Author Archives: dkane47

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

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.

Artificial Intelligence That Helps Teachers

I’ve been told by the internet that artificial intelligence is taking my job, or something. I’m not personally very worried right now but what do I know, I’m just a simple human. I do have a request, though. Presumably, before AI can do all of the things a teacher does, it will be able to do some of the things a teacher does. That sounds awesome! I would love AI to help me with some of the more mundane and tedious tasks of teaching. Here is a list of a few things I would appreciate help with:

  • Grade. Grade stuff for me! Not even everything, but help with the basics. I can already have a computer grade multiple choice stuff easily, but AI could help grade short answer questions and justifications.
  • Identify common errors. I give AI a set of student work, and it tells me “lots of students had trouble with single-digit percents, you should spend more time on that topic.” Interface between different software. I use DeltaMath for practice, then drag those grades over to Google Classroom where we communicate with students. Then those grades get imported into SchoolRunner where we have our gradebook. Along the way I modify a few grades based on particular circumstances — students on a special plan, excused absences, things like that. This process takes a surprising amount of time. I would love to push a single button and have all that happen for me.
  • Automate tasks that are easy to forget, like excusing a student on medical leave from assignments for a specific period of time.
  • Take attendance! Get me an AI that can take class attendance, that would be awesome. Especially if it consistently remembered to change attendance for students who arrive late.
  • Transcribe and summarize a lesson for a student who is absent.
  • Write recommendations. Some of the writing should be mine, but an AI that is optimized to write a recommendation by pulling that student’s grades, classwork, participation, and more could get me a great start.
  • Write a basic sub plan when I’m sick. Nothing crazy, just put a few tasks together, make copies, post it all to Google Classroom, and email it to the relevant people.

Too often articles at AI only want to use the word “disrupt” and tell us how the sky is falling. That might be true. But I’d like to imagine a world where AI helps people do the jobs they do better and improves our quality of life, rather than only displacing humans. I think it’s possible.

What else could AI help teachers with? And if our tech overlords are reading this, could you try building some of these tools for us?

What I Mean When I Say “Blogging is Dead”

I’ve run my mouth on Twitter a few times in the last few months saying something along the lines of “so sad that math teacher blogging is dead.” Inevitably a few great folks stop by to say, “hey blogging isn’t dead I still blog! Plenty of teachers still blog! Stop making it sound like no one blogs.” Foot in mouth. Awkward.

I shouldn’t say blogging is dead. What I really mean is that the blogosphere is dead. And someone reading this is probably getting mad so I will be more specific. Here is what feels different about blogging today compared with blogging ten years ago:

  • Blogging stood alone. Ten years ago lots of blogs had a blogroll in the sidebar — a list of other blogs they recommend. If you clicked those links you’d find 10-20 or more other blogs, most of which were posting anywhere from once a month to once a week. You could find hundreds of blogs just by clicking through blogrolls and links in blog posts. Twitter was important, sure, but blogging culture could exist on its own in a way that it doesn’t anymore.
  • My feed felt alive. I have had a Feedly account for a long time to track math teaching blogs. I have one group of blogs that are my favorites, and a second larger group that’s just every blog I’d come across up until about 2018 when I stopped using that tool regularly. I just opened it up for the first time in at least a few months. There are still people blogging — but the vast, vast majority of my 60-ish favorite blogs are dormant. And of the 500+ other blogs I follow, the feed is dominated by about 10 blogs that are consistently active. I haven’t picked through carefully but most of the others look dormant as well. I don’t look at my Feedly because it’s just not as alive anymore.
  • The conversation centered on blog posts. I learned recently that the Twitter algorithm changed at some point to reduce the visibility of external links, so people stay on the platform rather than taking conversation elsewhere. I have no idea whether that’s a cause, but regardless conversation on Twitter today is mostly about tweets, not about blog posts. Similarly, it’s much less common for blog posts to reference other blog posts. Blogging thrives when blogging begets blogging.

There’s other stuff. Blog posts did a better job of giving context so we talked past each other less often. People were less worried about building a brand. The virtual filing cabinet phase was amazing. Twitter Math Camp and a stronger NCTM conference scene fostered connections that are hard to maintain solely online. Great activities served as anchors that got shared over and over again. Other stuff I’m not thinking of.

Anyway, I’m sorry I keep saying blogging is dead. Lots of people are still blogging. Blogging is great! I’m blogging right now. I love it. But I’m still sad that it’s not what it once was. Blogging is dead, Twitter is dying, and I don’t know what’s next. But all this doesn’t discourage me. I love blogging. I have no plans to stop. Join me!

