I sometimes think about three possible levels of analysis:

1. The universal level, which I think corresponds to your idea of simplicity: e.g., "Care deeply for your students, and seek to understand their lives." Ideas at this level apply to all teachers everywhere.

2. The level of the society or the community: e.g., "Black students in the U.S. have a shared experience of..." or "First-generation college students have often been told..." Ideas in this level may apply across cities, states, or even countries, but we can also imagine teaching environments where they wouldn't be super-relevant. (E.g., the sociology of U.S. racism is less relevant if you teach in China; the hallmarks of first-gen U.S. college students are less relevant if you teach 2nd grade in Finland.)

3. The level of individual classrooms and students: e.g., "My student Dan was telling me he struggles to find a quiet place to do homework because..." Ideas at this level apply to nobody but me (and maybe a couple of my co-teachers).

It seems like conferences and PD often aim for the middle level. Which makes sense. The universal level can devolve into platitudes; the individual level can devolve into isolated experiences; the societal level aims at things we all share in common, and can thus (maybe!) talk about fruitfully.

The trouble is that the societal level is, as you're pointing out, the most complex and flow-chart-y. "Because of [complex historical pattern], we find [complex sociological pattern], which means that [statistical generality] unless [exception created by different historical pattern]..." That's how I hear your concern about overly intricate PD. This societal level tends to call for a lot of specialized historical and sociological vocabulary.

I'd speculate that the best PD touches all three levels. "Here's a universal value we share. Here's a story of how I struggled to live out that value with a particular student or class. And here's a takeaway about the ways we all (in this society) can reach for that value."

I think Mandy's suggestions (e.g., "What if we thought about teaching math like rough draft and revising?") can work in this way. The speaker articulates a universal reason this might be a fruitful metaphor; teachers share individual experiences in light of that metaphor; and by this process, we all build a shared (societal) understanding of how the metaphor might (or might not) help guide us.

LeWitt uses simple geometric structures to construct beautiful mathematical art forms.

We’ll learn to use simple pedagogical structures to construct beautiful mathematical learning experiences.

Also from my syllabus:

Essential Questions

How do I design transformative mathematical learning experiences for my students?

Big Idea

Teaching is infinitely complex, yet designing math learning experiences with simple consistent classroom structures and routines allows for transformative mathematical learning and ambitious teaching to take hold.

So I don't think teaching is simple at all given we have 15-35 free variables to account for all the time in addition to numerous constraints imposed on the system, but I absolutely agree there are 1. perverse incentives to present at conferences (and publish) across education, and 2. we need to be able to simplify without reducing, much like knowing the scales for musical improv.

When our school had a first year principal who was a long time ELA teacher, they asked me if there is anything in particular that they should look for in math classrooms where they would be unfamiliar with the curriculum. Since I knew they would be super overwhelmed with a million other things, I wanted my advice to be as simple as possible and so I said to always gauge how much students were genuinely thinking. Does the teacher "elevate student thought" was the tagline that is slightly more succinct but not as nice as your mantra. The hope was that the space was inviting enough that students felt comfortable sharing their takes so it elevated out of their own brains, that student opinions were valued and elevated to the center of the class, and that they had room to grow and elevate the rigor they could engage in throughout the year.

Side note: any tips for writing an NCTM proposal in a day? Also, shout out Joey and Cici for Play with Your Math!

"Elevate student thought." It begs all kinds of questions immediately (as does mine) but offers a much more durable & reliable compass needle than "try these 12 things in your classroom ... " etc.

Hi Dan! I love the thinking you are inviting here. You always have such a unique and insightful perspective on things and I appreciate the consistent element of surprise you keep in your newsletter :)

With respect to where I’m at, I think the work of a teacher, much like that of an artist or a writer, is to teach you how to /notice/. And for the greater part, appreciate the act of doing so in the process.

Incredibly simple, but also incredibly complex because as Sister Corita Kent frames it in her book, Learning by Heart, looking is not the same noticing. Looking is the beginning of noticing, but to notice something, is to see elements of its detail, interact with it, and learn how to connect it to a bigger whole. So how do I design a lesson that gets one to notice? Or equally, in your words, offer a type of noticing that “invites, celebrates, and develops student thinking”?

But also, who should be considering this question more, the teacher or the curriculum designer? I think when you have access to a really, really, really good curriculum it helps to free up a lot of space to do all that tinkering with crafting the actual delivery of the lesson. Similarly, when I think of the necessary ingredients that makes an “expert teacher” expert at simplifying things such that they are noticed by students, it almost always starts with the material they spent years collecting, adapting, and fine-tuning. (There is a lot of other necessary ingredients, but that’s a key one, I think).

