Take out the bit about racism. Those who know something about the philosophy of math, including Timothy Gowers, said sometimes 2 + 2 = 5. Nobody on twitter says anything about my blog, but I had three posts about the topic. The one linked to by my name quotes Felix (1960), a member of Bourbaki and a precursor of New Math, I guess.

Hillary Putnam has a paper called something like, philosophy of math: why nothing works.

sometimes 2 + 2 = 5

In an extremely academic sense this is correct, but it's a dumb hill to die on nonetheless.

I made the mistake of looking at some of the blather on twitter about this a month or so ago when it was a thing. No one looked good.

That whole handout seems like "we've given up on a lightbulb ever lighting up in our students and instead we're getting more and more desperate in how to cram processes into their brains." It drives me crazy. But also, I'm not in that teacher's shoes. If my job depended on getting the bottom 5% of students to some minimum proficiency maybe I'd also have students singing row your boat? I'm glad I'm not them.

3 was me.

Mathy people - what do you do about the bottom 5% of your classes?

3 was me.

Mathy people - what do you do about the bottom 5% of your classes?

Math in junior high was really just beating your brain into a new way of thinking. Maybe it was worse for be because my son was offended on an emotional level by algebra.

The first graphic is great: never too early to get them started on commutative diagrams...

Those graphics really are something. When they reach the level of those joke flowcharts from XKCD, be sure to post them here.

Mathy people - what do you do about the bottom 5% of your classes?

Invite them to office hours. If they're willing to dedicate the time, I can get them to understand enough to pass the class. If they don't put in a lot of extra hours, they're not going to pass the class.

Also, given that the second graphic includes a song, let me be the first to link this.

Those who know something about the philosophy of math, including Timothy Gowers, said sometimes 2 + 2 = 5.

You do this, according to the link, by redefining what "=" means.

But then I would counter by arguing that 2+2 never equals anything except 4, and back up my statement by saying that it is entirely correct as long as you allow me to use my own definition of "never" which is "hardly ever" (Porter, Corcoran et al, cited H. Pin., 1878.)

I especially enjoy the arrow pointing left that says "move to the right", and an arrow pointing right that says "move to the left".

11: You do this, according to the link, by redefining what "=" means.

It is not clear to me that anybody knows what "2", "+", "=", or "5" mean or what it even would mean for these symbols to be defined. I cannot link from this computer but I do have a couple of other blog posts in the weeks after that one to annoy you.

The way the twitter conversation went is something like some (pure and applied) mathematicians and analytical philosophers were, like, cool, people want to talk about my subject. Some brought up IEEE STD 754, or had screenshots from Excel. Others reacted. They wanted these woke post modernists, who must all be into critical race theory, kept way from teaching their children.

It is not clear to me that anybody knows what "2", "+", "=", or "5" mean or what it even would mean for these symbols to be defined.

Oh man. This is not my hill to die on. But I think you should talk to a guy named Bertrand Russell.

I showed this to SPouse who writes district curriculum on these exact topics and she was appropriately appalled. It's a good example of "traditional" vs (gasp) Common Core, where the former is just memorize this procedure and you'll figure out why while the latter tries to start with understanding why.
The trivial examples of 2+2 are different bases or clock systems (not necessarily for 5 but for answers besides 4) and everyone says Oh of course except for those cases that's obvious so it's really a question of what you consider an obvious exception.

I guess I don't understand the mindset that leads from

"Maybe there are biases in the way we teach math that should be addressed"

to

"2 + 2 = 5, you racists! Sick burn!"

14.2: I would, but he already died on this hill.

I always assume "2 + 2" is about bunnies, so the answer to 2+2 is that it depends on how many are male and how many are femaile.

16: Did anyone you're reacting to say that?

18: a 2 is basically a bunny doing a headstand, so that works. 2 + 2 = either 22 or anything from 222 to 22222222, depending, as you say, on the male/female mix.

I do have a couple of other blog posts in the weeks after that one to annoy you.

