A student, Melissa, in my SAT class, shared her way of thinking about a well-known problem that appeared on this blog awhile back as part of a larger investigation. You may recall the post about the ratio of the areas of circumscribed and inscribed figures.

I told her I would celebrate her method on my blog, which would definitely be viewed by less than or equal to a million people a day! One of my students asked me why I don't simply write a book about all these ideas and I replied, "I'm thinking about it." He asked what the title would be and I replied, "What I've Learned From My Students."

Here's the problem/investigation/challenge/activity///// for you or your geometry students:

Consider the diagram above. Assume that it depicts two squares. The smaller inscribed square is formed by joining the midpoints of the larger square.

(1) Explain why the area of the inscribed square is one-half of the area of the larger.

Easy so far...

(2) Consider the circle circumscribed about the smaller square, i.e., it passes through its 4 vertices. Explain why this circle is inscribed in the larger square. This requires that you show that the 4 sides of the larger square are tangent to the circle. [By the way, most students would assume this is obvious from the diagram, but ...]

Not impressed? Deja vu all over again as Yogi would say? This is how Melissa demonstrated that, for a given circle, the area of the circumscribed square is twice the area of the inscribed square. Is this the method you have seen or used yourself? Consider that she used no variables, didn't plug in particular values for the dimensions, etc. She drew the diagram and basically said that the diagram proves itself! A proof without words, so to speak (that might even make Sidney Kung proud). I congratulated this young lady and the class applauded. Interesting how students immediately recognize someone's brilliance...

## Tuesday, October 30, 2007

### The Power of Visualization: Another View of Inscribed vs. Circumscribed

Posted by Dave Marain at 6:31 PM 14 comments

Labels: circles, geometry, inscribed, ratios, spatial sense

## Saturday, October 27, 2007

### Alec Klein Interview Part III - The Final Chapter

It's been awhile but here's the final installment of the long phone interview I had with Alec a couple of weeks ago. If you need to refer back to Part I or Part II or the preview for background, click on those links. For those visiting for the first time, Alec is the author of the recently published and critically acclaimed book, A Class Apart, about Stuyvesant High School in NYC, a highly exclusive specialized public school for talented math and science students. Again, I wish to emphasize that Alec responded extemporaneously to these questions, most of which I prepared ahead of time. I enjoyed the spontaneity of the dialogue but I'm not sure I would use the phone interview again, considering the arduous task of transcribing each single word. The following is an excerpt of the last half hour of the interview. Alec's responses are reproduced as accurately as I was able to pick them up.

Key: MN: MathNotation K: Alec Klein SHS: Stuyvesant HS

I'll start by backtracking to Alec's last remarks from Part II:

K: In other societies, teachers are held in much higher regard. I'm thinking of countries like Japan for instance, and it would be nice if we could adopt a little bit more of that in this country. Unless you see them up close doing what they're doing, you don't really appreciate just what a commitment it is to be a teacher, the hours, not just teaching in the class, but after school and it's really heroic work, I think.

Beginning of Part III

MN: I recently spoke to a Korean mother the other day and I asked for her opinion about her child going to SHS and what the other mothers have been discussing. Mothers make these decisions in the Korean culture. Here was her response -- she said some of the parents really want their children to go there, they help them prepare for many years, but many now are concerned about the pressures, the cheating, the drugs, and all of the other pieces they feel are negative. Do you feel this situation has been exacerbated in the past 20 years since you went there? Do you see it as escalating?

K: Yeah, I do. I think high school has gotten a lot tougher, a lot more competitive, there's a lot more pressure. I think that being an adolescent is a lot more complicated. There's an increased emphasis on, for instance, standardized tests in today's world than when you and I went to school and, as a result, these kids are spending enormous amounts of time taking tests, preparing for those tests. It's sort of an unending battery of tests they have throughout high school and on some level that is just a fact of life that's changed and I think that beyond that it seems that it's just tougher. It seems that there are more kids vying for fewer spots, whether it's to get into SHS or to get into colleges, but it seems a lot tougher.

