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Oxford researchers are taking part in an international study to film the teaching of quadratic equations for secondary school pupils. The hope is that lessons will be learned on how to bring out the best in pupils learning about mathematics.
Over the next few months, video cameras will appear in secondary schools across England that have chosen to take part in an international study to observe maths lessons focused on quadratic equations. Researchers from the University of Oxford have joined forces with the Education Development Trust to undertake the study in England, which will involve up to 85 schools from different parts of the country. The research team has to enlist 85 teachers and around 1,200 pupils, so they can analyse video footage of different teaching practices and pupils' responses to assess what works best. Schools in Oxfordshire will be among those approached about taking part in the pilot.
The research project is led by Education Development Trust, working with Dr Jenni Ingram and Professor Pam Sammons from the Department of Education at the University of Oxford. They will analyse how pupils' attitudes toward quadratic equations are linked with their progress and results, and observe how teachers' attitudes and methods affect outcomes.
Dr Ingram said: "We believe this study will improve our understanding of the relationships between a range of teaching practices and various student outcomes, including their enjoyment of mathematics, mathematical knowledge and engagement with learning."
Or you could watch Khan Academy.
(Score: 2, Troll) by Dunbal on Tuesday March 28 2017, @04:42PM (31 children)
The hope is that lessons will be learned on how to bring out the best in pupils learning about mathematics.
There is nothing more repetitive than quadratic equations. A monkey can be trained to solve quadratic equations. There is the formula. Plug the shit in. Turn the handle. WOW. You want to bring out the best teach students about calculus and how you can turn weird curvy shapes into solid volumes, how you can figure out the area under an irregular curve and more importantly WHAT THIS CAN MEAN IN THE REAL WORLD from economics to pharmacology. But hammering away at a 5000 year old formula is uninspiring. Oh but it's really easy to teach and grade, that's for sure.
(Score: 3, Insightful) by ikanreed on Tuesday March 28 2017, @05:19PM (12 children)
Which is why I think the quadratic formula should be banned from education until college math.
We're incorrectly teaching kids that math is boring, rote, be-the-calculator work. I think that math of that sort has no value to anyone in society.
You aren't going to make any scientists or engineers out of making math boring. You aren't going to keep the boring, rote rules in anyone's heads past their final exam, just in case they need to solve a quadratic equation or determine the third side of a right triangle 10 years later.
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @05:41PM (4 children)
Banning the quadratic formula does nothing helpful. It requires you to either go through the painful process of completing the square, you have to either engage in some rather exotic math or you have to restrict the quadratics given to ones that don't require irrational or imaginary numbers for roots.
Good math students turn as much of what they learn into memorized results as possible and having a surefire way of doing the problem changes the focus from being able to do the work to being able to do the work with the least amount of actual time and energy. The biggest difference between me now and when I was younger is that I've learn so many little tricks and such that I don't spend anywhere near as much time trying to reinvent the wheel or struggling as I know where to look a lot of this stuff up and how to use similar questions to work my way through the one I'm working on.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @01:18AM (3 children)
Good math students turn as much of what they learn into memorized results as possible
People who are good at math actually understand the material on a very deep, intuitive level and don't just memorize facts, although memorizing facts is still an option. Don't confuse being 'good at math' with being a human calculator or someone who can quickly solve problems; they are not necessarily the same at all, and the vast majority of the time are not the same. Most A+ students of math understand next to nothing but have memorized much.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @03:53AM (2 children)
Not really, I've been doing math professionally for the better part of a decade. And the more math I do, the more I've come to understand that the real difference between the people who struggle and the people who flourish is the sheer size of the memorized library of functions and results as well as the willingness to push for a solution.
And no, I'm very familiar with the difference between being a human calculator and being good at math. I wouldn't be able to improvise and solve problems of types I'd never been presented with if I hadn't developed a great deal of mathematical thinking.
It's just like any other skill, if you aren't relying upon a great deal of stored knowledge, you're not going to be fast or efficient. If you're constantly having to re-invent the wheel you're just not going to get anywhere.
(Score: 1) by shrewdsheep on Wednesday March 29 2017, @08:39AM (1 child)
I've been doing math professionally for the better part of a decade.
Care to elaborate? This smacks of self-declared superiority. What I suspect is that you lack in reflective capabilities. As a matter of fact, intuition is key for mathematics. A deep intuition makes it much easier to master and memorize the wider field of mathematics. If it is easy for someone to reproduce a mathematical fact from other such facts mentally, it becomes much easier to memorize. You may or may not be better in mathematics than people around you. If it is the case and you indeed know more mathematical facts, you fail to see that the underlying reason would be a better intuition.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @03:56PM
You've got that completely backwards. The basis for intuition is a large amount of memorized patterns. There's a process here. If you haven't already matched the equation against a list of things you already know how to do, and things that look similar to what you're wanting to do, then you're wasting a ton of energy and leaving yourself open to edge cases which only come up occasionally.