I think you are overestimating how interesting this entire issue is. I didn't even read all of the one you linked to and it looked like it was quite short. I got distracted by something Crom was doing and couldn't be bothered to go back to it.

19:

Arthur Chu, almost literally. I guess it's my fault for reading the tweet.

"I got distracted by something Crom was doing" s/b "I got distracted, by Crom!"

Math pleases you, Crom... so grant me one request. Grant me 2+2! And if you do not listen, then the to the decimals with you!

Crom comments: Stop trying to make "Fetch!" happen, ajay. It's not going to happen.

And if you do not listen, then the to the decimals with you!

The Dewey decimals!

Perhaps your dog has difficulty hearing you. Check its ears for wax buildup.

25 should of course be a comment about Crom having forgotten the answer to the Riddle of Stick.

The only good part of the 2+2=4 twitter discourse was the poll about whether 2+2=4.0.

(Answer: it depends on how the type theory you're working in deals with coercion. 2+2 is an integer, but 4.0 is a real number. Are integers real numbers? Or is there just a rule for turning an integer into a real number?)

22: Do you mean in this thread? https://twitter.com/arthur_affect/status/1289499027273179136?s=21

Or someplace else?

Because I don't see him at all saying that 2+2=5 is a sick burn on racists. He's commenting on the fact that a bunch of conservatives got bent out of shape by a mathematician talking about what it could mean for 2+2 to equal 5.

30: Not that thread, somewhere else.

Anyway, I'm happy to concede that probably no one literally said the words "2 + 2 = 5, you racists! Sick burn!".

My real point was that, if someone is genuinely interested in drawing attention to potential sources of bias in secondary school math teaching, making a big show of how people who think that 2 + 2 = 4 are naive because under the right weird set of assumptions 2 + 2 = 5 is about the most useless possible way to go about it.

I didn't think you meant he said it literally, I was just surprised. I didn't follow this closely, but my sense of the dispute was (1) mathematician makes fairly anodyne academic point that if we see someone saying something like 2+2=5, it's best practice to figure out what their definitions and assumptions are before dismissing it, because it is possible for a statement that looks that weird to be meaningful or useful (2) bunch of conservative twits throws a tantrum about how this is literally Orwellian (3) people like Chu come in to defend the first guy. I don't see how your version fits into that, although I guess I might if I saw what you were talking about.

The song itself seems alright, if we're teaching processes rather than principles. The flowchart just seems to unnecessarily complicate things.

I liked this sidebar on math concepts.

I find that irrational number problems are easier to solve it you assume i=2.

34: That was a nice article, thanks!

I guess -1=4?

Math is so much simpler once you've achieved enlightenment and realized that we are all 1.

With everything?

And like one and one don't make two
One and one make one

40 is great.

I always thought they said "In life, one and one don't make two..." Who knew.

OMG we were all assigned unconscious bias training at work and the intro video says "Here's an example. I'm going to give you a math problem and I want you to say the answer, out loud, as fast as you can." As soon as she said that sentence, before the problem, I said 4. Of course the question was 2+2.
Unfortunately since I'm remote today I was the only one in the room to be impressed by myself.

What was that supposed to teach you about unconscious bias?

I memorized the quadratic equation to the tune of "Row, Row Your Boat," so maybe it's just a thing.

The point being made was that the large majority of information processing is based on previously experienced situations or learned facts, so we often have an immediate answer (which may be correct or incorrect) when presented with a question and don't have to bother thinking about the specific situation.

44: It proved he was racist.

Shouldn't you have said 5?

Let's be clear about our terms. If 2 := S(S({})), (+) := lambda x. lambda y. if(x = S(x')) then S ( (+) x' y) else y (and all the newly introduced terms are defined in the obvious ways), and (=) is the primitive equivalence relationship between identical objects, then yes, 2 + 2 = 4 evaluates to true. You let me know what truth means, though. If that was too airy-fairy, here's the deets.

Of course, it's much more often that 2 + 2 = 8 or 2 + 2 = 7 than 2 + 2 = 5, but it does happen.

