I’ve created a poll about whether people know how to extract square roots by hand http://boards.straightdope.com/sdmb/showthread.php?t=833173
And why is it any different than typing “2+2” when that’s needed?
The answer has to do with things like frequency, simplicity, and convenience.
What we’ve been discussing is which of three categories various sorts of calculations should fall into:
(1) Things that people (students) should have memorized and be able to recall
(2) Things that people (students) should be able to figure out, to calculate for themselves in their head or with pencil and paper
(3) Things that people (students) should do on a calculator or computer.
Just because some things fall into a certain one of these categories, doesn’t mean everything does.
I’d say that which category something falls into depends on
(A) How often you’d need to use it
(B) How simple or fast or convenient it is to do it one way vs another
(C) Whether some special benefit (like a deeper understanding of underlying concepts) is gained by doing it one way vs another
2+2, and other problems like that, come up often enough that it is a significant time-saver to be able to do it in your head. Factoring and square roots both come up much more rarely… but I’d have to say that, of the two, square roots are the more common. So if anything, that’d suggest that we should be teaching manual extraction of square roots rather than manual factoring.
And of course we shouldn’t just teach “Ok, today we are going to learn how to factor 2nd order polynomials. Type this into your calculator ‘factor (2x^2 + 9 + 6)’. The answers are the roots. That’s how you factor 2nd order polynomials. Now, lets move on to square roots”. Students still need to learn plenty of things about factoring: They need to know what it is, and what it can be used for, and why it’s relevant. And maybe they should even learn a little about how it can be done. But how it’s done isn’t really the important part.
I don’t understand this at all. I’m not trying to be argumentative, I honestly don’t understand. It seems like this point of view can be used for almost everything that is taught in school. Math, History, Computer Science, etc.
A few examples:
“Why do I need to know how to program a sorting algorithm? I can just look one up”
“Why do I need to know the dates of the Civil War? I can just look them up”
“Why do I need to know how to calculate derivatives? I can just have Wolfram solve it for me”
Again, I’m not trying to be an ass or arguing. The concept of not learning how to do things is school is honestly baffling to me.
72 has more factors so it’s usually easier to work with in your head, but there’s another reason that it’s used instead of 69.3.
69.3 would be correct for more compounding periods, but if you’re talking about for example an 8% annual increase, not compounded but after one year it’s 8% bigger, then the the actual doubling time is 9.0065 years. For most rates of growth in the range that people commonly deal with, 72 is closer than 69.
Examples of rates of growth and (growth times doubling time):
2% 70.006
4% 70.692
6% 71.374
8%: 72.052
10%: 72.725
12%: 73.395
With computers is is critical to know why you are using a method, or you will most likely end up with a solution that has a time or space complexity that will become quadratic or exponential, when you want to use a linear solution for anything that isn’t trivial when possible.
I spend more time memorizing the Big 0 complexity of solutions than remembering answers they offer.
Here is a site that is a good reference on the “costs” of these calculations.
This is why they make you program a bubble sort in interviews, to make sure you know what to avoid, or more importantly in what cases you need to avoid it.
I agree with Chronos.
Also, keep in mind, the things I’m saying here are with respect to those math classes which are generally made mandatory for everyone to take, regardless of interest, out of some sense that everyone needs to know them.
I just don’t think everyone needs to spend a significant portion of their life tortured with manual polynomial factorization drills or whatever, and my observation, as a teacher and just as a person in the world, is that it doesn’t really accomplish much of lasting value for most people except make them hate “math” as, rightly, a bunch of tedious trivial nonsense.
Even as a mathematician, this skill has never been necessary; conceptually, there’s not much to it (maybe two things end up multiplying to give this; run through possibilities one by one and see if any work…). But to actually do the factoring? I’d go with a computer every time; it’s precisely the sort of tedious heuristic or brute-force task computers are good at and humans aren’t. There’s no point trying to be the John Henry of mindless algorithm execution in running 54931 through the sieve of Eratosthenes or whatever.
But, again, the things I’m saying here are with respect to those math classes which are generally made mandatory for everyone to take, regardless of interest, out of some sense that everyone needs to know them.
For those people who are particularly interested in training themselves on particular niche skills, they should of course have the opportunity to learn and to practice whatever they want, whether it’s practicing factoring 54931 in their head or practicing sewing clothes by hand or drilling guitar arpeggios or learning the general theory of unique factorization domains (there’s a whole wonderful theory of how polynomial rings over a unique factorization domain are themselves unique factorization domains, by combining the fact that polynomial rings are even Euclidean domains if taken over a field with the fact that the GCD of the coefficients of a product of polynomials is the product of the GCDs of their coefficients separately…) or learning how to build guitars or whatever.