MN: Ok, I agree with that. One of my fellow bloggers teaches in another specialized high school in the city. I asked him for some ideas for questions I might ask you. Two of his colleagues graduated from SHS and the only question they had was this: ""Is there something specific to the culture of SHS that makes the pressure and cheating greater than other specialized high schools or is it basically the same all over?"

K: Well, I think that the pressure at SHS is pretty intense. It's hard for me to say whether it's exactly the same at Bronx Science or other schools but clearly when you create a school that only admits 3%, you're beginning right there with a population of students who are already quite motivated and whose parents, in many cases, are very involved in their lives. It's not surprising that it becomes a pressure cooker in that environment because they're all used to being the star from their middle school days, then, suddenly, they're in a high school full of stars, so the pressure can be pretty intense. I think that even the students say that the cheating problem is in large part a result of all that pressure they feel to get ahead, to stand out, to be number one, and it's an unfortunate problem they are constantly wrestling with -- how do they deal with this cheating problem. I don't think they've come up yet with the answer at SHS. I think cheating is a problem throughout schools and, on some level, I blame the parents because it starts at home. If the kids have the kind of right grounding, they will come to understand that it doesn't pay to cheat, they're not getting anything out of that. By the time they get to high school that kind of foundation should be in place.

MN: Let me talk about the pressure, Alec. Pressure comes from many sources. From my own personal experience in my life as well as in the students' lives with whom I've worked, there is intense pressure to succeed, to compete, all of those same things you're talking about. How much of this is their feelings of self-worth, and how they feel that acceptance and love from mom and dad are tied up with their grades?

K: It's a really good question. I came across several examples where the problem wasn't the student, it was the parent who was putting so much pressure on their children to be perfect. Teachers told me about receiving telephone calls and emails from parents, asking them "What happened with my child's grade?" The teacher would say, "What do you mean, they got a 97," and the parent would say, "What happened to the other three points?" I also witnessed a student who faced the burden of incredibly high parental expectations. There's one student who came home with a report card grade of 94.86, her overall grade point average, which incidentally is a pretty good grade point average, but it was slight drop from 95.29, less than half a percentage point, and she was punished, even though the drop to 94.86 didn't actually count for anything since that report card was an interim report card which didn't count toward her overall grade point average on her transcript. It was a progress report. To me, that begins to get to the question of where does this come from and where do they begin to feel that pressure. I think it begins at home. There are shades of gray here. In the case of Marina and other students, they are the products of immigrants who have come over to this country in pursuit of the American Dream and they see their children as a key to that future, through education, and so a lot is expected of them. It becomes easier to understand that pressure when you understand that, in many cases, these are folks who came to the United States with pennies in their pocket and are trying to establish a better life.

MN: I'm going to read an excerpt of Anna's review on Amazon. She graduated a couple of years ago from SHS:

"It is true that at SHS, teachers go out of their way to help and encourage their brightest students. This is apparent in the wide array of course offerings and the abundant resources and support offered to those who excel in math or science. Brilliant students are rewarded as they should be, particularly at a school like SHS. That's part of SHS's promise and to a select few it delivers. Let's not forget that this is a school equally obsessed with its own image as it is obsessed with its most talented and successful students. But what about those students who are mediocre in chemistry. There are some you know. Ok in Physics, decent in Calculus. Teachers at SHS rarely reach out to a student unless the student is at one of the two extremes, either soaring or failing. Students who receive marks in the 70's or 80's are sometimes chided but mostly ignored. Without top grades many are discouraged from pursuing higher level courses even if the student shows a strong interest that doesn't necessarily match his or her GPA. This is a school that emphasizes success not learning."

So, I read that and I feel this is a young lady that may have a little bit of a chip on her shoulder but I don't want to discount her point. Are you surprised by that or do you feel, like in most schools, the kids in the middle don't get the attention.