Anyway, yeah, it was a fun opportunity to talk about philosophy of math (and particularly that of mathematical foundations, and the various logical programs and their failures), and that there is value in questioning even obvious truths, and that's going to rub up against our cultural preconceptions. Of course, you can't square a circle--unless you aren't restricted to the ancients' cultural hangups about the right way to do things, in which case it's super easy. (I'm vaguely recall that one of them, Apollonius maybe, understood this, but I don't feel like looking this up.) But still, it was quite silly because of all the talking past each other; 2+2=4 was really four or five conversations where everybody wanted to talk about their thing and didn't care what anyone else was saying.

So everyone was right and everyone was wrong, but as a classical logician I'm not equipped to express that.

The series 1, 2, 3, 5, 8, 13, ... satisfies the Peano axioms.
s(1) + s(1) = 2 + 2 = 5. I find this a point for Russell against Hilbert. Matt Y likes to recommend Benacerraf's What numbers could not be. This is a point against Russell.

I'm sure many know more than me, but it is not complete ignorance that leads me to say that I do not know what it mean to define these symbols.

The first image is like one of those overloaded DoD PowerPoint slides. There are okay tricks in there that just get swamped by too much stuff.

But the second image...huh? Wouldn't a number-line approach be simpler? And comprehensible?

50: Of course we can just choose to use different symbols to represent the same concepts; you've assigned the symbol 5 to S(S(S(S(0)))), you aren't using anything Fibonacci-like to make that trick work. (And of course that's a valid point, and feeling vaguely po-mo, really pisses off the conservative ignoramuses.) But for the purposes of this silly game, I think it's more fun to come up with some vaguely plausible mathematical object where 2+2=5 holds that isn't isomorphic to regular arithmetic but still attempts to preserve some character of the original numbers. I was using the simple model where values are capped above some ceiling, e.g. imagine adding voltages and any input is clipped to be between 0V and 4V for the output; then 2V+2V and 4V and 5V result in the same output value, so we might say that they're equal. (Equality, equivalence, it's all just a matter of perspective.) In this case the system preserves the comparative bigness of 5. 2+2 can equal most other small integers by using an appropriate modular arithmetic; 5 is an annoyingly stubborn exception due to the small difference.

I'm sure you know more than me; Bourbaki quickly went from fascinating me to boring me, but it's been a while since I've thought hard about this sort of thing.

I don't think it's that bad -- for a kid who's more verbal than spatial "add if the signs match, subtract if they don't, use the sign of the bigger number" is a perfectly doable way to do addition. You'd want the kid to understand a number line, but I could imagine a kid who would have a harder time keeping left and right straight who'd do better relying on verbal instructions.

To this day, I still do "greater than" and "less than" by saying to myself that the alligator eats the bigger number. I'm not sure what that says about my verbal versus spatial abilities.

Is this helpful: https://twitter.com/tothelostmonth/status/1310762651803287554

32.1 fairly anodyne academic point that if we see someone saying something like 2+2=5, it's best practice to figure out what their definitions and assumptions are before dismissing it, because it is possible for a statement that looks that weird to be meaningful or useful

This is definitely a fairly anodyne academic point, but it could get confused with the idea that if someone's kid is asked to add 7 and 5 and gets 13 rather than 12, that's okay because it's just base 9, which is totally used in other cultures. Even though no other culture uses base 9 and neither does the kid; i.e. this kid is using base 10 and getting the answer wrong.

49 there is value in questioning even obvious truths, and that's going to rub up against our cultural preconceptions.

Sure, there's a value in questioning even obvious truths, but if that truth is the claim that 2 and 2 make 4, any preconceptions you rub up against aren't going to be cultural ones, in the sense of varying from culture to culture. No culture accepts the claim that 2+2=1 (which is a false one, even though you can make a different and true claim, modulo 3, using the string of symbols '2+2=1').

I have no idea how one fixes the math achievement gap but there has to be evidence about early math exposure that's similar to the "number of words heard by kindergarten" metric. I was at the grocery store with Pebbles when she was two and a half, and we were doing the "let the toddler help with the apples and count them as they go in the bags on this endless shopping trip" routine and a woman stopped me to praise my parenting and explain that she's a kindergarten teacher who has students who can't get the one-to-one correspondence of counting.

