Moving to Substack

This will be my last post on the WordPress version of my blog. I’m moving to Substack! In fact, I’ve already imported all of my old stuff and posted my first new piece over there.

I wrote a few months ago about how much I miss the blogging culture that we had ten years ago. I love writing and I will probably always write. I value every commenter who has pushed my thinking over here (in fact I have a piece in the works inspired entirely by a recent commenter). But the reality is that Twitter is where the community around blogging has been the strongest for a while, and teacher Twitter is mostly dead. I miss the community of teacher bloggers that once existed. I have no idea if it will ever exist again, but I think Substack is the best chance to make that happen. Every other possible replacement for Twitter just feels like a bad ripoff of the original. Substack is designed for long-form writing, and building community around long-form writing. That’s where I learn the most, so I’m going to give it a shot.

I will keep posting what I write to Twitter as long as there are a few teachers over there willing to chat and challenge my thinking. But I’m also hoping to build some community on Substack that is more focused on blogging and talking about blog posts using the tools they have.

If you’d like to read my writing you can click over to fivetwelvethirteen.substack.com, and either check back to see what I’ve written or subscribe and get my writing in your inbox once a week or so.

On Review Days

“Review days are a waste of time — if the students know it, they don’t need a review day.”

I hear versions of this sentiment sometimes, and I think the sentiment reflects a misunderstanding of how learning works. Learning is not a process that is either successful or unsuccessful. Some folks seem to believe that if they learned it, they will remember it, and if they didn’t learn it, they won’t. From that perspective review days seem like a way to help students do well on a test without actually helping them learn.

But learning is a dynamic process of acquiring, consolidating, rehearsing, and retrieving. Anything we want students to learn — whether it’s fluency with a skill, understanding of a topic, or developing a productive disposition — happens gradually over time. Anyone who learns something once and never uses it again will forget it. Some math skills come up over and over again and get practiced naturally, but some don’t. Of course review days can play a role in supporting long-term retention.

To be honest, the model of learn something –> review and take a quiz at the end of the week –> review and take a test at the end of the month –> review and take a midterm/final at the end of the semester is a decent first attempt at a schedule for spaced retrieval. I’m not saying it’s perfect. I could write a long blog post about how to structure retrieval for the long term. But it’s not a bad idea, and I see that type of review get attacked as “only trying to help students pass the test” and “they should have learned it the first time” and “traditional.”

I see the same misconception about learning at play in standards-based grading. Learning isn’t binary. But lots of standards-based systems pretend it is, and try to stamp learning as successful or unsuccessful and then never circle back. I know plenty of teachers who are thoughtful about this type of long-term review within standards-based grading, but the basic premise seems to assume that learning is a binary that doesn’t change with time.

This isn’t a “traditional grading is perfect” post. It’s not a “don’t use standards-based grading” post. It’s also not a “everyone should do review days” post. I’m only saying that learning is dynamic, and if we want students to remember what we teach beyond the test we should expect and plan for forgetting. Part of that is planning in review. If I give a test and lots of students get something wrong, that’s a failure. It’s a missed opportunity to refresh their memory of that concept. If I review the concept the day before and they get it right that doesn’t mean students will definitely remember the concept forever, but it’s way more helpful than not reviewing and watching students get it wrong.

An Argument Against Cold Calling

I’ve seen more debates than I can count about cold calling students, or calling on students regardless of whether their hands are up. But many of the debates feel philosophical rather than practical. Someone says, “I could never cold call, what about students with anxiety in math class? I don’t want to put students on the spot and make them feel dumb in front of the class,” or something along those lines. Then someone else says, “cold calling helps me see whether students actually understand because I’m calling on a random sample of the class. It sends a message that every student’s ideas are worth sharing and learning from while holding everyone accountable.” I want to critique that second argument. I’m sympathetic to the philosophical argument against cold calling — I am a lifelong stutterer — but I’ve also used cold call at times. I’ve become more skeptical of the benefits it’s supposed to have, and I want to explore that skepticism here.