Anything at all, for people who are interested. But let’s examine how much it accomplishes to try foisting everything upon everyone ahead of time, just on the off-chance they may someday care to have it. How many people does it actually stick with, and pay off for, for all the resources that go into it?
Yes, I too know 0.69 by heart. Not by conscious memorization, but because I saw it over and over. I can’t tell you very much about ln(3) without sitting down and working it out.
Do you actually think it’s useful to know the exact dates of the Civil War? Why? Wouldn’t it be better to spend time in history class learning history, instead of lists of dates?
Is this question to me or to others? If it’s to me, I have exactly this complaint about the style of history classes when I was younger, which seemed to spend an awful lot of time on drilling and testing memorizing exact dates of battles and shit (stuff which, again, in the modern world will never be necessary, lookup just a button press away; one may question how necessary exact date rote knowledge ever was, mind you…), instead of actually analyzing history. Luckily, standard general history classes did get somewhat better later on, in a way which standard general math classes did not.
I don’t mind having the dates mentioned, mind you. Go ahead and mention the exact dates, of course. I also don’t mind kids being told how one could factor a large number (mindlessly march along and test for divisibility by everything…; there’s nothing to it, conceptually) or whatever. It’s making these into things to drill and test where it becomes problematic, where it takes on far too much bothersome focus.
(Of course, there’s a difference between testing knowing rough dates and order vs. testing knowing exact dates (April 12!), which I could translate into analogous math things that matter and don’t as well)
I wanted to include this somewhere in a previous post, but didn’t quite find the place, so I’ll just toss it out there, ad hoc, into the aether:
I could make a guitar player of anyone, any idiot, given enough time and coercive authority. Give me 12 years with a child in a cage, and I’ll have them moderately virtuosic whether or not they like it (I’d need a little time to learn guitar myself; still, we’d get there). But I’d prefer to let people do what they want, and if they’d like to practice guitar later or never, let them practice it later or never.
One problem with that is that we learn our best during our school years. We also have no idea what we actually want to do with the rest of our lives when we are 10.
Teaching a bit of everything that could be useful depending on what path you decide to take is easier than trying to catch up on learning from scratch once you have made that decision.
Yeah, but I don’t know how to play guitar, do I?
We learn best when we are interested. A few skills are so ubiquitously vital that they are worth jamming in even against childish resistance (for example, literacy and fluency with the basic concepts of arithmetic). Most aren’t. If I stopped random adults on the street, I wager 9 out of 10 would be such that having drilled (if they had drilled) polynomial factorization in their adolescence (as opposed to later when interested, or never) had made zero lasting benefit to their lives. And if I asked them to factor the extremely factorable 2x^3 - 7^2 - 7x + 30 by hand, I suspect even many STEM folk would fail miserably (Cardano’s method notwithstanding…). No biggie. Their lives aren’t the worse for wear.
It’s not reasonable to try and cram in by age 18 or 22 every scattershot thing which might be useful in students’ further lives, against students’ own wishes. Why should the acquisition of knowledge and skills be prix fixe rather than a la carte? Even if it were a reasonable goal, even recognizing the good intentions behind it, the attempt just doesn’t work; people remember the things they care about, and forget the rest.
Everyone requires a certain amount of baseline skills forcibly ensured, and a certain amount of compulsory exposure to what else is out there in the world is useful. But that doesn’t mean everything we’ve come to traditionally treat as a mandatory baseline skill is in fact worthy of being so treated. And one great guide is to see how much is actually accomplished in trying to get everyone to learn such-and-such; how much bang we actually get for our resource bucks. The calculation does not seem to come out in the black for a lot of these algorithm running drills that pass for “math”.
When students tell me they hate math and it’s pointless, they’re right to hate it and find it pointless, because all they’ve ever seen is hateful pointless stuff. They recognize the truth; though they’ve been yelled at their whole lives for being insufficiently invested, they can see through the bullshit. They’ve talked to adults who remember little from math class, and seen the future for themselves, and figured the whole game out. (I also feel similarly about plenty of non-math stuff as well, but math is what I have most familiarity with.)
Look, I’m not saying I have the world’s ear, that I can actually enact the changes I’d like to enact. But I’d like to at least acknowledge the reality of what is and isn’t useful in what we do, even if I can’t change it.