The Calabat hates online school. They're learning mental math strategies which are fine but a hard sell for a kid who has lots of basic math facts memorized. "Kate added 4+3 by doing 4+4-1. What did Kate do wrong?" "She didn't know it was seven?"

What did Kate do wrong? It's been a long day but that seems all right.

Kate should have broken the four into 3+1, then recognized the double three, and added 1.

Kate is playing some kind of constrained video game where she loses health points for every unit she travels along the number line. I hope she gets out okay!

I could imagine a kid who would have a harder time keeping left and right straight who'd do better relying on verbal instructions.

That's fair.

4+3. The kid should double the three then add 1? So off to multiplication and order of operations. I can see it as a method though I'm more 4+3=7. I guess when it gets harder I tend toward something like 4+7=10+1=11.

Kate didn't do anything wrong. Just not the method being taught. The curriculum teaching that some equivalent methods are wrong, that's not good.

56.last I don't really disagree with you, but I do agree with the importance of what weight you're putting where. There's something interesting in how you wrote it: the first time, you acted as if the utterance 2+2=1 embeds a specific truth about arithmetic on the positive integers, while for the second weird case involving modular arithmetic, you felt the need to quote it and treat it just as symbols. The first usage was really a tuple of the string and the cultural context necessary to interpret it, while for the second usage you were more explicit in giving the context. That's a great way of showing that cultures often prefer a particular reading of statements, with other potentially meaningful readings marginalized, using a setting where essentially no one has emotional stakes (and if they do, doing some self-investigation into why they're so het up over it might be helpful). If I'm trying to discuss Gauss's Disquisitiones and you kept on belligerently interpreting my statements as non-modular because "no culture accepts the claim," it'd be rightly annoying. I'm halfway to convincing myself this is a useful way to teach cultural relativism, or at least not being a dick about people having different basic low-level assumptions, to the sort of person who loves STEM and hates humanities. (Who am I kidding, those people almost always have no interest in math as practiced.)

62.last: Agreed. Number sense is critical and shaming kids for doing it the "wrong way" is counterproductive. I'm fine with teaching or at least exposing them to a preferred fast way so they have that in their toolbox, but other ways shouldn't be discouraged unless and until they become problematic at building higher concepts. (The Calabat is in Utah, right? Unlike Heebieland, Utah is Common Core, so it's a surprising question.)

59: I don't know anything about contemporary math education, and I was always a good memorizer, so I don't entirely understand why breaking 4 into 3+1 is an improvement. I feel like we had to do something like that, only everything was supposed to be multiples of 10. When I had to memorize stuff that wasn't covered in public school to take a private school entrance exam, I found that I was able to do arithmetic so much more quickly.

Trapper Keeper folders had handy multiplication tables printed on them which helped a lot.

9: I want to be a fly on the wall in your office hours and see how it works, how much time you are talking, how much listening, what you're offering them in the way of tools. When I try to help a (college) student who does not have the number sense to see where a (mathy) answer comes from, I explain it, then wait while they think it over and I'm just hoping the light dawns. If it doesn't, I don't really have a good way forward. Explain again, wait again. Try to offer a real-world analogy, maybe.

I don't think it's that bad -- for a kid who's more verbal than spatial "add if the signs match, subtract if they don't, use the sign of the bigger number" is a perfectly doable way to do addition.

That mnemonic alone isn't bad, but the flow chart below it is just insane.

Also it's fine for a mnemonic to aid reasoning, but not to replace it. The goal should be to get kids to confidently believe that math is something to be reasoned out.

To this day, I still do "greater than" and "less than" by saying to myself that the alligator eats the bigger number. I'm not sure what that says about my verbal versus spatial abilities.

Argh. I don't think you're saying that to troll me, but I loathe the alligator-mnemonic with the fire of a thousand suns. It is sabotaging the math by making it seem mysterious instead of cooperating with the natural, built-in mnemonic inherent to the symbols!

