[kenberg]

Your example was fine, I didn't think it was intended as the absolute last word on anything. My point was solely that it actually is tougher than it looks to get these exactly right.

[gwnn]

How about "x=2 because 3+2=5 and because any first order polynomial can have at most one root, as per the fundamental theorem of algebra"?

[PassedOut]

 kenberg, on 2014-June-10, 14:00, said:

Your example was fine, I didn't think it was intended as the absolute last word on anything. My point was solely that it actually is tougher than it looks to get these exactly right.

No doubt it is difficult to get the questions exactly right. But now I have an idea of how the Common Core works, whereas I had none before. It seems to me that this, or something like it, is the only way to get a handle on how to improve educational performance across the US. I have a granddaughter now so I might have an opportunity to see some of the results...

[kenberg]

 PassedOut, on 2014-June-10, 19:29, said:

No doubt it is difficult to get the questions exactly right. But now I have an idea of how the Common Core works, whereas I had none before. It seems to me that this, or something like it, is the only way to get a handle on how to improve educational performance across the US. I have a granddaughter now so I might have an opportunity to see some of the results...

Congratulations!

[blackshoe]

 kenberg, on 2014-June-10, 11:42, said:

I know I have told this before, but it bears on the issue of reasoning and shows just how widespread the problem is. I was in a Burger King or its equivalent when, next line over, a woman paid for her order and got the change back for a 20. But she had given the cashier a 100, not a 20. The cashier looked in the drawer and agreed. The cash register announce the change for a 20, no one in the BK had any idea how to cope with this. No one knew how to reset the cash register. I barged in. Thankfully everyone agreed that 100 was 80 more than 20, and they then accepted my solution that the woman should receive an extra 80 in change. Whew!

I wonder what happened when they closed out the cash register at the end of the shift or end of the day or whenever they do that. I can imagine it reporting that the till was 80 bucks light. Well, perhaps somebody knew how to deal with that problem.

[jeffford76]

 blackshoe, on 2014-June-11, 14:01, said:

I wonder what happened when they closed out the cash register at the end of the shift or end of the day or whenever they do that. I can imagine it reporting that the till was 80 bucks light. Well, perhaps somebody knew how to deal with that problem.

Why would the till be light? $80 it didn't know about got put in (the $100 instead of the $20), and $80 it didn't know about got taken out (the extra change).

[kenberg]

 blackshoe, on 2014-June-11, 14:01, said:

I wonder what happened when they closed out the cash register at the end of the shift or end of the day or whenever they do that. I can imagine it reporting that the till was 80 bucks light. Well, perhaps somebody knew how to deal with that problem.

Light why? I'm not seeing the problem here.

Since I am typing, let me bounce back to the problem about credit cards:

Quote

Which of the following does NOT occur when using a credit card instead of currency?

a. possible service fees
b. accrued interest on unpaid baalnce
c. increased spending limit if overspending occurs
d. less cash carried and an expedited checout process

Zel observed that the point was "The children are taught that overspending with a credit card has consequences and they cannot rely on the card company raising their credit limit if they get into debt. You can therefore tell this is the right answer even if you do not agree with the others."

Yes, probably he is right about this. Basically it is a propaganda question. I believe we had many such questions in high school. I remember learning that we always were to wear a tie and coat when applying for a job. The book said so, and this was therefore the correct answer. Mostly these questions taught me (or confirmed what I had already concluded) that you should ignore a lot of what you are told by adults. However, if we really wanted to propagandize, I would suggest that students learn that if they ever have spent so much on their credit card that the spending limit is an issue then they need to take a very serious look at their spending habits. Financial emergencies happen, but usually not to fifteen year olds and even with adults the emergency is far more often bad planning.

Really I have no idea from experience what the correct answer is, and I would be skeptical of anything someone told me, especially if they had a view that they were pushing. I would be seriously worried about any child who did, from experience, know what the correct answer is. .

One of the many stupid things that I have done in life: When I moved from Minnesota to Maryland I had to change my driver's license. No problem right? So I turned in my Minnesota license and asked for a Maryland one. Oops, there is a written test, multiple choice. Example: You are driving 50 mph and put on the brakes. How many feet does it take to stop? When I took my Minnesota exam some twelve years earlier, I had memorized all of this stuff. But now? Of course it is simple. The answer is the largest number that appears. The whole thing is a joke. You do not actually have to know anything except how to take a multiple choice test. Many many tests go that way. I got an A in my very boring undergraduate psychology course. I never came to class except for the multiple choice exams. No doubt psychology is an interesting subject. I accept this on faith, the course would never lead one to that view. One of the questions (really): You feel a tight band around your head and believe people are following you. This is an example of a. Normal behavior ... I forget b c and d, but I managed.