Here are three things I bet most proponents of cold call will agree with. The first benefit of cold call is that the teacher can better check for understanding by hearing from a random member of the class rather than someone with their hand up. This way, the teacher can figure out when students don’t understand something and respond accordingly. The second benefit of cold call is to send a message that every student’s ideas are valuable. This holds students accountable for paying attention, and encourages students to get used to sharing their ideas and having their ideas interrogated. Third, it’s not something to do every once in a while. If you’re going to use cold call, spend time building a culture where sharing mistakes is valued and use it often so students know what to expect.

The check-for-understanding argument is the one I’m most skeptical of. It’s true that calling on a student at random (or targeting the question to a specific student I want to hear from) gets me more information than calling on a raised hand. But let’s say there are 25 students in the class. I went from getting one unreliable piece of information, to one more reliable piece of information. A better way to check for understanding is to use a whole-class response system. These days I’m partial to mini whiteboards so I can have every student write down their answer and hold them up for me to see. Then I’m at 25 pieces of information. Mini whiteboards aren’t the only answer to whole-class responses. Desmos activities are a great way to quickly see what everyone thinks. I also use finger voting on multiple choice questions. And having students write down their idea or answer on paper and circulating also works well — while I won’t be able to see 25 students very easily, I can look at 10 answers in a few seconds without much trouble, and 10 is a lot better than 1.

My other issue with the check-for-understanding argument is that cold calling a student and hearing a misconception isn’t very actionable. I have no idea in that moment how many other students have that misconception. I could bounce the question around the room, but that takes more time and it can feel pretty painful if it is a common misconception. And of course that one student could get it right but be a false positive, either getting lucky or happening to be one of the only ones who got a question right. I try to think about checking for understanding in terms of hinge questions. A hinge question is the hinge in the lesson: do I move on, or do I need to circle back and spend more time on an idea? Cold calling gives me a few dozen kindof unreliable hinges, and it’s hard to know what to do in response to any given question. Instead, I want to pick a small number of possible hinge points ahead of time, plan for a whole-class response system in those moments, and make sure I have reliable information to make that decision.

The second argument for cold calling is that it holds students accountable and sends a message that every students’ ideas are worth sharing. And yes, sure. But cold calling isn’t the only way to send that message. I’ve found that mini whiteboards also do a great job. And if a student isn’t engaged, calling on them can seem like a “gotcha” rather than a support. I have lots of students who struggle with engagement. Cold calling is only one of tons of different things I can do to try and engage them, from having a private conversation, to praising positive work, to warm calling to build confidence, and more.

I try to be intentional about how I invite student participation. Cold calling means I have to respond to lots of wrong answers and misconceptions in front of the class. Those moments can feel fraught. Sometimes they are valuable learning experiences for everyone. But other times they seem to drag, or a student freezes in the moment and I have to navigate what to do next, or they end up confusing students whose understanding is fragile. I want to share student ideas, but giving think time and sharing an interesting idea more intentionally can build confidence with much less risk.

I also find the whole conversation about cold call overly focused on verbal questioning. Verbal questioning is one part of math class, but I find students’ written work to be much more important for learning than verbally questioning through problems. If I want to break a problem down into pieces then verbally questioning can be a good choice, but mini whiteboards work as well. And big chunks of math class should involve students doing math, solving whole problems from start to finish. That necessarily needs to involve written work. How I respond to that written work can involve cold call, but I want to know what student thinking looks like before I start calling on students so I can plan for a productive conversation.

These days I rarely cold call. I use popsicle sticks sometimes, but only for low-stakes, low-floor questions like “what do you notice” or “which one doesn’t belong” that don’t have right or wrong answers, and where I want to get some quick participation from a cross-section of students. I don’t find benefits in cold calling students in “what do you do next?” or “and the answer to #3 is what?” type situations. This comes back to my point about the time necessary to build a positive culture around cold call. I don’t see many benefits, and it’s not worth the effort of trying to create a space where cold call is positive for my classroom culture.

Learning Is Slippery

I wrote last week about my new formative assessment structure and how it played out in my lesson on the triangle inequality theorem. I felt really good about that lesson and wrote the blog post that same night. The next day we did a few problems involving that same concept, and I was surprised at how many students who had been successful with the lesson didn’t remember a day later. We spent some more time going over it, and on Thursday I did a quick quiz thing including the triangle inequality theorem. Again, a bunch of students who had been successful earlier in the week weren’t. And at the same time, two students who had struggled in the initial lesson crushed the quiz.

