Some of you wanted to know how to solve the math problem I posted yesterday.
Here’s my worked out solution!
Hope it makes sense 🙂
Yes, but you didn’t literally show all your work. You just skipped showing all the commutative, associative, and distributive rules being applied. It’s a lot more tedious if you really show all your work. You should start off like this:
h(x) = -2x
f(x)= x^2 – x + 1
f(h(x)) = (-2x)^2 – (-2x) + 1 = (-2x)*(-2x) + 2x + 1 = (-2*x*-2*x) + 2x + 1 = -2*-2*x*x + 2x + 1= (-2*-2)*(x*x) = 4x^2 + 2x + 1
I don’t have to look. You rickrolled your students, didn’t you?
I was never able to get the link to the video to work. What is it?
Secret word was Bubbles.
We could go full-on rigour and require proofs for the addition of natural numbers! 😀
It’s so obvious when you see it right in front of you like that…
I’m disappointed. I was hoping that you might have some clever non-brute force ways to get the coefficients. (I mentioned in the last thread how you can get quick data in some ways). Frankly, I don’t think giving this sort of complete brute force problem to students is that helpful. If they are honors students they can probably already do algebra pretty reliably. At that point one should be giving them interesting problems. I would give as examples for about the same level:
1) Let f(x)=x^2+x+1. Find a polynomial g(x) such that f(g(x))=x^4+3x^2+1
2) Factor x^4+4.
3) Let f(x)=x^2+x+1. Find a polynomial g(x) with integer coefficients such that f(g(x)) factors into two irreducible polynomials of the same degree.
4) For each of the following polynomials if it is factorable, factor it into irreducible factors. If you think one of them is not factorable explain why not: x^4-1, x^2-x+1, x^4+1, x^3+2.
These seem like better problems for honors level students. They have to do real algebra but require some amount of thinking to get them.
I am afraid I am going to come of as rude and arrogant, but I want to say this anyway. I am along time lurker and I normally love what you do. This time, however, I am afraid I wish you did not give a problem like this to your students. I am a mathematician, and I think you are doing more harm than good.
Most people already think of mathematics as something arid and ugly, as a set of rules and algorithms handed down from on high to be followed blindly. Just computations. At the university level we have to undo the damage of years of high-school education that reinforce this point of view, and we have to try to work with students who refuse to learn anything other than mindless manipulations.
Your proposed problem does not accomplish anything. It does not show anything pretty. Students who cannot solve it will just be frustrated and, even after being solved the solution, will wonder what the point is. And they will be right. Students who can actually solved it won’t have learned anything (because they were already good with the computational aspect, which is the only thing you care for here). Contrast this with the problems that Joshua Zelinski proposed above. Those are clever and cute, require an understanding of the concept (rather than mindless manipulations), will teach something even to an advance student, and will leave a student who solves it with a warm, fuzzy feeling inside. (That is how you plant the seed for a new mathematician.) Even a student who tries hard and fails will learn something by being showed the solution afterwards.
On the other hand, if you insist on proposing this problem, at least offer a sample solution that uses some shortcut due to a hindsight. For example, use associativity of composition of functions to do a shorter calculation: it is easier to first compute g(g(h(x))) and then insert that in the middle; or notice that the sequence g(f(g(f(x))) appears in the middle, so you can save some time by computing g(f(x)) first and then composing it with itself.
Finally, there is one way to do this problem entirely by hand (without a calculator) but bypassing much of the tediousness. We know that all the coefficients appear in your list. We can reduce all the coefficients modulo 20 and keep only numbers between 0 and 19. Then, after all the compositions, look back to your list and identify what the numbers actually were. There is a bit of ambiguity, which means we will have to try a couple of urls, but it is worth it to simplify the problem severely. If this had been your intended solution, then I would not be complaining.
Thanks, Hemant for posted the solution. If I ever got a cheat sheet, I always wrote down what to do in this problem. My brain will simply not remember the diff between f of g, and g of f.