Tricks, and What To Do About Them

Dan Meyer wrote a great piece about what to do when students arrive to class using a trick to solve a problem that you want them to understand conceptually. Dan is teaching a lesson about negative numbers using the metaphor of floats and anchors, but students tell him that “two negatives make a positive” without referencing the metaphor. Dan is worried (as he should be) that, without some conceptual guideposts, students will struggle to remember when that rule applies, and when it does not.

I wrote a comment but realized I had more to say, and this post is my attempt to connect a few different ideas that Dan’s post got me thinking about.

I want to try and understand this situation from the perspective of a student who has a hard time in math class. I think, from that perspective, one common response to this situation isn’t a great idea, and I’d like to share an alternate approach that is under-appreciated in the online math education world.

My Story

My wife taught a course on map and compass skills last weekend. Despite working in outdoor education for a long time, my compass skills are not that great. While talking through some compass skills I used a trick and discovered the limits of that trick.

The hardest thing about working with a compass is the declination. Maps are oriented to “true north” at the north pole. Compasses point to magnetic north, which is somewhere in northern Canada. Where I live in Colorado the declination is 8 degrees, and when working with a compass sometimes you have to add 8 degrees, sometimes you have to subtract 8 degrees, and sometimes your measurement is good as is. A good understanding of compass use means being able to adjust for declination in different contexts.

Isogonic Lines Show The Pattern of Magetic Declination

Here’s my trick. Two common tasks when using a compass are going from the Map to “Features” in the world around you, for instance taking a bearing to a peak (measuring the angle on the compass) on the map, and then using that to figure out which feature around you is that peak. Map to Feature, MF, Moldy Feet. Moldy feet are bad, so I subtract 8 degrees from my map measurement to get my features measurement. The reverse: Features to Map. I take a bearing to a peak I can see, and then use that to figure out which peak on the map I’m looking at. Feature to Map, FM, Foot Massage. Foot massages are good, so I add 8 degrees from my feature measurement to get my map measurement.

I applied this rule to a few different situations. My wife was very patient with me working slowly through the process but I got the bearings right each time. But then I stumbled across a situation I didn’t know what to do with. I tried to orient the map, which means figuring out which way to turn the map so north on the map matched up with north in the world. But I was confused. Was this map to features, or features to map? I couldn’t figure it out. This was frustrating! Things had been going so well. My knowledge stopped there, and I struggled to figure out what to do or feel confident that what I was trying would be right.

My Proposal

The situation I was in reflects how our students feel when they use a trick. I often see teachers talk about how they try to squash tricks like FOIL, or two negatives make a positive, or keep change flip. I understand why they feel that way. But if someone came up to me and said “that moldy feet/foot massage thing is a trick! Tricks are bad! Don’t use tricks!” I would feel frustrated. That trick was all I had! If you tell me the trick is bad, you’re telling me I know nothing about compass declination and I’m useless. That’s not a good feeling.

The core of the issue with tricks is that they expire. They work in some situations but not others, like my moldy feet/foot massage trick failed me when I tried to orient the map.

So instead of telling students “tricks are bad,” I propose we show students the limits of that trick, and help them understand why the trick stops working where it does. This might seem obvious. But there’s a huge difference between telling a student “don’t use FOIL, tricks are bad” and showing them its limits with a thoughtful set of problems.

More Specific

Craig Barton, in his book How I Wish I’d Taught Maths, introduced me to the idea of “Same Surface, Different Deep” (SSDD) problems. The idea is to give students a set of problems with the same surface structure — at first glance, you might think they require the same math to solve. But their deep structure, or the math needed to solve them, is actually different in each case. These problems give students a chance to practice deciding which method applies to which problem, and better understanding where on method ends and another begins.

So back to Dan’s situation. When I confront a thorny classroom challenge like this one I often find it’s best to take a step back, move on to something else if at all possible, and revisit the topic the next day with fresh eyes. Here is a set of problems I might give students the next day:

Write a number sentence for each problem and solve it:

  1. Lin doesn’t have any money in his bank account. He pays Mai $4, and he pays Diego $8. How much money does Lin have?
  2. Lin doesn’t have any money in his bank account. He takes 4 friends out for sandwiches, and pays for their sandwiches which are $8 each. How much money does Lin have?
  3. Lin doesn’t have any money in his bank account. He spends $4 buying candy, then makes $8 from Diego for walking his dog. How much money does Lin have?
  4. Lin doesn’t have any money in his bank account. He spends $4 per day for 8 days. How much money does Lin spend in total?
  5. Lin doesn’t have any money in his bank account. Lin’s friends are willing to loan him $4 each. He needs $8. How many friends does Lin need to borrow from to get the $8?
  6. Lin doesn’t have any money in his bank account. He spends $4 per day. How much money will he have in 8 days?
  7. Lin doesn’t have any money in his bank account. He’s been spending $4 per day. How much money did he have 8 days ago?