I definitely agree with you that teaching needs to be simple. As one of the student intern teachers of the nation, I would say that if teaching is a totally hard and complex profession, then I would not choose of doing it as my career for the future. However, I admit that it is hard, and it does get tough sometimes such as when students do no pay attention in classes, or parents would complain of certain ways that you teach are not useful their children and so on. On the other side, I also think that this is a profession that involves teachers' intelligence for classroom management and content teaching, but together if you can practice as much as possible and knows that every student is different, and we cannot use the same way to teach every child, then the job will be simple. In my eyes, teaching is simply a method for teachers to help students to learn, tell students to distinguish what is good and bad, and how students can develop into a good person. Therefore, I think that teaching is simple and complex because it involves many different aspects and factors for an effective teaching to exist.

I love graphs! And this is a good one, but I don't think it really takes into account the multifaceted nature of teaching, and therefore what a multi faceted presentation would look like. Take a teacher, who is arriving at a conference with a thorough knowledge of the multitude of facets and the ones they are interested in. Let's say there are four. They have classroom management, which at this point is complex and easy, helping students solve equations, which is simple and easy, they are looking to choose a new textbook with new teaching methods, which is hard and complex. Oh and they are trying to love their neighbor like themselves, simple and hard.

What would this person wanting to get out of a session? Well they would want to move their position in some way or another. In terms of the textbook goal. The person would want to have a session that narrows down the choices for adoption, making the complex task a little easier. The person might also need to go to a session that complicates their understanding of solving equations, taking something that was easy and making it complex (perhaps through gaining a deeper understanding of how to value student work or something like that). In these two cases the teacher has moved in opposite directions from their sessions and has grown as a teacher because of it. If 'teacher growth', and not just 'simplification of teaching' was the goal, then sometimes things need to be complicated.

Simplification is one type of growth, but complication can lead to another kind of growth, and can be an important takeaway from a conference too.

I like that perspective, Carl. "It's more complicated than that," can be the necessary message for a teacher with the simple view.

My sense, however, is that the incentives around conference speaking (and maybe knowledge production under market neoliberalism more generally?) encourage all of us to specialize and complicate rather than broaden and simplify.

There simply wasn't much connection even to existing frameworks like Principles to Actions, whatever their flaws. For whatever reason, it feels necessary for all of us to generate our new fractalized way of seeing the same terrain. Which seems tough for teachers to me.

I always appreciate hearing your post-conference reflections. I haven't fully processed my own conference experience yet, so I appreciate this prompting.

I wasn't at NCTM for long, having just spent days at NCSM.... but I think I know what you mean. As a speaker, I think it's enjoyable to offer an idea -- what if we thought about teaching math like rough drafting and revising? what if we thought about teaching math like teaching yoga? -- and then think aloud a bit and invite dialogue during the session and beyond. Simple concepts turn out to be complex when I wrestle with them with others. And then people surprise me and help me as they share how they think about the ideas and I get to keep learning with them. As an attendee, I love when speakers share something that they're still wrestling with and treat the session as an opportunity to think with people.

👋 An idea I have in response is that it takes a lot of skill as a speaker / writer / thinker-type to keep the audience focused on the 10,000 foot view of "simply put, this is why we're here" while also exploring implications, tangents, thought experiments, etc.

I actually find teaching quite easy. Though, your point that there are a lot of competing priorities is one of my big complaints about what we place on teachers shoulders.

I am in the wonderful position of being able to volunteer teach and not having to do any of the s*it work! And it's amazing. And teaching is easy! And I love it, when I get to:

"invite, celebrate, and develop student thinking..."

So, I think teaching is easy to teachers with time and space and energy. Without those things, it's hard. All the things are hard, when you are saddled with competing priorities and not enough time.

Teachers need time and space!

I love the scope of your article. I love the movement of easy/hard to conference speakers to teachers to teaching. I struggled with the transition between the major points. Specifically: I didn't follow the transition from hard/easy -> conference speakers -> teachers. I think that's because you generalize "conference speakers," when in fact, I needed to understand, they are "conference speakers who are experts on teaching math." Given this, I can infer that you expect they are speaking about teaching, and, when they have conflicting priorities, they miss the mark. Also, your title is perverse incentives but the thing I took away was "what teaching is..."

BTW, I had the same experience this summer with Computer Science Teaching speakers at CSTA!

I sometimes think about three possible levels of analysis:

1. The universal level, which I think corresponds to your idea of simplicity: e.g., "Care deeply for your students, and seek to understand their lives." Ideas at this level apply to all teachers everywhere.