But really, intuition is the result of having a lot of these facts memorized and being able to apply them in ways that aren't immediately obvious. Adding 0 and multiplying by 1 are particularly common examples.
(Score: 4, Insightful) by DannyB on Tuesday March 28 2017, @06:08PM (3 children)
In high school I found the quadratic formula and other "turn the crank" problems to be interesting. Programmable calculators were then quite new. And microcomputers programmed in BASIC. I developed a brief interest in types of problems that had a systematic solution. Adaptable to either calculators or microcomputers. I quickly moved on. But one problem that I continued to have an interest in through college and into the 1980's was simultaneous linear equations. Partly because they had a lot of practical application. Example: like in DC and AC Circuits with all the Thevenizing and Nortonizing. Naturally this led to extended study on my own time of playing with matrix addition, scalar multiplication, matrix multiplication, and matrix inversion. Building software that would allow expressing a matrix as a variable, and write matrix multiplication as A = B * C, seemed like quite a leap in the early 1980's. But at that time I hadn't gotten my hands on Lisp yet. :-)
My point is there's nothing wrong with problems like the quadratic formula, as long as you don't obsess on it. Like this one history teacher that was obsessively obsessed with the civil war and gave it a disproportionate amount of coverage while I wrote BASIC code in my notebook.
If you eat an entire cake without cutting it, you technically only had one piece.
(Score: 2) by ikanreed on Tuesday March 28 2017, @06:16PM
If they're anything like the history teacher I had who was similarly obsessed with the civil war:
1. It was very important to them to establish that there were good guys and bad guys in the war.
2. The slave owners weren't the bad guys.
3. It was an international history course and the US was supposed to be just a tiny sliver of the curriculum.
4. Reenactment society stories in place of actual history.
5. Incredibly creepy towards 15 year old girls.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @04:37AM (1 child)
> But at that time I hadn't gotten my hands on Lisp yet.
The BEST moment in programming is when you realize an obstacle can melt before a good tool and careful expression.
(Score: 2) by DannyB on Wednesday March 29 2017, @02:00PM
In Lisp, many problems are solved in two steps:
1. Build a new language in which the problem is easily expressed.
2. Easily solve the problem in that language.
Example: I want to find a solution to some type of puzzle / game. Let's say, Sudoku. Or Unblock Me. Or some other game. Build a board type, and game pieces. Build proper operators that allow manipulating a board and pieces. Use a search, like breadth first search, to seek a solution. Or select different search algorithm to explore the graph of legal moves to find solution. Fine tune your search with rules specific to the type of problem. Extra credit: don't stop when a solution is found; structure your code such that it returns a lazy list of solutions. Pulling the next solution from the lazy list continues the search. Use a newer lisp like Clojure to make a lot of this easier so you don't have to build basic parts like lazy lists, immutable and persistent data structures, etc.
If you eat an entire cake without cutting it, you technically only had one piece.
(Score: 1, Interesting) by Anonymous Coward on Tuesday March 28 2017, @07:17PM (1 child)
They need to find a better way to teach them. If it's a matter of matching patterns to known rules, then find a way to encode the patterns and explain how to "read" the patterns to find out what to plug into what. If it's truly "mechanical", then find a way to mentally automate the mechanical processing in step-by-step algorithm that a student can "run" in their head. Look at the equation, convert it into the pattern language, look up the "from" matching pattern in the (memorized) pattern mapping table, then get the "to" that maps to the "from" pattern, and apply the "to" pattern.
The problem is that you'd have to introduce a pattern language, sort of like reg-ex's for math, and that would probably confuse most students even more. I kind of created such meta languages myself to help me memorize and process math rules, but I suspect a meta language that works best for me won't necessarily work on others. People think differently such that it's hard to make one-size-fits-all pattern language. (These roll-your-own pattern languages got me through school, by the way.)
Learning how to learn is one of the trickiest tasks of education because people think so differently. A technique that favors one group will slow another.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @03:58AM
If you really want to understand math, a great exercise is to write you're own generator programs to generate classes of equation. Even just writing a program to take two random points and create linear equations for all three standard forms of equation will do wonders for your understanding of how the equations work.
I dare anybody here to do that and come back claiming they don't understand how the equation works or what the pieces do.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @01:15AM
We're incorrectly teaching kids that math is boring, rote, be-the-calculator work.
Then we would have to ban everything but basic math, because schools aren't teaching any kind of math properly.
