[kenberg]

Basically, I am arguing for realistic goals. Or so I claim. Perhaps someone can convince Fred the future plumber that he should study the derivation of the quadratic formula, perhaps someone can convince me I should read Shelley. I really hope that the country's future does not depend on either of these events occurring.

[nige1]

IMO, the educator's task is difficult. Most of what you learn, you won't use again. But it's hard to decide which bits will be useful.

Aitken devoted years of practice to become a calculating prodigy -- which he regretted when sophisticated mechanical and electronic calculators became widely available.

Hardy was proud that there could be no practical application of his work on number-theory -- which is now a foundation of cryptology.

Some students in Europe and America still learn computing and engineering skills -- which they may not be able to use -- now that many such jobs are being exported to the Middle and Far East.

Topics without apparent practical application aren't necessarily useless. You need to practice reasoning -- to learn how to think -- useful for solving problems in novel contexts.

There's also the appreciation of beauty for its own sake.

What students are capable of understanding is equally controversial. But average students attain amazing mathematical proficiency, in some countries.

[kenberg]

Another few words, and then I will try to shut up. Of course it is not simply about deriving the quadratic formula or reading Shelley, or Keats, or whoever. For me, the point is this: As noted in an earlier post, there are a lot of people out there who cannot work out how much change should be given fo a 100 if the cash register has erroneously been asked to calculate the change for a 20. This is a serious lack of ability, making a person totally dependent. But most people do not need the quadratic formula to live a successful life. Nor do they need to know the binomial theorem or the factor theorem, also on the cited pdf.

Overreaching has consequences. What usually happens is that an industry develops that creates data to show that students have an advanced level of understanding due to some great program or other, and really the students have managed to make scripted responses to very tightly scripted questions. Quantitative reasoning is very important. It always has been. Maybe I am fantasizing, but I believe every kid in my 8th grade (I was 12-13, to clarify for the non-American readers) class could have figured out the problem with the change. Most of these kids did not come from college educated parents. If a lot of 13 year-olds cannot do that today, we should ask how come.

Often you hear about "back to basics". Often that means teach rote responses to drill questions. Some memorization is useful, I would say it is important, but surely it is not enough. Understanding is crucial. Often quite simple problems, making change and moving up from there, is a much better approach to reasoning than deriving the quadratic forumula. Not everyone is going to college, and even for those that do we need to think about this some. Take journalism. Read any paper, even the best, and you find that quantitative issues are often badly mangled .This is not because the journalism major does not know how to derive the quadratic formula, it is because s/he goes on autopilot whenever numbers come up. The reporter, in most cases, could do better, I have faith that this is so. Bur numbers come up, s/he copies them down and puts them in the story as best s/he can, and simply closes his/her mind to reflecting on whether what is written makes any sense at all.

Not at all am I suggesting that mathematical theorems should not be taught. But we cannot do everything. I didn't play football. I didn't learn to play the tuba. I never acted in the theater. Some never learn to derive the quadratic formula. It's ok.

[Elianna]

kenberg, on 2014-June-12, 11:38, said:

Not at all am I suggesting that mathematical theorems should not be taught. But we cannot do everything. I didn't play football. I didn't learn to play the tuba. I never acted in the theater. Some never learn to derive the quadratic formula. It's ok.

I know that your point is not all about deriving the quadratic formula, but I do have to say that I "teach" this in that I teach "completing the square" as a method to solve quadratic equations, and then I have students do it with symbols to get a general formula (I usually have to help them on the combining fractions step). So basically my students have had the opportunity to derive the quadratic formula before it's taught to them (in my class. To be perfectly honest, they actually learned the QF in Alg 1 before they get to me in Alg 2, but they're so caught up in what they're doing that they don't realize what they're looking at until we get to the square root part).

I tend to be rather protective of keeping high expectations of all students, probably because of where I work: 90% of our students receive Free or Reduced Lunch, and most of them were on track to fail out of HS. Our goal is to get them to-and-through a 4-year college. Many of our students struggled through school not because of any learning disability but because very little was expected of them, and so they expected little of themselves. We spend much of their 9th and 10th trying to convince them that they are capable.

