## Re: Not worth the paper it's printed on.

1

Isn't that a pointlessly abstract thing for high school math anyway?

Posted by: Moby Hick | Link to this comment | 09-11-15 7:25 AM
2

It would be in an advanced class. But to teach something which so boldly violates the principles of logical deduction is awful.

Posted by: heebie-geebie | Link to this comment | 09-11-15 7:27 AM
3

P1: Knowing how to do logical deduction is good
P2: Teaching is an effective way to convey knowledge

C1: You shouldn't teach things that boldly violate the principles of local deduction.

Posted by: Moby Hick | Link to this comment | 09-11-15 7:30 AM
4

This is stupid to teach in the first place. It's better to say nothing than to be wrong. I'd even argue that teaching bijections in the first place, even for finite sets, is silly--just teach the properties that make functions functions, don't worry about into/onto/etc.

Posted by: dalriata | Link to this comment | 09-11-15 7:52 AM
5

So, is a lot of mathematics accomplished by sneaky little bitchery, or was it just Cantor?

Posted by: foolishmortal | Link to this comment | 09-11-15 7:53 AM
6

On the other hand, I don't mind not giving a proof for something that's too beyond their skill level and instead just showing a few examples to build up intuition. But you have to be right.

Posted by: dalriata | Link to this comment | 09-11-15 7:53 AM
7

I don't understand how they made the error. Was it that since every rational number is the ratio of two integers and there are an infinite number integers, there are 2 times infinity rational numbers and thus no one-to-one correspondence?

Posted by: Moby Hick | Link to this comment | 09-11-15 7:56 AM
8

4: presumably the students already understand what a function is. What's so crazy about the concepts of 1-1 and onto?

Posted by: heebie-geebie | Link to this comment | 09-11-15 7:58 AM
9

Sorry, infinity times infinity rational numbers.

Posted by: Moby Hick | Link to this comment | 09-11-15 7:58 AM
10

In my defense, infinity squared is equal to infinity times 2.

Posted by: Moby Hick | Link to this comment | 09-11-15 8:00 AM
11

8: What are they going to use it for? They'd learn it and then immediately forget about it without building anything on it. I do think in some cases one-to-one and onto are taught at the same time as functions, and that's a waste.

10: Only if you're talking about cardinals.

Posted by: dalriata | Link to this comment | 09-11-15 8:04 AM
12

I don't think I know what not-cardinal numbers are. It really has never come up.

Posted by: Moby Hick | Link to this comment | 09-11-15 8:07 AM
13

What are they going to use it for?

Most of them are never going to strictly remember anything. Why should most people to have an algebra class? (I don't necessarily think they should.) The point is to teach people how to think abstractly and how to read symbolic notation, and that they have the skills to be able to teach themselves a basic concept when it arises later in life. Onto and 1-1 are reasonable concepts towards that end - not too difficult, require logical reasoning, can be used for interesting things.

Posted by: heebie-geebie | Link to this comment | 09-11-15 8:07 AM
14

Everybody should use algebra. I don't actually understand what "onto" or "1-1" mean and I've done fine with math.

Posted by: Moby Hick | Link to this comment | 09-11-15 8:10 AM
15

Maybe they didn't have "sets" when I was in high school.

Posted by: Moby Hick | Link to this comment | 09-11-15 8:11 AM
16

And understanding will work better if they do something with it. If it's "oh here's a silly little property that we will never talk about again" you've wasted their time more than you would by using some other abstract symbolic notation.

Posted by: dalriata | Link to this comment | 09-11-15 8:12 AM
17

In high school, the closet we ever got to logic was programming. Apple Basic taught me about life.

Posted by: Moby Hick | Link to this comment | 09-11-15 8:14 AM
18

Probably one of the authors of this book once took a class in college where they learned about countability and totally misremembered what they learned and didn't bother to look it up.

Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 09-11-15 8:14 AM
19

14: "onto" means a function is surjective: f:A->B is onto if forall b in B exists a in A such that f(a) = b. That is, everything in B is in the range of A. "1-to-1" means that a function is injective: forall a_1, a_2 in A f(a_1) = f(a_2) -> a_1 = a_2. That is, it maps each value in the domain to a unique one in the codomain (B).

If f is surjective, |A| >= |B|. If f is injective, |A|

(If I'm off on any of this, I'm on a conference call and rather distracted.)

Posted by: dalriata | Link to this comment | 09-11-15 8:16 AM
20

Gah, html. If f is injective, |A| ≤ |B|. If it's both--bijective-- then |A| = |B|. This is an important thing to prove, and when you learn other kinds of structures to put on sets you can show that two sets are equivalent up to that structure and thus can be thought of as the same.

Posted by: dalriata | Link to this comment | 09-11-15 8:17 AM
21

Hey, funny moments in math-related nepotistic string-pulling. A friend of my [currently presidential high-school-student daughter]'s is the daughter of a guy Pause used to work with, teaching math at the college where kids from their high school get to take classes. My daughter, the math prof's daughter, and their cohort are taking calculus at the college this fall. The father apparently looked over the available sections, and told the school to put his kid's classmates generally in sections X and Y, rather than Z, as having preferable instructors.

Nothing like having inside connections.

Posted by: Emmy Noether | Link to this comment | 09-11-15 8:21 AM
22

Nice! If only I were still there so I could find out whether I made the list of good teachers.

Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 09-11-15 8:23 AM
23

You just need a set of family and friends that map onto the set of professors.

Posted by: Moby Hick | Link to this comment | 09-11-15 8:23 AM
24

And understanding will work better if they do something with it.

Sure, if they spend one class on it, then it doesn't do much. But it's no worse than spending a day on double-angle trig formulas or something, in isolation. If you're doing a unit on functions and sets, then sure, tease these concepts apart.

In general, my belief is that you're teaching a process, not content.

Posted by: heebie-geebie | Link to this comment | 09-11-15 8:25 AM
25

I habitually refer to the daughter as "your friend, the Genetically Engineered Soviet Superman". What with the academic/athletic prowess, the family tree including the odd Olympic athlete, and the general large healthy pretty blondeness.

Posted by: Emmy Noether | Link to this comment | 09-11-15 8:27 AM
26

I agree with that. I strongly suspect that this is barely being used after initially teaching it and will mostly just confuse and frustrate students.

Posted by: dalriata | Link to this comment | 09-11-15 8:27 AM
27

The best way to frustrate students is to not let them pee.

Posted by: Moby Hick | Link to this comment | 09-11-15 8:28 AM
28

Holy crap (to the OP).

Posted by: snarkout | Link to this comment | 09-11-15 8:29 AM
29

I found that I made a comically shocked face upon reading the OP. Eyebrows raised as high as they would go, eyes wide, mouth actually hanging open. Zowie.

Posted by: redfoxtailshrub | Link to this comment | 09-11-15 8:32 AM
30

I strongly suspect that this is barely being used after initially teaching it and will mostly just confuse and frustrate students.

In this instance, I hope you're right.

Posted by: heebie-geebie | Link to this comment | 09-11-15 8:35 AM
31

5: Yes.

Posted by: MHPH | Link to this comment | 09-11-15 8:36 AM
32

What would you use a not-cardinal number for?

Posted by: Moby Hick | Link to this comment | 09-11-15 8:36 AM
33

What would you use a not-cardinal number for?

I'll answer your question second, but first I'm going to get dressed.

Posted by: nosflow | Link to this comment | 09-11-15 8:39 AM
34

No rush.

Posted by: Moby Hick | Link to this comment | 09-11-15 8:40 AM
35

30: I don't think it would be useful confusion and frustrating.

