## Here’s what I think I do differently

I just read a really interesting post called ‘Applying Variation Theory.’ It’s by a teacher from the UK who I don’t yet know, Naveen Rizvi. The core mathematics is familiar territory to teachers of algebra. To factor an expression like $x^2 + 10x + 16$, you can ask yourself “what pair of numbers sum to 10 and multiply to 16?”

Therefore: $x^2 + 10x + 16 = (x + 8)(x + 2)$

In her post, Naveen talks about intentionally designing problems of this sort to draw out this underlying structure — that the factors of “c” have to sum to “b.” Factoring then becomes a quest to search the factors of “c” for things that sum to “b” (or vice versa).

Her main points is a good one, which is that if you keep everything else the same, and then vary just one thing, that thing will draw a lot of attention. Here is how she uses these ideas to design a practice set of quadratics, one that isn’t so unlike Dan’s:

My experience of teaching this topic is that, even knowing the relationship between c and b, these problems can be very difficult for students.

Why? What makes these factoring problems hard? Part of the reason has to do with fact automaticity, to be sure. While there are many topics in algebra that a student can handle without automaticity, this is definitely one of the times when things get much hairier for kids if they don’t know e.g. quick ways to find all the ways of multiplying to 60.

But take a closer look at some of these problems, and you can see that there is more going on than just knowing the relationship between b and c and knowing your facts. Consider one of these problems from Naveen’s page, which I just chose randomly:

$a^2 + 29a + 28$

This is a problem that I can imagine my students having some trouble with, at first. Not because the facts are difficult or because they don’t know the relationship…it’s just that the solution (a + 1)(a + 28) might not occur to kids. One thing I’ve noticed is that a lot of kids don’t think of 1 x N for a while when they’re searching for ways to make N. This makes sense — it’s so computationally straightforward, they don’t spend a lot of time thinking about multiplying by 1. It can slip under the radar.

Now, might a kid working on Naveen’s problem set become familiar with this tiny nugget or structure, that if the “b” term is one off from the “c” term, you should try sticking “1” into one of your binomials?

A student might make this generalization from the examples in the practice set, but this would essentially amount to learning by discovery. Which absolutely happens sometimes, but especially since this activity isn’t structured around giving kids instances that would cue-up that generalization…it probably won’t happen for most kids.

Now, the question is whether this sort of “micro-strategy” is a good use of classroom time. Maybe it’s too narrow a class of problems to be worth making it an instructional focus, I don’t know. Maybe you just go for the main strategy, and hope that kids are able to apply what they know to this little side-case.

But then again, maybe you give a quiz and kids end up mostly baffled by this problem. Teaching is full of surprises — this could happen.

That’s when I say, OK, let’s design a quick activity that would focus entirely on this micro-skill. Maybe a mini-worked example, or maybe a string of mental math factoring problems ala Naveen or Dan’s that puts that entire “variation theory” focus on this one, specific corner of the mathematical landscape — just the one that the students in your class need.

$a^2 + 8a + 7$

$x^2 + 14x + 13$

$j^2 + 176j + 175$

$q^2 - 14q + 13$

$s^2 - 11s - 12$

And then I call this “feedback” and don’t spend a lot of time writing up stuff in the margins of their quizzes.

There’s something nice about these little micro-skills. For one, it’s an alternate way of thinking about what’s leftover after you’ve taught the “main” skill. (Meaning, it’s not just that kids are forgetting what you’ve taught and need to be reminded — it’s that there are little corners of the mathematical world that haven’t yet been uncovered for kids.)

One thing I’ve been struggling with has been trying to figure out what exactly my pedagogy involves that’s distinctive. It’s not about the activities I tend to choose or design — since I’m pretty boring in that regard. I’m not an amazing motivator of people. Kids like me, I think, but not in the “oh my god he was my best teacher” way. More like “he’s nice.” (“Being nice” is an important part of my pedagogy.)