2. The level of the society or the community: e.g., "Black students in the U.S. have a shared experience of..." or "First-generation college students have often been told..." Ideas in this level may apply across cities, states, or even countries, but we can also imagine teaching environments where they wouldn't be super-relevant. (E.g., the sociology of U.S. racism is less relevant if you teach in China; the hallmarks of first-gen U.S. college students are less relevant if you teach 2nd grade in Finland.)

3. The level of individual classrooms and students: e.g., "My student Dan was telling me he struggles to find a quiet place to do homework because..." Ideas at this level apply to nobody but me (and maybe a couple of my co-teachers).

It seems like conferences and PD often aim for the middle level. Which makes sense. The universal level can devolve into platitudes; the individual level can devolve into isolated experiences; the societal level aims at things we all share in common, and can thus (maybe!) talk about fruitfully.

The trouble is that the societal level is, as you're pointing out, the most complex and flow-chart-y. "Because of [complex historical pattern], we find [complex sociological pattern], which means that [statistical generality] unless [exception created by different historical pattern]..." That's how I hear your concern about overly intricate PD. This societal level tends to call for a lot of specialized historical and sociological vocabulary.

I'd speculate that the best PD touches all three levels. "Here's a universal value we share. Here's a story of how I struggled to live out that value with a particular student or class. And here's a takeaway about the ways we all (in this society) can reach for that value."

I think Mandy's suggestions (e.g., "What if we thought about teaching math like rough draft and revising?") can work in this way. The speaker articulates a universal reason this might be a fruitful metaphor; teachers share individual experiences in light of that metaphor; and by this process, we all build a shared (societal) understanding of how the metaphor might (or might not) help guide us.

"I'd speculate that the best PD touches all three levels."

Very enthusiastically endorsed.

People learn math when they think and interact. The teachers role is to provoke, nurture, nudge thinking and manage (minimally) peer interactions.

Like it. Now I'm ready for your talk about x, y, or z smaller thing.

From my syllabus:

Sol LeWitt @MASSMoCA

LeWitt uses simple geometric structures to construct beautiful mathematical art forms.

We’ll learn to use simple pedagogical structures to construct beautiful mathematical learning experiences.

Also from my syllabus:

Essential Questions

How do I design transformative mathematical learning experiences for my students?

Big Idea

Teaching is infinitely complex, yet designing math learning experiences with simple consistent classroom structures and routines allows for transformative mathematical learning and ambitious teaching to take hold.

So I don't think teaching is simple at all given we have 15-35 free variables to account for all the time in addition to numerous constraints imposed on the system, but I absolutely agree there are 1. perverse incentives to present at conferences (and publish) across education, and 2. we need to be able to simplify without reducing, much like knowing the scales for musical improv.

When our school had a first year principal who was a long time ELA teacher, they asked me if there is anything in particular that they should look for in math classrooms where they would be unfamiliar with the curriculum. Since I knew they would be super overwhelmed with a million other things, I wanted my advice to be as simple as possible and so I said to always gauge how much students were genuinely thinking. Does the teacher "elevate student thought" was the tagline that is slightly more succinct but not as nice as your mantra. The hope was that the space was inviting enough that students felt comfortable sharing their takes so it elevated out of their own brains, that student opinions were valued and elevated to the center of the class, and that they had room to grow and elevate the rigor they could engage in throughout the year.

Side note: any tips for writing an NCTM proposal in a day? Also, shout out Joey and Cici for Play with Your Math!

edited Oct 7, 2022"Elevate student thought." It begs all kinds of questions immediately (as does mine) but offers a much more durable & reliable compass needle than "try these 12 things in your classroom ... " etc.

Hi Dan! I love the thinking you are inviting here. You always have such a unique and insightful perspective on things and I appreciate the consistent element of surprise you keep in your newsletter :)

With respect to where I’m at, I think the work of a teacher, much like that of an artist or a writer, is to teach you how to /notice/. And for the greater part, appreciate the act of doing so in the process.

Incredibly simple, but also incredibly complex because as Sister Corita Kent frames it in her book, Learning by Heart, looking is not the same noticing. Looking is the beginning of noticing, but to notice something, is to see elements of its detail, interact with it, and learn how to connect it to a bigger whole. So how do I design a lesson that gets one to notice? Or equally, in your words, offer a type of noticing that “invites, celebrates, and develops student thinking”?

But also, who should be considering this question more, the teacher or the curriculum designer? I think when you have access to a really, really, really good curriculum it helps to free up a lot of space to do all that tinkering with crafting the actual delivery of the lesson. Similarly, when I think of the necessary ingredients that makes an “expert teacher” expert at simplifying things such that they are noticed by students, it almost always starts with the material they spent years collecting, adapting, and fine-tuning. (There is a lot of other necessary ingredients, but that’s a key one, I think).

