Over the last few years, I’ve introduced several new aspects to my teaching: standards-based grading, student essays, prompting class discussion with questions on slips of paper, explorations with Desmos, assignments through Google docs, and so on. Some of these have had real positive effects, and I definitely believe in continuing to try new things. However, this year I’ve decided to focus on just one element of my teaching, which is **to engage every student in every class**. This means not worrying about all the potential newness, paying attention to what happens each time the class meets, and figuring out from those observations how to make every class meeting productive for everyone. This doesn’t mean I won’t try new things, but I want my focus to be on student engagement rather than experimentation.

Here are a few specific things I think this entails:

*More preparation before the semester begins.*I’m doing more work ahead of time to prepare my classes than I have before. Usually I make sure my syllabus has an outline of the topics in rough chronological order, a description of when homework is due and exams will be given, and a litany of other policies and expectations. Then, during the semester, I choose homework assignments as we go along and follow the schedule with some fluidity, which means lots of time spent figuring out just what the next class can cover. I want my time outside of class to be more reflective. That is, instead of emerging from class and picking a homework assignment that goes with what we did, I want to have time to think about what each student did during the class and what might encourage them next time to be even more involved in the work. Instead of spending prep time picking topics, I want to look at the topics already before me and think about how each student might connect with them. (Writing standards is already a big help towards this: when I consider what skills I want the students to demonstrate by the end of the semester, it forces me to balance the material, on a global scale, in terms of importance and time invested.)*More peer-instruction methods, like think-pair-share.*In other words, I should talk less (but PI is the positive formulation of this principle). How many answers can the students generate on their own? While some might think having students come up with the answers rather than providing a nice clear explanation myself would take more time, I am thinking of the fact that even in my “good” classes any explanation I give usually has to be given multiple times, because not everyone is focused at the same time. The next level would to be see how many*questions*the students can generate on their own before they start coming up with the answers, and I have that goal in mind. Nothing like trying to answer your own question to keep you engaged!*Effective use of silence.*I have absolutely no problem with periods of silence in my class. If nothing else, stopping the flow of information for a few moments now and then underlines the message that “class is not an info dump”. But I want to be sensitive to what*kind*of silence is occurring. The best kind is when you know there’s cogitation going on: the students are faced with a new idea or a collision of ideas and are trying to sort it out in some way they can enunciate. But there’s also the kind where everyone is just so baffled and lost that they can’t come up with answers, questions, or anything else. And sometimes in the silence you sense that the students know the prompt they’ve been given is banal, and responding to it proves nothing other than that they’re not literally asleep. I want to be attuned enough to know which is happening. Even better, I want the students attuned enough that they can tell me which is happening and whether the period of silence is worthwhile.*Finding and using “low-floor, high-ceiling” activities.*These are the kind of things anyone can get excited about. A student who is floundering should have something to grasp on to. A student who has mastered the material so far should have somewhere to grow. One way to do this is to have a whole bunch of questions of increasing “difficulty”, and I’ve used that tactic, but it conflicts with some of these other goals. In particular, someone who has trouble getting started on the list might feel at the end of class like they’ve failed if they don’t get to all the questions, and someone who rushes through and gets to the end might get the sense that there’s nowhere further to go. Moreover, when I ask more questions it leaves less room for students to ask theirs. I guess what I’m saying is that tracking down these types of activities is hard, and defies the way in-class activities are often done in calculus. (Possibly the objection I have to many traditional types of calculus problems, like Optimization and Related Rates, is that they have such a high floor and low ceiling. They’re basically puzzles, aimed at a particular level of understanding, which means they’re fun for some but not really broadly useful for learning.)*Being more deliberate about formative assessment.*This might be the hardest one for me, and yet I think it’s key to the whole endeavor. It’s easy to have a sense of how a few particular students and the class as a whole are doing. It’s easy to grade a quiz or a test and look over the results to draw conclusions about students’ understanding (a.k.a., summative assessment). It’s harder to come up with ways that encourage students to work independently, take risks, and also produce something concrete I can assess and provide feedback on. So I’ll be mining the math-twitter-blogosphere for ideas on a variety of ways to make formative assessments!

## 4 comments:

Hi Joshua,

What classes are you teaching? Just calculus? How big are your classes?

Also, what is your strategy for helping students generate questions? Do you have an idea of whether/how you will support that?

Finally, do you have an example (or better yet, source) of low-floor high-ceiling problems? I am not good at coming up with these.

Bret

Bret: This fall I’m teaching three different classes (for the first time! another reason to have lots of preparation done before the semesters starts!): precalculus, calculus, and analysis, probably with about 20, 30, and 25 students, respectively. I’ve taught the latter two at my current school, but it’s been a long time since I taught precalculus.

One way I think I’ll get students to generate questions is to be very explicit about it, particularly in class reflections (e.g., “What questions do you have about what we did in class today?”). Then, as they get practice asking questions outside of class, move that questioning into class. I also plan to ask “What do you notice?”-type questions, and after they make some observations, teasing out the “Why is that?” from them. This means I need to know what kinds of things interest them.

As for low-floor high-ceiling problems, I think they can be considered alongside the “ladder of abstraction”, as well as the way Dan Meyer describes taking textbook problems and making them more mathematical, less puzzle-like, by stripping away the superfluous scaffolding. One thing I’m considering is, when a problem involves working with a table of data (like populations or temperatures), having them collect that data themselves via Wolfram Alpha or some other source. Then they can even have some choice in what data set they use, and as they’re collecting it, they have the possibility of observing some patterns before they begin the formal analysis. I see those as low-floor characteristics. Finding a high ceiling is almost easier, because we can almost always see a way to push problems further in interesting ways.

Hi Joshua - love your post. Re Low Floor, High Ceiling Tasks - have you seen this great website by Jo Boaler et al: http://youcubed.org/teachers/

Really great stuff.

Engagement and participation are two things I am always working on, and I applaud your focus on it. Right now I am reading Invisible Children by James Pye, written 25 years ago, which specifically addresses those children who find ways to melt into the background. As I read it, I'm a little saddened to find how much of it describes things that go on in what I have always thought of an active, engaged classroom. I like your strategy for explicitly asking for students to generate questions, especially if they can write them down. I'm looking forward to reading more about your classroom this coming year.

@hermathness: Thanks for your comment! I'm taking Jo Boaler's "How to Learn Math" class for teachers over the summer, so I had heard of youcubed.org. Looks like mostly elementary and secondary ed topics right now, but I'll keep an eye out for college-level activities.

While I feel like I've improved greatly in the feedback I give, I want to do better at "early intervention" in my classes, specifically reaching out to students who are not benefitting from the class as much as they might. Since most of the time they're already present in class, it seems more effective to engage them during class time rather than to ask/require them to come to office hours (although hopefully they would come to my office once they're more involved in the class as a whole).

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