If anyone is interested in reading about the educational inequities in the US I recommend reading "Savage Inequalities" or "Shame of the Nation" by Jonathan Kozol.

[Zelandakh]

Elianna, on 2014-June-12, 22:20, said:

I tend to be rather protective of keeping high expectations of all students, probably because of where I work: 90% of our students receive Free or Reduced Lunch, and most of them were on track to fail out of HS. Our goal is to get them to-and-through a 4-year college. Many of our students struggled through school not because of any learning disability but because very little was expected of them, and so they expected little of themselves. We spend much of their 9th and 10th trying to convince them that they are capable.

I also received Free Lunch tickets as a child as well as missing an important year of schooling (age 13-14) and do not see this as a good reason to fail; even les of a reason for teachers to give up, which in truth many do. Your approach is clearly right and the last sentence is particularly relevant. It was often the case when I was tutoring that the issue they had with maths was more one of a mental block than a lack of ability. Building up confidence and showing them that they really do have the ability is absolutely key to the improvement here. Doing this one-on-one makes for an interesting mix of training in actual maths techniques and psychology.

This is another of the areas in which having creative solutions to problems was an advantage - sometimes tailoring an alternative solution method to the student is a better idea than trying to fix the method the teacher has used. It is hard not to have confidence in a method you both understand and get a 100% success rate with.

[kenberg]

I am struggling with all this, I like to think I have an open mind. Some thoughts.


A word about free, or subsidized, lunches. I started hs in fall 1952. A hot lunch was available for everyone at a very reasonable price. Twelve cents, if I remember correctly. Of course this was in 1952 monetary units. I think a White Castle hamburger (very small, a person needed several) was about twelve cents. A movie, once you were over age 12, was thirty-five cents. I think everyone had twelve cents, and for that matter most of us had jobs and could produce our own twelve cents. Anyway, if we as a society could support twelve cent lunches for everyone in 1952 it seems we should be able to do so now, with adjusted prices. Never mind having the kid prove he is poor, it's a little embarrassing at least to some. Btw, Coke was not sold in school. I am referring to the cola.

Added: Thinking a bit more, i believe the minimum wage in 1952 was 70 cents an hour. Now it is a little more than ten times that. So make a lunch cost a buck twenty or so. A basic lunch. I sometimes bought extras. My money, my extras. This connection between cash and options can be a useful lesson for kids to learn. There would be no need for evaluators to check up on eligibility, no concern about fraud, just a basic lunch for a buck twenty and the kid could buy extras. Not coke. Kids could have debit cards, redeemable only at the lunch counter, to eliminate worries about shake-downs. This seems workable to me. Except the soft drink lobby would of course protest to the state legislature...

Back to the course. I can't recall exactly where Algebra 1 left off and Algebra 2 began, but the way you describe your Algebra 2 tracks well with what I recall.

When I started High school I was put (no one asked me) in the "college prep" track. I took Algebra and Spanish, my frined Fred took Shop Math. Some of the girls, quite a few actually, also were in college prep taking Algebra and Spanish. If I wanted French, I had to wait a year, we had one language teacher who in the year I began taught Spanish 1 and French 2. No other languages were available. The girls who were not in college prep took (I am so sorry to say) "home economics" where I suppose they learned how to adjust a recipe for 4 so that it would serve 6, but as far as I know they were given no math at all.

OK, after Algebra 1 there was a year of Geometry with Mrs.Swann. Axioms, definitions and proofs. Voila!! It was easy, it was interesting, and it was great fun. I think everyone in college prep took Geometry, most of them with much less enthusiasm than I had for it There were several sections.

Algebra 2: Even those in college prep were not required to take this. There were two sections, taught by Mr. Berger I guess maybe 25 students per section . As I mentioned, my graduating class had something like 208 students. 200+ anyway, and the + was a single digit. So maybe something like 25% of the class was in Algebra 2. Incidentally, I was also taking a course in Metal Shop that year. I believe I was the only student taking both Algebra and Metal Shop.