Stuff like this is a fact of teaching. Learning is slippery. Teachers think something went well but it doesn’t stick like we want it to. Students surprise us. There’s no way to know for sure what a student has learned. I’ve wrestled with that idea a lot this year. I don’t think there’s anything wrong with the system I’m working on right now. I’ll keep using it. You can see me wrestling with the slipperiness of learning here as well, where I try out a next day system to see what students remember the day after a lesson, rather than while it’s fresh in their minds. But the reality is there’s no perfect answer. I can’t peer into students’ minds. I can do my best, and recognize that uncertainty is part of the job.

Now here’s what seems like a non-sequitur, but these ideas are connected for me.

I worked for six years at a semester school. It’s a school that students apply to attend for one semester, typically their junior year, for a “study away”-type experience in high school. The school was private and had most of the elements of a typical private school, but drew students from both public and private schools.

While I was there I noticed a conspicuous bias related to public and private school students. When a public school student succeeded, people attributed their success to the innate abilities of the student. When a public school student struggled, people attributed their struggles to a lack of preparation at their public school. When a private school student succeeded, people attributed their success to the quality of their school. When a private school student struggled, people found something to blame it on — our own shortcomings, the student’s parents, anything but the school they came from. You might be saying to yourself, “well maybe the private school students were just better.” I ran the numbers. The average grades students received were exactly the same. No difference. Grades aren’t a perfect measure of learning, but it seemed clear to me that it was bias and not a real difference in preparation.

There’s a big placebo effect in education. Learning is slippery. It’s hard to tease out cause and effect. But humans are constantly seizing for causal explanations. We seek out evidence that confirms what we believe, and there’s so much contradictory evidence of learning that it’s not hard to convince ourselves. It plays out in the micro level, where I convince myself that a lesson went well or a specific student learned something. And it plays out at the macro level, when we make inferences about what works and doesn’t work for a student in their education as a whole.

All this is a good reminder about the importance of humility. Teaching is hard. We know some things about what works, but there’s way more that we don’t know.

A Handy Little Formative Assessment Structure

Here’s a model I’m using for many of my lessons now. It’s nothing revolutionary. I’ve stolen all the pieces from people online or colleagues at my school. But the pieces fit together really well and have solved some challenges I’ve struggled with this year.

My lessons begin with a Do Now and some sort of number sense routine. Then is the “instruction” part where students learn new things. I use the Illustrative Math 7th grade curriculum as the backbone, though I modify many lessons or substitute in a Desmos lesson or something of my own creation. At the end of each lesson students do a bit of practice on DeltaMath (an online math practice website). But I’ve found that many students struggle with the transition from instruction to independent practice. I’ve had a hard time designing something in between that sets students up for success, helps me respond when the whole class is struggling with an idea, and identifies students for individual support.

Here’s what I’m trying now that feels like it does all of those things. After the instruction piece I have each student grab a mini-whiteboard and a marker. I ask them a series of questions, have them answer on the mini-whiteboard, and simultaneously hold up each answer for me to see. If students do well we move on. If they struggle with one concept I give a bit of extra instruction and some more practice, or step back and offer more scaffolding. If I see a specific misconception we stop and discuss it.

Then, for their last problem, I tell students to solve it, show me their answer, and when I check them off they can move on to DeltaMath. Any student who gets it right moves on. I pull up a chair with students who are struggling and talk them through a few examples and help them get started with DeltaMath.

For today’s lesson I substituted the excellent (free) Desmos lesson Can You Build It? for the IM lesson we were on. Students explored the Triangle Inequality Theorem and got some practice applying it in different contexts. At the end we pulled out mini-whiteboards and I gave them a few problems. Overall students did really well, with the exception of problems where the two shorts sides add up to exactly the long side, like 3, 5, and 8. We took some time to discuss this in partners and go back to the visuals in the lesson. Then I gave them one final problem: write down one set of three sides that can make a triangle, and one set of three sides that can’t. Most students got it. A few made small mistakes that they corrected and moved on — if they were students who had done well with the earlier whiteboard practice I could feel good about moving them on. Some students were still struggling, and I helped them with the problem and got them some momentum with the independent DeltaMath practice. This is really common! I often hear teachers use 80% correct as a threshold for a successful lesson. But what do we do about that 20%? This structure answered that question for me.

