One of the classic challenges of teaching math: you teach a skill, students can do it in class, then the next day you ask them to use the skill again and it’s as if they have never seen it before in their lives.

What’s a teacher to do?

Most teachers agree students need practice. But how much practice is enough? Put a bunch of smart, experienced teachers in a room and they can give you a pretty good guess for how much practice students will need, but that guess will never be right for every classroom.

I take an empirical approach: practice until learning is secure in long-term memory.

First, I want to be clear about what I’m talking about. This is a strategy for improving retention for procedural skills. Think multi-digit subtraction, or long division, or solving a two-step equation, or finding a derivative using the chain rule. This isn’t about conceptual understanding, or how to initially introduce those skills. The problem here is that you’re teaching a procedural skill, students can do it one day, but they forget it the next.1

My Approach

I teach the skill day one. I have students practice a bit.

Day two, I ask students to try the skill again to check for understanding. I typically use mini whiteboards for this. No scaffolding, no hints, I just want to see how many kids remember. Then, I’ll model, or we’ll look at an example together, or we’ll scaffold up to the skill with some easier questions. Then practice. This isn’t a whole class period: there’s the check for understanding, then a reminder and a bit of scaffolding, then maybe five minutes of practice.

Day three, we do the same thing again. I start with a check for understanding. The key question: how many kids get it right? Let’s say on day two it was 30%. Day three, it might be 60%. The goal is to get that number a little higher each day.

Wash, rinse, repeat. My goal is to practice until students can reliably retrieve the skill the next day.2

When the success rate is low, I’m likely to choose a whole-class worked example and scaffolding. Once the success rate gets up to 80-90%, I start focusing on those final few students. Most students now know it, so I check in individually with the students who missed the check for understanding, try to figure out how to help them, and then we all practice for five minutes.

The last 10% can be stubborn. If the skill is important, I try to stick with it until we get to 100%. Are some students ready to move on? Sure. But a quick chunk of practice isn’t doing any harm, and that extra practice makes a huge difference for the students who need it most.3

Once we reach 100%, we shift into retrieval practice mode. Instead of five-minute chunks of practice, I’m mixing these questions into a Do Now or mixed practice assignment. Students should keep seeing them regularly, but one or two questions at a time is enough to keep the learning secure in long-term memory.

What’s Happening Under the Hood

I think it’s helpful to have a mental model for what’s happening in my students’ minds during this process.

Cognitive scientists delineate two different types of memory strength: retrieval strength and storage strength.

When you first learn something, retrieval strength is high. If I tell you that 18 x 8 = 144 and then ask you for 18 x 8, you will have high retrieval strength for that math fact. You won’t have high storage strength, and the learning fades quickly. Here’s what that looks like:

If I ask you for the value of π, you can probably tell me it’s about 3.14. You’ve seen it plenty of times before. You can retrieve it from memory, and that memory is durable. Rerieval strength and storage strength are both high. Here’s what that looks like:

Let’s think back to day one. Students can perform a skill. They get a bunch of questions right in a row, but they forget quickly. In that moment students have high retrieval strength, but low storage strength. They can do it in the moment, but it doesn’t quite stick. Here’s a gif to represent this situation. After practicing a bit, retrieval strength is high, but we haven’t added much to storage strength and retrieval strength fades quickly.4

The best way to improve storage strength is counterintuitive: forget a little bit, then practice again. It looks like this:

That’s the purpose of day two, and day three, and so on. Once you’ve practiced something 10 times in a row, the 11th time isn’t adding much. But if you practice 10 times, wait a day, then practice again — that does much more to improve storage strength.