For infinite numbers, cardinals and ordinals don't look the same anymore. Ordinals capture order, and how e.g. the rationals in [0,1] don't look like the natural numbers. For finite numbers there's only one possible order for each cardinal so you can elide the difference. But yeah, pants before shoes.

Posted by: dalriata | Link to this comment | 09-11-15 8:43 AM
36

Actually, instead of the rationals in [0,1] let's say the set of numbers of the form m+(n-1)/n:

1/2, 2/3, 3/4, ... 1 1/2, 1 2/3, 1 3/4, ... 2 1/2, ...

Since those are actually well-ordered in the usual ordering. Their ordinal is omega_0 * omega_0, right?

Posted by: dalriata | Link to this comment | 09-11-15 8:48 AM
37

Maybe whoever wrote this textbook was confused by the kerning on the two lines.

Posted by: snarkout | Link to this comment | 09-11-15 8:55 AM
38

Maybe this is really testing who will challenge the teacher. Anyone who does loses a hall pass.

Posted by: fake accent | Link to this comment | 09-11-15 8:59 AM
39

||
I'm checking in for a minor hand surgery, should I ask them to guarantee all services are being provided in-network or is that just being a little bitch who reads too many healthcare scare stories? (I have pretty much the most common insurance carrier in the state.) And would they even be willing to tell me if I ask?
|>

Posted by: SP | Link to this comment | 09-11-15 9:08 AM
40

39: In one of the recent healthcare scare stories I read, the person asked if the provider/service was in-network, was explicitly told "yes", and then found out later that it was actually out of network (when they got the bill).

Posted by: AcademicLurker | Link to this comment | 09-11-15 9:15 AM
41

I asked, admitting person shrugged and said, "Yeah I think so." Hooray for US healthcare! Now I'm paying my \$250 copay.

Posted by: SP | Link to this comment | 09-11-15 9:19 AM
42

She also couldn't tell me whether I already had a referral on file, or whether I needed separate referrals for the hand surgeon consult vs the procedure.

Posted by: SP | Link to this comment | 09-11-15 9:23 AM
43

Just be sure to write "No gonads to be removed" on your inner thigh and lower abdomen.

Posted by: Moby Hick | Link to this comment | 09-11-15 9:28 AM
44

Yeah, I tried asking that before I got my neck decancerized, and there wasn't anyone in the doctor's office who was willing to admit they knew anything reliable about what was covered or whether they were in-network. Turned out okay, but my choices were to take a leap of faith or walk out with my neck still all cancery.

Posted by: LizardBreath | Link to this comment | 09-11-15 9:28 AM
45

43: Making sure you don't have any moles or anything that might look like a comma between "No" and "gonads".

Posted by: LizardBreath | Link to this comment | 09-11-15 9:29 AM
46

45 LB beat me to it.

Posted by: Barry Freed | Link to this comment | 09-11-15 9:31 AM
47

I so wish I could use the time machine to put heebie in charge of my math education. Either I would have understood, and therefore likely enjoyed, it or apparently would not have suffered through algebra. Sounds dreamy!

SP's current dilemma and the bullshit LB and HG went through are making me so grateful for Kaiser, though. That is truly horrendous.

Posted by: dairy queen | Link to this comment | 09-11-15 9:33 AM
48

Castration is anyways in network.

Posted by: Moby Hick | Link to this comment | 09-11-15 9:33 AM
49

I volunteer at the public library's homework help center. Last week I helped a fifth grader whose math homework included the problem "draw all possible rectangles of area 18." (Follow up questions: are any of them squares? Is 18 a perfect square?)

Posted by: | Link to this comment | 09-11-15 9:35 AM
50

49: "Sure, just as soon as you give me an infinite amount of paper."

Posted by: Ginger Yellow | Link to this comment | 09-11-15 9:39 AM
51

With sides of integral length, obviouslam.

Posted by: nosflow | Link to this comment | 09-11-15 9:41 AM
52

Oh, that one is an obvious editing error (left off "with sides whose lengths are whole numbers"). Most kids will understand what was meant rather than what was said, and anyone sophisticated enough to understand the problem will also immediately understand what was meant. It's not ideal, but it's not doing anyone any harm.