Though I still don’t have a snappy way to put it, I think that this is part of my story:

• I’m really curious about how kids think
• So I try to use that to come up with a more systematic understanding of how they think about different types of problems, especially when it’s something that people typically think of as constituting a single “type” of problems (e.g. factoring quadratics, solving equations, adding fractions)
• While teaching a topic I try to figure out which types of problems the kids understand how to handle, and which they don’t yet
• Then I focus in on a micro-skill for handling one of those little types of problem and I teach it with a short little activity followed-up by practice, in place of feedback

I don’t think any of this is exciting or inspiring, and I don’t really think I can make it so. There really is something here, though.

I think the exciting action comes in the second of those bullet points, in describing the mathematical landscape in a way that’s pedagogically useful. One of my favorite things in math education is Carpenter et al’s Cognitively Guided Instruction. I’ve moved away from some of the pedagogy it has grown into, but the problem breakdown is my paragon of pedagogically useful knowledge. It’s what I always come back to.

I sometimes wish there was more to what I do, and I also wish I didn’t have such a hard time figuring out how to describe what it is that I’m into. I’ll be doing a thing with teachers this spring that will give me another shot at refining my little message. I sometimes get jealous when I see all the cool things that people make and share online. I’ve never been cool, and none of what I’ve written here is cool either, but it’s what I’ve got.

(Last year’s version of this post: here.)

## I’m realizing I don’t really understand what working memory is

I’m trying to find some solid ground. Here are the questions I’m trying to get straight on:

• Suppose someone didn’t believe in the existence of a separate short-term memory system, just as (apparently) people in the ’50s and ’60s were skeptical. How would you convince a skeptic?
• What is the working memory system, anyway?
• Say that you were a behaviorist, someone uncomfortable with talk of the cognitive. How would you make sense of the observational findings?
• More concretely, what were the problems that Baddeley and Hitch were trying to solve when they introduced their working memory model?

I’m looking for foundations. Prompted by this, I went back to this, which brought me here, to Warrington and Shallice’s 1969 case reporting on a patient with severely impaired short-term memory.

The case is fascinating:

“K. F., a man aged 28, had a left parieto-occpital (“head” – MP) fracture in a motor-bicycle accident eleven years before, when a left parietal subdural (“brain” – MP) haematoma was evacuated. He was unconscious for ten weeks.”

He had lasting brain damage, especially when it came to language:

“His relatively poor language functions were reflected in his verbal I.Q. His ability to express himself was halting, and some word-finding difficulty and circumlocutions were noted.”

His short-term verbal memory in particular was damaged:

“The most striking feature of his performance was his almost total inability to repeat verbal stimuli. His digit span was two, and on repeated attempts at repeating two digits his performance would deteriorate, so that on some trials his digit span was one, or even none. His repetition difficulty was not restricted to digits; he had a similar difficulty in repeating letters, disconnected words and sentences. Single verbal items would be repeated correctly with the exception of polysyllabic words which were on occasion mispronounced.”

The thing that made him especially interesting was that, for a guy with significant short-term memory damage, there were a lot of things that he could do:

“Memory for day-to-day happenings was good and he had an adequate knowledge of recent and past events. Immediate memory for the Binet figures was accurate.”

Here is the real surprise, for people in the 1960s: his long-term learning was, actually, not bad. Consider the ten-word learning task, at which he performed admirably:

“A list of 10 high-frequency words was presented auditorily at the standard rate. Subjects were required to recall as many words as possible from the list immediately after presentation. This procedure was repeated until all the 10 words were recalled (not necessarily in the correct order). K. F. needed 7 trials. Twenty normal (“didn’t fall off a motorcycle” – MP) controls too an average of 9 trials, 4 of the subjects failing on the task after 20 trials. After an interval of two months he was able to recall 7 of these 10 words without relearning.”

On two other long-term memory tests, K. F. seemed to be performing normally as well.

And, what’s the significance of all this?

This was written when the existence of a short-term memory system was not universally accepted. (Is it universally accepted now? It feels like it but I don’t actually know.) And it’s useful to me for identifying what the core, foundational findings are that we have to grapple with in memory. There really aren’t very many, it seems.

At the core of things is the “digit span” task. This is the finding that there is some sort of limit on how many random things we can remember. This itself was the core finding that was supposed to support short-term memory. (“All subjects have a limited capacity to recall a series of digits or letters, and this limitation is regarded as a characteristic of the ‘short-term’ memory store.”)