(Score: 4, Interesting) by kazzie on Tuesday March 28 2017, @05:20PM (3 children)
Yeah, it's an old formula. So let's pay attention to the history. Why was solving quadratic equations important to people so long ago? Who was the Mathematician Al Qwarizma, who is the namesake of the Algorithms we use in mathematics and computing? Not to mrntion the ability to use software such as geogebra to allow pupils to play around with coefficients of quadratic equations, and see what happens to the resulting curve without having to plot and graph it all by hand.
(Today I taught a physics lesson on Newton's laws of motion which included a brief stint of Latin: having looked at the title page of Newton's Principia where he first published the laws. Having decoded the meaning of the title, we discussed why he'd choose to publish in Laton instead of English. Confext and history brings things to life!)
(Score: 2) by Dunbal on Tuesday March 28 2017, @05:47PM (2 children)
Yes and this can be mentioned in passing. Not turning it into the core of a curriculum. Science and technology have moved on from there. We no longer live in Pythagoras', Euclid's or even Al Qwarizma's world. Here's the formula. This is how it was derived. This is what it does. This is how you work it. Do examples 1-10. NEXT.
(Score: 3, Insightful) by art guerrilla on Tuesday March 28 2017, @11:45PM
i would prefer to live in kazzie's world...
(Score: 2) by meustrus on Wednesday March 29 2017, @12:55PM
Hm, so you would like to take an interesting intellectual opportunity to broaden students' minds and turn it into an assembly line for rote memorization? I'm sorry, but that's the kind of thinking that has led us to where we are now: massive numbers of students having difficulty with algebra because they just don't "get it". It's not that they can't plug numbers into a formula. Anybody can do that. It's that nothing has clicked in their minds to make the formula interesting enough to actually learn.
If there isn't at least one reference or primary source, it's not +1 Informative. Maybe the underused +1 Interesting?
(Score: 3, Interesting) by melikamp on Tuesday March 28 2017, @05:24PM (6 children)
(Score: 1, Informative) by Anonymous Coward on Tuesday March 28 2017, @05:38PM (5 children)
Students struggle with it when it's not being taught correctly. I struggled with it because the schools here were using guess and check. I had to discover why all the pieces work the way they do on my own, or not get it at all. But, it's not freaking complicated even if you're going to factor it out. ax^2 +bx +c where b can be split into two factors of ac which gives you something like this to factor ax^2 +bx +dx +c and you should always have the same ratio between b/a and c/d.
Otherwise, you've got the quadratic formula or completing the square as backup. But, the quadratic formula is hardly difficult to understand. You can derive it by completing the square on the generic equation ax^2 + bx + c =0 and solving for X. Otherwise the -b/2a is the horizontal shift of the vertex from the base function and the square root of b^2 - 4ac is the distance from the axis of symmetry to the roots. Hence if it's zero yo have one repeated root and if it's negative you have only imaginary roots because it doesn't cross the X axis.
(Score: 3, Funny) by VLM on Tuesday March 28 2017, @06:45PM (1 child)
You can derive it by completing the square on the generic equation
When I took algebra, admittedly decades ago, this was implied as the pinnacle of the class, the purpose for taking the class. And we got to write the whole proof down and took a fill in the blank test to make sure we really understood that the biggest accomplishment of algebra is applying about a page worth of rules can numerically solve an entire class of equations. Not solve one particular equation. Not solve some members of a class of equation. But every member of that class can be solved at the class level.
Which is actually pretty amazing if you didn't already know.
Where my algebra class fell down was in not extending the argument a wee bit to explain why there's no similar simple equation for, like, 9th degree polynomials. So trying to rush things and pack more work in resulted in the most interesting point being missed, that some classes of equations can be algebraically solved at the class level and some entire classes of equations cannot.
Its a hell of an argument against sex education in schools. If the same people developed the sex ed curriculum as developed the algebra curriculum then somehow sex would be the most boring, tedious, and pointless experience that a human can have, and people would take great joy in telling each other proudly how incompetent they are at it "I donno I just push buttons randomly until it works" "without electronic/mechanical help I can't do anything" "One kid does all the work and the rest just watch" "I can't estimate I just plug and chug" "I do my homework by asking the 4chan anons for the answers" "My mom can't do it so she can't help with my homework" Well that's beginning to sound creepy. Hilarious but creepy. Anyway school can just suck the life right out of the whole topic or hobby. Likewise I'm glad there's no programming classes at my kid's K-12 school district, I can't imagine a better way to ruin programming for life for kids than to make it a topic of K12 education.
(Score: 2) by meustrus on Wednesday March 29 2017, @12:57PM
I dunno, that sounds like an argument for sex education in schools to me. Sex is not the pinnacle of human experience, after all, no matter what advertisers would rather you believe.
If there isn't at least one reference or primary source, it's not +1 Informative. Maybe the underused +1 Interesting?