Some, quite a few, kids were on a non-college track. I think that the school did pretty well by them. At a recent hs re-union, the first I ever went to, I was chatting with a guy that I at first couldn't place. But then I remembered him as the guy who was absolutely fearless on the trampoline. He was not in college prep, but he did just fine in life.
Also, there were the kids gong on to college, but not in STEM (as we now call it) fields. And then there were the ones with me in Algebra 2, and Trig/Solid Geometry the following year.

My high school experience was definitely a mixed bag. I took engineering drawing (called mechanical drawing then) from a guy with an alcohol problem. By last period, when I was supposed to be there, he rarely had a clear idea of who was there and who wasn't, and I usually wasn't. But some classes were quite good.

My own kids: My older daughter has a Ph.D., my younger daughter never went to college. Both are successful. My father finished 8th grade, both his parents were dead by then. I grew up in comfortable, thought certainly not rich, surroundings. I fully absorbed the cultural expectations of the time: A guy (yes, expectations were different for men than for women) is suppose to grow up to be self-supporting, how he does that is his business.

I know times have changed. I know college grads make more, but I am skeptical about reading too much into that. My younger daughter is doing what she likes (part owner/manager of a boarding kennel for dogs) and is successful at it, I think if she went to college she would have been studying to do something she didn't like. I worry that we sometimes are taking a pony that could grow up to be a perfectly fine horse, and putting him in a program designed to turn him into a moose. This may not work well.


As I said, I regard this as very important, far too important to be adamant about in any direction. I am truly interested in various experiences here.

[gwnn]

Completely unrelated to this thread but for some reason I find it funny:

"If mathematically you end up with the incorrect answer, try multiplying by the page number."

[kenberg]

gwnn, on 2014-June-13, 06:15, said:

Completely unrelated to this thread but for some reason I find it funny:

"If mathematically you end up with the incorrect answer, try multiplying by the page number."

We employed a similar trick in physics. If an answer turned out to be absurdly small or absurdly large, we would, somewhere in the middle of the calculations where we hoped no one would question why, either multiply or divide by the permittivity of free space.

This usually took care of the problem.

[hrothgar]

Few quick comments from a hiring manager at a tech company.

1. It is extremely difficult for us to find qualified candidates.

We actually have programs in place to work with college freshman and sophomores to help them develop the foundation skills that we're going to be interested in three years down the pike. (There's a pod of four interns in my department alone)

2. The primary skill that we require is the ability to master new things fast. The primary reason to learn things in high school and college is to get experience learning new things, not mastering any given set of facts / skills. (The odds that anything that they're doing today / tomorrow is going to be directly applicable to their jobs in five years time is somewhere between slim and none)

3. Coupled with this, we want people who are excited that they are going to be challenged by new things, able to communicate effectively, and who work and play well with others.

With all this said and done, here are some of the stuff that gets used on a day to day basis

1. Programming and scripting

An awful lot of the work folks do in tech involves manipulating data. Folks need enough basic competency in scripting to be able to grab data from a variety of sources and shove it into a database. Python, awk, and SQL are today's equivalent to Excel / Lotus

Being able to write programs to munge this data and generate graphical user interfaces to display and manipulate data is also critical

2. Applied math, with a focus on probability and statistics, queuing, calculus, and optimization

3. Systems level thinking (can you take a complex system, represent it as a set of nodes and graphs, and identify feedback loops and lags

4. Storytelling (can you use narrative to convey information)

[Zelandakh]

hrothgar, on 2014-June-13, 07:24, said:

2. The primary skill that we require is the ability to master new things fast.

This is one of the easiest skills in the world to find. It is directly linked to the intelligence factor, i, so you need to be giving potential candidates IQ/aptitude tests. It so happens that this is also the skill I most stress in interviews and the rectuiment assessment I had for my first full-time IT job included such a test. The test came before the interview and the interviewer told me he was confident it would be the highest score for the day so I was feeling pretty good after that!

The point is that these tests have a high success rate for finding candidates with your primary requirement. Then you just have to weed out the ones that are unsuitable for other reasons.

[hrothgar]

Zelandakh, on 2014-June-13, 08:45, said:

The point is that these tests have a high success rate for finding candidates with your primary requirement.

I have interviewed at a number of top companies in the US.
I have never seen a single one administer any kind of standardized test.

