New math of the 70-ies. Yikes, or nostalgy ?

[vanesch]

"new math" of the 70-ies. Yikes, or nostalgy ?

Maybe I'm committing a crime towards my oldest son (10 y.o.), but I (with difficulty) got my hands on some "new math" handbooks from the 60-ies (from Georges Papy, the "new math" gourou in Belgium), and I try to teach it to him at a leisurely pace.

I was trying to find any positive material on that new math thing from the 70-ies, but no matter how hard I google, it seems to be encountered with an almost unanimous "yikes" apprehension.

Nevertheless, I was brought up with "new math" and I absolutely loved it. I still get a thrill when I browse through the pages of those old books (back then I had other books, I don't remember which ones). Not so long ago I got back into contact with an old schoolmate of mine, and she told me (to my great surprise) that it had been sheer horror to her (I remember she was a good student so this surprised me).

Just to be clear: the "new math" I'm talking about was the re-foundation of high school mathematics based on (naive) set theory, including a revision of Euclidean geometry in set language. The main concepts were relations and functions (defined as specific kinds of relations) between sets, and algebraic structures (groups, fields, rings). I have to say that I have had enthusiastic and rather good high school math teachers. To me, the way that that material was brought was not only crystal-clear but also rather exciting, and that's the impression I would like to leave to my son.

But apparently, to most if not all people that have something to say about those "new math", it was a total failure that "destroyed" a whole generation of students.

Do people have comments on the new math thing and was it a good or a bad thing according to you ? Was it sad or justified that it was taken away (is it really taken away btw ?).

 

[lisab]

I was brought up with 'new math', too. I loved it! I think it made me a better visual thinker.

I was disappointed to see the math they taught my daughter. Very little of the 'fun' concepts that you and I enjoyed, but lots and lots of word problems.

 

[phinds]

I was in engineering at college in the 60's so I never experience it myself, but I remember always hearing it referred to back then as "content free" nonsense.

 

[Andy Resnick]

Do people have comments on the new math thing and was it a good or a bad thing according to you ? Was it sad or justified that it was taken away (is it really taken away btw ?).

It's hard to answer this- I guess I had "new math" in elementary school (K-6, ages 6-12 or so), but it was just "math class" and since I only went through school once, there's no way to directly compare alternate approaches.

My kids are being taught "lattice multiplication", which is new for me- so I guess that's the new new math :). In fact, I very recently discovered there are multiple algorithms for adding, subtracting, etc. Who can say if one is 'better' than another?

I think it's misguided to divorce content from the instructor at this early stage in education- a great teacher can teach a lot of material very effectively (and propel the student forward), a poor teacher can mangle the most basic concepts (and set the student back).

 

[vanesch]

It's hard to answer this- I guess I had "new math" in elementary school (K-6, ages 6-12 or so), but it was just "math class" and since I only went through school once, there's no way to directly compare alternate approaches.

That's right of course, you can only learn a thing once for the first time

My kids are being taught "lattice multiplication", which is new for me- so I guess that's the new new math :). In fact, I very recently discovered there are multiple algorithms for adding, subtracting, etc. Who can say if one is 'better' than another?

I had to google it, and found this:
http://www.coolmath4kids.com/times-tables/times-tables-lesson-lattice-multiplication-3.html

It is "standard" multiplication, but turned over 45 degrees

I think it's misguided to divorce content from the instructor at this early stage in education- a great teacher can teach a lot of material very effectively (and propel the student forward), a poor teacher can mangle the most basic concepts (and set the student back).

This is very true. However, I always wondered what was this vehement reaction and this almost uniform "yikes" that I read when I'm looking up "new math", while I simply loved it. At the time I didn't realize of course that it was "different" math than before (as you said, because of the "only once" aspect of learning something for the first time). The only thing I noticed was that when I went to university, apparently, "math sophistication" actually went DOWN a bit from what I was used in high school. But I remember that it was very clear in high school - although I recently learned that not all my co-students liked it the way I thought they liked it (like me).