Take an equals sign, =. It's a balance, Now pinch one side to make it smaller, and the other side opens up. Now it's an inequality, and the pinched side designates the smaller side, and the larger side designates the larger side. There is nothing to memorize!

AND!! "The alligator eats the larger number" almost distracts from the comparison-of-two-numbers altogether! Under that logic, you could have an isolated number with an alligator opening up to eat it, because it has chosen the largest number around, like a max function or something.

63: I don't think we have any substantive difference of opinion, but I don't think my phrasing (putting my earlier second instance of "2+2=4" in quotes and not the first) should be taken to have any sinister chauvinistic agenda behind it either. If I'm speaking to a classicist and say that no culture accepts that Aristotle lived in the 20th century, and add that you can make a different and true claim, about Aristotle Onassis, using the string of words "Aristotle lived in the 20th century", I shouldn't be taken to imply that the quoted string is universally more properly used to talk about the philosopher than the shipping magnate, i.e. that I'm privileging the first meaning. I'm just relying on a context where the first usage can be taken as read, and the second possibility needs to be highlighted. If we were hanging out with Onassis, or he were otherwise the more salient person, the reverse would hold.

That said, I agree that it's probably worth avoiding (to the extent that it's possible and reasonable), even the appearance of treating as natural what is conventional, just for the sake of not annoying people.

Correction to 63: "2+2=4" s/b "2+2=1"

The Calabat hates online school. They're learning mental math strategies which are fine but a hard sell for a kid who has lots of basic math facts memorized. "Kate added 4+3 by doing 4+4-1. What did Kate do wrong?" "She didn't know it was seven?"

I have a feeling they get this kind of shit in in-person school too, to be fair.

Remote school has been eye-opening though, in forcing me to acknowledge how much drudgery is built into their work day. And this is with many, many worksheets being swapped out for educational apps.

On the one hand, I strongly support going slowly through math and just getting super firm on the basics. They're now making algebra the norm for middle school here, instead of just for accelerated students, and I think this is truly stupid, misguided, and will sabotage kids in the longrun. Just give their little frontal cortexes and extra year to develop! It's so terrible when someone's first time through a course is a blur, and then they are rushed through quickie-reviews at the beginning of every math course thereafter, and they end up only knowing the quickie-review version and being haunted by mistakes and the sense that they don't really know why they're doing the things they're doing. (For example, if you've just had quickie-review algebra, you mis-categorize things a lot. You don't really know why, in certain situations, you get all your "x" terms on one side and your numbers on the other, whereas in other situations you get everything on one side so that it =0, and proceed from there, because nobody set a really firm context that one is for linear equations and one is for quadratics.)

BUT. The flip side of this is that Rascal is spending kindergarten learning to count, when he regularly adds 3 digit numbers in his head in order to figure out various yu-gi-oh card implications. Or Ace and Calabat analyzing what Kate did wrong, when they were following that kind of reasoning at age 2.

One last anecdote: yesterday, Ace was zoom-interviewed by two undergraduate students who are pre-service teachers. The packet of questions was fantastic, and the students follow a script. They're supposed to be learning how to gently get a kid to reveal their thinking, and ask follow up questions that put the kid in a situation to consider an inconsistency in their thinking, and give them time to think about it and consider it, etc. (They were mediocre. There was an opportunity where Ace had a pair of contradictory beliefs about measuring, and they missed the opportunity to pause and dwell in the moment and follow it through.)

ANYWAY. Ace was hilarious. She was holding court like a kindly, polished professor, explaining her answers in polished lecture form, without any "uh"s or awkward pauses, speaking in unnaturally complete sentences. It was so great. I surreptitiously recorded a bit of it. Pausing at times to hold up her work, like you would do if you were teaching the material, and pointing out key moments and why she'd done it the way she had. It was hysterical and Jammies kept losing his shit and having to duck behind the kitchen counter to muffle his laughter.