Way too much fear of exposure to liability suits.

[nige1]

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

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. 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.

Wikipedia gives a cute derivation -- multiplying both sides by 4a -- resuscitated by Larry Hoehn:





In my era, school-teachers used a more intuitive way of "completing the square" -- which Elianna probably uses, and which, IMO, is a better introduction.

The study of pattern needs no practical justification (although there's plenty). Mathematics is queen of the arts and handmaiden to the sciences. Most of us will never understand much of it, although we're still in thrall to its beauty.

[kenberg]

The really still involves "completing the square", it's just a slightly different square. Adding b^2 to 4a^2 x^2+ 4abx gives 4 a^2 x^2+ 4abx+b^2 which is the square of (2ax+b). But yes, we were taught to start the process by dividing by the coefficient of x^2. The variant form you give has probably occurred to many (to me, for example) but it's mostly a matter of taste, I think.

I think an important part is to do what Elianna does, namely to solve some specific problems, say 3x^2+5x-7=0, by completing the square and then, later, take on the general case. In this given specific case it probably seems more natural to start by dividing by 3 rather than by multiplying by 12. Once you do a few that way, then it probably is most natural to divide by a rather than multiply by 4a in the general case. But either way works.

Multiplying by 4a is probably spiffier, dividing by a is probably more natural.

[barmar]

I spend a good deal of my free time on the web site StackOverflow. This is a Q&A site where (mostly new) programmers post programs they're having trouble with, and more experienced programmers like me help them fix their problems.

What I've discovered is that there are huge numbers of people who are trying to program computers (mostly creating web applications) who don't know how to think logically. They do almost everything by copying bits of code from tutorials. They don't understand how things work, they're just cutting and pasting. Then when it doesn't do what they were hoping, they're totally lost. The excuse they always give for not being able to work things out is "I'm new to all this". This is a lame excuse, in my opinion -- programming is not rocket science, it's just a somewhat more detailed form of procedural thinking that any intelligent person should be able to do. Of course you make mistakes (I do), and it takes experience to learn all the ins and outs, and some people like me have a particular aptitude towards it (just as there are some people who are naturally better musicians). But these people seem unwilling to try to understand how the tools they're using work -- they're like someone who decides to build a car from scratch, but is unwilling to learn how an engine works.

As Richard said, the important thing that students need to learn is how to think, how to solve problems, etc. I don't think I've ever had to solve a quadratic equation since I left school, nor have I done any calculus. But when I learned how to do these things in the first place, I learned useful mental processes. I was in grade school in the 60's and early 70's, the big thing then was New Math -- they' were replacing all the rote learning of previous generations with learning how to work things out from basic principles. What I see in all these SO newbies is people trying to do programming by rote, and that's not going to work.

[kenberg]

A question: It is my understanding that some states have "adopted Common Core", and some states have not. In practical terms, what does this mean? I see statements that Common Core does not actually impose anything on the states. This is usually meant as a rebuttal to conservative claims that the feds are taking over educational policy. Ok. But just what does it mean for a state to "adopt Common Core"?. What does it commit them to doing?

I am asking because I really have no idea of just what it means to "adopt Common Core".

[blackshoe]

There is at least one state (one of the Carolinas, I think) which had adopted Common Core, and has now rejected it. I know no details, I just saw a headline in passing somewhere.

As for what it means to adopt it, I would guess the state looks at it, decides "this looks good, let's do it", and then implements whatever parts of it they like - and does not implement anything they don't like.

[kenberg]

blackshoe, on 2014-June-14, 08:08, said:

There is at least one state (one of the Carolinas, I think) which had adopted Common Core, and has now rejected it. I know no details, I just saw a headline in passing somewhere.

As for what it means to adopt it, I would guess the state looks at it, decides "this looks good, let's do it", and then implements whatever parts of it they like - and does not implement anything they don't like.

Huh! I am getting this really strong deja vu feeling that we have discussed this before and in exactly the same way, even up to the "one of the Carolinas".. At any rate, you probably have it about right.