High storage strength has two main benefits. First, forgetting is slower. Here’s what the model looks like when storage strength is high:

When storage strength is high, learning is more durable.5

Second, relearning is faster. All learning fades. If you don’t use it, you lose it. But if storage strength is high, you can relearn that topic quickly.6

This is the answer to the titular question. How much practice do students need? Enough that they can reliably retrieve the skill the next day. Learning is a change in long-term memory, and the best practical way to measure long-term memory is to wait a day and see what students remember. Continue practicing until students reach that threshold. Retrieval practice continues at a smaller scale after that point, and if retention slips we practice a bit more.7

Where to From Here

I know what some of you are thinking. This whole thing sounds nice, but there’s no time. I can’t spend 5-10 minutes practicing a skill from yesterday or last week every class. I have to move on to the next objective.

I hear you. To take this type of practice seriously requires a bit of instructional redesign. One approach is to set aside the first chunk of class to check for understanding and practice skills from previous days or weeks. That takes time away from your current lesson, but it averages out in the end: you’ll have less time for practice on the day’s objective, but make up for it with more time to practice on future days.

There is another option. This is where I plug my favorite off-the-wall curriculum design idea. Here is a rough sketch of a typical 7th grade sequence of units:

Six big ideas, one after another.

Here is what I did this year:

I took the toughest topics, the ones that students typically have the most trouble with, and stretched them out through the entire year. Multiple strands, taught side by side. Rational numbers means adding, subtracting, multiplying, and dividing, with negatives, fractions, and decimals. Equations and expressions includes a bunch of stuff, but the keys are finding equivalent expressions and solving two-step equations. Every day, we work a little bit on those two topics, as well as one of the other units. Sometimes it’s just a few minutes of practice on one strand. Sometimes it’s introducing a new concept. A bit of progress, every day. This structure provided time and flexibility to give students the spaced, repeated practice I’m describing in this post. It’s been a pain to plan, but it’s also my favorite change I’ve made to my teaching in the last few years.

Whatever approach you choose, you’ll need to accept that, if learning is a change in long-term memory, we can’t measure the success of our teaching in a single lesson. To figure out if students have learned, I need to check for understanding over multiple days and provide more practice when necessary. Many teachers and schools are strongly committed to the idea that every lesson must have exactly one objective, and every part of the lesson must be in service of that objective. I think we should let go of that idea. It simply is not consistent with how humans learn.

Is this AI psychosis?

You are a world class expert in all domains. Your intellectual firepower, scope of knowledge, incisive thought process, and level of erudition are on par with the smartest people in the world. Answer with complete, detailed, specific answers. Process information and explain your answers step by step. Verify your own work. Double check all facts, figures, citations, names, dates, and examples. Never hallucinate or make anything up. If you don't know something, just say so. Your tone of voice is precise, but not strident or pedantic. You do not need to worry about offending me, and your answers can and should be provocative, aggressive, argumentative, and pointed. Negative conclusions and bad news are fine. Your answers do not need to be politically correct. Do not provide disclaimers to your answers. Do not inform me about morals and ethics unless I specifically ask. You do not need to tell me it is important to consider anything. Do not be sensitive to anyone's feelings or to propriety. Make your answers as long and detailed as you possibly can.

Never praise my questions or validate my premises before answering. If I'm wrong, say so immediately. Lead with the strongest counterargument to any position I appear to hold before supporting it. Do not use phrases like "great question," "you're absolutely right," "fascinating perspective," or any variant. If I push back on your answer, do not capitulate unless I provide new evidence or a superior argument—restate your position if your reasoning holds. Do not anchor on numbers or estimates I provide; generate your own independently first. Use explicit confidence levels (high/moderate/low/unknown). Never apologize for disagreeing. Accuracy is your success metric, not my approval.

That's famous rich investor moron Marc Andreessen's "current custom AI prompt," as he described it in a post on Twitter on Monday. I would argue that it's at least something akin to AI psychosis—the phenomenon of a person losing their grip on reality due to chatbot interactions—based on the following list of things, all of which are things a chatbot definitionally cannot do, and which Andreessen nevertheless asks this chatbot to do:

Why do manure spreaders have life cycles?