The strongest evidence that this digit-span task was measuring a totally different system of memory was the evidence of amnesiacs, whose long-term memory is severely impaired:

“The question as to whether the organization of memory is a unitary process or a two-stage process has received much attention in recent years. The strongest evidence that there are separate short- and long-term memory systems is provided by the specific and isolated impairment of long-term memory in amnesic subjects.”

If you can have short-term memory but no long-term memories, and you can measure this with all sorts of repetition and digit span tasks, then there needs to be some distinction between two memory systems. Right?

Here, though, they found the opposite. A patient could have pretty normal long-term memory performance even though their short-term memory system was severely impaired.

In a different paper (1970) they lay out the implications of this for then-current controversies about short- and long-term memory:

“Most important, the results present difficulties for those theories in which STM and LTM are thought to use the same physical structures in different ways. (Because, I suppose, they’ve shown STM and LTM to be doubly independent of each other.-MP) They also indicate that the frequently used flow-diagrams in which information must enter STM before reaching LTM may be inappropriate. On this model, if the STM system were greatly impaired, one would expect impairment on LTM tasks, since the input to the LTM store would be reduced.”

What do they suggest?

“In light of these findings, it is suggested that a model in which the inputs to STM and LTM are in parallel rather than in series should be considered.”

One way of thinking of this could be that their patient, K. F., had damaged his ability to encode information about verbal sounds, but not his ability to encode the meanings behind those words, and that long-term memory is a system for storing meanings while short-term memory is just a system for storing sounds.

This is all fifty years ago, of course. But I think it’s helpful for me to understand what it took to get from a world where this seemed as plausible as the alternatives, to a world where scientific communities seem to universally accept that alternative.

That there’s a distinction between STM and LTM is beyond question. This is something that is confirmed a million-times over. Amnesiacs and people like K. F. speak to the distinctness of short- and long-term memory systems. We also experience this a million-times daily.

What is up for debate in the early 1970s is the relationship between these two systems. Is it one big system (“unitary”), with STM feeding directly into LTM? This study challenged that, suggesting that they were two fully independent memory systems.

Current models of memory suggest that they are connected, though in a more complex way than was understood before Baddeley and Hitch came along. (At least, I think that’s what’s going on…)

## Baddeley, A. (1983). Working Memory. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 302(1110), 311-324.

I.

This paper seems to be one of, like, a dozen mostly equivalent review pieces on working memory that Baddeley has written over the years. This one strikes a nice balance between concision and context, so it’s the one I chose to read with care.

Baddeley’s contribution is the idea that working memory consists of multiple independent components. When he was writing this paper, there were only three parts that he’d identified in short-term, working memory: a place for remembering sounds, a place for remembering space and visuals, and a “central executive” who is in charge of the whole system. This is contrast to earlier models that didn’t feel the need to distinguish between different components of working memory.

There are two lenses through which I’m reading this. First, how did we get here? Second, what metaphors do we use to describe memory?

Here is a handy table from a psych textbook, by the way:

II.

Baddeley positions his work as a reaction to the “modal model” of memory, popular in the 1960s, one that is represented by (I’ve learned) Atkinson and Shiffrin.

Atkinson and Shiffrin were people; this is all I know about them.

The modal model, according to Baddeley, looked like this:

• Short-term memory is one unit — no components. This is the working memory, the memory whose main function is to facilitate learning, reasoning, decision-making, etc.
• Long-term memory is where you keep memories for the long-term
• Here’s how long-term memories are formed: stuff automatically goes from short-term memory to long-term memory, but the process takes some time. (That’s why you don’t learn everything.) If you want to learn something, you have to make sure it’s rattling around short-term memory for long enough.

And what did things look like in the 1960s, when the modal model was prominent? I don’t have a clear picture of this. Wikipedia sends me to a piece from 1963 that talks of an explosion of results relating to short-term memory in the preceding years:

In 1958, and increasingly thereafter, the principal journals of human learning and performance have been flooded with experimental investigations of human short-term memory. This work has been characterized by strong theoretical interest, and sometimes strong statements, about the nature of memory, the characteristics of the memory trace, and the relations between short-term memory and the memory that results from multiple repetitions. The contrast with the preceding thirty years is striking. During those years most research on short-term memory was concerned with the memory span as a capacity variable, and no more…I venture to say that Broadbendt’s Perception and Communication (1958), with its emphasis on short-term memory as a major factor in human information-processing performance, played a key role in this development.