: I want to be a fly on the wall in your office hours and see how it works, how much time you are talking, how much listening, what you're offering them in the way of tools. When I try to help a (college) student who does not have the number sense to see where a (mathy) answer comes from, I explain it, then wait while they think it over and I'm just hoping the light dawns. If it doesn't, I don't really have a good way forward. Explain again, wait again. Try to offer a real-world analogy, maybe.

If they have a really specific question, I will answer that question, but after that, I'm going to ask them to work a problem and see where they get stuck. If they're just a little stuck, I'm going to ask them questions and have them continue to hold the pencil. But if they're really stuck, then I'm going to be the one who holds the pencil. Then I ask, "ok, what should I write next?" but I can use the pencil to draw their attention to various parts of the question. Like, if they have no answer to "what should I do next?", then I can back up and ask, "What are we trying to do in this step again?" while I circle the initial conditions and the prompt to solve for t. "How are we going to put these pieces together?" etc. Then build up to where they're ready to tell me what to write next. After we get through a problem with me holding the pencil, I'd pass the pencil back to them and have them work a problem, using our problem as a template.

69: I didn't think it was chauvinistic, I thought it was interesting given the context. Yeah, I don't think we disagree; I've mostly just taken this as an opportunity to be whimsical and frustrate people I'm ideologically opposed to, because I can be a jerk in that way.

Also I have a secret skill, which is that I've learned to write upside down, which is great for office hours and working together on a single sheet of paper, across a desk.

The trick is to start off the problem with the paper facing you, writing right-side-up, like normal.

Then flip the paper around. Any time you come to a "3" or an "e" or a "7" or "s", you scan the paper for that symbol where you wrote it right-side up, and then you just copy it, to get the orientation correct. This turns out to be something you can do really fast and accurately. (If I come to an "s" for the first time, I either guess, and guess wrong, and the students smile in response, or guess right, and keep going. Or you can nudge the paper sideways enough that your rightside-up brain kicks in.)

I loathe the alligator-mnemonic with the fire of a thousand suns.

Likewise. I remember being taught it and thinking "no wait, the big gap is at the side with the big number. Why go on about this alligator?"
Of course it helped that exactly the same symbols are used in music for crescendo and diminuendo, with the big end meaning "loud here " and the small end "quiet here" and no one mentioned alligators when I learned that.

I used to draw little teeth in the symbols when I was bored.

68: That's crocodiles.

Fortunately, SAS lets you use GT, LT, and similar.

Obviously, if there aren't two numbers, the alligator will wait. They know it's rude to start before the rest of the table has food.

74: Comity - I thought it was interesting too.

The open side is on the side with the bigger number because it represents the arrow of time. The only natural motion is falling, and so < and > symbolically represent the natural progress towards lower heights.

The reason the alligator is a dumb mnemonic is that it's half a mnemonic and half that thing where you pretend to your kid that the spoonful of mashed potato is an aeroplane. (It's a floor wax and a dessert topping.) The idea is that if you bring in alligators the kid will forget for a second how bored they are with the whole exercise.

73 and 75 thanks - actually not so different, even the upside down writing, except that I'm more practiced with this stuff in my core areas instead of in math, I guess, so I'm probably bad at building up to a concept from where they're starting, in math. And I hadn't thought to look for other upside down esses!

83: And it worked. To this day I both eat regularly and do math involving inequalities regularly.

Give me a Mobes till he is seven...

57.last is so infuriating. The whole point of this kind of number sense exercise is to play around with numbers and realize that there's not a "right way" and a "wrong way." I'm not sure how much of this is driven by the teacher or the person writing the exercise failing to understand the point, and how much is driven by the logistics of the "assign an insane amount of homework that needs to be graded extremely quickly" paradigm. If each student can answer the question their own way then you can't grade quickly. (See also the emphasis on "simplification" rules like no square roots in the denominator.)

Sorry that was me.

The "no radicals in the denominator" rule is so bizarrely sacred. I don't understand why high school teachers are so obsessed with this.

So bizarre... I mean the skill of moving around where square roots (or other roots!) are is important, but it's just as common that you want to get rid of square roots in the numerator! For example, if you want to use the limit definition of the derivative to compute the derivative of square root. But no, everything has to get straightjacketed into a strictly unidirectional series of "simplifications" to a unique final answer. Argh...