K: I sense that's a heartfelt response and I don't want to discount it either. You know, I think she makes some good points. Number one, that there are students who fall through the cracks at SHS. It's a big school, there are 3000 students in a 10-story building. I would say though that the book that I wrote includes those same students as well, not just the high achievers. I'm thinking for instance of Jane who was not a high achiever, at least in terms of her grades at SHS. She's obviously a gifted individual and very talented but who did fall through the cracks and suffered from a debilitating drug addiction, and whose grades were mediocre. She's one of the central figures in the book. So the point of the book was not to only highlight the Romeos of the world and Romeo was one of the most gifted students I encountered and he was one of the central figures in it as well. The idea was to show some of the range of different kinds of students at a place like this. I think that SHS is a tough place to navigate wherever you are, precisely because it's big, big in population and big in physical size, and there are all these kids jostling for attention and seeking the best teachers. It's not an easy place. I think it's a difficult place and I would agree with that. I would say that's something I tried to incorporate into the book as well.

MN: You did and I think Jane is a perfect example of that.

K: It's such a tragedy when kids like that fall through the cracks. It's unfortunate, it's a fact of life in any large high school, gifted or not gifted. You know, high school's not easy.

MN: She also talks about the teachers, Alec. She thought that the best teachers teach the best classes. She also felt the top classes were only for the top students who had to make a cutoff grade, but not for the others, so those didn't get the best teachers.

K: You know I think there's some truth to that but not completely. There were some amazing teachers who were just beginning their careers at SHS, so therefore had some of the very basic courses and they were really gifted. Then there were other teachers who were known to be great teachers, the students really vying to get into those classes. So I think it's a mixture. At SHS, like any other school, there are good teachers and bad teachers. At SHS, they draw from the same pool as the other schools in the system, so you get a pretty wide range. But it's true that to some degree the school draws some teachers who want to teach students who are willing to learn and then there are some teachers who are thinking it's going to be a vacation because the kids are so motivated to learn - they don't even need to teach them. You're going to get both. I think most of the teachers are pretty darn good.

MN: I want to get to my last couple of questions because we're coming up to an hour. Do you believe, and this is central to the whole issue of gifted education, that the prizewinners, the scientists, the leaders who've graduated from SHS would have gone as far as they have if they had not attended a specialized high school?

K: I've actually received emails and notes from people who've graduated from SHS over the past 60 or 70 years. I got one today from a graduate of 1939 and he said, and I'm quoting his email:

"I would never have gotten the Ph.D. in Physics were it not for the challenges at SHS."

It's not uncommon in that people have expressed to me the fact that their experience at SHS propelled them in a positive way and I think the achievements of a lot of the alumni, the four Nobel laureates, and many others who have been trailblazers in science, medicine, public service, industry are pretty amazing actually. I don't think it's an accident.

MN: I appreciate that. That's a great answer. Alec, do you believe that deep down the parents are more attracted to SHS by its name and prestige and increasing the chances of their child getting into the best colleges, or do you really believe that many want their children to have the challenge?

K: I think it's a little of both. I've certainly encountered parents who were drawn to the brand name that SHS represents and I think there is sort of a frenzy among the parents to get their kids into brand-name colleges like Harvard and I think a little bit of it is misplaced. I've come to the conclusion that, even though I myself went to an Ivy League, I don't think it really matters where you go to college. It's what you do after that, how you apply yourself, whether you chase your dreams. I think that a lot of the parents are caught up in that. I've also met a lot of parents who really believe their children will get the best education at SHS because the standards are high there, it's tough and rigorous and the curriculum is designed to be so and a lot of the parents are drawn to the school because of that promise.

MN: A couple of more, Alec, and thank you for staying on. Do you believe that the existence of specialized schools is more of a reflection on the quality of local public schools or that children who are uniquely talented deserve a unique education?