Thanks. I appreciate learning this.
I’ll try not to make this too long:
Plane fires missile (for example, the Shrike HARM missile, an air-to-ground anti-radar missile). Missile flies down range, but misses target. We want to know why/how. To know that, we have to know as close to exact as possible the missile’s path.
Cameras line both sides of the missile range. They each have a fixed focal point, the co-ordinates of which are known. They are synchronized so that they all take each frame of film at the identical time (for the tests I was working on, usually at 5 fps). So you take the film from a camera, put it up frame by frame on a screen, and set two crosshairs on an agreed part of the missile (usually the nose). The machine then reads where on the frame the nose is located. Each frame has two dials, which show the azimuth (bearing on the compass) and elevation (number of degrees up/down from horizontal) the camera was pointing at that time. With these pieces of date, we can plot the reasonably precise direction from the focal point of that camera to the nose of the missile frame-by-frame.
Now, you take one camera’s data from one side of the range, and the data from a camera on the other side of the range. You calculate where the two lines intersect. That spot is the place where the nose was located at that point in time. How do we know this for a fact? Angle-side-angle triangle congruence, of course. The “side” is the known length of the distance between the cameras. The angles are the angles between the side we know and the directions to the nose of the missile. By ASA congruence, the triangle we “draw” from these calculations is congruent to the triangle that existed at the time the frames were shot.
Seismologists do the same thing when hunting the epicenter of an earthquake. They can know the direction of the earthquake from the data recorded by a seismograph, and if they have that data from two such locations, they can then triangulate the epicenter. Of course, data can be imprecise, so you prefer to use more than two, and narrow the area of imprecision with the added data. In the case of the missiles, we usually had 5 or 6 cameras’ worth of data, so the imprecision could be narrowed considerably.
While I can’t relate directly to other industries, a lack of basic math skills is one of the main barriers I have with hiring individuals in the computer industry. In my case (anecdote and not data) the lack of qualified US citizens is the only reason I have resorted to the difficulty of recruiting foreigners workers.
While terms like “machine learning” or “AI” may seem trendy or irrelevant the reality is that most of the new innovations in computer absolutely require the fundamentals of algebra like polynomial factorization.
And I am just talking about basics like the distributive property to factor a monomial out of a polynomial, which has been a challenge to find in my industry for all of history.
As an example, in machine learning the use of a Sigmoid function is almost universal, and the inability solve for a monomial is critical.
And yet it is hard to find job candidates that can even use FOIL to multiply two binomials.
When adults need to go back and learn basic algebra rules, which are often one of the hardest parts of Calculus it becomes a serious barrier.
And that only catches them up with what we needed a decade ago, now n-rank tensors and matrix math is becoming critical and is even a higher barrier.
In the IT world the traditional system admin role is contacting in numbers, and fields like networking which have resisted automation are changing.
I will avoid writing a huge post detailing the reasons but being a component programmer is going to be a requirement for almost all tech workers soon and this is highly dependent on basic math skills. Even writing queries, or program logic resembles these trouble shooting skills at a basic level.
While I can’t comment on other industries in GQ with confidence I would think that erroring in the direction of over-education is the safest solution for our future. Math is like playing the guitar or basketball, it is primarily a learned skill and requires practice.
Ask yourself how long a mirror must be to show your entire self in it? Assume for an easy answer that you are exactly 6’ tall.
Most students assume the mirror must be 6’ tall to do this. They are influenced in their thinking by having seen floor-length mirrors, usually on the backs of doors, or in clothing stores. They are, of course, wrong.
Ask yourself whether or not the distance you are from the wall on which the mirror is hung matters in answering the question? If you are like most students, you will think the answer must be “yes.” You, again, would be wrong.
Simple application of congruent triangles can prove the correct answers to these two questions. Used to be fun to have a group project on the subject, where groups try to figure out the answers without having a mirror around to help, then get to use mirrors of various lengths at various distances to see how they did, and refine their thinking. 
Another way of thinking about it, instead of via congruent triangles, is that a mirror is a portal into a mirrored universe. To see yourself in the mirror is like seeing your evil twin on the other side of the mirror, exactly as far away from it as you are. The mirror being halfway between your eye and your twin, how tall will your twin’s image on the mirror be?
That’s pretty awesome, thank you for posting it. But, again please don’t take this as argumentative, but when you say “You calculate where the two lines intersect”, do you mean you manually do it, or a computer does it for you?