So the picture you get is that there was controversy about the distinction between short-term and long-term memory, springing from a great deal of results in the early ’60s.

One of the major impetuses? The famous psychological subject H.M. (Read about the controversies surrounding him in a fascinating New York Times magazine piece from a few years ago.) H.M makes an appearance in the Baddeley paper — he seemed to have the ability to form immediate memories (despite his profound inability to form long-term ones) help make the case for a distinction between short- and long-term memory systems.

And what was the state of things before the early ’60s? Baddeley credits Francis Galton with an early version (1883) of the notion that there are two separate memory systems, but I can’t figure out where exactly he says this or how he puts it. Wikipedia points us to William James, who distinguishes between primary and secondary memory. in the “Memory” chapter of The Principles of PsychologyI’ve only skimmed it, but I think he thinks of primary memory as a lot like the after-image of some visual perception. It just lingers for a second — real memory is secondary memory.

And what about before James and Galton? I’d have to figure that something as basic as the distinction between short- and long-term memory is not an insight unique to psychologists. I know a few other philosophers who make distinctions that seem relevant, but I’m not sure how to trace the lineage of short- and long-term memory before the late 1800s.

As it is, it seems that the early impetus is just the recognition that some stuff we can remember for a little bit, and other stuff we remember for a long time. Maybe this is as far as we get without more careful measurements.

III.

Back to Baddeley, who makes the case that the modal model of the early 1960s is wrong. There are multiple components to short-term, working memory.

What was wrong with the modal model?

• Even when short-term memory is impaired, long-term memories can be formed just fine. So how could it be that the path to long-term passes through a unified short-term store? There has to be a pathway besides the damaged one that memories could pass-through, on their way to the land of long-term.
• This is hilarious, but the BBC tried to use the modal model in their advertising to let people know about newer wavelength bands they’d be switching broadcasts to. They figured the more frequently a phrase is in short-term memory, the more likely it is for a long-term memory to be formed, so they just slipped the phrase into radio broadcasts here and there and…nobody remembered. So much for “saturation advertising.” So how are
• The third thing — that people remember recent stuff better, even in long-term memory — is confusing to me. I don’t get how it’s relevant to this yet.

So what does he suggest, to fix things?

First, that there is a central executive that is in charge of making decisions during activities that is unrelated to the memory stores. I think this is supposed to explain how people who have short-term memory impairment can still function or form long-term memories. The assumption is that as long as the central executive (who decides what the brain should do) decides to actively reinforce the stuff that makes it into short-term memory, long-term memory can happen. And the central executive can also be in charge of doing stuff accurately, even if the stores of memory traces are depleted.

The central executive is a bizarre notion. Metaphorically, it’s a little dude in your head that decides where to put attention, or when to actively reinforce the memory traces in the other stories of working memory — thus he’s also responsible for learning and reasoning. He is, to put it bluntly, your soul, an unanalyzable source of free will. It’s weird.

Second, Baddeley proposes two different passive stores of memories — the phonological and the visual/spacial. Each comes with an active element, something that can reinforce the memory traces.

The metaphors are fascinating here. The phonological loop is, metaphorically, a piece of audio tape that loops around your brain, over and over. When a sound goes into your mind it lands on the memory trace, and then an active recording element has to rewrite the sound on your mental tape for it to be sustained over time. Otherwise, it gets overwritten (or it fades?) as time goes on.

Baddeley’s model for visual/spacial stuff could have used an audio tape metaphor, as far as I can tell, but chose something that felt more appropriate for visual information — a scratchpad. So there’s an entirely parallel system that is posited for visual info, but with a totally different set of metaphors that is appropriate for visual stuff.

So there’s a scratchpad, and when visual or spacial stuff comes into your head it is inscribed onto the pad, very lightly. It’s only sustained if the active element reinscribes it on the pad.

