Sure, it doesn’t need to be useless (does he say that, really?), but it needn’t be useful either.
In any case, most of K-12 math is useless for most people.
I hadn’t noticed that before. Neat.
Anyway, if any of the non-mathematicians in this thread need another reason to doubt Lockhart’s insistence on the beauty of uselessness, just look at the number of great mathematicians who applied their math to physics, astronomy or engineering: Newton, Gauss, Euler, Archimedes, Lagrange, Laplace, Hilbert, and many more.
I don’t know that he says it outright (Hardy, who influenced him, does), but that’s always the impression I get when reading the essay. For example
Yeah, sometimes you’re just playing with concepts that are only in your head. But, a number of important mathematical results have come from people wondering how gravity works. You wouldn’t suspect this if you just listened to Lockahrt.
It just means manipulating the equation until you have a perfect square on one side. An example (admittedly very easy, because you can factorise it at a glance anyway):
x[sup]2[/sup]-2x-8=0
x[sup]2[/sup]-2x=8
x[sup]2[/sup]-2x+1=9
(x-1)[sup]2[/sup]=9
(x-1)=3 or -3
x=4 or -2
Once you’ve had to use this method lots of times on more complicated quadratics, you can derive the general formula by completing the square on ax[sup]2[/sup]+bx+c=0 at the drop of a hat.
Oops. I made a double post
Basically, take half of the coefficient of x, square it, and add it to both sides.
EDIT: After dividing my the coefficient of x[sup]2[/sup] of course.
Note: this is not really directed at Thudlow. These are feelings and experiences I have had with math. I envy those to whom math comes easily. I don’t think you all know how truly difficult it can be for others.
No, this is not true. Everything in math does NOT make sense, unless you accept underlying premises (or principles). “Imagine a unit circle with a circumference of 1” was how my Trig class started. Since I can imagine circles (I had no idea why it was called a “unit” circle; I still don’t and I don’t care) of any circumference, I lost interest that first day. Not that I had a lot to begin with. Arithmetic, is very useful, as are some basic equations such as distance=rate x time, but other than that, it is not useful or logical to me. WTF is the quadratic equation FOR? I can stare at a word problem for hours and not know where to start in solving it–I may not even know WHAT they want me to solve for. I can figure out area and perimeter–and that’s about it.
It’s not that I don’t see the beauty in solving a math puzzle–I can appreciate that, but the rules ARE arbitrary IMO. Really–we have to balance an equation? Why? WTF is a constant and why do we use it and who decided what would be a constant? How does it matter? Can we solve things without it? What does 3.14 have to do with anything in my world (not that I mean to say that all math must be personally useful or significant, it’s just that no one ever explained WHY we needed the damned pi to begin with–I KNOW the Ancient Greeks “discovered” it–AND?) Who says the product of two negative numbers is a positive one? That does NOT make logical sense at all. Surely multiplication should work the same on the “other end” of the “range” of numbers, that is, -4 x-4 (should)=-16?
I wish I excelled in math. I wish it came naturally to me. It doesn’t . I struggle. Never ask me to do anything remotely complicated with money (as in collecting cash or splitting a bill more than 3 way). I can’t do it well or easily or correctly. I cannot figure out word problems–I do not know what the writer of the problem is looking for after about 5th grade level word problems. I have a graduate degree and know I am intelligent, yet I cannot do basic HS math. I can (and enjoy) graphing x,y stuff, but (another example) I cannot identify which is the divisor, the dividend or the quotient. No clue. These are just examples.
In studying for the GRE, I decided (since I had scored so low on a sample pretest that my score did not register on the scale provided which means I got less than 11 correct, IMS), to just follow their damned rules and not try to make sense of it. I did manage to score what would be a low C (I don’t recall my score offhand)–thank God I didn’t need a good math score to get into grad school!
Yes. From the 5th grade student teacher who told that since I was so smart in English etc, that I MUST be good in math, too (this was the 1970s) to the 6th grade teacher who had to take all our time to help a special needs kid (long story) and so my questions were not answered, to the 7th grade math teacher who was batshit insane and used to turn red with rage and throw chalk/erasers and protractors at us, to the HS ones who said I really didn’t “need” math or the one who drank and suffered from the shakes in Trig class–I had my share of suck ass teachers. I also had ones who really tried to get some of this stuff to stick.
That is intriguing.
Sorry is this seems rather intense. I have intense feelings about math. I am frustrated and resentful that I cannot seem to acquire better math skills–not that I’m trying on a daily basis, mind. 