But the general criticism of "too abstract" and "they can't solve simple problems" or even "contentless jargon" seems totally misplaced, so I wonder where the "hate" comes from.

True, I got a teacher that was himself involved in the "new math" movement, so he was very enthousiastic about it, and mastered the subject very well. Maybe that made the difference (as you point out). Maybe many teachers didn't feel at ease with it and if you don't "feel" the material yourself you can't teach it.

I still remember in 11th grade that he introduced first the general definition of a topological space, then as a "special application" metric spaces, and as an even more special application, R^n. But this seemed all clear and natural. We even played around with non-hausdorff and discrete topologies to get a feeling for what it was about. We had a generator of neighbourhoods (like balls), and from there we could deduce the whole topology spanned by the generating set. We did this for fun on small sets with a finite number of elements. (like on http://en.wikipedia.org/wiki/Topological_space )When I tell this now to people, they stare at me as if I went to school on Jupiter or something, while it was like "playing games" back then.

 

[Andy Resnick]

I had to google it, and found this:
http://www.coolmath4kids.com/times-tables/times-tables-lesson-lattice-multiplication-3.html

It is "standard" multiplication, but turned over 45 degrees

The math-education folks I talked to about it uniformly love lattice multiplication- they all say it's good prep for algebra (to quote one- "Lattice multiplication is one of the coolest, particularly since it is easily modified to be used as a fool-proof way to multiply two polynomial expressions.")

Some of the other algorithms they told me about include:

Austrian Subtraction Algorithm http://www.math.osu.edu/~snapp/108/108hw3.pdf
Mayan Multiplication http://nullbloggers.blogspot.com/2007/09/mayan-multiplication.html
Treviso Algorithm
Russian Peasant Algorithm
Scaffolding Algorithm

Honestly, I'm sure what to make of all that- especially since by 6th grade everyone is expected to have graphing calculator (!) :)

However, I always wondered what was this vehement reaction and this almost uniform "yikes" that I read when I'm looking up "new math", while I simply loved it.

Who knows- people can react very irrationally when confronted with change. Just think about the reaction to "New" Coke :)

 

[Sankaku]

I went to elementary school in Canada in the late 1970s and I remember math being pretty standard stuff - no 'new' math for me. A rural school where most kids were more interested in Hockey did not inspire math teachers to push the boundaries much.

But the general criticism of "too abstract" and "they can't solve simple problems" or even "contentless jargon" seems totally misplaced, so I wonder where the "hate" comes from.

Personally, I think a lot of the criticism comes from people who use Math as a yardstick to measure kids by. I suspect much of it comes from non-mathematically sophisticated parents. "It is harder to measure where little Johnny is in the 3rd grade hierarchy, so we can't tell if he will get into Harvard."

That said, did your whole math class respond well to your teacher's style? I would guess that different approaches may work better for different students. If little Johnny hates math and doesn't want to think abstractly, then they might think plugging through rote questions will give him enough skills for shop class.

I would hope that having an inspired teacher would prevent anyone from hating math in the first place. However, I doubt all new-math teachers were as talented as yours. Maybe there was just too big a spread in results? Good teachers did amazingly, bad teachers did even worse than usual?

I still remember in 11th grade that he introduced first the general definition of a topological space, then as a "special application" metric spaces, and as an even more special application, R^n. But this seemed all clear and natural. We even played around with non-hausdorff and discrete topologies to get a feeling for what it was about. We had a generator of neighbourhoods (like balls), and from there we could deduce the whole topology spanned by the generating set. We did this for fun on small sets with a finite number of elements. (like on http://en.wikipedia.org/wiki/Topological_space )When I tell this now to people, they stare at me as if I went to school on Jupiter or something, while it was like "playing games" back then.

Wow - that is amazing. I wish I had gone to school on Jupiter!

 

[vanesch]

That said, did your whole math class respond well to your teacher's style? I would guess that different approaches may work better for different students. If little Johnny hates math and doesn't want to think abstractly, then they might think plugging through rote questions will give him enough skills for shop class.