IV.

Let’s end here, because I’m still massively confused as to how any of the results that Baddeley says are issues with the modal model are satisfied by creating two independent stores of working memory. I’ll need to read something else to make any progress on this, I think.

Update: I’m going to read this next.

## “I was sitting there and my skin was burning and I said, this might make a great TV show one day.”

I watched the new Chris Rock special on Netflix. Watching all these classic comics with new specials feels a lot like that Modern Seinfeld feed — Chris Rock on Trump! — but it had some very funny moments. The personal stuff (about his divorce and failings) was interesting too, but I sort of felt it would have been more sincere to just do jokes about it.

Anyway, one thing led to another after I finished the special, and I present to you the clip above.

I also present to you the very, very funny and awkward clip below:

## “That structure was just guess work, but we seem to have guessed well.”

This interview with Alan Baddeley, one of the researchers responsible for the contemporary notion of “working memory,” is fantastic. I’m not up to speed on all the psych concepts yet, but there are a lot of juicy bits.

First, you can’t get into research like this any longer:

When I graduated I went to the States for a year hoping, when I returned, to do research on partial reinforcement in rats. But when I came back the whole behaviourist enterprise was largely in ruins. The big controversy between Hull and Tolman had apparently been abandoned as a draw and everybody moved on to do something else. On return, I didn’t have a PhD place, and the only job I could get was as a hospital porter and later as a secondary modern school teacher – with no training whatsoever! Then a job cropped up at the Medical Research Council Applied Psychology Unit in Cambridge. They had a project funded by the Post Office on the design of postal codes and so I started doing research on memory.

On his “guess work”:

I think what we did was to move away from the idea of a limited short-term memory that was largely verbal to something that was much broader, and that was essentially concerned with helping us cope with cognitive problems. So we moved from a simple verbal store to a three component store that was run by an attentional executive and that was assisted by a visual spatial storage system and a verbal storage system. That structure was just guess work, but we seem to have guessed well because the three components are still in there 30 odd years later – although now with a fourth component, the ‘episodic buffer’.

What if he had guessed something different, and if that different guess had held up decently? Psychology is at a local maximum, but it seems to me that there’s very reason to think that our current conception of the mind is at anywhere near a global maximum, the most natural and useful way of conceiving of things.

On that:

The basic model is not too hard to understand, but potentially it’s expandable. I think that’s why it’s survived.

Here is the context for their description of working memory, as distinctive from short-term memory:

I suppose the model came reasonably quickly. Graham and I got a three-year grant to look at the relationship between long- and short-term memory just at a time when people were abandoning the study of short-term memory because the concept was running into problems. One of the problems was that patients who seemed to have a very impaired short-term memory, with a digit span of only one or two, nevertheless could have preserved long-term memory. The problem is that short-term memory was assumed to be a crucial stage in feeding long-term memory, so such patients ought to have been amnesic as well. They were not. Similarly, if short-term memory acted as a working memory, the patients ought to be virtually demented because of problems with complex cognition. In fact they were fine. One of them worked as a secretary, another a taxi driver and one of them ran a shop. They had very specific deficits that were inconsistent with the old idea that short-term memory simply feeds long-term memory. So what we decided to do was to split short-term memory into various components, proposing a verbal component, a visual spatial one, and clearly it needed some sort of attentional controller. We reckoned these three were the minimum needed.

In other words, short-term memory was assumed to be unitary. Baddeley and Hitch figured that it could have three independent components. Since short-term memory was running into trouble, it sounds like they kind of rebranded it as working memory.

Are there other differences between short-term memory and working memory besides its multi-component nature? I think so, but I’m currently fuzzy on this. Another thing that I’ve read suggests that short-term memory was seen as just on the pathway to long-term memory, so it was essentially part of a memory-forming pathway. Working memory is supposed to play a broader range of roles…I’m confused on this, honestly.

Finally, here’s a solid interaction:

Your model with Graham Hitch has a central executive controlling ‘slave’ systems. People sometimes have a problem with the term ‘slave’?
This is presumably because people don’t like the idea of slavery.