I definitely agree with you that teaching needs to be simple. As one of the student intern teachers of the nation, I would say that if teaching is a totally hard and complex profession, then I would not choose of doing it as my career for the future. However, I admit that it is hard, and it does get tough sometimes such as when students do no pay attention in classes, or parents would complain of certain ways that you teach are not useful their children and so on. On the other side, I also think that this is a profession that involves teachers' intelligence for classroom management and content teaching, but together if you can practice as much as possible and knows that every student is different, and we cannot use the same way to teach every child, then the job will be simple. In my eyes, teaching is simply a method for teachers to help students to learn, tell students to distinguish what is good and bad, and how students can develop into a good person. Therefore, I think that teaching is simple and complex because it involves many different aspects and factors for an effective teaching to exist.

I love graphs! And this is a good one, but I don't think it really takes into account the multifaceted nature of teaching, and therefore what a multi faceted presentation would look like. Take a teacher, who is arriving at a conference with a thorough knowledge of the multitude of facets and the ones they are interested in. Let's say there are four. They have classroom management, which at this point is complex and easy, helping students solve equations, which is simple and easy, they are looking to choose a new textbook with new teaching methods, which is hard and complex. Oh and they are trying to love their neighbor like themselves, simple and hard.

What would this person wanting to get out of a session? Well they would want to move their position in some way or another. In terms of the textbook goal. The person would want to have a session that narrows down the choices for adoption, making the complex task a little easier. The person might also need to go to a session that complicates their understanding of solving equations, taking something that was easy and making it complex (perhaps through gaining a deeper understanding of how to value student work or something like that). In these two cases the teacher has moved in opposite directions from their sessions and has grown as a teacher because of it. If 'teacher growth', and not just 'simplification of teaching' was the goal, then sometimes things need to be complicated.

Simplification is one type of growth, but complication can lead to another kind of growth, and can be an important takeaway from a conference too.

I like that perspective, Carl. "It's more complicated than that," can be the necessary message for a teacher with the simple view.

My sense, however, is that the incentives around conference speaking (and maybe knowledge production under market neoliberalism more generally?) encourage all of us to specialize and complicate rather than broaden and simplify.

There simply wasn't much connection even to existing frameworks like Principles to Actions, whatever their flaws. For whatever reason, it feels necessary for all of us to generate our new fractalized way of seeing the same terrain. Which seems tough for teachers to me.

I love the quadrant presentation. What age student are we addressing here? Teaching is not simple.

I always appreciate hearing your post-conference reflections. I haven't fully processed my own conference experience yet, so I appreciate this prompting.

I wasn't at NCTM for long, having just spent days at NCSM.... but I think I know what you mean. As a speaker, I think it's enjoyable to offer an idea -- what if we thought about teaching math like rough drafting and revising? what if we thought about teaching math like teaching yoga? -- and then think aloud a bit and invite dialogue during the session and beyond. Simple concepts turn out to be complex when I wrestle with them with others. And then people surprise me and help me as they share how they think about the ideas and I get to keep learning with them. As an attendee, I love when speakers share something that they're still wrestling with and treat the session as an opportunity to think with people.

👋 An idea I have in response is that it takes a lot of skill as a speaker / writer / thinker-type to keep the audience focused on the 10,000 foot view of "simply put, this is why we're here" while also exploring implications, tangents, thought experiments, etc.

edited Oct 6, 2022I actually find teaching quite easy. Though, your point that there are a lot of competing priorities is one of my big complaints about what we place on teachers shoulders.

I am in the wonderful position of being able to volunteer teach and not having to do any of the s*it work! And it's amazing. And teaching is easy! And I love it, when I get to:

"invite, celebrate, and develop student thinking..."

So, I think teaching is easy to teachers with time and space and energy. Without those things, it's hard. All the things are hard, when you are saddled with competing priorities and not enough time.

Teachers need time and space!

I love the scope of your article. I love the movement of easy/hard to conference speakers to teachers to teaching. I struggled with the transition between the major points. Specifically: I didn't follow the transition from hard/easy -> conference speakers -> teachers. I think that's because you generalize "conference speakers," when in fact, I needed to understand, they are "conference speakers who are experts on teaching math." Given this, I can infer that you expect they are speaking about teaching, and, when they have conflicting priorities, they miss the mark. Also, your title is perverse incentives but the thing I took away was "what teaching is..."

BTW, I had the same experience this summer with Computer Science Teaching speakers at CSTA!