[Fluffy]

 gwnn, on 2014-June-10, 14:52, said:

How about "x=2 because 3+2=5 and because any first order polynomial can have at most one root, as per the fundamental theorem of algebra"?

This is similar to how I'd solve it, this might sound weird but for some reason when I saw the equation I tested to substract numbers from 5 untill they gave me 3 instead of substracting 2 from 5, obviously took me a small fraction of a second anyway, but still a bit weird.. Kenberg explained why this is not right in theory, but I also know about first order polynomials.

 gwnn, on 2014-May-26, 03:30, said:

Linguists* describe how people use language in their day-to-day lives, logicians describe how people ought to think and speak in an ideal world that is nowhere to be found in our galaxy. Why is it unfortunate that lawyers (who work with real people in real life, not robots from outer space) use language as other people do? I agree, though, that sloppy wording is in no one's best interest.

Lawyers, at least on my country, do not use the same language as I do. They might share grammar and some words, but it is far from the same language.

[blackshoe]

 jeffford76, on 2014-June-11, 14:20, said:

Why would the till be light? $80 it didn't know about got put in (the $100 instead of the $20), and $80 it didn't know about got taken out (the extra change).

You're right. Disregard my last.

[kenberg]

If you are up for it, there is a long pdf on the math standards at

http://www.corestand...h_Standards.pdf


Browsing, I saw something that triggered a memory:

Quote

Use the method of completing the square to transform any
quadratic equation in x into an equation of the form (x – p)2 = q
that has the same solutions. Derive the quadratic formula from
this form.

In fact my high school teacher, without previous warning, called on me to come to the board and derive the quadratic formula referred to above. [For the many who don't remember and or don't care, this gives a formula for solving the equation a x^2 + b x +c=0]. I did it. I believe there were, perhaps, two other kids in class that could have done it. My 1956 high school graduating class was 208, give or take a couple. So I ask: Will the typical high school graduate in, say, 2020 be able to derive the quadratic formula? Is this what we are thinking? We might also ask if he should. I walked to school with Fred. Fred took "shop math" as it was then called, and planned to become a plumber. I am not saying that Fred could not have learned the derivation (if you put a gun to his head), although I doubt it would have made much sense to him and I am positive he would never have any use for it.

Readers of this thread, those not engaged in mathematics, might ask themselves if they can derive the quadratic formula and, if the answer is no, ask if it has held them back in their career.

[nige1]

 Elianna, on 2014-June-10, 13:36, said:

Historically, the ACT did not penalize incorrect answers, and the SAT did on the multiple choice sections. (Both of these are used as college entrance exams.)
I know that the old California Standardized Tests did not penalize, but I am not sure about the new tests. What I do know is that as they are adaptive, my guess is that students cannot skip questions, and so they do not penalize per se for missed questions, but they adjust future questions based on results of the previous questions, so if you miss the middle-of-the road questions you will start getting easier questions, but have less of an opportunity of getting a higher score.
There are two companies that are making tests for common core, and states have signed up for them in roughly equal proportion. California signed up for Smarter Balanced (SBAC), and I forget the other one.
I know that the example I came up with was not great, but on my behalf, I thought of it in 5 minutes. If someone wants to see actual sample questions, they can go to the smarter balanced webpage, and try out the sample test: http://sbac.portal.a.../practice-test/

Thanks Elianna. The tests seem like a good idea to provide feedback on student attainment and teacher proficiency.

[mike777]

1) Note that teacher tenure in public schools ruled to violate the law in Calif of all places.
2) Many of us confuse education and scholarship, they are not the same.
3) the arrow of discovery
4) the belief that university knowledge generates economic wealth, consider changing the direction based on empiricism.
5) In education consider nonlinearity (convexity) The gains are large in relation to potential side effects.
6 Charlatan and the academic..to use education in medicine as just one example. Medicine misleads people because for so long its successes were prominently displayed, and its mistakes are literally buried.
7) encourage autodidact inclined students
8) Many of us often very often accept what experts say as true because they are experts, not based on testable evidence or theory. This is often true in my favorite sport, baseball.