K: Well I think that's one of the key questions: Is it a good idea to have a school like this. As I've noted, I don't think I'm in a position to make that call. As I said, I'm a journalist, I'm the generalist who's an observer in this area and I think it's up to the educators to try to work that out. I would just add that the irony about SHS is that it was founded a century ago as a manual training school for boys, meaning that it was designed for boys to learn how to do carpentry and, in fact, I went back to the old yearbooks in the archives to the beginnings of the school, in the early 1900's and, in the yearbook it says the school was obviously not intended to prepare students for college. The idea was to help boys to use their hands. Obviously that is no longer the case! Now SHS is viewed as a vehicle to get into the very best colleges. Almost every single student goes to a 4-year college and about 1 out of 4 gets into an Ivy League school. Aside from that, it's co-ed, which is not the way it was founded. So it's sort of an indication of about how the ideas in education have changed and evolved from generation to generation. A notion that there are these specialized high schools for the gifted and talented is not really necessarily in vogue at the moment. What is in vogue is something quite different. As I understand it, there's an emphasis on schools that perhaps may have a focus on a particular subject but the admission is not usually based on a single test as it is for SHS. I think SHS continues to exist and prevail because of the success it has achieved over the decades and as long as it continues to do that it will probably stand the test of time. But it's not necessarily a popular thing now to have schools like it.

MN: Well, Alec, I know that as the journalist you don't want to take strong positions, particularly you don't see yourself as an education specialist. However, I did certainly come away from reading the book feeling the positives far outweighed the negatives. Is that a fair evaluation?

K: I think that's fair. My experience last year, working on the book, I felt that it was a good place, SHS. To go beyond that, it's a special place. But I am troubled by the practice of segregating kids by whatever is so-called measure of gifted and talented. I'm just not sure that's the way to go, but, on the other hand, I'm not really sure what the alternative is to that and I'm sure it's something that will be debated for years to come.

MN: Well, the Academy in Hackensack, NJ, does have an entrance exam which is fairly rigorous, but they also interview students I believe and they look at their record in school as well. They also have to be recommended by a teacher.

K: That's actually gotten a lot more attraction in education circles because it takes into account other factors. Some schools also factor in diversity. It sounds like that's been gaining more adherence than what what SHS does.

MN: Interestingly, I'm thinking of contacting Terence Tao. He's won the Nobel Prize in mathematics. They don't actually call it Nobel, they call it the Fields Medal. He, at the age of 8, entered high school and was taking Calculus by the time he was 10 or 11 and got his Ph.D. from Princeton by the time he was 21. He is now a full professor at UCLA in his twenties, and he's considered one of the greatest mathematical minds of this generation. I mention that, because I'm really interested in asking him if he went to a specialized high school. I believe he comes from Australia, so there might not have been a SHS there. I'm going to be following up on this, I just want to let you know.

K: I'm curious what he'd have to say.

MN: Yeah, I'm curious about his opinions about whether he believes these talented kids deserve special programs. Alec, again, I want to thank you. Of course, I wish you the best with your book. I personally enjoyed the book greatly -- it struck home with me, because of my own personal background.

K: I appreciate that. I know you understand a lot of these issues more than most people.

MN: Alec, take care. I'll be in touch...

Posted by Dave Marain at 7:55 PM 7 comments

Labels: A Class Apart, Alec Klein, book review, gifted education

## Tuesday, October 23, 2007

### And vs. Or: Developing Deeper Understanding in Algebra

While some are awaiting the remaining chapters in Alec Klein's interview, I know the rest of my readers are wondering why I've gone into early hibernation. I'm actually doing some independent mathematical research apart from anything on this blog but I will not bore you with the details at this time. That may change (i.e., I may bore you later!).