I don’t think you get the point here. That might actually be interesting. 🙂
That’s all gibberish to me, and, I suspect, to the vast majority of college-educated, intelligent people. Yet, somehow, we manage to live happy, productive lives. How is that? It’s like learning to solve a Rubik’s Cube in 30 seconds. Impressive, but so what?
If this was the only type of problem I ever gave, I’d agree with you. But it’s not. It was a one-off challenge, only for the kids who want to test their own algebra skills.
What fraction of them do you think just used a TI-83 or Wolfram Alpha or something similar?
They might have but they had to show me the work, so shortcuts wouldn’t have helped.
I solved it! It took me three hours in between dealing with customers at work, but I actually got the answer right. That’s kind of a big deal for someone who gets panicky about math 🙂
For us math nerd types, give us one math problem a week to solve. 🙂
Let people come up with innovative, interesting ways to take shortcuts and “cheat” to make it easier (or faster) to solve. That is also a useful skill (in addition to being able to correctly work through tedious calculations).
No. It isn’t like learning to solve a Rubik’s Cube quickly. (Although there’s interesting group theory related to that.)
Being able to manipulate polynomials at a basic level is something lots of people need to do. If one wants to be a physicist or an engineer one needs to be able to do this. If one wants to be almost any other type of scientist, or one wants to be an accountant, or an actuary one will need to be able to do algebra.
If one isn’t in any field like those but wants to just understand scientific results beyond the basic headlines one needs to be able to understand algebra. For example, you may have heard about the recent reports of faster-than-light neutrinos from CERN http://en.wikipedia.org/wiki/OPERA_neutrino_anomaly. Do you want to understand that? If it turns out to be wrong (as it likely will) do you want to understand more than just the headline? If it turns out to be right do you want to understand the implications for our world around us?
This isn’t the only example. Simply understanding the modern map of the world around us requires math. Do you want to understand why planets travel in elliptical orbits? Well, that takes calculus and for that you need a lot of manipulation of polynomials.
Or do you want to understand the counter-intuitive result that if I have around 30 people in a room it is more likely than not that two of them will share the same birthday? http://en.wikipedia.org/wiki/Birthday_problem
Having the patience or ability to reliably solve the sort of problem that Hemant Mehta is not necessary for having an understanding of the world around you. But having the ability to do smaller similar problems certainly is. And having such work be “gibberish” to you certainly puts limits on how much you can understand almost any science beyond headlines. You might be productive, and you might be happy, but to say that people who can’t do this are educated might very well be an abuse of the term.
Do you think that science is one of the most important methods we have of finding out about ourselves and our universe? Do you think science is important to answering the big questions and making the big achievements that matter for humanity? If so, not being able to do basic algebra is like a deeply religious Christian who hasn’t read most of the Bible. Yes, that applies to most Christians, but I suspect that most people here would to that as a problem with Christians (or for that matter many other religions with large religious texts where the adherents don’t know them that well).
Science is our best hope for long-term survival of the species, it is the only hope we have for extending lifespans, stopping existential risk threats, traveling to other planets, and meeting other life if it is out there. And to understand science at even a basic level, one needs math.
You might be productive, and you might be happy, but to say that people who can’t understand this are educated might very well be an abuse of the term.
Seems a bit unfair. I cannot understand algebra (I still struggle with basic arithmetic), but I consider myself an educated person. I was on the Honor Roll in high school. I graduated from college. It was easy for me to get A’s in every subject but math (and chemistry). It’s fair to say that I am not educated in math, but I don’t think it makes sense to say that I’m not educated at all.
1) There are 2 solutions: g(x) = x^2 + 1 or g(x) = -x^2 – 2.
2) x^4 + 4 = (x^2 + 2i)(x^2 – 2i) = (x + sqrt(2)(i – 1))(x – sqrt(2)(i – 1))(x + sqrt(2)(i + 1))(x – sqrt(2)(i + 1))
I don’t have time for the others, have to get back to work…
I have to agree entirely with Alfonso. And i don’t think it’s a good excuse that you rarely give that kind of problem.