[mike777]

 kenberg, on 2014-June-11, 19:43, said:

If you are up for it, there is a long pdf on the math standards at

http://www.corestand...h_Standards.pdf


Browsing, I saw something that triggered a memory:



In fact my high school teacher, without previous warning, called on me to come to the board and derive the quadratic formula referred to above. [For the many who don't remember and or don't care, this gives a formula for solving the equation a x^2 + b x +c=0]. I did it. I believe there were, perhaps, two other kids in class that could have done it. My 1956 high school graduating class was 208, give or take a couple. So I ask: Will the typical high school graduate in, say, 2020 be able to derive the quadratic formula? Is this what we are thinking? We might also ask if he should. I walked to school with Fred. Fred took "shop math" as it was then called, and planned to become a plumber. I am not saying that Fred could not have learned the derivation (if you put a gun to his head), although I doubt it would have made much sense to him and I am positive he would never have any use for it.

Readers of this thread, those not engaged in mathematics, might ask themselves if they can derive the quadratic formula and, if the answer is no, ask if it has held them back in their career.

Ken I remember something, a conversation when I went back to MBA school in the middle of my life.

In having a chat with the head of the marketing dept she said that knowing the words, words we never use in normal life and their meaning can help you master a subject.


For example in your post:
"derive the quadratic formula"

If in math class we are taught the words, "derive" and "quadratic formula", I bet many more of us could do it.

Funny enough just using the words in a lecture in not teaching us but I bet you of all people knew that which makes you a wonderful math teacher.

IN my algebra I class I would often very often get really pissed off because I could see the teacher was moving far to fast over basic beginner concepts and losing most of the class the rest of the week. The story of many of my classes in education

As an older student in my MBA classes I knew most of the words, I had heard of and used many of the words taught in my classes. Those who had not were left behind.

I remember one girl sitting next to me on the first day in my first class with ten color coded pens to write everything down.

I wrote less than nothing which was not best.


what I learned most important that I needed a business writing class which I took on the side on my own. I strongly recommend this. If anything on these forums I love it is the writing of posters. Second is how posters explain math in clear terms, such as Ken, Helene, Richard and others.

the rest was flash,,,lots of flash and nonsense.

[WellSpyder]

 kenberg, on 2014-June-11, 19:43, said:

Readers of this thread, those not engaged in mathematics, might ask themselves if they can derive the quadratic formula and, if the answer is no, ask if it has held them back in their career.

I use maths to some extent in my career - I'm an economist - and I was one of those to whom maths at school always came fairly easily. But I don't think I could derive the formula - and indeed I don't actually remember its derivation being discussed at school, although I suppose it must have been. In fact, I happened to read a derivation only a few months ago and thought it was pretty neat, but it hasn't stuck.

As an aside, I could certainly still recite the formula, and indeed understand and apply it, but I'm not convinced that if I couldn't do this it would have held me back in my career either!

[helene_t]

You mean the roots in a second degree polynomial? It was somehow derived in the 1G class, that is children aged 16. All I remember is that it involved writing the polynomial as A(x-alpha)(x-beta), which by the way was generalized to n'th degree polynomials although we obviously didn't get the root formulas for higher than 3rd degree. I am not sure how stringent it was proved that alpha and beta are the only roots.

I have used the formula on a few occasions in my work. The derivation of that particular formula I haven't used but a general ability to derive formulas is essential in my job.

[Zelandakh]

The formula he means is -b +- sq rt(b^2-4ac) all over 2a, helene. Something that stuck in my head from that time. Incidentally, learning quadratic equations was the spark for learning to be creative in finding alternative solutions to a problem, something that has been beneficial to me many times. The benefits from learning sometimes go beyond the things directly being taught; sometimes we also need to inspire students with the possibilities of more advanced maths. Otherwise we may as well just teach everyone mental arithmetic + using a calculator + Excel spreadsheets + the latest smartphone maths app. That is going to be more than enough for a significant portion of the population.

[kenberg]

I often look back to my own experience to decide on whether something sounds reasonable, and it was with that in mind that I mentioned the derivation of the quadratic formula. it is on the pdf that I linked to and seems to be an objective. So I asked myself if it seemed reasonable based on my memories of high school. I am skeptical. I will comment on a few comments.

First from Mike:

Quote

what I learned most important that I needed a business writing class which I took on the side on my own.

I absolutely totally agree. Far more useful for most people in most careers.


