I have started doing some tutoring on the side for one of the college students who helps at  my children’s elementary school afterschool program.  The young woman [I will call her Joanne] who needed help has impressed me over the years with her dedication to her work, and to the children.  Upon seeing her talking to another parent about a hard math test coming up, I let her know I was a math teacher and I’d be happy to help her.  We didn’t get together until recently, but working with her for just an hour was fascinating.

Joanne is taking what is essentially an Algebra I class at her college.  She has attempted three other math classes and has either dropped them or failed them, and she needs to pass one math class to earn her degree.  Her advisor suggested she try a self-paced course, which happened to also be computer-based, using the ALEKS program.

As we began to work on expanding polynomials, I shared a few techniques that seemed to help, at least at that moment.  Based on some of the comments she made, I realized that she didn’t really understand what an expression like “2y^6″ meant.  If students don’t understand the building blocks, and why they are useful, how can we expect them to make sense of multiplying one polynomial by another?  Our assumptions can be risky, and mostly they are risky for the students in our classroom.

The other thing that has stuck with me that Joanne mentioned was about her former math teacher.  She sees her high school teacher at her job, and that teacher has chided her about her math struggles, saying” Joanne, you know this stuff!  We did all of this together in class.”  Again, this is an issue of assumptions (which include mine as well).  What did Joanne understand when she studied these ideas the first time?  If she did really comprehend all aspects of her work at the time, what makes it so hard the second time around?  If nothing else, it points out how rarely anyone is being asked to use some of the skills we emphasize in high school.  How can we make the work that we do more relevant?

All of these ideas coalesced in my mind this weekend.  I was out in the city, and overheard two people talking about the incredible sales going on.  One person was accurately estimating what the sale price of something would be if it were 40% off.  They talked through their technique, and reached a correct answer.  Then, the person turned to their friend, joking and saying, “Math!  It’s finally helping me!”  This makes me sad.  I’m not quite sure what to do with that feeling, beyond keeping it in my mind every day.



I’m in the midst of conducting a few teacher evaluations, so as the department chair, I’ve been sitting in on a bunch of math classes.  I find a week of time when I can visit one particular course, so I can get a sense of the flow of the class, how a given day feels, and look more longitudinally.  Last week, I also participated in Instructional Rounds, a new form of school-based professional development that we are trying out as an institution.  On that day, I visited a Religions class, an English class and an art class, for 20 minutes each.  On a somewhat related thread, I have been teaching a colleague’s class for about two weeks while they are out on short-term leave.  It’s sort of like observing a class because I’m trying to figure out who they are, and how I can help them.

Overall, these visits have been really enjoyable.  Visiting the art class was probably my favorite, but I did realize that even just spending 20 minutes in a class gives me plenty to observe and a lot to think about.  Here are a few of my takeaways after being in others’ classes:

  • It’s invigorating and refreshing to watch my colleagues teach, regardless of discipline.
  • I can easily empathize with students who might have an entire class day with teachers talking at them.  I also find myself thinking about the student who is less inclined to participate in class discussions.  That person could spend the whole day not speaking, and not engaging, depending on the activities planned in each class.  I need to make sure my class engages all of my students at least some of the time.
  • A one-on-one private conversation with a student is often the best first step to take to understand someone.  I found myself giving the advice on multiple occasions to talk directly with the student whose actions are frustrating to you; step away from the heat of the moment, and ask some questions about why they are acting the way that they do.  Then, be prepared to listen.  How does that kid who is dominating the class discussion want you to notify them that they need to take a break?  Empower the student to navigate whatever situation you are in, and work on a solution together.
  • Returning student work promptly is always a good idea, and will always be appreciated.
  • Teachers doing all or most of the talking does not result in a good class experience.
  • Teachers need to be paying attention all of the time as to what is going on in their classroom.  They won’t see everything but they will see a lot if they look.  This makes me of a think of recent post from @dcox21 on When the Activity Isn’t Enough.  He wrote about watching students actively doing a Marbleslides activity on Desmos, and observing what we can learn about how they approach the challenge of finding the correct equation.  We all need to keep watching, and keep learning.