The challenge for the bright kids should especially be interesting, fun and have nice insights, because that’s what will make them want to dig deeper in the wonderful mathematics.
Please ! Do that for them, i wish it had been done for me.
Regarding 2, there’s actually a way to factor it over the integers.
Would you consider someone educated who couldn’t find their home country on a map? That’s about the equivalent level here. Not solving the problem that Hemant gave, but simply understanding what is being asked for. That’s not much at all.
Well, I don’t understand the problem or what is being asked for. It might as well be written in Farsi. Some people struggle profoundly with certain subjects, yet they excel at others. I often cringe at the spelling errors made by FA commenters, but I wouldn’t call them uneducated simply because it’s easy for me to memorize the correct spelling of a certain word. Those people likely felt stupid when they continually got bad grades on their spelling tests, just like I felt humiliated and upset when I sat through algebra class without being able to understand a thing that the teacher said.
I’m not denying that math is useful. I’m glad there are people in the world who can understand math. I’m just not one of those people, and it’s not a matter of simply trying harder or being exposed to it more. I had good, very patient teachers. I got excellent grades in other subjects. I’m not uneducated in those subjects. I readily admit that my math skills are at the elementary school level, but it certainly doesn’t seem fair to label everyone who can’t understand mathematical equations as uneducated across the board.
(Tried to write a reply apparently didn’t go through so trying again).
I’m not using the term “uneducated” to mean “uneducated across the board”. The person who can’t find their country on a map isn’t necessarily uneducated about everything, but there’s a severe gap in their education. Similarly, the people on FA who you mention who can’t spell are demonstrating a lack of education. I suspect in most of their cases they are just being lazy. But if they genuinely can’t spell then there is a gap in their basic education.
I took your original comment to mean uneducated in general. If that’s not what you meant, then I don’t disagree with you. Of course I have a gap in my education. I’m not sure I would agree that this is basic education, though. The problem in question strikes me as quite a bit more advanced than being able to find your own country on a map. Algebra isn’t even introduced for many people until high school.
Nevertheless, I genuinely can’t understand algebra, just like some people genuinely can’t remember the correct spellings for words, no matter how much studying they do. For some of us, it’s not laziness. The main reason I responded was that these threads brought back a lot of bad memories for me, particularly of being told that I wasn’t trying hard enough when I was giving 100% and still did not understand what I was being asked to do.
You may actually suffer from some kind of innumeracy akin to dyslexia. And I’m sorry that that wasn’t recognized–after spending months and years trying to grasp it with evident effort. But you also need to understand that mathematicians like Joshua and myself are not attacking people like you. We’re not denying that there are people whose brains are so wired that they simply cannot fathom basic algebra, the way some people’s brains cannot grasp basic spelling.
But we face a daily tide of people whose innumeracy has nothing to do with their capacity and everything to do with the (lack of) value placed in mathematics in their home, school, and work lives. Every time someone finds out I’m a mathematician, unless they’re studying math or (some) science themselves, there is a better-than-even chance they’ll throw out “I’m just not a math person,” or “I just never got numbers.”
Imagine that. Imagine that *every time* you spoke with someone who didn’t study English or History, you had a reasonable expectation that they would say “Oh, I just never got letters” or “I’m not really a talkin’ kind of person.” Imagine not just laypeople, not just students, but whole departments of University faculty *filled* with people who were almost proud of their illiteracy. So while I can accept that despite the ability of competent and caring teachers, and your own best efforts, you cannot understand algebra. What I cannot accept is the fact that nearly every person in North America who considers themselves educated could not do so. Those are about whom Joshua is speaking–people who’ve been failed by our society (and by themselves) in one of the best intellectual endeavors people have ever undertaken.
I’d be willing to bet that you just have a mental block
with using symbols and could actually be quite good at basic algebra if you had
it presented in the right way. I tell my kids that algebraic manipulations just involve easy to understand things like the following:
1. doing the same thing to the top and bottom of
2. doing the same thing to both sides of an equation (or
3. adding zero
Of course these things are done in a special way and with a purpose in order
to simplify the expression or move or gather like symbols on one side.