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @11:12PM (2 children)
First, for context: I'm 3 to 4 standard deviations above the mean for intelligence, so perhaps an IQ of 145 to 164. (as of age 17)
My algebra class started off with simple stuff. I guess you would call it "guess and check" these days. I was really good at that.
We were then supposed to learn how to complete the square, but I never learned it. I was too good at guessing. I can guess with fractions, imaginary numbers, whatever. I still don't know how to complete the square. (Huh? Eh, I'll just put the answer.)
There we got to the quadratic formula. I memorized it like you'd memorize a jingle or poem, or like the alphabet. I don't forget it. Even before I got around to programming it into my TI-62 calculator (with unreadable non-English instructions) I was flipping it around in my head to avoid excess keypresses. Why use parentheses if you don't need them?
Having learned the formula, I never found a need for completing the square. The formula works fine... though I can still guess faster for typical school problems. I'm older now, with my mind resistant to new things, so I'll probably never learn to complete a square. Oh well.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @12:32AM
The only time you ever have to use completing the square is if you're wanting to convert the standard quadratic equation to the vertex form of the equation. Otherwise, you'd be foolish to complete the square as the quadratic formula is just the completed result of the process.
As far as guessing goes, that's your teachers' faults for not giving sufficiently difficult problems. If you've got a string of decimals like you commonly see come up in physics questions, you're not likely to be able to just guess the answer. Similarly, if you've got 3 or 4 digit coefficients that don't reduce, you're not likely to be able to just guess those either.
The quadratic formula itself is great because it will give you an answer no matter what quadratic equation you're given. It's just not always the most efficient way of getting there.
As for the parentheses, they're definitely not optional. The parentheses involved with completing the square are there for your protection. They save you from creating this thing that you then have to factor. They also exist as a way of handling nonstandard coefficients on the squared term.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @01:23AM
I'm 3 to 4 standard deviations above the mean for intelligence, so perhaps an IQ of 145 to 164. (as of age 17)
But apparently too ignorant to have realized that IQ comes from the social 'sciences' and has never been rigorously proven to be significantly related to one's intellect. IQ correlates with several things our mouth-breathing societies consider important, but again, there is no proof that those things are excellent indicators of one's intellect.
(Score: 2) by DeathMonkey on Tuesday March 28 2017, @05:50PM
You're right.
Maybe someone should do some sort of study to see if there's a better way to teach it.
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @07:18PM
I usually just substituted y=x-number so that the middle term went poof. Fuck formulas.
(Score: 3, Interesting) by looorg on Tuesday March 28 2017, @08:02PM
The formula (PQ, or if you like x=-(p/2)+-sqrt((p/2)^2-q) isn't the preferred or thought solution method for quadratic equations anymore. It has a lot of issues in that the students dont know what they are really doing and instead they just plug the numbers into the formula and then hit calculate and that is it. So they have usually no way of knowing if this is even reasonable or not. They could graph is out and see and that could be ok to. The method used today as far as I have been told is factorization (or completing the square or whatever they refer to it as to). The problem with using the formula is that the students forget or turn things around and then they can get crazy results, was there a - here, did they forget to use to () in all the places etc.
It has a lot of benefits in that it becomes easier to see the solution(s) when you divide it up. Also it has the added benefit on working on cubic equations so you don't have to memorize and use Cardanos Formula since that is to horrible to work with it's not even funny.
I was thought the formula when in elementary school and I didn't understand it at all except that you inserted numbers and it gave me the solution. There was no explanation as to why it was the way it was. Factorization in that regards makes a lot more sense, certainly so if you are not using a calculator. I wasn't thought factorization until university and the method makes a lot more sense when you are not allowed to use a calculator on exams, Only engineers use calculators, mathematicians don't.
(Score: 2) by maxwell demon on Tuesday March 28 2017, @08:31PM (2 children)
Imagine you would teach football the way math is generally taught: The teacher tells you all the rules of the game, and then lets you do exercises like "if player X comes with the ball from the left, and player Y is in that position, and player Z is in that other position, and opponent players are in positions A, Band C, what should player X do?" You are also doing exercises about scoring rules, and the rules of the tournaments. At no time the students are actually watching the game, let alone playing it. I guarantee you the common opinion of the students would be: Football is the most boring stuff you can think of.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @11:31PM
Coach: "I want you to belt the other team right in the asymptote! Bend their spines into a parabola with high coefficients!"
(Score: 2) by NotSanguine on Wednesday March 29 2017, @02:22AM
Football is the most boring stuff you can think of.
Yup. Much more so than math.