Or why are we using radical notation at all, and not just teaching fractional exponents from the get go? It doesn't need to be anything complicated - literally just the 1/2 situation instead of square root could deepen students' understanding of a lot of this stuff.

Isn't is easier for the teacher to grade if everyone needs to give the same answer to be correct? I'm guessing the teaching materials are designed to accommodate teachers who aren't very much ahead of the students.

Right, 92.last is 87.last, I think it's a pretty plausible explanation.

Yes, I missed that. Sorry.

The labour-saving value for markers of having a simplification process that leads to a unique answer is I think just a side-effect of a more high-minded, erm, rationale. Which is that rationalising the denominator

[in e.g. (6-sqrt(5))/sqrt(8)]

is necessary in order to get to a sort of normal form expression

[(3/2)sqrt(2) - (1/4)sqrt(2)sqrt(5)]

in radicals with rationals as coefficients. One advantage of the process being that you can now compare various original non-normal expressions for equality by comparing their normal forms. Correspondingly, there's a gain in intuition about the sort of mathematical object you're dealing with when you have the original expressions.

I don't disagree with anything in 95, but I think it's mistaking one application of a trick for the trick itself. The trick is that you can get rid of square roots by "multiplication by the conjugate" or more generally by some kind of Galois-theory norm-type calculation. One application, as you say, is that Q[\sqrt{n}] has a Q-basis given by 1, \sqrt{n} (and various generalizations to other number fields). Or, along the same lines but lower level, that you can divide complex numbers. But this trick is used all over the place and not always in denominators! The power rule in calculus is a really notable example and for fractional powers you're using this trick in the numerator not the denominator. Using the trick itself is the relevant skill, normal form under certain circumstances is just one particular application of that technique.

Which is that rationalising the denominator

[in e.g. (6-sqrt(5))/sqrt(8)]

is necessary in order to get to a sort of normal form expression

[(3/2)sqrt(2) - (1/4)sqrt(2)sqrt(5)]

That's not how I'd do it myself, fwiw. I'd say:

start with (6-sqrt(5))/sqrt(8)

Break it into two fractions:
6/sqrt(8) - sqrt(5)/sqrt(8)

Clean up the sqrt(8):

6/2sqrt(2) - sqrt(5)/2sqrt(2)

Reduce the first, and then call it done.

3/sqrt(2) - sqrt(5)/2sqrt(2)

BUT more importantly, I have no preference between that final form and the original form. Students need to move smoothly between various forms, but there's not one that's intrinsically more useful.

Honestly I probably would't have messed with it in the first place.

Oh sure, I agree with both 96 and 97 in that training students to rationalise the denominator as a sort of Pavlovian reflex is a bad idea. Students need to be able to move back and forth between different equivalent expressions, and if you do happen to want to rationalise the denominator, the trick you use involving the conjugate can be used differently in other contexts. All I was thinking was that what we now often find taught in corrupt form as rote instruction (substituting 'do this or else!' for 'this gets you to a form you often want') might have a less mindless underlying motivation. Which is that of getting the student to be able to see the 'unsimplified' expression involving radicals as equivalent to an expression where certain radicals are treated as a sort of basis (as per 96). I might easily leave the last expression in 97 as is, and generally wouldn't rewrite 1/sqrt(2) as (1/2)sqrt(2) but it seems to me that the heuristic advantage to the student of knowing you can do the rewrite to a form without radicals in the denominator is that, to the extent that radicals are mysterious relative to the rationals, it's easier to imagine the result of multiplying the mysterious thing by a half ('I get half the mysterious thing') than the result of dividing 1 by the mysterious thing.

(Sorry that I'm not always able to reply quickly enough for a proper conversation, btw.)

One of the points of rationalizing the denominator is that it makes it (relatively) easy to consolidate the sum/difference of multiple irrational fractions into a single fraction with a common denominator, which may lead to additional cancellations in the numerator.