I was teaching another math lesson – being a substitute of sorts for the day. Simple enough, just do what’s in the book. It’ll make things easier.

Lesson books are dry – even when they have voice (still trying to figure that out, perhaps another topic). Somehow I have to say the words that the book says and the kids will (magically) say what the book says they will respond. There also be several moments to tell them to be quiet, and not very much time for them try and formulate what you’re trying to say. I don’t have a problem with direct instruction – and I’m not more experienced as a teacher than the writers of lesson books or curriculum, but I have instinct, and I have some training.

Instinct took over. The lesson seemed to go better. Just start teaching! (I thought) We started exploring several things at once, after all, this was the culminating topic of the unit. And we were on a roll. Then time nearly ran out, and we hit a snag. I wasn’t able to explore the final part – perhaps the most integral part of the topic. Everything else was going smoothly. I had flow and the students were responding. Everyone was talking, thinking, and learning. But I ran out of time.

I didn’t follow the sequence. I didn’t do the steps. I didn’t follow the timeline. Response here and response there. But they learned something. But I missed something. I didn’t prepare for this lesson, and thus didn’t see (or didn’t know) what would be the most challenging part – at the end.

Is it time to follow the book? Is it time to plan? Stepping in for the moment in someone else’s classroom, there might not be a choice. But there might still be room to make it my own.

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Like you, I find time to be a nemisis in the classroom environment at times. Working in mathematics with difficult concepts and sometimes exceptionally short periods of time to launch the concept, work on strategies, check for student understanding and still have a gradual release of the topic is a daily challenge, especially in a middle school environment where everything must transpire in 50 minutes. Working with the text helps with planning, but deviating seems critical as well as you watch for student understanding. Planning yet being flexible in the approach seems to work well. Personally, I try to keep my eye on the clock so I know how far off the projected lesson has gone, but ultimately understanding is the essential aspect of learning; there will always be more time the next day and another lesson that will go much faster than intended at another time.

You’re raising really important questions here! And what I hear you and the above commenter narrowing upon is the importance of supporting students in developing a conceptual understanding of the mathematics they are learning. I’m so curious to know more about that moment where you felt like you “started teaching!” Where you instincts took over. What do you think it was that helped the lesson turn and go better here? I notice you say that students were thinking and talking! Wow! There is a range of challenges and tensions you’ve raised. In math ed people talk and worry about following pacing guides and “covering” topics and going deeper and achieving understanding. I have heard thoughtful educators argue all sides. Those who advocate for pacing guides think carefully about making sure all children have access to the full range of mathematical concepts they should encounter in a given amount of time. It is an equity issue if some children never get to that final chapter on, say, algebra, and therefore miss those experiences. The other side of the worry is that a pacing guide forces teachers to move on and cover topics without the depth needed for thinking and learning. As if covering those topics means students learned them. What are your ideas about depth vs. breadth?

I think, for me at least, I can prepare or ready a lesson so much because the students will always react in ways you can’t imagine. I have only known the students for a few months, but even people you have known forever can surprise you. I think I felt like I “started teaching” when the students, and I were not only interacting with me, but each other. It felt like a teaching zone (briefly).

As far as breadth vs depth, I believe you need both. What I’d really like to explore is real math problems, not just word problems, manipulatives, etc. I believe you need both to solve bigger problems that can be approached multiple ways. I’m also all for pushing toward math as problem solving, less as hard math (meaning solving abstract equations using multiple methods). And in the real world, people use tools to solve things, but they still have to know how to set up equations and why.

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