If the equation isn’t balanced, then it’s false. 2 + 3 = 5. 2 = 5 - 3. But if we only take the 3 away from one side, then we get 2 = 5, which is false. A constant is simply a number that doesn’t change. X + 2 = 5. 2 is 2. It doesn’t change. Pi may not be useful to you, but it is for many people for many things. Even I use pi from time to time. For multiplying a negative, switch it to English: ‘I did not not eat the pie.’ You either ate it, or you didn’t. If you didn’t not eat it, then you did. So a negative times a negative is a positive.
I am not a math person. I feel your frustration, believe me. I struggled. But if you put things into language terms, it can help.
That makes perfect sense, and now I remember we did learn that in school.
Believe it or not, the equation “distance = rate x time” is the most compelling reason why “negative x negative = positive.” At least, in the world as I see it. I can try to explain:
Suppose there is a highway that starts at your city with mile zero. That is, if you leave the city along this highway, you first see mile marker 1, then mile makrer 2, etc. Suppose you’ve left your city along this highway, travelling at 60 mph the whole time, and right now you are passing mile marker 180. Your destination is another 200 miles down the highway.
The first question: where will you be two hours from now? Using your equation above, distance = rate x time, so distance = 60 mph x 2 hours = 120 miles. You’re already at mile marker 180, so in two hours you’ll be at marker 180 + 60 mph x 2 hours = 180 + 120 = 300.
The second question: where were you two hours ago? Again, using your formula, we get 60 mph x 2 hours = 120 miles. But this time, you’re 120 miles back, so you were at mile marker 180 - 120 = 60.
Now, in question one we wrote “two hours in the future” as “2 hours”. It stands to reason that “two hours in the past,” the opposite of “two hours in the future,” would be represented as “-2 hours.” Just using the fact that “-” is the opposite of “+”. So using your distance equation, we should have 180 + 60 mph x -2 hours = 180 + (-120) = 180 - 120 = 60 miles marker. With me so far?
The central idea is that positive and negative are directions. For questions three and four, you are now coming back home from your destination. But you’re still travelling at 60 mph, and still passing mile marker 180.
The third question: where will you be in two hours? Same reasoning in question 1, you’ll be 120 miles ahead of where you are, but you’re travelling the opposite direction of the mile markers, so the marker you’re passing will be 180 - 120 = 60. Since you’re travelling the opposite direction, the “rate” you want to use is -60 mph. So the equation is 180 + -60 mph x 2 hours =180 - 120 = 60.
The fourth (and final) question: where were you two hours ago? Obviously, you were at mile marker 300. The equation must be 180 + -60 mph x -2 hours = 300. And the only way that this works out is if -60 mph x -2 hours = +120.
And that’s it. All the rules about multiplying signed numbers come about from the formula “distance = rate x time”, and applying some common sense.
REA Problem Solvers
Parabolas. They describe trajectories of objects in a vacuum…which must be corrected of course for real world friction etc. but hey. Also, parabolic microphones, parabolic mirrors…I think a lot of telecom stuff WRT satellites must depend on them.
Surely multiplication should work the same on the “other end” of the “range” of numbers, that is, -4 x-4 (should)=-16?
[/QUOTE]
You’ve posed many good questions…I want to leave some for others but I’m allowing myself one more. 
Donald Trump’s daughter said she was walking down the street with him one day, before he made it big, and he saw a homeless guy. He said something like, “That man is worth $100M more than me.” The guy may have had a personal net worth of $0, but Trump was in the hole, big time.
- You have $5 in one pocket, $5 in another. $5 x 2 =$10.
- You owe friend A $5, same to friend B. -$5 x 2=-$10.
Two friends owe you $5…treat that as both positive like in example 1 if you like. But if you treat a debt as a negative, and you don’t owe it to two people (they owe it to you), then it’s like you owe it to -2 people. -$5 x -2 people=$10. You’ll be $10 to the good either way you calculate it.
I just want to offer my sympathy to the OP.
Back in secondry school (high school), I scored highest in my year in a maths exam. I didn’t even feel like maths required any effort, it just came naturally.
But I didn’t take my maths further than high school level, and now need to do so.
And in the intervening 14 years, it seems my brain has atrophied!
I’m slow to pick up maths concepts, and quick to forget what I’ve learned.
Quadratic equations are not for any one thing, it’s simply a name given to second order polynomial equations, distance=rate x time is a first order polynomial equation. The thing about d = rt is that it assumes the speed is constant when in reality it rarely is, so we might want to add a constant acceleration in there. An object’s speed(assuming constant acceleration) at any given moment is equal to its acceleration * time + initial velocity. So the rate in the first equation is r = at+v[sub]Initial[/sub].
This means that d=(at+v[sub]Initial[/sub])t = at[sup]2[/sup]+v[sub]Initial[/sub]*t , and now we’ve turned distance=rate x time into a quadratic equation describing the distance travelled by, say, an object falling in a vacuum.
This video has a fairly good explanation and rigid proof.