Posted by: LizardBreath | Link to this comment | 09-11-15 9:42 AM
53

49

oh good god

Also, I agree with heebie that onto and one-to-one are useful concepts for high-schoolers to know, especially if the alternative is Yet Another Quadratic Formula Problem.

Posted by: Trivers | Link to this comment | 09-11-15 9:43 AM
54

When I was in school I point blank refused to solve any polynomial using the quadratic equation. It offended my sensibilities. I more or less got away with it.

Posted by: foolishmortal | Link to this comment | 09-11-15 9:58 AM
55

ms bill (a math/theology double major in college) sometimes uses "one to one" and "onto" in conversation comparing sets of non-mathy things. I have tried unsuccessfully (and perhaps incorrectly) to invoke the analogy ban...

Posted by: bill | Link to this comment | 09-11-15 10:07 AM
56

I guess I'd be okay with it if they relate it to invertability of functions and why the plus/minus square root thing is different. I'm just skeptical that happens in practice instead of just checking off a mandated curricular item. This is probably entirely personal--I absolutely hated having to learn little things that didn't integrate into my general body of knowledge.

Posted by: dalriata | Link to this comment | 09-11-15 10:20 AM
57

54: I mostly did that, too. I found it hard to memorize and didn't really help me understand anything. Completing the square (or occasionally factoring) for me.

Posted by: dalriata | Link to this comment | 09-11-15 10:24 AM
58

I was always a factorIng man myself. I guess my problem was I never really understood why the QE worked, and it seemed like cheating to use it when I didn't.

Posted by: foolishmortal | Link to this comment | 09-11-15 10:58 AM
59

Completing the square *is* the quadratic formula (ie if you started with ax^2+bx+c and completed the square you'd get exactly the quadratic formula) so there's not much reason to bother with the quadratic formula if you understand completing the square.

Posted by: Unfoggetarian: "Pause endlessly, then go in." (9) | Link to this comment | 09-11-15 11:03 AM
60

(Forgot an "=0" in there.)

Posted by: Unfoggetarian: "Pause endlessly, then go in." (9) | Link to this comment | 09-11-15 11:04 AM
61

59: I guess I'd say that completing the square is how the quadratic formula is derived, but the mechanical process is different. Alternatively you could say there's not much reason to bother with completing the square if you can memorize the quadratic formula, which is also reasonable--it's usually faster to apply, after all. I just found it unsatisfying and my memory has always been atrocious.

Posted by: dalriata | Link to this comment | 09-11-15 11:10 AM
62

Done! It was just the one doctor and three nurses so I doubt there's any issue with coverage.

Posted by: SP | Link to this comment | 09-11-15 11:18 AM
63

And now I will attempt to bike back to work with one hand full of lidocaine.

Posted by: SP | Link to this comment | 09-11-15 11:20 AM
64

Just take the lidocaine now instead of holding it for the whole trip.

Posted by: Moby Hick | Link to this comment | 09-11-15 11:21 AM
65

Silly, SP spent the last 10 years building up an immunity to lidocaine.

Posted by: JP Stormcrow | Link to this comment | 09-11-15 11:32 AM
66

40

Things like that are so rage inducing. Like, there should be a penalty for straight up lying. (Yes, I know it's not actual lying, but gross incompetence that leads to telling someone the wrong information is the same.) Like, seriously, it should be on par with medical malpractice. I can't think of a single other industry in which we allow ourselves to be so poorly treated when it comes to cost and cost transparency.

I now have public insurance, which is kind of amazing in how open and straightforward everyone is in what I have to pay. Instead, I get the receptionist and the doctors telling me right off the bat, "so, we'll do services x,y, and z for free, and then you have to pay \$30 for service w, is that ok?" with option w being a slightly more extraneous service that I don't actually need.