Here are three things I like about this sequence:

It helps me address class-wide misconceptions, like the example above trying to make a triangle with lengths of 3, 5, and 8. I can often spot these types of things earlier in the lesson, but mini-whiteboards are a great way to systematically check everyone’s understanding. Then I give a bit more practice or scaffolding where it’s needed.

It helps me support individual students who need help. Those students often hide in other formats. Mini-whiteboards help me see who needs help and hold me accountable for getting to everyone with the final problem. Without that final problem I would often know students are struggling, but it’s hard to keep track of who got which questions wrong in a sequence and then remember to check in with them. Having me personally check each student’s final problem before they move on holds me accountable.

It’s like an exit ticket, but I respond the same day. I’ve used exit tickets a lot, but most of the time I don’t actually do much to adjust my instruction based on what I learned. I typically have the next day’s lesson planned — with my current schedule I don’t have any prep periods between my last period teaching 7th grade math one day and my first period teaching 7th grade math the next day. I don’t want to stay up late re-planning all the time if I can address it in the moment. And too often when I do follow up I would only do a quick discussion or error analysis of the exit tickets before moving on, rather than catching it with mini-whiteboards handy and time to give students a little practice after some extra support.

None of this is revolutionary. Mini-whiteboards are old technology at this point, I stole the check-this-problem-with-me-to-move-on from the teacher next door, all stuff that’s been done before. But it fits together in a way that works for me. This process provides accountability for students, surfaces real-time evidence of their understanding, gives me a chance to do something about it, and holds me accountable for supporting every student who is having a hard time.

Different Reasons to Check for Understanding

I have worked a lot on my “check for understanding” skills the last few months. It’s hard! I have a lot more learning to do. But one thing I understand now that I didn’t before: a check for understanding is only as good as what I do with that information.

At a basic level, I check for understanding to see whether students understand what I taught. If they don’t understand it, I reteach it. But the more I reflect, the more I see different uses for a CFU:

Foundational knowledge. Do students remember and understand what they’ve previously learned related to this topic? Are those pieces of knowledge secure, or do I need to reteach before diving in to new ideas?

Conceptual knowledge. Do students understand the concept?, Do I need to explain it a different way or provide a different experience to help more students understand it?

Procedural knowledge. Can students solve problems of this type, or do they need more practice to make it stick?

Transfer. Can students apply what they’ve learned in a new or unfamiliar context, or do they need more opportunities to understand where this concept does and does not apply?

Each type of CFU leads to different next steps: reteaching foundational knowledge, offering new explanations or experiences to support understanding, providing more procedural practice, or promoting transfer. I find this distinction helpful because I can’t do all four of these at once. They happen at different places in the learning progression, they require different tools and different questions, and they lead me in different directions.

Why I’ve Been Writing So Much About Procedural Fluency

I’ve written a lot recently about procedural fluency, both why it’s important and how to help students acquire it. I keep getting responses along the lines of, “there’s a lot more to math than procedural fluency.” I agree! There is.

Here’s the thesis of this post in three sentences: There’s a lot more to math than procedural fluency. But procedural fluency is a legitimate and important part of math that is foundational for lots of other valuable mathematical learning. Doing procedural fluency well — helping more students get there more quickly — sets students up for success and leaves more time for everything else that’s important.

One reason I’ve been motivated to write about procedural fluency is that I’ve seen a marked shift away from emphasizing fluency in the last ten years that I’d attribute largely to the Common Core. I’ve lost track of how many times I’ve heard someone say, “now with the Common Core we teach for understanding,” as if we didn’t before. The Common Core emphasizes conceptual understanding, absolutely, but it says to pursue conceptual understanding, procedural fluency, and application with equal intensity. Somewhere along the way the idea became that if we just teach concepts better procedural fluency will happen on its own. Or maybe it’s that because the internet, Google, 21st century skills whatever whatever students don’t need fluency anymore. Or maybe it just became less cool to talk about fluency and more interesting to try and get better at teaching conceptual understanding. These ideas are most prominent in the online teaching world but they’ve had a huge influence in regular schools as well.