If you are able to be fair and consistent (in doing the
same thing) to two people you know, and then generalize the concept to two abstract people, then you can do algebra.
Aw, thanks. I thought I was (mostly) over it by now. It’s been over 15 years since I was in high school, but it’s amazing how those feelings of inadequacy and helplessness can come flooding back.
I understand, and while it’s true that not everyone has an extreme case, it does seem that more people have trouble grasping advanced mathematics than many other subjects. At least that’s my perception. Maybe larger numbers of people are able to understand it, and they’re simply not being introduced to it properly. That’s always a possibility.
By the way, I was tested for learning disabilities several times, and no one could ever find anything wrong. I suppose it would have been easier if I had gotten a diagnosis because then there would have had an actual reason why I couldn’t understand what was going on. Instead, it seemed like no one quite knew what to do with me. I was pulled out for remedial classes and individual tutoring as early as fourth grade, but it never made any difference.
I know you mean well, but being told that I had a “mental block” was honestly one of the most frustrating things to hear. If I have one, I don’t know how to get rid of it. The main problem is that I don’t understand equations. I don’t understand what they mean, and I don’t know how to solve them. It’s like looking at a foreign alphabet. I can do simple arithmetic on paper, but that’s as far as my ability goes. I had three years of algebra in high school, and I never got any closer to being able to understand or memorize equations.
Long time lurker and probably a little late to comment, but I fully agree with Alfonso.
I actually enjoyed the problem because it gave me an excuse to try out symbolic programming (it took me a few frustrating hours to figure out Sympy) , but I don’t see any way this would benefit a student. I should know, am a highschool student myself. Sort… of.
I love mathematics and plan to study physics, but these kind of problems are left to the computers. They are just mindless computations. For those who want to test their skills or find it fun to do immense mathematical calculations, there are thousands of problems they can find by simply googling, or they could just as easily make up some big numbers and play with them. For everyone else, it just probably comes off as frustrating, boring and pointless, and they’d be right. As Gprano said below, it isn’t an excuse if you rarely give such kind of problem.
Indeed, one of the problems with highschool is that it focuses on mechanical manipulations of problems than actually understanding what they really mean (not suggesting that you do that, it’s just my personal experience).
Yeah, being told that you have a “mental block” isn’t productive; it identifies that YOU have a problem, not just that a problem exists, and making it personal just compounds the issue.
I had the same problem learning (well, trying to learn) Spanish for six years- I couldn’t find the proper way to wrap my brain around the subject material. Being told “This isn’t that difficult” didn’t help, either; it just made me feel bad as well as stupid.
And I was originally going to try drawing some fancy analogy between algebra and rubic’s cubes, but a) this isn’t the place for it, and b) I’d probably butcher the analogy worse than a blind sous chef in a china shop, and leave you even more confused and frustrated. I’ll just say that I feel your pain, just in different subjects.
Oh, yes, I got a lot of “This isn’t that difficult” along with a healthy dose of “What don’t you understand?” which really weren’t helpful at the time. Looking back, I feel bad for my teachers and tutors, who would patiently and painstakingly explain something, only to have me say that I didn’t understand it. I could never articulate what I didn’t understand because it wasn’t just one thing; it was everything.
Ironically, I loved Spanish. I was in Spanish Honor Society and tutored many of my classmates. Spelling, grammar, and vocabulary have always been easy for me, but I knew lots of people who struggled mightily with foreign languages in high school. I’m of the opinion that we all have different strengths and weaknesses. Not everyone can become a mathematician, just like not everyone can become a professional artist or learn to speak Cantonese fluently. Rather than beat ourselves up about it, we should focus on the things that we know we’re capable of doing well.
Ohhh, rickrolling them (but still including the secret word) would have been perfect!
FWIW – I’ve been looking at the problem, and the solution, and trying to decide if I *could* have solved it. I think I could have, but to say it would have taken all freaking night is an understatement. Because of this, and given that the two other folks in my department were not in today, in retrospect it is a good thing I did not even try.