No, no, you're not thinking; you're just being logical. --Niels Bohr
(Score: 2) by Joe Desertrat on Wednesday March 29 2017, @08:09AM
There is nothing more repetitive than quadratic equations. A monkey can be trained to solve quadratic equations.
Even fiendishly difficult ones?
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @05:07PM (20 children)
Teach kids to define abstractions, encode their thoughts with these abstractions, and then apply logical transformations on those abstractions. That is what is important.
If you can do that, then you can do quadratic equations; the converse, however, does not hold.
(Score: 3, Interesting) by ikanreed on Tuesday March 28 2017, @05:22PM (18 children)
Hell, on that subject, teach more kids what a logical converse is, and fewer what the Taylor expansion for a sine function is.
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @05:44PM (11 children)
Taylor series are from calculus, how many kids are actually being shown that without having had at least pre-calculus? Also, logical converse is something from statistics which is one of those things they like to sneak into algebra even though it's not given adequate time and energy for students to grasp. Around here, they no longer expect that of pre-college students as it doesn't actually help with math.
If they're showing kids things that are taught in calculus 3 before they've graduated from high school there's all sorts of other things that need to be fixed first and avoiding the quadratic formula would be rather low on the list.
(Score: 2) by ikanreed on Tuesday March 28 2017, @05:58PM (1 child)
I, personally, first got it in 10th grade pre-calc as a magic formula just in case I ever needed to calculate a sine function by hand. It was a complete waste at the time, because it was seriously just rote bullshit without the context of what a Taylor expansion meant.
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @07:36PM
That's normal, the memory comes from understanding and connection. Magic math is easily forgotten or messed up.
(Score: 2) by Dunbal on Tuesday March 28 2017, @05:59PM (8 children)
If they're showing kids things that are taught in calculus 3 before they've graduated from high school there's all sorts of other things that need to be fixed first and avoiding the quadratic formula would be rather low on the list.
I have studied the human body for 25+ years now. From basic biology classes to pre-med to medical school to residency to continuing education and work experience. I'm pretty much an expert in how humans are put together and how they fall apart, and even how they can be patched together again for a while. Yet you (who for argument's sake aren't in the medical profession) own a human body and you seem to work it just fine. You have a pretty solid empirical knowledge of what your capabilities and limitations are as well as knowing how to care for and maintain that body (although you might not necessarily apply it).
The point I'm making is that somehow math teachers assume that just because there is a logical progression in mathematics, every successive step built on the previous one, more or less, then the teaching of mathematics MUST assume the same sequence and every single step must be taught. Mathematicians fail to grasp the fact that the human brain is still by far the greatest abstraction tool around. You don't NEED to suffer through geometry and trig to understand calculus, for example. Just like you're perfectly fine living in and taking care of your body without knowing absolutely all the gory details, someone can learn advanced or interesting concepts - CONCEPTS - without slogging through all the background work.
If you're going to choose a career in math, then you have to go back and re-visit all those concepts. But for the general population gloss over the boring stuff and focus on the interesting stuff. That's how you'll get people to learn. But this is also much harder to evaluate since it becomes qualitative instead of quantitative. It's much easier to put a tick on 5.2 than to grade an essay, for example.
Therefore students of mathematics tend to suffer because of the laziness and lack of imagination of math teachers. Not only that, but those who end up liking mathematics are the ones who either enjoy suffering, or enjoy making others suffer....
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @07:06PM (7 children)
There's a transition that has to take place. These are foundational skills, the students absolutely have to know this stuff and there's not much point in making them work for it. It's far more important that they develop a positive experience and some necessary skills.
Later on is the time to have students generalizing and improvising. It's also a matter of time, There's no real time to learn this incorrectly and then relearn it.
As far as the body goes, the understanding of the involved processes is rapidly changing, but how a person feels doesn't change unpredictably.
(Score: 3, Interesting) by Dunbal on Tuesday March 28 2017, @07:45PM (6 children)
the students absolutely have to know this stuff
No they don't. Anymore than a high school biology student has to know immunohistochemistry. Right now we have a system that purports to teach the "foundation" skills you mention and the reality is we are graduating students who cannot even add or subtract. There's something seriously wrong here when you claim you want to stick to the current model which is demonstrably broken. How much benefit are students getting from doing trig or quadratic equations when division is a challenge without a calculator and fractions and percentages are some mysterious exotic country? Focus on the REAL basic stuff like arithmetic, and everything else can be glossed over or skipped entirely unless taken as electives.
There's no real time to learn this incorrectly and then relearn it.
Every other science does this. You don't learn PV=nRT in high school. You learn the 3 gas laws separately.
(Score: 2) by bob_super on Tuesday March 28 2017, @09:45PM (2 children)
> Focus on the REAL basic stuff like arithmetic, and everything else can be glossed over or skipped entirely unless taken as electives.