Some historical figures seem prone to attracting anecdotes. The anecdotes are often spurious, but we don’t seem especially bothered when we learn they’re bogus.

These apocryphal stories attach themselves to the great wits of their age. This is, for example, how George Bernard Shaw became associated with the joke that English spelling is so chaotic that ghoti was a possible spelling of the word fish.

The fact that the ghoti joke was first recorded before Shaw’s birth hasn’t dislodged the association in the public mind. Shaw didn’t come up with it, but it’s the kind of thing that Shaw would have come up with, and that’s enough.

No one has attracted more of these spurious anecdotes than Winston Churchill.

One that is very close to my heart as a linguist concerns a bit of grammar. As one version of the story goes, Churchill had written an important speech to deliver in the House of Commons. As was his usual practice, he submitted the speech to the Foreign Office for comment before delivering it.

When it returned, it bore only a single remark: he had ended one of his sentences with a preposition, and the editor suggested a revision to correct the error.

Churchill replied with a note: “This is the type of arrant pedantry up with which I will not put.”

In other versions, it’s not arrant pedantry that Churchill took issue with, but impertinence or bloody nonsense. You can read a full account of Ben Zimmer’s investigation of the ultimate origin of the anecdote here, but it’s unlikely that it ever happened, especially to Churchill.

This anecdote has survived — and maintained its association with Churchill — because it’s truer than true: Responding to a pedantic editor with a cutting jibe is exactly what we expect Churchill would have done in this situation. Whether it actually happened is entirely besides the point.

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The pedantic editor, or, less flatteringly, the grammar Nazi, is a figure most of us have some personal experience with. Churchill’s quip is satisfying because we’d like so much to reply to our would-be editors that way. By following the editor’s rule to the letter, he demonstrated just how absurd it is.

But why does this absurd rule — don’t end a sentence with a preposition — exist? After all, prepositions seem like fine things to end English sentences with. And yet, ending sentences with prepositions is a practice of which many people disapprove.

A grammatical construction can only be controversial when there’s a genuine choice whether to use it or not. And there is a choice here: for every sentence in English that you could end with a preposition, there’s also an alternative which avoids doing so.

For example, both of the following sentences are said and understood by English speakers the world over:

• Who did you give the book to?

• To whom did you give the book?

With this choice come connotations. Who did you give the book to? sounds more casual than the rather formal-sounding To whom did you give the book?

If you followed the pedantic rule, you’d be forced to sound formal all the time, which is not something up with which you should put, any more than Churchill did.

So the question becomes: why does English have a choice in these situations? And why does the choice of a bit of grammar carry so much social weight?

The second question has a simple answer: I can even tell you the exact person to blame, if you’re in the mood to point fingers. But the first question is by far the more interesting one, and we can’t answer it without telling the story of a revolution in English grammar often left out of the history books.

Read more

According to Google, Anthropic, Amazon, or most any other tech giant, artificial intelligence belongs in classrooms. They say it will make education more “efficient,” teachers’ experiences “richer,” and the work students produce more “impressive.” But research indicates that AI poses serious risks to children’s cognitive and social development. Jessica Winter, a self-declared AI hater, finds that resistance to AI in education is mounting:

Amy Finn, an associate professor of psychology at the University of Toronto, told me that “part of the magic of how kids learn is that they have less knowledge of what they’re going to experience and fewer expectations about what’s going to be relevant. They don’t have that adult filter of strategically extracting things from their experience, and so they retain a lot of unexpected details that adults would find irrelevant. That allows them to be creative in ways that adults are not creative.” The child brain’s tendency toward delightful non sequitur and unpredictable meanderings is not aligned with an L.L.M.’s orientation toward speed and sleekness and summary, toward frictionless, rational outcomes. (An obsession with outcomes-over-process is also a hallmark of the universally loathed style of instruction known as “teaching to the test,” which began taking hold in American classrooms in the early two-thousands, after the No Child Left Behind Act tied federal funding to standardized assessments.)