I don't know why, but I felt like I couldn't generate these silly little tangents with all the stress of the previous administration. I was just always paralyzed by the daily insanities. It's so nice to have crises spaced slightly further apart.

That's impostor syndrome talking, heebs. Your silliness is an ever-flowing spring.

Heuristics are necessary to function in the world. As you age, you become both humbler and wiser as you learn that your rules are riddled with exceptions.

Or, like most old people, you don't see that, and you become narrower and dumber.

Old people are the worst. I say: never trust anyone over 60. But I came to that opinion by modifying my heuristic from last year, which was, "Never trust anyone over 59."

I'm not sure I understand what transference of knowledge means. Is it applying what you learned in one context to another context? If so, isn't that also a form of generalization -- assuming that what works in one context will work in another context?

I thought transference was when you're fucked up in a certain way but rather than admit it, you accuse everybody else of that.

So the Heebie of today is a direct product of the analogy ban? Is that it?

6: I literally almost put in a few sentences about how it forced me to grow as a thinker and writer. I loved arguing by analogy so, so much.

Is it applying what you learned in one context to another context? If so, isn't that also a form of generalization -- assuming that what works in one context will work in another context?

Yes and yes.

I think when the Ed people say that students are awful at transference, they mean it on the fairly small scale: "I learned fractions under Ms. X in 4th grade, and now when we're building on that material with Mr. Y in 5th grade, I'm mystified and can't retrieve all the information that I had access to, although if I walked across the hall and sat in my old seat and was given those worksheets with the blue mimeographed ink, it would come back to me."

But what I'm saying is that transference and generalization are the same phenomenon, and they're an early stage of learning, and narrow transference is the hardest part of generalization.

SPLITTERS!

8.2: It really sounds like they are confusing transference and remembering. Maybe I just don't believe that they would be able to do the fractions if they were back in their old classroom.

I think the idea is that changing the context wrecks people's memory - and perhaps that it wasn't that well-learned in the first place, although they could comply.

3. Abbie Hoffman said never to trust anyone over 30 so you're giving yourself some leeway there. Lenin said all politicians over 35 should be shot. But he was still a politician in his early 50s, when, to be fair, he was shot. Clearly an early example of Soviet inefficiency.

Didn't Lenin die of a stroke?

He was shot and that didn't help his overall health.

Right, I'd forgotten about the assassination attempt some years prior.

Grothendieck was definitely a lumper.

Transference is also hard for machine learning, maybe once people work out how to help the algorithms do better we can then help each other.

Seems wrong. Brains use wider integration as they age as a way to compensate for declining functional capacity.

16: well it isn't so hard as to be impractical; lots of the most advanced models have a final layer slapped on and then used for specific tasks

16: well it isn't so hard as to be impractical; lots of the most advanced models have a final layer slapped on and then used for specific tasks

Math-ed transfer / generalization researcher here. Your observations make sense. Children generalize incorrectly all the time. Adolescents and adults also do this, but it's less obvious and we don't study it as much. In the math world, one thing we've learned is that transfer is happening all the time but it's often not the kind of transfer we want to see. Moreover, traditional transfer studies don't capture it well, because those studies tend to investigate only one pre-determined, narrow type of transfer. In other words, they ask, "Did the participants transfer (this one specific thing I wanted them to transfer)?" rather than, "What did they transfer?". When you ask the first question, the answer is often no. When you ask the second question, it turns out that students are transferring all the time but it's often not what we wanted them to transfer.

Much of what I've studied throughout my career is the second question, i.e., what do students construe as similar across representations, tasks, or problem contexts? And more generally, what characterizes the processes of mathematical generalization, and how can we help students generalize in more mathematically productive ways? Anyway, within this tradition of research, generalization is seen as a specific type of transfer.

When you ask the second question, it turns out that students are transferring all the time but it's often not what we wanted them to transfer.

I can definitely remember having a teacher where the main information I carried from one term to the next was that this teacher was an asshole. But that wasn't math.

17: That seems interesting, but I don't understand it. (I am in the declining functional capacity stage of life.) What does "wider integration" mean in this context?

Wait. We have math education lurkers?

23: Decade+ long lurker, first time commenter!

Welcome.

Be grateful we lost the fruit basket. It was a terrible fruit basket.

24: I am 100% positive we know people in common!

It's harder to know uncommon people.

Not here, it's not.

Anyway, having tried and failed to explain various bits of algebra to a teenager, I have more appreciation for the difficulty of the task.

28: Much more difficult to know very rare people. Or at least that's what I'm transferring from my D&D knowledge.

27: I have been thinking that too, especially if you work where I think you might (which sounds super creepy, sorry, but I love your blog!).

32: Aw, thanks!!

Depending on how long/closely you read, you may think I work in a big department with lots of people you surely know. I'm friends with a lot of them, but actually teach at a small school in a nearby town.

Also, your research sounds super fascinating.

So what's the answer - how can we help students transfer in the right ways? Is it just a matter of making them reason it out for themselves, so that the next time they come across it, they can reason it out again? Intuitively, I feel like that's how it worked for me. Or is there something else that affects it a lot?

17: I mean the line of studies showing compensatory use/connection across brain structures, e.g. https://www.sciencedirect.com/science/article/pii/S1053811920304456

17: I mean the line of studies showing compensatory use/connection across brain structures, e.g. https://www.sciencedirect.com/science/article/pii/S1053811920304456

32: Ah, okay, I think I was mistakenly thinking you worked in that big department (and I do know all of the math-ed folks there).

34: As for helping students transfer well, this area of research doesn't have a well-established set of findings that crosses domains, so the answers for, say, function reasoning in algebra is different from geometry is different from another domain. We've found that identifying what students do transfer as a first step is useful, because it helps us understand what they're taking as structure vs. surface features, and adjust instruction accordingly. In algebra, for instance, helping students think about functions from a continuously covarying quantities perspective is really useful for productive transfer when they get to calculus.

I saw a funny instance of transfer from chemistry a while back. We had given the participants a task with a rectangle that added area from one step to the next in a way that created an L-shape. (So say you've got a 3 cm (h) by 5 cm (w) rectangle, then you add a 1 cm by 5 cm horizontal rectangle to the height, and a 3 cm by 1 cm vertical rectangle to the width, and keep doing that). We asked the students to determine the area of the nth figure and one kid distinguished the area of the original rectangle as P and the combined area of added pieces as V for valence, saying, "Let's do V for valence because that's one word I know for outer ring." It was such a nice example of how weird and unpredictable - but sometimes useful - some of the students' generalizations can be.

The classic example of transference of knowledge is when the teacher explains to Vinnie Bobarino how to do a math problem and the transferable lesson that Vinnie derives is that the answer to every multiple choice question is C.

37: that is super interesting! thank you.