Posted by: Buttercup | Link to this comment | 09-11-15 11:37 AM
67

Not knowing what is in network isn't even incompetence in some cases. The insurance companies have different policies and little incentive to be clear and plenty of reasons to bill whatever they want to bill just to see what they can get paid by somebody else.

Posted by: Moby Hick | Link to this comment | 09-11-15 11:41 AM
68

59,61: Thanks, that's been bothering me for twenty years.

Posted by: foolishmortal | Link to this comment | 09-11-15 11:44 AM
69

The insurance companies have different policies and little incentive to be clear

I don't get why state regulators don't crack down. (Yeah, yeah, I know, government can't do anything right, regulators are always captured by the industry, and so on. But there are some contexts where regulation works okay, and I don't know why this isn't one of them.) Insurance is supposed to be heavily regulated by the state, and this seems to be a context where 'incompetence' that turns into effectively deceptive practices in the insurers' favor is SOP. It seems like an easy target.

Posted by: LizardBreath | Link to this comment | 09-11-15 11:45 AM
70

I have not looked into it, but my guess is that it is because the shittiest insurance providers are cheaper and the government buys a lot of health insurance.

Posted by: Moby Hick | Link to this comment | 09-11-15 11:50 AM
71

69: I bet it's the complexity of how they screw consumers that makes it difficult to simply crack down. Plus capture.

Posted by: Minivet | Link to this comment | 09-11-15 12:31 PM
72

I know you know more about the area than I do, but 71 does not sound nearly cynical enough.

Posted by: Moby Hick | Link to this comment | 09-11-15 12:36 PM
73

Anyway, I don't think anybody set out to make things happen the way they do now, but I think at least somewhere along the line somebody looked at the alternatives and decided that rationing by confusion and assholiness was the politically preferable option.

Posted by: Moby Hick | Link to this comment | 09-11-15 12:43 PM
74

I was at a thing last night that turned out (surprisingly to me) to have a number of health care/insurance marketing people. They were kind of like, um, marketing people. For everyone's benefit I avoided having any substantive discussion with any of them.

Posted by: JP Stormcrow | Link to this comment | 09-11-15 12:47 PM
75

They're like marketing people who know that convincing a subset of their customers to buy from their competitors will give them profits.

Posted by: Moby Hick | Link to this comment | 09-11-15 12:49 PM
76

Knowing the quadratic formula is useful for some other related bits of knowledge. What's the axis of symmetry of a parabola? It's the quadratic formula without the +/- part: x = -b/2a (because it's halfway between the two roots, if they exist). Need to know the number of real roots without computing exactly what they are? Just look at the sign of the discriminant: b^2-4ac. If it's positive, you have two real roots. Negative, no real roots. Zero, you have one duplicate root. One consequence of that is that if a and c are both non-zero with opposite signs, you know at a glance there are two real roots without further computation.

Also, that sign of the discriminant formula thing comes up again if you are trying to quickly determine the shape of a (possibly rotated) conic section, though that's generally an advanced topic in precalc classes these days. From the general 2nd degree polynomial formula Ax^2 + Bxy + Cy^2 +Dx +Ey + F = 0, once again you look at the sign of B^2 - 4AC (assuming A, B, and C are not all zero). Negative: ellipse/circle. Zero: parabola. Positive: hyperbola. If you've memorized your quadratic formula, it's much easier to pick that one up. And knowing how to complete the square is really useful in wrestling one of those formulas into standard form if the conic section in question has been translated away from the origin.

What good is knowing all this stuff? If nothing else, it's probably worth a few extra seconds and points on the math section of the SAT (the first stuff, that is - they don't cover most precalc stuff on the general math section of the SAT). That's the "teach to the test" answer - the other answer is that it makes it easier to remember and connect to other bits of knowledge, and because it can be fun.