To some extent, it’s a tool that you need in the toolbox for tackling some tougher math. I’ve been taking a course in applied mathematics, and I’ve seen quadratics turning up all over the place, such as in how to work out the behaviour of a weight on the end of a spring. This is important if, for instance, you want to work out how stiff the springs in your car ought to be and how much damping to give them. You want to smooth out the bumps in the road, but you don’t want your car to still be bouncing up and down half a minute after the bump.
You have to balance an equation because that’s what an equation is - two things on opposite side of an equals sign. It’s like putting stuff in the pans of a balance scale. Once they’re balanced, if they’re to stay balanced then you must do the same to both sides - whether that means adding a pound to both sides, doubling both sides, or whatever. Nothing arbitrary about it whatever.
A constant is a “magic number” that can be derived by experimentation. How long does it take to boil a pint of water with a 1kW kettle? That depends not only on the mass of water (sixteen ounces in US measure, I believe) but on how hard it is to heat water - a certain amount of heat must be put in for every degree change in temperature. If you were to fill the kettle with pure alcohol, the answer would be different. The difference between a pint of water and a pint of alcohol, in this case, is the constant known as specific heat - there are many others.
Can you solve things without it? Yes, but you have to reinvent the wheel each time - if you don’t know the constant, you can’t solve the equation (specific heat times mass times temperature increase equals amount of heat to put in), so you have to measure instead.
Or how much does a copper bar expand when heated? There’s a constant for that - proportional increase per degree, known as “coefficient of expansion”.
Pi is a constant - the circumference of a circle divided by the diameter. That’s always the same number no matter how big or how small the circle. And if you want to be able to work out how fast your car engine needs to turn to propel yourself at a certain speed - not necessarily your problem, but the designer’s - then knowing about pi is the least of your worries.
And another example. Let’s assume you and I have bank accounts with overdraft facilities, so our bank balances can go below zero. Let’s assume your daily pay from me is the only thing presently affecting your bank balance. Now…
Today you have $0. I pay you $5 per day. How much money do you have in five days’ time? 5 days x $5/day = $25.
Today you have $0. I charge you $5 per day (that’s the same as paying you -$5 per day). How much money do you have in five days’ time? 5 days x -$5/day = -$25.
Today you have $0. I pay you $5 per day. How much money did you have five days ago? -5 days x $5/day = -$25.
Today you have $0. I charge you $5 per day (that’s the same as paying you -$5 per day). How much money did you have five days ago? -5 days x -$5/day = $25.
When I win BIG, I shall laugh heartily.
------->has math anxiety.
Congruency and betweenness…(I’m learning the very very very basics of geometry, with triangles. Boy, people are fascinated by triangles.)
Thank you all for your responses. I can’t say that your explanations clicked with me, but I do appreciate the effort. 
It is nice to know what quadratics are for, but they were not a big problem for me in math. I LIKED being told how to solve an equation and then doing it endlessly. It is how that particular formula was created and how it was “discovered” that eludes me–I also cannot seem to figure out just when to use what formula in RL. I am a simple person. 
What gets me about math is 1). you’re supposed to (somehow) remember all these different formulae and drag them out when needed and 2). you’re supposed to know which formula to use when. As if… chance is a fine thing!
I really like d=r x t because it makes sense to me and teenage me could tie it to something concrete (like a road trip). Other things are more abstract and therefore more problematic for me (and others, I’m sure): sine and cosine; base 8 or base 2; prime numbers (which I find interesting, but have no clue if they’re useful in any practical sense)… I can’t tell you what all I find perplexing because I have forgotten what most it is called. It’s been 30 years since HS math.
I do appreciate that many disciplines use pi. I still don’t know WHY they use pi or how pi is so special to our understanding of our world, but that’s ok.
Answer me this: why in some calculations do we suddenly have to multiply by 1000 or stick 60 in, especially when they cancel one another out? (not that 60 and 1000 cancel one another out–I’m using 2 different examples to make one point).
Like this formula for calculating IV drips (now done by the computer in the IV pump, but still…)
(dose/volume x 1000)/60/kilograms= dose/cc.
There are other formulas, but I cannot find them just now–I used to have them all in a small book that fit in my pocket and I used them religiously (and accurately) to calc drip dosages and drug dosages for pts. I just never understood how those particular formulae were the right ones, if you follow me.
I’m not familiar with those particular formulas, but I’m guessing that they’re conversion factors. (Grams and kilograms, for example, differ by a factor of 1000, and second and minutes differ by a factor of 60.)
You might be able to use dimensional analysis to make sense of such formulas.
Here are some quotes about math that you might appreciate.
Bertrand Russell, Mathematics and the Metaphysicians, collected in Mysticism and Logic.
James M. Barrie, Quality Street, act II.