So here is my message to everyone who responds to my writing on procedural fluency with a variation on the “but there’s more to math than procedural fluency.” Yes! There’s so much more. Come to my classroom! You’ll see routines that get students thinking mathematically in different ways — contemplate then calculate, which one doesn’t belong, fraction talks, number talks, visual patterns, and slow reveal graphs, among others. You’ll see explorations, not because I think exploring is always the best way for students to learn but because I think exploring is something every student should experience in math class. You’ll see students learning about mathematicians inspired by Annie Perkins’ Mathematicians Project. You’ll see three-act tasks. You’ll see arguments and debates. You’ll see multiple representations. You’ll see fun tangents and challenges and more.

All of those things are important parts of mathematics. I think they should be part of more math classes. There are lots of ways to think mathematically, and we should expose students to all of them. But one legitimate way to think mathematically that is harder than many of the others is developing procedural fluency in a skill. The importance of fluency is often invisible to teachers simply because we already have it — the fact that it is invisible is exactly what frees up our working memory for other types of mathematical thinking.

One might get the impression that I think math class should be all fluency all the time But fluency is only truly necessary when a skill is foundational for something in the future. When remembering it rather than having to stop and think gives a significant advantage, it’s worth becoming fluent. That’s not every skill! That’s not even most skills. In 7th grade it looks something like this: finding unit rates, finding x percent of y, solving one-step equations (these are all skills from prior years that we build on and should continue practicing in the 7th grade context), adding, subtracting, multiplying, and dividing integers, and combining like terms. That’s it! Now another teacher might come up with a slightly different list, but my point is that procedural fluency isn’t necessary every single time. If we are thoughtful about when it’s important and do it well we can save time for all that other stuff while setting students up for success in the future.

And developing procedural fluency takes time. If it’s done poorly it takes lots of time. If it’s done poorly it’s pretty unpleasant for students. So I’m writing about procedural fluency not only because fluency is important, but because teaching for fluency is hard and worth doing well. Doing fluency well means that students have the foundation they need to expand their mathematical horizons. Doing fluency well means being thoughtful about when fluency is important and when it’s not, so we prioritize the topics that are truly foundational and don’t stress about the topics that aren’t. Doing fluency well means doing it efficiently, structuring and spacing practice rather than brute-forcing through endless blocked worksheets. And when all those pieces are in place there is more time for everything else that’s valuable in math class in addition to procedural fluency.

Procedural Fluency

Our latest position statement on Procedural Fluency in #Mathematics is out now.

Read the full statement here: https://t.co/fNbXf0kY3v pic.twitter.com/ITVVnSGHiV

— NCTM (@NCTM) March 22, 2023

This is garbage.

First, it’s poorly written and confusing. Second, NCTM is terrified of memorization. I understand memorization is often misused but it is also an important part of learning math. NCTM needs to provide leadership on what doing memorization well looks like, not put their heads in the sand. Third, it’s a great example of how to manipulate research. For complex and contentious topics like procedural fluency it’s possible to cherry-pick the existing research to make it say whatever you want it to say. That’s what NCTM is doing here.

I’m not going to provide lots of citations to things that back up my claims in this post. Finding some random study that justifies my teaching opinions doesnt make me right. I think it’s a useful exercise to write out my thoughts on procedural fluency and try to capture some better guidelines for teachers. Both research and my personal experience tell me that procedural fluency is important, and that memorization is one component of it. Here is my current understanding of procedural fluency:

Carefully evaluate when procedural fluency is necessary. Fluency is an important goal for skills that are foundational to future math learning. Fluency in skills like multiplication facts and one-step equations will make future learning easier by reducing the demand on students’ working memory. Fluency is less important for topics like the triangle inequality theorem or the quadratic formula.

Conceptual understanding and procedural fluency develop together; don’t worry too much about which one comes first, and recognize that each supports the other. I wrote a bit here about one way to decide where to start, but the important thing is not to be dogmatic about concepts before procedures and to teach both iteratively.