My hope is that, by working on and discussing these problems, students will see the limits of their trick. Yes, sometimes two negatives make a positive. But sometimes they don’t. And negatives do lots of other stuff too. These problems give students a context to understand why sometimes two negatives don’t make a positive, and contrasting cases to practice seeing the boundaries of when their rules work and when they don’t. Then, I take the problems that students struggle with and provide some more practice. Maybe we use mini-whiteboards to get some quick feedback on what they understand and what they don’t, then practice some more.

I’ve had a positive experience every time I’ve used this SSDD approach. I give students some problems, and tell them “these problems all look similar on the surface, but they require different math to solve. Your job is to figure out which math to use for each problem. Good luck!” Math is hard, and solving a bunch of random problems feels confusing. Here, a few things stay the same each time. That gives students more mental space to focus on the differences between problems, and learn what I want them to learn.

The Most Important Thing

If “two negatives make a positive” is all a student has to go on, and I tell that student “two negatives make a positive” is a bad thinking tool, I’ve told that student all the knowledge they brought to class is useless. An important part of teaching is meeting students where they are, seeing the knowledge they bring to class as an asset, and building off of it. If someone came up to me and told me my moldy feet/foot massage trick is bad, I would feel frustrated. But when I encounter a situation my trick doesn’t help me with — and I recognize that my trick is limited — I have an opportunity to learn something new. That learning should be connected to what I already know. In my situation, I spent some time figuring out how orienting a map is connected to the map/features distinction I already had some understanding of. I built off of that knowledge, and ended up with a deeper understanding of the trick I was already using.

Some students bring the trick and an understanding of the math beneath it. If I attack the trick, they just nod along and keep using it because they know when it applies and when it doesn’t. But some students only bring the trick. If I attack the trick I’m telling them that their mathematical knowledge is wrong or inadequate. By giving students a chance to see where the trick works and where it doesn’t, I say, “yes, your knowledge has value, also let me add on some new knowledge so what you know is more powerful and more flexible.” That conversation is a great starting place to help students understand why two negatives make a positive in some situations but not others. You can apply this same approach to other concepts. Frustrated with FOIL? Give students a bunch of stuff to multiply — monomial/monomial, monomial/binomial, monomial/trinomial, binomial/binomial, and binomila/trinomial. Where does the trick work? Where does it fail? Why? That trick is valuable — it’s helpful for a specific slice of problems, and if I can help students understand where that slice begins and where it ends, I’ve been successful. To revisit my compass example, my trick works! It’s helpful for some specific situations. But when I expand my knowledge and see why it ends where it does, I end up with a more flexible and connected understanding of what’s happening under the hood.

Conceptual Understanding

Teachers talk about conceptual understanding, but I often feel confused about what conceptual understanding is, exactly. I think most of conceptual understanding is simply knowing where a method is useful, and where it isn’t. Procedural understanding is all the different methods we use to solve problems. Conceptual understanding is knowing which procedure(s) to apply. The best way to develop conceptual understanding is to compare and contrast different problems to understand why a procedure works here and not there. That’s what SSDD does. Tricks can have a place, as long as I help students see where tricks expire and give them tools to supplement their tricks.

A Caution

I want to reiterate my main point. Many math teachers hate tricks with a passion. We see all the ways they expire, and see students mindlessly keep-change-flipping their way through every problem with a fraction. It’s tempting to try and squash every trick we see. But tricks are a big chunk of the mathematical knowledge some kids bring to class. If we delegitimize that knowledge we risk alienating exactly the students who are often alienated in math class.

When I realized I didn’t know how to orient the map using the compass I felt frustrated. I was tempted to give up. And I have a lot of things going for me! I’m a reasonably mature adult. I have a lot of other knowledge about maps and the outdoors. I was motivated to learn. And I still wanted to give up. For lots of students, the moment we address a trick can feel fraught. They often feel dumb in math class. They struggle to see why math class matters. Trying a tough math problem feels uncertain and confusing. Being an asset-based teacher means seeing all knowledge that students bring as an asset, not only the knowledge that fits into our preconceptions of what a good math student looks like.