Well, as I said, I recently got in contact with a former (good) co-student (after 25 years of so) of whom I thought she liked it and actually she told me she hated it (she has been herself a math teacher ). So maybe I was the only one who got excited back then and took for granted the politeness of my co-students as being their liking of the material

That said, your comment would level down just ANY material at ANY level, except for the most rudimentary skills like learning to read or learning to count. After all, if little Johnny hates geography, you might think giving him the map of the local public transport network will give him enough geography skills for life ; if little Johnny hates history, then just listening to his grandpa's stories will give him enough sense of personal history to do well in life, if little Johnny hates sports, then making him walk to the nearest bus stop will give him enough physical ability to do well etc...

All this is true enough. If little Johnny will sell vegetables in a small store in his native village, there's a lot of material that he doesn't need to learn at school, and which will bore him stiff. The problem is that if we adapt the general schooling program to little Johnny's needs, which are modest as we see, we screw up the education of little Kevin, who might have enjoyed being fed harder and more interesting material and who would have had fun living up the challenge.

Wow - that is amazing. I wish I had gone to school on Jupiter!

The point is that it was in fact EASY. The way it was done, it sounded natural, evident, clear, and with almost a feeling of "I could have come up with that myself" (which was of course not true !).

I still remember that we enumerated the 29 different topologies you can define over a set of 3 elements in the classroom. At home I wrote a program that counted the number of different topologies on 4 elements (it is 355) on my old CP/M machine.

 

[Math Is Hard]

I was "new-mathed" in grades 1 and 2, but then in grade 3 my family moved to a new city where they had traditional math curriculum. I became an instant loser. Everything was how about fast you could add and subtract, and tests were in the form of "beat the clock".

Many years later, I became involved in database work where set math was important, and I took to it like a duck to water. It was odd to me that I could debug queries that were troublesome to senior programmers and DBAs, but I think it had everything to do with the set intuitiveness that was built into me in my early years.

 

[Borek]

I have a feeling we (in Poland) have seen this "new math" thing at some moment, but it was well after I left the classes where it was taught.

Edit: Marzena remembers the same, but neither of us is able to pinpoint it in time.

 

[Sankaku]

That said, your comment would level down just ANY material at ANY level, except for the most rudimentary skills like learning to read or learning to count.

I totally agree. I didn't think is was a good argument, I was just trying to think from the perspective of the critics. After further reflection, I believe my other idea might be more salient. The material might have been too dependent on the skill of the teacher. A good teacher (like yours) would be fabulous, but a poor teacher would have disastrous results.

Unfortunately, I think that much of the school curriculum is designed so that poor teachers (and/or hamstrung teachers) can't mess it up too badly.

 

[vanesch]

Unfortunately, I think that much of the school curriculum is designed so that poor teachers (and/or hamstrung teachers) can't mess it up too badly.

That's a PoV I never thought of but might be extremely relevant actually. The problem to be adressed is not Little Johnny's limited abilities, but teacher Big Joe's limited abilities. Maybe indeed, the right school program is one that is stupid-teacher-proof.

 

[eumyang]

Just to be clear: the "new math" I'm talking about was the re-foundation of high school mathematics based on (naive) set theory, including a revision of Euclidean geometry in set language.

Are the books authored by Dolciani (the first editions, not the current ones) considered to be "new math" books? I remember looking through the old 60's editions and I saw that Chapter 1 on all of the books had to do with set theory.

 

[marcusl]

Good grief, new math crippled me! Being unfortunate enough to live in a "progressive" school district in a university town, I was the guinea pig for every P.o.S. new math program dreamed up by some well-meaning nutcase professor--Yale, SMSG, you name it and I suffered through it. Moved to LA in 6th grade and was put in a remedial class because I was wobbly at addition, and couldn't multiply or divide! Had to work hard all through school and college to overcome my disadvantaged background. Thank goodness for that move or I'd still be unable to function at a grocery store and probably would be flipping burgers somewhere.