From Well Spyder

Quote

I use maths to some extent in my career - I'm an economist - and I was one of those to whom maths at school always came fairly easily. But I don't think I could derive the formula - and indeed I don't actually remember its derivation being discussed at school, although I suppose it must have been. In fact, I happened to read a derivation only a few months ago and thought it was pretty neat, but it hasn't stuck.

As an aside, I could certainly still recite the formula, and indeed understand and apply it, but I'm not convinced that if I couldn't do this it would have held me back in my career either!

My son-in-law is an economist and I am sure he knows the formula, I think it is likely he can still derive it although he is over 50 and probably has never seen any reason to do so. He was a math major as an undergraduate. When my daughter turned 40 I asked her if she remembered it, just the statement not the derivation. She sort of did, although not exactly.
She has a Ph.D. and a successful career, but as far as I know she has never had occasion to solve 3x^2i5x+17=0, much less to solve a x^2 +bx +c=0 in terms of the parameters

From Helene:

Quote

You mean the roots in a second degree polynomial? It was somehow derived in the 1G class, that is children aged 16.

Yes. And the age matches my age. I was 15 when I started this class, 16 when I finished..

Zel:

Quote

The benefits from learning sometimes go beyond the things directly being taught; sometimes we also need to inspire students with the possibilities of more advanced maths.

I agree with this, but I doubt it woould have dome much for my friend Fred. As far as I know, Fred went on to a successful career as a plumber. Now if Fred wanted to move up in the plumbing business, Ibet that course in business writing along the lines mike suggests would have been far more useful.



Now a few more comments: There is a reason our teacher, Mr. B., called on me. After i finished, he announced that everyone in class should be able to do that. I am sure he knew, just as I did, that most could not. But student interest has a great deal to do with all of this. I read ahead. I went to a Saturday class at the University taught for free by professor Paul Rosenbloom, a great mathematician and a real character. In the summer when I was 16 I attended a summer course in Physics at teh University. No credit for it, no tuition, the Prof just let me sit in. BUT I was ot on the honr role for my high school when I graduated the following year. Perhaps I will someday read A Tale of two Cites, I have been told I might like it. But I am 75 and if I am going to do this, maybe I should do it soon. I am sure I will never read the poetry of Shelley. I spent a reasonable amount of time in after school detention for this and that. I liked math, I liked physics. Along these lines, a friend Billy took high school physics with me. He was interested in cars, and his high school adviser told him that he would therefore like physics. At least once during every class, Billy would ask "What does this have to do with cars?"


it seems to me that getting a sixteen year old to study something that he has no interest in is very difficult. You can ram some of it down his throat, but expecting Fred the future plumber to seriously work on the derivation of the quadratic formula is not going to be easy.

[Zelandakh]

 kenberg, on 2014-June-12, 06:38, said:

it seems to me that getting a sixteen year old to study something that he has no interest in is very difficult. You can ram some of it down his throat, but expecting Fred the future plumber to seriously work on the derivation of the quadratic formula is not going to be easy.

Having done some private tuition along the way I have some experience of this. For example, when I needed to teach a football crazy boy about geometry I created a question where it was necessary to find the angle to shoot a penalty into the top corner of the goal (if the ball went straight, which admittedly is a major simplification). The point was that the question immeiately piqued his interest whereas the same question without dressing it up would have been a chore. It is obviously harder with a class of 30 or so to find questions that are interesting to everyone. If it were just Fred though, perhaps we could find something within fluid dynamics - the rate of flow in a pipe or something - that resolves to a quadratic and see if that works. There is always a way - which comes back to my comment earlier about creative solutions.

[kenberg]

Possibly so. My own experience: My younger daughter was having trouble with arithmetic early on. She loved board games, card games, etc. She was maybe 7 or so. So I made up a game and explained the rules to her. I got to the point where one number had to be subtracted from another. She stood up with hands on hips, announced "I am not playing any stupid subtraction game" and stormed out of the house. I think it's in her genes. Parents may differ on this, but I was proud of her. She loved snakes and would catch them and bring them home. Big snakes. She knew a lot about snakes. But subtraction didn't do anything for her.

[blackshoe]

There are several different ways to derive the quadratic formula.

As for Shelley, all I remember is "My name is Ozymandias, king of kings; look on my works, ye mighty, and despair!; Nothing else remains." Something engraved on the remains of a statue found in the desert, iirc.