[kenberg]

There is the fundamental question: What do we want kids to learn as they grow up. borthgar cope with new situations) , barmar (master procedural thinking), Elianna (realize you can do more than you think) and others have spoken to this, and I do not mean that my one phrase summary adequately reflects their views. I aree with all of this, most particularly with Elianna.

I am reading Elizabeth Warren's A Fighting Chance. I really like it. Her mother empasized the importance of marriage. Young Betsey included marriage in her plans but she also had other ideas. Long ago a teacher told me I should not study so much (!!!) since no girl wants to Mrs. Einstein. I dismissed her, the teacher, as a well meaning idiot but I also came to realize that girls were subjected to this sort of advice over and over again, at least when I was young. There are a number of other aspects to her early life that I very much recognize. Her father had a heart attack, mine had a stroke. But mostly, what I see as most important, is the realization that the circumstances of your birth do not define you. Neither she nor I have any left over baggage from childhood to gripe about, that's not the issue, but there comes a time when a kid can say "Hey, I think I could do that". Some of this has to come from the kid, but help can be helpful. Elianna, and many other teachers, seem to get this. I hope, El, you are not embarrassed by the salute, but yes, this is right.

[Elianna]

kenberg, on 2014-June-14, 07:01, said:

A question: It is my understanding that some states have "adopted Common Core", and some states have not. In practical terms, what does this mean? I see statements that Common Core does not actually impose anything on the states. This is usually meant as a rebuttal to conservative claims that the feds are taking over educational policy. Ok. But just what does it mean for a state to "adopt Common Core"?. What does it commit them to doing?

I am asking because I really have no idea of just what it means to "adopt Common Core".

Most states (I first typed "all states" but realized that I don't know that for sure) have historically had "standards" that students are expected to learn. They usually were created (somewhat) independantly, and stated things like "Students solve a quadratic equation by factoring or completing a square." However, what was in certain courses differed from state to state. I'm told that courses labelled "Algebra 1" in some states differed on whether they included quadratics! (This is big to me because in CA, quadratics are very big in Algebra 1).

One of the goals of common core was to create a common basis of terminology so that similar courses in different states cover similar material.

For Math, there are two main parts of the common core:
1) Subject specific standards
2) Common Practices

The subject specific standards are by grade level for K-8, and then in bundles (my terminology) that districts can organize as they choose. Districts can choose to continue with the "traditional" sequence of Alg 1, Geo, Alg 2 or switch to a curriculum that incorporates parts of all three over three years (called "Integrated Math", which is more like they do in other countries, I've been told).

The common practices address habits of mind and are:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

These common practices are expected to be reinforced at every level.

You can find California's version of the common core here:
http://www.cde.ca.gov/be/st/ss/

Basically, it's how California suggests braking up the main strands, and has the two suggested pathways for HS. It's rather confusing to read though. Also on that site you can scroll down to see CA's previous math standards.

[kenberg]

I clicked on the math standards for California. 162 pages. Whew. But of course that's what is needed if you are going to set standards for a full program. I think the main conclusion is that it would take some work to prepare an informed opinion. Very much it depends on interpretation and execution.

Maybe fifteen years or so ago Becky and I were part of a large effort to formulate high school standards in Maryland. These were to be minimum standards required for graduation. The group consisted primarily of high school teachers and I believe we worked very well together. These were no doubt some of the best teachers in the sate and they had very solid ideas. They also had high hopes for what could be done, and I was worried that reality might bite back. It did. As it became clear that setting such standards would result in significant challenges to get kids graduated, especially among the disadvantaged, there was some re-thinking. We were no longer involved at that point. I once asked my daughter how these standards played out with my granddaughter and she wasn't too sure just how, but said that Kathryn, the granddaughter, had taken some exam in seventh grade (age 12) that apparently took care of it. I am not particularly bragging here. My granddaughter was a decent, not spectacular, student, but so were most of her classmates. I would be very surprised if the standards had any noticeable effect on any of them.

It's a really tough problem. As I see it here, the good high schools are much better than the one I went to in the 1950s, and the bad ones are a great deal worse. The result is that standards that would challenge my granddaughter and others at her high school would totally wipe out almost everyone from some of these other schools. If there is a solution, I don't see it.The common core standards I see appear to be strong, I am interested in how they fare.