Introducing new math concepts for fun, and as a supplement to rather than replacement for fundamentals, might not harm your child--especially since you are a professional. Just be careful!

 

[turbo]

I was in engineering school well before "new math" came out, so I missed all the fun.

 

[mathwonk]

I was brought up on old math and I was good at it, but did not understand anything since nothing was explained. New math came out just after I entered college in 1960 and I loved the high school materials that accompanied it, as they explained why things were done the way they were.

Now I understand the lament of one person here who had weak basic arithmetic skills, as those should of course precede any new math study.

In my opinion the remarks about teachers are also relevant. I believe the new math program foundered on two counts:

1) the excellent SMSG books from Yale were not widely used because of objections from the commercial publishing world, which substituted very inferior "new math" books for them;

2) due to a lack of funding most teachers were not trained in this new material and hence could not handle it well in many places.

If I had been taught this material in my tenmnessee high school after mastering arithmetic in grade school, instead of sitting bored with my head on my desk for 4-5 years, I believe I would have succeeded much better in college where I had to compete with boys from Exeter and Andover and NY or New Trier schools with qualified teachers.

By the way it is not true that most of the SMSG books featured large scale use of sets and other abstract baloney. The excellent mathematical methods in science volume by george polya does not mention them, and they occur only briefly in the book on matrices, and the word "sets" occurs even in my old 1957 algebra book by welchons and krickenberger.

I agree some harm may have been done by new math books written for elementary schools that included superfluous and trivial set language, but i would still suggest the matrix algebra book for a young student today.

for plane geometry i would strongly recommend euclid over the smsg version of birkhoff's metric geometry scheme.

 

[xdrgnh]

New Math is crap, the topics in new math were counter productive for making a more technologically advanced country. How can learning about sets and topology at a early age foster the prerequisite for calculus and other math that is actually used in for engineering. Teaching just the theory of addition, multiplication and division doesn't mean that student know how to actually add or multiply. That was the main problem with "New Math" it only taught the theory with no emphasis on simple computation skills.

 

[Sankaku]

...the topics in new math were counter productive for making a more technologically advanced country.

Yes, because we all know that the whole purpose of math is to make "a more technologically advanced country."

 

[mathwonk]

"New Math is crap,"

xd, have you actually read any of the smsg books?
would you share with us which ones and what your precise objections to them are?

what are your views of:

calculus of elementary functions?
elementary functions?
first course in algebra?
geometry?
intermediate mathematics?
intro to matrix algebra?
math for junior high?
analytic geometry?
calculus?
geometry with coordinates?
studies in mathematics?

do you know who lynn loomis is? robert walker? or any of the other authors of these books?

i hope the word 'crap" does not convey your full understanding of the topic.

 

[xdrgnh]

Actually if you look at the context of how New Math started then yes it's purpose was only for the technological and scientific advancement of the United States. The reason why New Math was created was so that we could have better students to compete with the Soviet Union, it had barely anything to do with the love of pure math. Mathwonk I never heard of the SMSG and I just did a search on them and this is what I found out " was an American academic think tank focused on the subject of reform in mathematics education. Directed by Edward G. Begle and financed by the National Science Foundation, the group was created in the wake of the Sputnik crisis in 1958 and tasked with creating and implementing mathematics curricula for primary and secondary education, which it did until its termination in 1977." pretty much the math they advocated to be taught in schools failed to help us compete with the soviets. The books you listed I never heard of but it doesn't matter, New Math is New Math AKA crap which failed at it's original purpose. Here's a quote from Feynman about it "If we would like to, we can and do say, "The answer is a whole number less than 9 and bigger than 6," but we do not have to say, "The answer is a member of the set which is the intersection of the set of those numbers which is larger than 6 and the set of numbers which are smaller than 9" ... In the "new" mathematics, then, first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worth while teaching such material".

 

[mathwonk]

xd it is interesting that you hold such strong opinions when you basically do not know anything about the subject under discussion. the most generous summary i can give is to say that although you know absolutely nothing about either the new math materials or the movement, nonetheless you have read some opinions of other people that convince you the movement did not meet its goals, as defined by you.