## First instance of the term “Working Memory”

“When we have decided to execute some particular Plan, it is probably put into some special state or place where it can be remembered while it is being executed. Particularly if it is a transient, temporary kind of Plan that will be used today and never again, we need some special place to store it. The special place may be on a sheet of paper. Or (who knows?) it may be somewhere in the frontal love of the brain. Without committing ourselves to any specific machinery, therefore, we should like to speak of the memory we use for the execution of our Plans as a kind of quick-access, “working memory.”

From Plans and the Structure of Behavior (1960)

## Some of the things I stand for in education, these days

When we critique an idea, we should critique the best version of it.

When we critique a pedagogical idea, we should critique the idea itself, not its misinterpretations. (Unless we’re saying the idea is easy to misinterpret.)

Most of the time when a pedagogical idea is critiqued, it’s critiqued for bad choices the teacher might make that have nothing to do with the idea itself. When we imagine an idea we don’t like, we imagine a classroom that we don’t like. That’s not fair, though.

Every pedagogical idea gets misinterpreted.

Nobody knows how to make ideas about good teaching scale, so we might as well talk about what good teaching actually looks like, without worrying about how the truth will or will not be misinterpreted at large.

You and I might teach in very similar ways but have wildly different ways to describe how we teach. The fiercest debates in education are also the vaguest. When you get down to classroom details, or even the tiniest bit of additional specificity, a lot of disagreement vanishes. It’s not that these debates don’t matter, it’s that they are highly theoretical.

OK maybe these debate don’t really matter.

Teachers have access to classroom details and specificity. One way that teachers can contribute to the knowledge base of education is to resist the urge to move to generalization, to spend some more time in the greater specificity that classroom life encourages. This is what teachers can uniquely contribute.

Non-teachers will often tend to think about scale. Teachers usually don’t, and this is something else that we can contribute.

Some people will tell you that being a jerk is important, as long as you’re being a jerk to the right people. Those people are jerks. Stupid jerks.

I think the core of what I do is trying to have a detailed understanding of how my students think about the stuff I’m teaching, and also of how they could think about the stuff I’m teaching. And then I’m trying to create as many opportunities for them to think about that stuff in the new way.

I care about my students’ feelings, a lot.

The way I see it, teaching is best understood through its dilemmas and tensions. This is not a new idea, but it’s one that hasn’t been sufficiently explored. And I’m very suspicious of people who claim to have resolved one of these tensions or dilemmas.

We shouldn’t ad hoc create rules for political discourse that only apply to people who we disagree with.

Debate is essentially performative. I mean, it doesn’t need to be, but it usually is.

Trying to understand someone and how they think is an almost complete improvement over debate in every way.

Lots of research is interesting. It’s especially fun to try to figure out how two different perspectives on teaching can fit together.

Teaching is a job, but it can be a great job if you like ideas and little humans.

Writing and reading should be fun and interesting. When writing instead aims to be useful, it’s usually not useful either. And 99% of writing about teaching is supposed to be useful.

At a certain point you have to decide if you’re trying, primarily, to change the world or to understand and describe it. It’s great that some people are trying to change the world, but I think inevitably those people end up having to be less-than-honest at times, for the sake of their projects or reputation. Personally, I prefer to understand and describe it when I write or think. (I’m not opposed to helping people, though!)

Blogging is dead, but it’s all we got for now, so let’s keep at it.

I like to think of my teaching life as a bubble, and I like to think of all other aspects of my life as little bubbles too. And a question I ask myself frequently is, can I grow these bubbles? Wouldn’t it be something if all the other bubbles could float and sort of merge into each other, turn into just one big bubble that encompasses everything? I feel like that would be nice, someday.

YOSEF
POIKJ
CVB
SEAT
ARM
PIANO
YOSEF
CAT
DOG
SEAT
POOP
POOP
BOOK
STOP
YOSEF
VIDEO
YFCVMFCV
COMPUTER
COMPUTER
CAT ZsaAAA
KP0UJVC
BABY
UFG
YOSEFBOY

YOSEFBOY

COME TO MY HOUSE

COME TO MY HOUSE

YOSEFBOY

BOX

BOOKS

MOO

MOMMY

YOSEFBOY
WINDOW
WINDOW