The heart of math is knowing where a method applies. If students only know how to solve problems served up on a platter, “solve 1-39 odd using the quadratic formula,” they will never be able to solve a real problem outside of math class. Tricks are an opportunity to help students see the boundaries of where one method ends and another begins. But if the message from the teacher is “tricks are bad, you don’t know anything useful” then students shut down exactly as we want to engage them in the most important learning.

I don’t like tricks. They frustrate me, and they feel like an obstacle to the flexible understanding I want my students to develop. But my frustration doesn’t make it ok to tell a student their ideas are bad. Their ideas have value, and it’s my job to help students see that value, see what more they have to learn, and connect what they know to what I want them to know.

Direct Instruction, Discovery, and Research, Again

I wrote last week about the endless direct instruction vs discovery debate. Here is (I think) a more concise way to say what I was trying to say:

Telling people “research says direct instruction is better than discovery learning” is a bad idea. That statement is true in a specific sense. When research has compared a “direct instruction curriculum” with a “discovery learning curriculum,” the direct instruction one almost always wins. But direct instruction means different things to different people. Many teachers will hear that advice and teach boring, narrow I/we/you lessons and get stuck in that rut because they think what they’re doing is research-based. And there’s plenty of research that supports specific elements of discovery learning, but discovery learning is equally ambiguous and implemented differently in different places.

Let’s say you’re a direct instruction partisan. You want to convince more people to use direct instruction. A better strategy and a model for what this type of research communication can look like is to share Rosenshine’s Principles of Instruction. Don’t say “use direct instruction.” You have no idea what a teacher interprets that to mean. Instead, say:

  • Begin a lesson with a short review of previous learning.
  • Present new material in small steps with student practice after
  • each step.
  • Ask a large number of questions and check the responses of all students.
  • Provide models.
  • Guide student practice.
  • Check for student understanding.
  • Obtain a high success rate.
  • Provide scaffolds for difficult tasks.
  • Require and monitor independent practice.
  • Engage students in weekly and monthly review.

(Click the link to read more about what each of these mean in practice.)

One reason the direct instruction folks often seem to have the upper hand in these debates is because they have lots of stuff like Rosenshine’s Principles to draw on, stuff that says “this is what research says effective direct instruction looks like.”

Let’s say you’re a discovery learning partisan. Don’t tell teachers “research says discovery learning is better than direct instruction.” Sure there’s some research that you could interpret to mean that. But even if that was true it would still be unhelpful. It’s not specific, and teachers will hear that advice and let kids flail aimlessly forever or assume they’ll remember everything if they figure it out themselves. Teachers will think what they’re doing is research based and get stuck in that rut.

I wrote last week about how there are a bunch of ways discovery learning can lean on research. But what I think the discovery folks are missing is something like Rosenshine’s Principles of Instruction. Too many of these debates are only ideological, and they miss out on the practical side. We don’t need more “here’s why discovery learning is actually better because research misses this and that and the other thing.” We do need more “here are ten things you can do to help ensure discovery learning works.” Maybe I’m looking in the wrong places but I don’t see that type of stuff, and when I do see it there’s a lot of disagreement within the discovery world. I see a lot of agreement among the folks who advocate for direct instruction done well, and that often leads to a stronger argument.

Direct Instruction, Discovery, and Research-Based Teaching

Can research tell teachers whether to use direct instruction or discovery learning?

What research has to say about direct instruction vs discovery mostly falls into two categories. Category one is research like “we gave one group of classrooms this discovery curriculum, and one this direct instruction curriculum, and the direct instruction group did better.” There are a bunch of these, and they should give discovery learning advocates pause. But they’re also about specific curricula. It’s hard to draw a line from “this curriculum didn’t do as well in that study” to “I should use this teaching practice in class tomorrow.”

The second category is research that focuses on more specific pedagogies in the classroom. That’s the research I want to write about. I think “what does the research say about direct instruction vs discovery” is too broad to be useful to most teachers.

Here are some lessons from research that could be helpful for someone planning a discovery lesson:

Connect what students are learning to what they already know. Lots of discovery lessons begin by connecting what students already know to what they are about to learn. Beginning a lesson with that type of exploration can be a great way to prime relevant background knowledge. Successful teaching builds on what students know rather than treating the lesson as a completely new topic.

Trying something before you know how to do it can prime the mind for learning. This is called the generation effect. I don’t mean students should be searching for an idea for hours, but trying something before receiving instruction can lead to better memory than jumping straight into instruction.