In the meantime, since the central theme of this blog has always been developing student conceptual understanding, here are a couple of problems for you to consider giving to your Algebra 2 students (or beyond). Conjunction vs. Disjunction is often misunderstood by students and these ideas appear in so many contexts in mathematics, from absolute values to inequalities and beyond. Consider giving these as warm-ups, for review, practice for SAT's, etc. I'm not suggesting these are difficult or challenging problems. Their purpose is to promote deeper reflection on the part of the student. Students who have strong background and understanding will simply solve these quickly and not see why anyone would make a big deal over them. However, you may find other students who don't grasp the ideas as readily or have forgotten. Comparing/contrasting is a powerful heuristic when trying to develop a more profound understanding of mathematics...

1. If (a -4)^{2} + (b+4)^{2} = 0, what is the least possible value of a^{2} + b^{2}?

2. If (a-4)(b+4) = 0 , what is the least possible value of a^{2} + b^{2}?

(A) 0 (B) 8 (C) 16 (D) 32 (E) 64

Ask students to explain to each other, why the word 'least' is irrelevant in Question 1 but not in Question 2. Also, how does question 1 relate to circles?

Posted by Dave Marain at 6:33 AM 12 comments

Labels: advanced algebra, algebra 2, concept, SAT-type problems

## Friday, October 19, 2007

### Alec Klein Interview Part II

Here's the continuation of the interview. If you haven't yet read it, I strongly recommend you start from Part I, particularly the insightful comments offered by our readers. I invite others who have thoughts about the issues of gifted education and schools like Stuyvesant HS to share their reflections and personal experiences.

In part II, Alec gets into issues of teacher quality and his own personal experiences at SHS. He also recounts how the book came to be and how a journalist views his role in telling a story.

Key:

MN: MathNotations

K: Alec Klein

SHS: Stuyvesant HS

We'll overlap a bit from Part I for continuity...

MN: Oh, I read all about that and I was very impressed that Mr. Jaye is a very special person.

K: He is. He's one of the great educators and he's also one of the secrets that made SHS so successful, precisely because he didn't follow the rules. Instead, he found a way to break the rules to get the students the kind of education they need. In the case of Milo, he let him in the school even though he hadn't taken the entrance exam and let him take precalculus. He hired a math genius who did not have a degree because he recognized that this individual was also a gifted teacher.

MN: Right, and now he's left and taken this guy with him to Bergen Academies.

K: That's correct and, in fact, I just saw Danny Jaye, and his school won a major award, I think it was from Intel, for being one of the most, if not the most, innovative high schools in the country. And Danny Jaye is the principal of that school now and it goes to show what happens when you put great educators in great roles.

MN: I agree. Alec, let's run down some of these questions. If you could have done some things differently, would you have reconsidered some of the chapters or do you feel you might have shifted your focus somewhat?

K: That's a good question. In any writing that I do, I always feel like there's always something I could have done better, something that I could have done differently. There's no such thing as a perfect book.

MN: Oh, of course. I mean, were there chapters that never made it to the final cut?

K: Oh no. I was going to say, having said that, I strongly believe this is the best thing I've ever written and it's the best piece of journalism that I've done. I think it probably took about seven years off my life - I worked so hard on it. It was a magical experience in the sense that I worked hard on it but everything came together in terms of the reporting and the chapters all kind of coalesced and it was just fortuitous that it happened that way. But it was actually a smooth writing process because the individuals who I spent time with and focus on in the story are just so amazing. In many cases, they're larger than life, whether it's Danny Jaye, the math chairman, who breaks all the rules or Milo, the 10-year old prodigy who is just off the charts as well as others. They were all so compelling in their story lines that it was just a joy to both interview them and to write it. In that sense it was one of the most satisfying assignments for myself.

MN: Let me tell you why I'm asking this question. I never imagined I would be doing journalism at this stage of my life. Part of the blog that I'm doing now is interviewing luminaries in mathematics, the change agents for math education as well as the mathematicians. I interviewed one of the architects of the NCTM Standards and his views are controversial. I went out of my way not to editorialize at all, in fact, no follow-up of his comments. I let them just stand out there and I allowed other responders to carry on the debate. I'm bringing this up because I think you went pretty far to remain objective in this book and your background as a journalist enabled you to do that, and I admire you for that. Was it hard for you not to take a position and express views?

