problem type. In addition, problems are algorithmically generated, so seeing this exact problem again would be rare.

Finally, often 10 or 15 minutes into class, we’ll start on our new material. I usually cover two lessons per lecture, typically split into 20 minute blocks. I follow a direction instruction model:

  • Examples of supporting skills. Here, I refreshing links to old material, such as offering slides reminding students of the graphs of parent functions they’ve memorized up to this point, and the differences between translations, reflections, and rotations. This is a good opportunity for quick quizzes.
  • Tell them the general rule of the new item. Today, we’re learning about transformations and reflections of linear functions. f(x + h) –> Shift left h units.
  • Work through an example of the application of the general rule on the whiteboard, prompting student answers to calculations to keep them focused.
  • Offer exceptions to the new rule. How do you shift vertical lines? Horizontal lines? This is an opportunity for more advanced students to shine.
  • Explicitly detail worked models with exceptions. Notice that we did everything right in this problem, but this is an extraneous solution, because we had to square the equation to find the solutions!
  • Note common mistakes. Let’s list 5 ways people can make mistakes on these problems. OK, yes, definitely forgetting the negative sign. What else?
  • Provide faded, worked examples, like these. Note that I explicit list out steps, detail all the mini-calculations, and color-code as often as I can. I gradually fade out the mental calculations, and leave the bare skeleton to the end. Finally, I’ll just provide a problem and have students tell me how to do it. OK, what I do first? Mmhmm. And then? Next? Ah, ah. We missed something. What was it? Good catch, Caleb! Sarah, what should I do next?
  • Have students self-assess. I usually offer a three or four question quiz, like this:
    • I have the POWER! of knowledge about unlike fractions
    • I think I can, I think I can, I think I can
    • Nope. Nope. Nope.
    • I spaced out 10 minutes ago.

When students offer a negative answer, I often ask them to come meet me in a one to one session, after class.

  • Take any last questions. Often, a student doesn’t want to interrupt class, or look foolish, so they’ll wait until after I dismiss the class and ask me privately about what they don’t understand, or proffer other information.

After class, I make a note of any especially difficult problems, write emails about issues that have come up in class, and so on.

Not every class session will include every part of this routine, but most of my math classes do. This allows me to be responsive to student needs and still cover the material on my schedule. I cover every lesson in the textbook, every year. Students can’t learn what they’re not taught, and it’s my job to teach them.

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