 

[Sankaku]

New Math is New Math AKA crap

As someone just coming out of high-school, you have done a lot of teaching then?

Remember that the reason that politicians say we do something (catch the Soviets!) may not be the reason everyone else wants to do that thing.

 

[jhae2.718]

Are any of the "new math" books available online? The math geek in me would love to read them.

Obligatory:

 

[mathwonk]

i have asked this question here and gotten some good links. you might search posts here for smsg.

and tom lehrer was my calculus section man freshman year. he was excellent.

 

[tiny-tim]

did tom lehrer sing to you? :tongue2:

 

[twofish-quant]

pretty much the math they advocated to be taught in schools failed to help us compete with the soviets. The books you listed I never heard of but it doesn't matter, New Math is New Math AKA crap which failed at it's original purpose.

Ummmm... Soviet Union collapsed in 1991 because they couldn't match the US technologically.

Personally, I loved the 1970's "new math". I remember thinking in third grade "this is easy and fun, so why are they teaching this." It wasn't until I took a class in set theory in college that I realized how *deep* the stuff was.

Also, in elementary school, I was never very good at arithmetic. Still am pretty horrible at it. Being able to do arithmetic is a skill that I never found to be important.

 

[TheCool]

1) the excellent SMSG books from Yale were not widely used because of objections from the commercial publishing world, which substituted very inferior "new math" books for them;

2) due to a lack of funding most teachers were not trained in this new material and hence could not handle it well in many places.

Those are excellent points. You're absolutely right.

Below, I've posted links to the original SMSG ("New Math") high school texts from Yale. The first two cover Algebra and the third covers Functions (including exponential, logarithmic and trigonometric). There's also a geometry book in the series.

http://www.eric.ed.gov/PDFS/ED135617.pdf

http://www.eric.ed.gov/PDFS/ED135618.pdf

http://www.eric.ed.gov/PDFS/ED135629.pdf

These books can serve as valuable resources for anyone looking for a semitheoretical introduction to "precalculus". The books are well written and contain challenging problems that aren't out of reach for average students. No doubt, these are better than 100 percent of contemporary high school texts.

Still, I think the presentation could have been better. They're a bit verbose and leave too little to be uncovered by the student.

New Math is crap, the topics in new math were counter productive for making a more technologically advanced country. How can learning about sets and topology at a early age foster the prerequisite for calculus and other math that is actually used in for engineering. Teaching just the theory of addition, multiplication and division doesn't mean that student know how to actually add or multiply. That was the main problem with "New Math" it only taught the theory with no emphasis on simple computation skills.

I think it's acceptable---if boring---to emphasize rote memorization and drill at the elementary school level. Humans are retentive of information received as children. Most of us can recall our times tables, yes. By contrast, how many remember how to graph a rational function? Solve a quadratic equation? Even students who get A's in their high school math courses don't remember what they learned.

That is the problem. It's a waste of the student's time and the tax payer's money.

If students understood WHY math works, the context in which it originates, they could easily derive formulas and algorithms for engineering and science.

 

[mathwonk]

heres a little example of how understanding theory helps computation: the quadratic formula. "Everyone" knows that in an equation of form X^2 - bX + c = 0, b is the sum of the roots r,s and c is their product.

Moreover, if we know also the difference r-s of the roots, then we can find both roots from adding and subtracting the sum and the difference. I.e. (r+s) + (r-s) = 2r, and (r+s)-(r-s) = 2s.

But knowing both the sum r+s and the product rs of the two roots we can find at least the square of their difference:

i.e. (r+s)^2 - 4rs = (r-s)^2, as everyone knows or can check in two seconds.

Hence, since r+s = b and rs = c, we have that b^2 - 4c = (r-s)^2. So

r-s = ± sqrt(b^2-4c), and thus 2r and 2s = b ± sqrt(b^2-4c).