K: That's a good question. Actually, it was not hard. I've been a journalist now for almost 20 years. It's almost second nature and, when I'm reporting, all I'm focusing on is the individuals whom I'm interviewing, the information I'm trying to understand, the questions I'm asking, the story line I'm trying to follow. The primary concern is the story and, frankly, I don't think anyone really cares what my opinion is and I think that's appropriate. As a journalist, I think my job is to gather a story if you will, put it out there and let readers and policymakers and educators and others decide what they will of the story. When it comes to telling a story it's important you tell the whole story because, otherwise, if you leave something out, all you do is damage the credibility of the story. So, in that case, I wanted to make sure that I told both the good and the bad about SHS and it is a good and bad story in the sense that you have, on the one hand, these great achieving students who are really amazing, but there is sort of a dark side to a place like SHS, that includes rampant cheating, parental pressure that goes to some extremes, drugs can be a problem, issues of racial segregation within the school -- these are all things I covered in the book. I also go into issues of teacher quality. There are obviously a great deal of wonderful teachers in SHS and in fact I've really come to admire teachers. I'm just so thoroughly impressed by the job that they do. Like in all schools, not all teachers at SHS are the best and I address that. So the story for it to be true and accurate needs to show both sides and I made it very clear that from the beginning of the project that my intent was to tell a true and accurate story as I saw it and not to editorialize and not to shape it the way I wanted to but to let it shape itself.

MN: But, ultimately, Alec, you made the choice to take the entrance exam and go to SHS. What drew you to that school?

K: I think, back then, when you think about it, getting into SHS was almost like a lottery ticket. If you get into SHS, you are getting a free elite public education. If you don't get into SHS, your alternative was to go to a neighborhood public high school that maybe didn't offer the kind of education you wanted, or you'd have to pay to go to private school, which, as you know, is incredibly expensive. I don't know what the going rate is these days. Then as it is today, there was a lot at stake for the parents who are trying to get their kids into schools like SHS. If they get in, they save quite an educational bill. Now, back then when I took the test to get into SHS, I'm not really sure that I gave a whole lot of thought to the alternatives. I think I probably took the test assuming I was going to get in, not because I was so confident, but I don't think I really gave it another thought -- I would take it and get in! I don't know what would have happened if I hadn't gotten into SHS. It would have been a tough question for my parents because there are a lot of schools in New York and also, for that matter, a lot of schools that need a lot of work.

MN: What about Bronx Science and Hunter and Brooklyn Tech?

K: Those are other great schools and in fact in the book I note there are a lot of other great schools throughout the country. San Francisco has Lowell HS and Virginia has Thomas Jefferson HS, which is a fantastic school, and there are some schools that are not even exam schools that are also known to be among the best like New Trier outside of Chicago. There are some really good public schools and there are really some rotten ones too.

MN: But you didn't characterize yourself as a math-science nerd in those days, in fact you're an English person, right?

K: Well the thing is that when I entered SHS I was probably better at math than I was in English. I was truly better at math, pretty quick when it came to math. But I had the good fortune of having a lot of good English teachers who really encouraged me to pursue writing. Dr. Bindman and Frank McCourt, before he became a literary phenomenon. But he was a great encouragement and so were the other English teachers I had. Unfortunately, I had some bad math teachers at SHS so it didn't inspire me to continue in that direction. Who knows what would have happened if I had had different teachers.

MN: Teachers make a difference, Alec?

K: I think they're huge difference makers and don't get the credit for it they deserve. ... we don't pay teachers what they...

MN: There's no merit system.

K: In other societies, teachers are held in much higher regard. I'm thinking of countries like Japan for instance, and it would be nice if we could adopt a little bit more of that in this country. Unless you see them up close doing what they're doing, you don't really appreciate just what a commitment it is to be a teacher, the hours, not just teaching in the class, but after school and it's really heroic work, I think.