Don't come complaining when the industry hires Europeans and Asians, then...
(Score: 2) by Dunbal on Tuesday March 28 2017, @09:49PM (1 child)
I was under the impression that trend was to hire robots. Seriously, how much math does a burger flipper/retail shop employee need? Those who are motivated to learn will learn. Doesn't matter where they're from.
(Score: 2) by bob_super on Tuesday March 28 2017, @10:09PM
> how much math does a burger flipper/retail shop employee need?
Enough to learn how interest and inflation work, so that the math doesn't turn into "how much assistance do they need to survive?"
> Those who are motivated to learn will learn.
Very American of you, and very wrong.
Conversely, those not motivated to learn will turn into your problem, whether you need to pay for them, repair the damage that they cause, or deal with losing your job or retirement after they vote with their guts for lack of ever straining their brain.
(Score: 2, Insightful) by Anonymous Coward on Wednesday March 29 2017, @12:39AM
That's really not a fair analogy. I've got my undergrad in the natural sciences and spend my days at work trying to figure out how to get students that are struggling from point A to point B with the least amount of time and effort possible.
2nd degree polynomials are something that shouldn't be glossed over as they are the first exposure that students really have to numerous concepts and they serve as a reinforcing mechanism for all sorts of math. Polynomial multiplication and division, breaking up an equation into smaller pieces, having differing powers of exponents, coefficient management and even just basic masking techniques that get rather complicated later on.
I'm a huge fan of masking out the bullshit. Realistically, there's insufficient time to explicitly teach everything, but glossing over quadratic equations is really not the place to save time. They just show up too often and in too many contexts and have too many math skills to make that an acceptable decision.
What's more, students who don't see and do these things in that unit wind up struggling later on when those practices are put into place in more complicated expressions like exponential and trigonometric functions. Which both have a non-insignificant amount of overlap with quadratic functions.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @04:50AM
> You don't learn PV=nRT in high school
Uh. I got the base laws and then piped up that they could be comingled because of sharing V and didn't that mean that temperature depended on pressure (roughly) and got told to shut up. Then next year we did indeed get the full ideal gas law.
n=1
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @03:45PM
What are the 3 gas laws? I only know pV = NkT (or pV=nRT, but that version I learned later in university). Honestly.
(Score: 1, Funny) by Anonymous Coward on Tuesday March 28 2017, @06:34PM (5 children)
It doesn't make any sense.
(Score: 3, Touché) by ikanreed on Tuesday March 28 2017, @06:49PM (2 children)
I don't know. Can you imagine one human being who would have a different perspective on how to use innocuous internet modding systems than yourself?
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @07:39PM (1 child)
You add absolutely nothing to OP's point, and you manage to do so less eloquently.
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @09:40PM
OK! That's it! I am modding your comment on the parent comment as -1 Uninteresting. I hope you will cherish this bit of information, and maybe some day boffins in the UK will video modding on SolylentNews in order to discover how to teach the maths.
(Score: 1) by khallow on Tuesday March 28 2017, @07:21PM
(Score: 2) by Dunbal on Tuesday March 28 2017, @07:50PM
Modding is just allowing other people to censor what you read anyway. I've always read at -1. I think I can spot troll/inflammatory comments for myself. Don't expect it to make sense - it's done by other people.
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @05:30PM
Precisely, that and basic mathematical literacy would go along ways. Only a small fraction of the math I use at work was ever taught to me, most of it is material that was added later because Wolfram Alpha was able to give step by step solutions to the work that was being done.
After you get used to it, working only on abstractions makes a huge difference in situations where you're coming in contact with work that you've never seen before.
(Score: 4, Interesting) by Azuma Hazuki on Tuesday March 28 2017, @06:18PM (4 children)
Math was always my weak point, the one lone B or even occasional C in a sea of A- and A and A+ grades. I think this is because I learn through explanation and doing, and just being told "this is the formula, plug it in and twist, and screw you if do it wrong" is...unhelpful. I, and I suspect many other people, want to know WHY it works, want to know what we can do with this, how it relates to the real world.
Calculus was another example. I didn't understand why something like "2x is the derivative of x^2" worked. No one explained *what* the derivative was or why we'd need to know it...until someone told me "Derivative is the rate of change of the quantity you're deriving. Acceleration is the derivative of velocity with respect to time, d(v)/d(t), because acceleration is how fast and in what way your velocity is changing. And velocity is the derivative with respect to time of position, d(p)/d(t), because velocity describes how your position changes. Acceleration is also therefore the second derivative of position with respect to time."
And *that* made the lightbulb come on.
We need to figure out how to do stuff like this everywhere in math. Make people see the actual cyphering as simply a means to an end, *not* an end in itself, as it so often seems to be in these classes.