I.e. r,s = (1/2)[b ± sqrt(b^2-4c)]. (almost) the usual quadratic formula.

 

[mathwonk]

i once (in 1960) invited tom lehrer to a party and offered to provide a piano but he said he did not do that anymore.

 

[tiny-tim]

heres a little example of how understanding theory helps computation: the quadratic formula. "Everyone" knows that in an equation of form X^2 - bX + c = 0, b is the sum of the roots r,s and c is their product. …

i prefer completing the square …

ax² + bx + c = 0

x² + bx/a + c/a = 0

x² + bx/a + b²/4a² + c/a - b²/4a² = 0

(x + b/2a)² = (b² - 4ac)/4a²

(x + b/2a) = ±√(b² - 4ac)/2a

x = (-b ± √(b² - 4ac))/2a

 

[Fredrik]

My kids are being taught "lattice multiplication", which is new for me- so I guess that's the new new math :).

I had to google it, and found this:
http://www.coolmath4kids.com/times-tables/times-tables-lesson-lattice-multiplication-3.html

It is "standard" multiplication, but turned over 45 degrees

The math-education folks I talked to about it uniformly love lattice multiplication-

I enjoyed reading about this, mostly because when I read it I realized that I have no idea how I was taught to multiply numbers in school. I have completely forgotten it. When I need to multiply 14 and 56 (the example from the page vanesch linked to), I certainly don't use "lattice multiplication" or whatever algorithm I was taught in school. I just do this in my head: $$14\cdot 56=\underbrace{10\cdot 56}_{=560}+4\cdot 56=560+\underbrace{4\cdot 50+4\cdot 6}_{=200+24=224}=560+224=784$$ Isn't this how every math nerd does it? Lattice multiplication and similar algorithms seem pretty useless to me, at least as long as we're talking about two-digit numbers. Maybe they're more efficient than my method when we're dealing with four-digit numbers, but I'm not convinced that this is a great reason to teach these algorithms. The method I use for four-digit numbers is to type them into the Google search box :tongue: in the upper right corner of my Firefox. I'm satisfied knowing that I could do it with my method if I wanted to.

 

[mathwonk]

the reason i prefer the way i showed is that it explains every step in the formula, i.e. why there is a 4 [because the difference between (r+s)^2 and (r-s)^2 is 4rs], why there is a square root [because you are finding the squared difference of r and s], why you add and subtract the square root [because you have r+s and r-s and want to eliminate r or s], why you divide by 2 [because by adding and subtracting you are getting 2r and 2s],...


tiny tim, why do you prefer the usual completing the square method? That was the method i was taught in high school and it always made it seem unmemorable, because to me it is unmotivated. i.e. it is clever but why do you do those things? i.e. how do you think of it? but i was never strong at algebra.

there are also constructions in euclidean geometry where the completing the square method is easier to translate into a geometric construction. such as euclid's construction of a regular pentagon, where he completes the square geometrically to solve the equation

X^2 = R(R-X). I.e. Euclid solves this equation geometrically by completing the square,

i.e. by writing it as X^2 + RX = R^2, he completes the square by bisecting segment R and writing it as (X+ R/2)^2 = X^2 + RX + (R/2)^2 = R^2 + (R/2)^2, then he solves it by constructing the right triangle with sides R and R/2, since then the hypotenuse must be (X + R/2), so we get X by subtracting R/2.

This is quite easy to do geometrically in a circle of radius R by constructing two perpendicular diameters, bisecting one radius, and connecting its midpoint to the point where the other diameter meets the circle. I.e. this segment is X + R/2. Then one easily copies R/2 onto this hypotenuse and copies the remainder as a secant X of the circle.

Then later he proves that X is the side of a regular decagon inscribed in a circle of radius R.

