I like this a lot, thank you for sharing! (even though I don't teach math 😅)
Over the years I have found myself breaking a big task into those "micro-skills" for students too. But a lingering thought is whether I'm just spoon-feeding them everything or not? Like is this even good for them in the long term? (my students are uni freshmen)
I definitely worry about that, and that worry makes more sense with older students. The balance I try to strike is being pretty structured with micro-skills, but also asking students to apply that knowledge in lots of different ways with less guidance (like the circle cutout problem in the post). I feel good about that mix.
Thank you for the clear and compelling explanation explication. Pretreating component skills is a very powerful aspect of teaching higher-order algorithms (and one with strong scientific support)!
"If we accept 80% success, we teach that final 20% a very clear lesson: even when they put in the effort, that effort is unlikely to lead to learning."
I would say two things: first, as long as I've been a teacher, I've had some students in my lowest-performing 20% who put in a lot of effort but still struggle with math. Second, yes most of those 20% don't put in a lot of effort. My argument, and my classroom experience, is that I can shift student motivation over time with the strategies I'm describing. It's not all students, it's not overnight, it's not always radical change, but it's real.
Not to say it's not worth advancing flexibility in subject delivery, but beyond a certain point you're running up against a law of diminishing returns. Sooner or later you just have to say those students have a brain geography better suited to different subjects.
See Dan Myers recent article on the necessity of friction for for learning and why AI tools that remove friction are counterproductive. https://substack.com/home/post/p-191304365
I think this ties in to your "effort leads to learning" idea.
I do quick, intentional, spaced practice based on upcoming needs just like you described with my students. The ones who put the effort in get it, with only one exception (she had a major SLD in math). Interestingly, as I work with a large variety of teachers (hundreds), almost nobody I work with does this. They do warm-ups, but it's the stuff in the curriculum, or a tpt worksheet.
They aren't thinking about "what do my students need to succeed in the upcoming lessons? What did we work on recently that they could use some spaced repetition with?" They are thinking, "my kids just don't have number sense or fact fluency, so we are going to keep practicing that stuff until they get it."
News flash... most kids who haven't gotten to a high level of "fact fluency"by high school aren't going to suddenly get there with the same exact approach that they have failed at for the last decade.
The warmup stuff drives me a little crazy. It's become really popular with the current generation of math curricula. While many of them can be perfectly fine to activate prior knowledge, they generally don't provide enough practice to make a difference for students who don't really remember something from a past year. There's a quick nod to that fact and then off to the rest of the lesson. This can be worse than nothing! It reminds students of something they don't really remember, but doesn't do anything to help them remember it!
Dylan, one of your greatest strengths is your ability to reflect deeply on your practice and articulate your process thoughtfully, connecting research to classroom application. You engage critically with new ideas, always sharing and justifying your perspectives with clarity and purpose. I’m continually impressed by how you manage to read widely, write insightfully, and teach with such intentionality.
I love this. To me this post is about intentionally building self-efficacy. I ran into a study a while back that strongly correlated student self-efficacy with math achievement on NAEP, and my hunch is that it is actually a positive feedback loop where developing stronger skills leads to more self-efficacy which then provides motivation to continue on with harder skills.
Yup that's a nice phrase for this idea. I definitely think it's a feedback loop, though the loop can become really brutal in the negative direction and hard to break out of.
This is terrific. I’m teaching Algebra I at a high school, and I’ve run into the problem of what happens when students don’t have the kind of experience that you described. So many have internalized that it doesn’t matter if they work hard, they just aren’t going to learn. Yesterday I was struggling to get students to engage with some visual quadratic patterns, and so many just wouldn’t even start. The most vocal were complaining about how I “wasn’t telling them what to do” when I had set up the tools they needed to solve the problems, but they weren’t willing to even start without being given the answers first.
My conclusion is that those students are probably acting rationally, given their experiences and the evidence available to them. I think our job as teachers is to give students experiences that show them learning is worth the effort.
Week of inspirational mathematics was designed to help students overcome this kind of (very reasonable, as Dylan noted) neurological conditioning. There is some good stuff in there, and some I don't like as much.
I feel pretty ambivalent about the week of inspirational maths. I use some of those activities, and some similar ones, but my experience is that a week of that stuff is totally insufficient for the problem we're talking about. I worry it becomes this throwaway week, and then back to regular math class without much to show for it.