I am "that girl" your mother warned you about...
(Score: 2) by Dunbal on Tuesday March 28 2017, @07:53PM
And *that* made the lightbulb come on.
And then you see something like KE = 1/2mv^2 and oh shi-
(Score: 2) by weeds on Wednesday March 29 2017, @04:33AM (1 child)
No one ever told you that dy/dx was a special version of delta y / delta x (slope) which you should have already known was rate of change? You had the worst calculus and analytical geometry teacher on earth.
Get money out of politics! [mayday.us]
(Score: 2) by Azuma Hazuki on Wednesday March 29 2017, @06:49AM
Oh, they mentioned "slope or instantaneous rate of change of X relative to Y" but that never struck me as anything but technobabble. It was only when someone pointed out some useful applications that things suddenly made sense.
I am "that girl" your mother warned you about...
(Score: 2) by urza9814 on Wednesday March 29 2017, @06:57PM
I had and noticed the same problem in college. Highschool we had some great teachers and I did pretty well, but by college you end up with "teaching" being synonymous with some foreign grad student mumbling into a chalkboard for an hour. I took basic calc in highschool, and I know what derivatives and integrals are and could probably manage to work my way through the simpler ones even today. Beyond that...shit, I don't even know the names of the concepts I was supposed to be learning in my second and third semester calc classes. The "class" was mostly formulas being written on a board with no explanation, and the book went no further than "If the problem looks like this, plug into this formula to get an answer"
In highschool calc, a friend and I would spend our free time trying to work out new formulas. For example, they taught us how to use an integral to calculate the volume of a solid formed by rotating a curve around the axis, and we spent months trying to work out a formula for rotating any curve around any other arbitrary curve. Including arguments about what exactly rotation even means when the axis of rotation is not a straight line. At one point during our highscool prom we both briefly abandoned our girlfriends to talk about that problem. And in one year I went from that level of obsession to complete disengagement. Because when you can visualize what the problem you're solving actually means, it naturally raises additional questions and challenges. But when all you have is a formula to memorize, it doesn't really lead anywhere.
(Score: 1) by khallow on Tuesday March 28 2017, @07:16PM (5 children)
On some of the comments here, quadratic stuff is really important in large part because it is the simplest example of nonlinearity. A lot of optimization and zero-finding techniques work with quadratic expressions which can then be extended in various ways to more general stuff with similar properties.
For an example I've been looking at, in the differential equations world, most ordinary differential equations (that is, equations of one variable) can be shoe-horned (via transformation of the solution function into a vector of functions) into the form of a Ricatti-like equation with the single derivative of a vector of functions of your variable equal to a quadratic expression of the functions plus a linear expression of the functions plus a er, "constant" term, a vector of pure coefficient functions which don't dependent on your unknown functions at all. All coefficients are fixed functions of the variable.
In other words, one can transform a nearly arbitrary differential equation of fixed order into a differential equation linear in the first derivative of a vector of functions derived from solutions of the first differential equation plus at most a quadratic expression of that vector of functions. General nonlinearity is transformed into quadratic nonlinearity.
(Score: 2) by Dunbal on Tuesday March 28 2017, @08:00PM (3 children)
quadratic stuff is really important in large part because it is the simplest example of nonlinearity.
I'm not arguing that it's not important. I'm arguing that it's irrelevant, unless you happen to be interested in a particular branch of engineering, science or math that deals with these kinds of functions. There is no justification for devoting a disproportionate amount of time to quadratic functions when most of your students can't figure out what 40% of a given number is. Topic should be covered? Yes in a general way, so that people who decide to later specialize in fields that require background have at least seen it before. But to make it a metric for teaching? Shame. In the current day and age bachelors' degrees are considered a mere re-hash of very BASIC high school stuff. Who the hell are we trying to kid? I have a daughter who graduated with very good grades with a degree in publicity and marketing. Her ARITHMETIC skills are appalling. And she's one of the good ones.
(Score: 1) by khallow on Tuesday March 28 2017, @08:39PM
There is no justification for devoting a disproportionate amount of time to quadratic functions when most of your students can't figure out what 40% of a given number is.
What happens if they can figure that out, say because someone is teaching the course effectively? It strikes me that no matter the subject, there isn't much point to a poorly taught class.
(Score: 2) by NotSanguine on Wednesday March 29 2017, @02:31AM
quadratic stuff is really important in large part because it is the simplest example of nonlinearity.