I’m sure there are many more items to add to this list, but these are my highlights for now.  When can you find 20 minutes to visit someone else’s class?  If you can make this happen, post something you learned or were reminded about by conducting that visit.

Good days

I recently read Zach Cresswell’s post on an ideal classroom.  I feel lucky because I’m teaching in what we are calling  a prototype room, as we think about new spaces for the math department at my school.  We have begun to find furniture and lighting, but I realized after reading Zach’s post that my class was not yet feeling like the class he described, and I wished it did.

It was pure coincidence that we were about to begin an axiomatic trigonometry multi-day lesson, where students would be working together and I would not be a prominent voice in the room.  These three days (especially the last two) made me so happy!  On day one, the first few proofs to do were fairly straightforward, so the students were working quietly and diligently on their own.  I didn’t like how it felt because I was hoping they would be talking to each other, and they weren’t doing that.  However, I realized afterwards that they didn’t need that help yet; they were doing fine solo.  Once they reached a harder proof, they immediately started talking.  For most students, that began on Day 2 of our in-class work on these problems.  Then, I started looking around (but forgot to take a few pictures) and realized I was seeing much more of the classroom that I hoped to have.  Even then, I couldn’t observe it all because I was deep in conversation for about 10 minutes at a time with one student at a time.  I did see students up at the board (of their own volition), talking through how to solve one problem.  Others were huddled (standing) in a group of about 6 or 7 people, listening to one person explain how to think about another question.  No one was using our two beanbags that day, but three girls were at the stand-up desk, and others moved around the room when they wanted to do so.  I had music playing through the projector.

I was already excited before I walked in the door to Day 3 of this unit.  In many ways, this was a repeat of Day 2, with new problems to consider.  A few new things arose:

  1.  My two students who were in the play and in the midst of run week apologized for not making it farther through the packet.  As I had already said to them, this was an experience and not the type of assignment where every student needed to get to the last question.  Whatever they could do in class would be the right amount.  I have to believe that this shift of expectations helped to make their week more manageable.  It also made me wonder if I could make this same statement about other sets of problems I ask kids to do.  Is it ever about doing all of the problems?
  2. One of my freshman began to shine.  I had seen his energy before, but others in the class didn’t yet know him well enough to see his “light.” As I looked over his shoulder, he definitely did not need my help.  Instead, he wanted to show me the proof he wanted to type up for homework.  They were going home that weekend with the assignment below:

Using LaTeX, type up three of your Axiomatic Trig proofs.  Include:

(1) a proof where you felt great about your final product, but you really struggled figuring out the proof

(2) a proof where you knew what to do, did it, and wrote a great proof


(3) a proof that showcases something else about this week for you.  In your .tex file, be sure to explain why/how you chose the proofs that you did, and what your third proof represents for you, and why you selected it.  How did this week go for you?  As a student?  As a learner?  As a way to solidify prior trig knowledge?  I really want to know what you think!  This is a journal entry-type thing but your writing will be with your proofs a .tex’d document.  

This student was so proud of his work on a particular problem that he knew he wanted to use that proof as his “3” choice proof.  It was fun to hear his exuberance and to be able to share that with him!

3. This same student had a chance, at my suggestion, to help two others in the class who hadn’t made it as far, and were struggling with one particular proof.  I love seeing a freshmen teach a junior something, and have that be a real, fruitful conversation.


So – as I had these two great days, I want to try and help myself remember what led to this moment:

  • letting go of being at the front of the room for any significant portion of class time
  • encouraging students to help each other, particularly building connections across grade levels
  • having students progress through material at their own pace, and only set the amount of total time spent on the unit (here, three days)
  • recognizing that students working quietly and productively can also be useful

Mostly, how can I provide just enough of a structure that there is something to explore, and then get out of the way so the students can dive in?