I like this a lot, thank you for sharing! (even though I don't teach math 😅)
Over the years I have found myself breaking a big task into those "micro-skills" for students too. But a lingering thought is whether I'm just spoon-feeding them everything or not? Like is this even good for them in the long term? (my students are uni freshmen)
I definitely worry about that, and that worry makes more sense with older students. The balance I try to strike is being pretty structured with micro-skills, but also asking students to apply that knowledge in lots of different ways with less guidance (like the circle cutout problem in the post). I feel good about that mix.
Thank you for the clear and compelling explanation explication. Pretreating component skills is a very powerful aspect of teaching higher-order algorithms (and one with strong scientific support)!
"If we accept 80% success, we teach that final 20% a very clear lesson: even when they put in the effort, that effort is unlikely to lead to learning."
How many of those 20% put in the effort?
I would say two things: first, as long as I've been a teacher, I've had some students in my lowest-performing 20% who put in a lot of effort but still struggle with math. Second, yes most of those 20% don't put in a lot of effort. My argument, and my classroom experience, is that I can shift student motivation over time with the strategies I'm describing. It's not all students, it's not overnight, it's not always radical change, but it's real.
Not to say it's not worth advancing flexibility in subject delivery, but beyond a certain point you're running up against a law of diminishing returns. Sooner or later you just have to say those students have a brain geography better suited to different subjects.
As a novice math teacher, I thank you for sharing this insight.
See Dan Myers recent article on the necessity of friction for for learning and why AI tools that remove friction are counterproductive. https://substack.com/home/post/p-191304365
I think this ties in to your "effort leads to learning" idea.
I do quick, intentional, spaced practice based on upcoming needs just like you described with my students. The ones who put the effort in get it, with only one exception (she had a major SLD in math). Interestingly, as I work with a large variety of teachers (hundreds), almost nobody I work with does this. They do warm-ups, but it's the stuff in the curriculum, or a tpt worksheet.
They aren't thinking about "what do my students need to succeed in the upcoming lessons? What did we work on recently that they could use some spaced repetition with?" They are thinking, "my kids just don't have number sense or fact fluency, so we are going to keep practicing that stuff until they get it."
News flash... most kids who haven't gotten to a high level of "fact fluency"by high school aren't going to suddenly get there with the same exact approach that they have failed at for the last decade.
The warmup stuff drives me a little crazy. It's become really popular with the current generation of math curricula. While many of them can be perfectly fine to activate prior knowledge, they generally don't provide enough practice to make a difference for students who don't really remember something from a past year. There's a quick nod to that fact and then off to the rest of the lesson. This can be worse than nothing! It reminds students of something they don't really remember, but doesn't do anything to help them remember it!
Well said -- I'm gravitating more and more toward the Pam Harris method with problem strings nowadays for that kind of work. See my list of resources: https://sites.google.com/lcmail.lcsc.edu/irmc2/tasks-practice?authuser=0#h.wdlbuktoa6xc or Pam's excellent book "Building Powerful Numeracy for MS/HS"
Dylan, one of your greatest strengths is your ability to reflect deeply on your practice and articulate your process thoughtfully, connecting research to classroom application. You engage critically with new ideas, always sharing and justifying your perspectives with clarity and purpose. I’m continually impressed by how you manage to read widely, write insightfully, and teach with such intentionality.
Thank you!
I love this. To me this post is about intentionally building self-efficacy. I ran into a study a while back that strongly correlated student self-efficacy with math achievement on NAEP, and my hunch is that it is actually a positive feedback loop where developing stronger skills leads to more self-efficacy which then provides motivation to continue on with harder skills.
Yup that's a nice phrase for this idea. I definitely think it's a feedback loop, though the loop can become really brutal in the negative direction and hard to break out of.
This is terrific. I’m teaching Algebra I at a high school, and I’ve run into the problem of what happens when students don’t have the kind of experience that you described. So many have internalized that it doesn’t matter if they work hard, they just aren’t going to learn. Yesterday I was struggling to get students to engage with some visual quadratic patterns, and so many just wouldn’t even start. The most vocal were complaining about how I “wasn’t telling them what to do” when I had set up the tools they needed to solve the problems, but they weren’t willing to even start without being given the answers first.
I've been in that position over and over.
My conclusion is that those students are probably acting rationally, given their experiences and the evidence available to them. I think our job as teachers is to give students experiences that show them learning is worth the effort.
Week of inspirational mathematics was designed to help students overcome this kind of (very reasonable, as Dylan noted) neurological conditioning. There is some good stuff in there, and some I don't like as much.
I feel pretty ambivalent about the week of inspirational maths. I use some of those activities, and some similar ones, but my experience is that a week of that stuff is totally insufficient for the problem we're talking about. I worry it becomes this throwaway week, and then back to regular math class without much to show for it.