I'm not arguing that it's not important. I'm arguing that it's irrelevant, unless you happen to be interested in a particular branch of engineering, science or math that deals with these kinds of functions. There is no justification for devoting a disproportionate amount of time to quadratic functions when most of your students can't figure out what 40% of a given number is. Topic should be covered? Yes in a general way, so that people who decide to later specialize in fields that require background have at least seen it before. But to make it a metric for teaching? Shame. In the current day and age bachelors' degrees are considered a mere re-hash of very BASIC high school stuff. Who the hell are we trying to kid? I have a daughter who graduated with very good grades with a degree in publicity and marketing. Her ARITHMETIC skills are appalling. And she's one of the good ones.
Okay, so your point is that since math is taught poorly, we should teach less of it, rather than try to improve the teaching of math?
You're a genius! You should run for your local school board or something.
No, no, you're not thinking; you're just being logical. --Niels Bohr
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @05:00AM
Back in the day, I represented my high school in a state-wide math competition. Now I work in a blue collar factory job. The people I work with aren't exactly math wizards. They're doing great if they can figure out a problem like this: We are scheduled to produce 10 palettes by 2PM. A palette has 30 units. We have two production lines that can each make one unit per minute. Are we on schedule?
Basically, I agree with what you are saying. If high school kids take a class where they are taught the quadratic equation, but after graduation they can't figure out how much to tip their waiter, it was probably a waste of time.
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @09:20PM
Just remember that there is only one true parabola [youtube.com]!
(Score: 2) by meustrus on Tuesday March 28 2017, @09:44PM (2 children)
Algebra != Quadratic Equation. But going through any algebra course would seem to mean the opposite. I can't think of a single time I've needed the quadratic equation. And you know why? Because for all of the importance they placed on memorizing it, they never once connected it to a real-world problem. Never said, "This here is what the equation is for. You can always look it up in your notes later if you need it. What you can't look up is whether this is what you need."
This is what we really need to be teaching instead: Math for Programmers [blogspot.com].
If there isn't at least one reference or primary source, it's not +1 Informative. Maybe the underused +1 Interesting?
(Score: 2, Interesting) by Anonymous Coward on Tuesday March 28 2017, @10:26PM (1 child)
You've never ran across it [maths.org] in real life [maths.org]??
(Score: 2) by meustrus on Wednesday March 29 2017, @12:51PM
I didn't say I hadn't run across it. I said I hadn't needed it. I am fully aware that it and other basic equations reflect many physical situations that we encounter every day. That doesn't mean that I stop what I'm doing and try to figure out how x=(-b[+-]sqrt(b^2-4ac))/(2a) fits the situation. I just don't. And even if I did, figuring out that relationship would be massively more important than being able to rattle off that equation without looking it up.
This is coming from somebody who is fascinated by trigonometry and calculus and does occasionally pause to consider how, for example, a three dimensional space is projected onto our two dimensional vision. I'm still unsure how to use calculus to find the volume of arbitrary shapes, due to the difficulty in fitting a real shape to an equation that can describe it.
If there isn't at least one reference or primary source, it's not +1 Informative. Maybe the underused +1 Interesting?
(Score: 0) by Anonymous Coward on Tuesday March 28 2017, @10:23PM
...the quadratic equation. Instead we memorized a lookup table.
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @12:20AM
A junior high school skill test.
Do the kids get what is happening or just the formula.
Can the find shortcuts for simple quadratic problems on their own?
Do they know when they have the right answer and not?
Whilst they are testing simple skills, how about counting and add and multiply tables?
It ain't calculus but it is a good start for pre-high school math.
(Score: 2) by kaszz on Wednesday March 29 2017, @06:39AM (3 children)
Find a cool technical project the kids want to have and let them build it from (relative) scratch. They'll find out the hard way they need math and then you can help out..
1 hour to motivate, 39 hours to harvest that motivation.
And of course all that test training got to go together with least common denominator education level. Start reasoning and show in the flesh real world examples of application.
(Score: 2) by meustrus on Wednesday March 29 2017, @01:01PM (1 child)
Or they go online and find somebody that already solved all the hard problems. Which is...also a useful skill, but presents a challenge for designing projects that actually accomplish what you want. The most likely outcome is also the worst possible: idiot teachers trying to punish online research.
If there isn't at least one reference or primary source, it's not +1 Informative. Maybe the underused +1 Interesting?
(Score: 2) by kaszz on Thursday March 30 2017, @03:04PM
Idiots will always mess up whatever they come in contact with :p
Anyway, one can usually find an online solution but rarely the exact solution you need to have. Just try to find a source code for DMA on microcontrollers to feeding them with external data at high speed. All trigger modes has to be correct, configuration, wiring etc. Then there's the chip errata....
(Score: 0) by Anonymous Coward on Wednesday March 29 2017, @04:00PM
When I took pre-calc we built solar cookers using the information we had about parabolas to place the focus where we wanted to heat the meat.
A bit more complicated would be building a parabaloid and then using it to talk with somebody on the other side of the classroom.