Use relationships and reasoning strategies to support memorization. For instance, students should not memorize 2*3=6, 3*2=6, 6/3=2 and 6/2=3 as four separate facts, but as a single relationship between the numbers 2, 3, and 6. For another example, students should be fluent in solving one-step equations — but they will never be perfect, and might use a trial-and-error reasoning approach to check their work when they encounter an equation they forget how to solve.

Strategies are an important complement to procedural fluency, but they can also impede memorization. For instance, if students only ever use trial-and-error strategies to solve equations, or only skip-count to answer multiplication fact questions, they are missing opportunities to retrieve from memory and develop fluency.

Spaced practice and interleaved practice are the best way to develop procedural fluency. Identify skills for which fluency is an important goal and make a schedule of spaced and interleaved practice, rather than blocking practice immediately after teaching a concept and then moving on.

Ensure retrieval is successful during practice to develop fluency. If students are always relying on reference sheets, a calculator, help from a peer or teacher, or other resources then they are not retrieving from memory and will not develop procedural fluency. To develop fluency, concepts and procedures must be successfully retrieved from long-term memory. I wrote a bit about that idea here.

Be humane. Math is hard. Developing procedural fluency takes time and can be a source of stress and anxiety. Be deliberate about where procedural fluency is an important goal. Provide support to students who need it. Make time for practice so all students have the opportunity to feel successful with a skill. Don’t shame students who need more time or struggle with accuracy. Be thoughtful about assessing procedural fluency in ways that don’t make students feel dumb.

Which Comes First, Procedural or Conceptual?

I’ve seen the procedural first vs conceptual debate first play out more times than I can count. Should teachers teach procedures first, concepts first, both in tandem, or something else? My answer is that it depends.

Here is a coordinate plane. The x-axis is the conceptual difficulty of a topic — how hard is the concept to understand? The y-axis is the procedural difficulty of a topic. How hard is the procedure to execute?

Every topic is a bit different! Multiplying fractions seems simple. If I tell a fourth grader that they multiply fractions by multiplying across they will probably say “sure, makes sense.” But ask a typical middle school math teacher why we multiply across and they will probably have a hard time. I know if I didn’t have time to prepare an answer I would struggle to give a clear and concise explanation of the concept behind that procedure. In this case the conceptual difficulty is much higher than the procedural difficulty.

Adding fractions with unlike denominators is complicated. Least common multiples are hard! Of course you can go with the product of the denominators, but then you end up with wasteful procedures for stuff like 1/5 + 1/10. Then you remember to only add denominators, and maybe simplify at the end. But the concept is simple! You can only add fractions with common denominators, because the pieces have to be the same size. Here the procedural difficulty is much higher than the procedural difficulty.

Here’s my rule of thumb: if the procedural difficulty is higher (quadrant II), teach the concept first. If students don’t understand why they can’t add fractions across they’ll think you’re wasting their time. If the conceptual difficulty is higher (quadrant IV), teach the procedure first. If I begin teaching multiplying fractions by trying to explain why the procedure works I’m likely to confuse students from the start. And since the procedure is simple I can start with the procedure — then leverage fluency with the procedure to help students understand the concept. If the two levels of difficulty are similar, it’s probably smart to teach both together, but the details depend on the topic.

I don’t think that my coordinate plane above should be canonical. They’re just my opinions, I’d love to hear where other teachers disagree. It’s a good exercise for teachers to put the concepts they teach on that diagram, and think about how their placement connects to how they are taught.

Novices, Experts, and the Knowledge Students Bring to Math Class

Here are two statements that seem like they might contradict each other but don’t:

Teachers typically overestimate the knowledge students have. Whenever I pick out skills x, y, and z that students will need to be successful with an upcoming topic and review and preteach them, I am surprised at how helpful that preparation is.

Teachers typically underestimate the problems students can solve. Whenever I ask students to try and figure something out, I am surprised at the variety and effectiveness of their strategies.

These can both be true! The distinction, to me, comes down to the idea of novices and experts.