Posted by: Dave W. | Link to this comment | 09-11-15 1:13 PM
77

76: All good points. The discriminant, especially its sign, does tell you a lot. That's why I compromised and eventually memorized the discriminant. Ironically, now that I work on math software I probably know the quadratic formula better than I ever did during my actual education.

Posted by: dalriata | Link to this comment | 09-11-15 1:20 PM
78

I sing the quadratic formula to my students to the tune of Pop Goes the Weasel.

Posted by: heebie-geebie | Link to this comment | 09-11-15 1:24 PM
79

X equals negative b
Plus or minus square root
Of b squared minus 4ac
All over 2a.

Posted by: heebie-geebie | Link to this comment | 09-11-15 1:25 PM
80

I mean, they know it already - I'm not covering new material - but they think it's funny.

Posted by: heebie-geebie | Link to this comment | 09-11-15 1:26 PM
81

That scans.

Posted by: Moby Hick | Link to this comment | 09-11-15 1:29 PM
82

That's a good mnemonic. Right up there with singing the dynasties of China to Frere Jacques.

Posted by: dalriata | Link to this comment | 09-11-15 1:30 PM
83

Doesn't it require a rest after "X"?

Posted by: Minivet | Link to this comment | 09-11-15 1:31 PM
84

I have my "French governments since 1789" song. Set to the "Five days of ... " of part of the "Twelve Days of Christmas."

Five French republics,
Four with no DeGaulle,
Three ousted kings,
Two emperors,
And a Vichy fascist regime.

Posted by: Moby Hick | Link to this comment | 09-11-15 1:34 PM
85

76.3: also, you can estimate where to bang in some nails to hold lath for a curved form for something. My undergrads tended to stare blankly at this, but then, they also stared blankly at any mention of knowing something for a job. Dunno. Good luck out there, kids.

(What did work was mention of death. Little bunny death, fishie death, shark death, even all the trees dead.)

Posted by: clew | Link to this comment | 09-11-15 1:34 PM
86

Speaking of parabolas, I found this video by Mathologer, "Multiplying Monkeys and parabolic primes," to be very well done. New to me. And here's a decent reddit thread on it.

Posted by: JP Stormcrow | Link to this comment | 09-11-15 1:46 PM
87

I learned that in high school except it was "radical" instead of "square root of"

Posted by: SP | Link to this comment | 09-11-15 2:17 PM
88

The OP example is not a one-to-one, because of poor kerning.

Posted by: Econolicious | Link to this comment | 09-11-15 5:21 PM
89

I mean, they know it already - I'm not covering new material - but they think it's funny.

I bet you are superfunnycute singing it.

Posted by: rob helpy-chalk | Link to this comment | 09-12-15 6:50 AM
90

I managed to get my students to remember the Ideal gas law with

PV = NRT,
Doo-dah, Doo-dah!
PV = NRT
Doo-Dah Doo-dah Day!

Posted by: togolosh | Link to this comment | 09-12-15 6:59 AM
91

Right up there with singing the dynasties of China to Frere Jacques.

Starting where?

Posted by: chris y | Link to this comment | 09-12-15 7:00 AM
92

All the mnemonics I can remember from school/university days are chemistry or biochemistry related. There is an entire Biochemist's Songbook, but the only one I can still sing any of is the tricarboxylic acid cycle set to Waltzing Matilda.

Posted by: Ume | Link to this comment | 09-12-15 7:19 AM
93

I know one for the components of a Big Mac.

Posted by: Moby Hick | Link to this comment | 09-12-15 7:24 AM
94

90: Our teacher taught us to remember the almost word "pervnert."

91: Starting with Shang, going to the PRC. It skips all the little ones and times of turbulence.

Posted by: dalriata | Link to this comment | 09-12-15 10:03 AM
95

I subscribe to the Straussian reading of the OP: "one-to-one and onto correspondence" is probably a subtle response to the school's likely abstinence-centric sex education program.

Posted by: arthegall | Link to this comment | 09-13-15 11:20 AM