Figure out whether students understand what they are learning and do something about it. Discovery lessons often involve students doing math for much of the lesson. Teachers who monitor student learning and adjust based on what goes well and what students don’t understand can do more to support their learning. This also connects to the first point — teachers can build from what students already know and see where their knowledge ends.

Here are some lessons from research that could be helpful for someone planning a direct instruction lesson:

Have students study worked examples. Studying worked examples is often more effective than a teacher explanation because it involves more active thinking from the learner.

Have students explain idea in their own words. This is called the self-explanation effect. Students who summarize what they are learning are more likely to remember it and be able to apply it in the future.

Make sure students are thinking. Memory is the residue of thought. Whenever a teacher is explaining something they should ask themselves, “what are students thinking about right now?” The more thinking students are doing, the more they are learning.

I have two points. Some elements of a discovery lesson can be “research-based” just as easily as an direct instruction lesson. Make sure you do a few reasonable things. Don’t let students struggle if they aren’t making progress. Monitor to make sure all students are learning and not only those who are usually successful in math. Don’t let students use inefficient strategies forever. Don’t assume students understand something just because they figured it out themselves. Don’t spend so much time discovering that you neglect practice and application.

And direct instruction can often look discovery-ish. The goal is to get students thinking as much as possible. That doesn’t often fit the “I do/we do/you do” model or other caricatures of direct instruction. Effective direct instruction is complex! And while there is broad research in support of direct instruction, it’s easy to screw up too. Don’t teach a lesson without connecting to prior knowledge. Make sure to check for understanding thoroughly. Don’t talk too much without asking questions and making sure all students think about the answers. Don’t let students passively observe but make sure they are actively thinking. Don’t narrow the curriculum to predictable I-do-one you-do-one teaching.

I’m really bored of direct instruction vs discovery arguments because they’re always too abstract to be helpful. In a given week I do a bunch of things that look like discovery, a bunch of things that look like direct instruction, and a bunch of things that don’t fit neatly into either category. I make decisions based on what I see working for my students. If teachers come into a classroom with the broad idea that “direct instruction is supported by research” or “discovery is not research-based,” that’s not nearly specific enough to make practical decisions like “how do I start class tomorrow?”

What Needs To Be Memorized?

I’ve written a few times recently about memory. One thing I want to emphasize again is that I’m not arguing math class should mostly be about memorization. There are some pieces of knowledge that it is helpful to commit to long-term memory. If we can do that efficiently, we can free up math class for doing lots of other things. If we don’t do that efficiently we set ourselves up for either unnecessary drudgery taking the long road to long-term memory, or frustrated students who don’t have the knowledge they need.

But one thing I didn’t address is what, exactly, needs to be memorized. I don’t have a clear rule for this, but I think an example is helpful:

I wrote in a previous post about how, for a long time, I couldn’t remember the difference between affect and effect. I spend a lot of time writing, and I use those words and often have to look them up. I didn’t have them committed to long-term memory, but that didn’t prevent me from writing at a high level or using those words effectively. However, one part of writing is getting into a “flow” state where I have a clear idea of what I’m trying to communicate and how to get it across. Having to stop to look up the difference between affect and effect can break the flow. I have to spend working memory resources on figuring out which word to use, and maybe push the idea I had out of mind and interrupt the flow of writing.

I have a decent vocabulary, so situations like that don’t happen too often. Occasional interruptions like mine with affect/effect are inevitable. It’s impossible to have everything I could ever need to know committed to long-term memory beforehand.

Here’s another example. For a long time I had trouble remembering that 8 x 7 = 56. That didn’t prevent me from solving complex math problems. I’ve heard lots of mathematicians and experienced math teachers mention how they never memorized a few items in their times tables. I still don’t know my 12s very well past 72. But here’s the thing. I have never met someone who doesn’t have a large part of their times tables in long-term memory who was also successful in a mathematical field. Not everything needs to be committed to memory. But the stuff we come across most often is important to remember to free up cognitive resources for other things.

Trying to memorize everything is a silly and unnecessary exercise. Endless retrieval practice can suck time away from all the other stuff that’s valuable about math class. But it’s important for me to take a hard look at the content of my course and identify the highest-leverage stuff, the knowledge that comes up most often down the road, or that is a small piece of larger problem-solving. And then it’s my job to get that knowledge into my students’ long-term memory as efficiently as possible. There’s more to math class than memorization — but getting the basics down sets students up for success with all the other problem solving and exploring I value, and getting the basics down efficiently frees up time for everything else.