An interesting result from research is that novices tend to learn more from direct instruction, while experts tend to learn more from exploration and problem solving. One issue with this research is that the distinction between “novice” and “expert” can feel fuzzy. When does someone go from being a novice to being an expert?

I find it helpful to see the distinction as being about how much knowledge students bring to a situation. Many parts of math are sequential. If a student comes to a topic lacking a foundational skill they may struggle to see the forest for the trees, need more direct guidance on where to focus their mental energy, and need a more structured learning progression. If a student has a lot of knowledge to bring to a subject they can be successful with less guidance.

Here’s an example. I’m teaching 7th grade inequalities right now. One piece of knowledge I might assume students have is fluency with the > and < symbols. They’ve seen them before, but they are never as fluent as they need to be. All students would benefit from a refresher of what the symbols are, what they mean, and how to use them in a few different contexts. If I skip this refresher I am setting students up to be novices. Remembering what the symbol means or puzzling through a new use of it will consume working memory. A big part of working with inequalities is connecting the idea of an inequality to what they already know about equations. If all of the foundational pieces are in place, students can come to the topic as experts because they bring a lot of knowledge and skills that they can apply in a new context. A bit of explicit instruction making those connections clear and they can do a lot more than I might expect. A student with that knowledge can be successful with less guidance and move more quickly to less structured problem-solving. Without the knowledge, students will struggle and need much more explicit and step-by-step instruction to move forward.

Here’s another example. A big topic in 7th grade is proportions. Most students arrive to the unit with tons of knowledge. They can often tell me that, if they drive for 2 hours at 50 miles per hour, they’ve traveled 100 miles, or that if they bike 16 miles in 2 hours they are traveling 8 miles per hour. That reasoning is a huge part of the proportions unit. I am setting students up as experts by drawing on what they already know. Where they are novices is formalizing that knowledge with precise mathematical language that they can then use to solve new problems. I might see students solving problems by finding a unit rate and assume they can apply that understanding elsewhere. Often they can’t, because they haven’t formalized their understanding in a way where they can apply it in an unfamiliar context. I can take their expertise, deliver some explicit instruction connecting it to the ideas of “unit rate” and “constant of proportionality” and help them expand what they can do.

My point is that novice and expert aren’t static labels that we can assign to students and leave in place for weeks or months or years. They are dynamic descriptions of the relationship between a student and what they are learning. When learning inequalities I can set students up to be experts by making sure the foundation is secure, then making clear the connections between what they already know and what they are learning. When learning proportions I can set students up as experts from the start by helping students recognize all the stuff they already know and can apply to problems. Then, I give formal mathematical language to what they already know and help them see how to extend it to new problems, building off of their expertise. I can’t assume they will absorb this language by osmosis; I need to be clear and explicit about it. In both of these situations students move back and forth between being novices and being experts. My instruction changes accordingly. In each situation it’s easy for me to overestimate the knowledge they arrive with, and easy to underestimate what they can figure out if I set them up for success.

Here’s a contrasting case. I also teach circumference and area of circles. This is a topic where I think students are best in the novice position. Circumference is a formula that doesn’t have much understanding behind it — the relationship is an empirical one, a pattern we’ve noticed and can use to solve new problems. That’s a very different type of formula than what students have seen before. The circle area formula does have understanding behind it, but the method of exhaustion necessary to see where the formula comes from is again a totally different way of understanding a formula than anything students have seen before. The unit is mostly about these two formulas, both of which are hard, and are hard in different ways. I do all the fun interactive stuff — we measure circles and find the constant of proportionality, and we count squares in big circles, and we use digital manipulatives to see how a circle can be rearranged into a rectangle. But I don’t pretend that those activities teach students for me. Some very clear, explicit instruction does the job — because students are novices at understanding formulas like these.

The structure of every topic is different. It’s easy to get lost in generalities during Twitter debates. One mathematical idea might set up a student as a novice early on, and then an expert later. Another idea might build off of a student’s expertise early but then move them into the novice role later on. These roles flow back and forth and blend together, and different students in the same room will fill different roles. There are lots more possibilities, and how a teacher approaches a subject affects this trajectory. I think the broader idea of novices and experts is helpful. But as soon as we start slapping those labels on students there’s a risk that we lose sight of what the label is actually describing.