 

[tiny-tim]

hi mathwonk!

the reason i prefer the way i showed is that it explains every step in the formula, i.e. why there is a 4 [because the difference between (r+s)^2 and (r-s)^2 is 4rs], why there is a square root [because you are finding the squared difference of r and s], why you add and subtract the square root [because you have r+s and r-s and want to eliminate r or s], why you divide by 2 [because by adding and subtracting you are getting 2r and 2s],...


tiny tim, why do you prefer the usual completing the square method? That was the method i was taught in high school and it always made it seem unmemorable, because to me it is unmotivated. i.e. it is clever but why do you do those things? i.e. how do you think of it? but i was never strong at algebra.

because it's a direct proof, involving only a b and c, while yours introduces r and s and proves a lemma about r and s before finally abandoning them and returning to a b and c

also because it makes it obvious where the ± comes from

also because (although the general proof works fine) you can use it with the given numbers almost without thinking …

eg x² + 58x +337 = 0

(x + 29)² +337 - 841 = 0

x = -29 ± √504 ​

 

[symbolipoint]

Mathwonk,

We could only imagine that you learned or were taught your r & s expressions for the Quadratic Formula when you were young, either in high school, or early in your college education, or both. Most of us were taught about Completing the Square to find the general solution to a quadratic equation as well as to memorize the general solution. We were also commonly taught a relationship between the two solutions to a quadratic equation and were given maybe three or four homework exercises with this relationship. This is why most of us are unfamiliar with those r & s formulas which you are comfortable with.

We know that we can derive the general solution by comparing with a general quadratic equation and a quadratic trinomial (so I say, but I forgot just how to do it). Most of us may remember choosing a symbolic term in the form of a square and adding this to both sides of an equation, starting from a quadratic expression. Also, some of us found or maybe were actually shown a visual method to demonstrate what Completing the Square means, using cutting and rearranging pieces from a rectangle to show a missing square corner. All the lengths of sides can be labeled with variables and we can derive what this missing square area piece is. So you see, we have good development in learning about completing the square to solve a quadratic equation. We can also use fairly simple algebra to convert between standard form and general form of the quadratic equation for our purposes. In standard form, we can easily graph the equation (that is, when we have a quadratic function).

 

[mathwonk]

a formula you can use without thinking makes me uncomfortable. I was taught the same completing the square way you were. I just didn't get it. I was never happy until I finally learned what the connection was between the roots r,s that we are looking for and the coefficients a,b,c.The mystery was cleared up for me when I finally realized that every quadratic can be factored as (X-r)(X-s) = X^2 - (r+s)X + rs = X^2 -bX +c, and that's what b and c mean. they are sums and products of the roots.

If you don't understand this you have no hope of understanding higher degree equations.

On the other hand if you do understand this you can also solve cubics.

E.g. noting that the roots of a quadratic are solved for as sums of other numbers, let's try that for a cubic, and let's put X = u+v and then expand

X^3 = (u+v)^3 = u^3 + v^3 + 3 u^2v + 3uv^2

= u^3 + v^3 + 3(u+v)(uv) = u^3+v^3 + 3Xuv.

Thus to solve X^3 = 3uvX + (u^3+v^3) we can take X = u+v.

Hence to solve X^3 = bX + c, we need to find numbers u,v, such that

3uv = b and u^3+v^3 = c.

Equivalently we need 27u^3v^3 = b^3 and u^3 +v^3 = c.

Thus we are given the product b^3/27 and the sum c, of the two numbers u^3 and v^3.

But by my solution of the quadratic, that means we know how to find u^3 and v^3,

namely by solving the quadratic equation t^2 - ct + b^3/27 = 0. I.e. then the two solutions are t = u^3 and v^3.

Then after finding u^3 and v^3 this way, we get the three solutions of X^3 = bX+c,

as X = u+v, where we let u be anyone of the three cube roots of u^3,

and we let v = 27u/b^3, for each value of u.

e.g. to solve X^3 = 3X +2, take u=v=1, and then X = u+v = 2.

to solve X^3 = 6X + 9, take u=1, v=2, so that 6 = 3uv and 9 = 1^3 + 2^3, and X = 1+2 = 3.

Try solving cubic equations, on the other hand, by "completing the cube".