Perhaps an appropriate place to stop for now. We have now reached the halfway point of the interview. To be continued...

Posted by Dave Marain at 10:13 AM 0 comments

Labels: Alec Klein, gifted education, interview, Stuyvesant HS

## Monday, October 15, 2007

### Educating Our Best and Brightest: Alec Klein Interview - Part I

Note: Part II is now posted.

As previewed earlier, Alec Klein, author of A Class Apart, agreed to an interview with MathNotations on Fri, 10-12-07. The interview was quite long, so I will break it up into parts. Alec gives our readers a detailed view into one of the best of the specialized high schools in our nation, Stuyvesant HS in NYC, a school so exclusive that only 3% of the students taking the entrance exam actually make it in. Several other outstanding schools are noted in his book as well.

This interview touches on the controversial issues regarding the educational needs of our nation's best and brightest math and science students. Alec raises many important questions in his book, eloquently and fair-mindedly presenting both sides of the debate. He does this while telling with sensitivity and compassion an essentially human story of exceptional students, teachers and administrators. He leaves it to the education policy makers to resolve the equity/excellence issues. The following is excerpted from our phone conversation.

"MN:" refers to questions or comments from MathNotations and "K:" refers to Alec.

"SHS" will be the abbreviation for Stuyvesant HS.

MN: I just wanted to start by thanking you. How are things going with the book and the tour and everything else?

K: It's been great. It's been received well. Quite good reviews and, from the editor, Simon and Schuster, sales are going well.

MN: And you can tell from Amazon that more and more people are writing and reading these reviews.

K: That's good to hear. It's always good to get feedback about the book and it's been overwhelmingly positive and that's encouraging.

MN: Oh, overwhelming! I do want to talk about Anna's review. I thought it was interesting, only because it's the exception to the others. But that's later.

MN: Alec, what motivated you to write this book about SHS, knowing that your main focus had been the business sector, the AOL issue?

K: I think it was a project of passion, it was something that I personally felt strongly about. I think it was a labor of love, to use the cliche. And on some level the idea for the book, the seeds of the idea, came about a few years ago when I was invited back to the old school to participate in a panel about corporate scandals which was the subject of my first book, Stealing Time. I had not been back to high school for about 20 years and I was sort of flooded by memories of the place and it also reminded me what a strange place SHS is and the fact that it's a public school basically packed with driven, high-achieving students many of whom are nerds and being a nerd is a badge of honor at Stuyvesant. It's kind of the alternate universe of high school, not your typical high school where your football captain is necessarily the popular kid. This is a school where students are actually proud of the fact that they study hard, that they do well academically and I thought that on some level it was a relatively unique school and yet it also tells the universal story of high school in the sense that many of the same issues that unfold at Stuyvesant unfold at schools across America, whether it's peer pressure or parental pressure, drugs, issues of intimacy, cheating, any number of issues. All those issues play out at SHS, so I think it has sort of this dual draw, a unique school but also universal in many ways. Apart from that, I thought it would just be a compelling narrative, a compelling story about the individuals in the school, the students and the teachers, and my hope was to be able to document that. And, while it's true that I am an investigative business reporter at the Washington Post, I like to think of myself as a writer and a journalist. I graduated from Brown University with a degree in English Literature so I'm not really sure if I was destined for business coverage exclusively. I think a good story is a good story whether it's about business or about high school or about anything else and I thought that was a good story.

MN: Alec, I've personally had the experience of teaching in some high-performing New Jersey high schools and privileged to teach both basic skills and the highest levels of advanced placement courses, computer science, calculus and so on. I'm only mentioning that because I have personally worked with a lot of students who are pretty close to the kind of students whom you're describing. Now, they may have more opportunities to be well-rounded with athletics and other extracurricular activities that SHS might not offer (although SHS does offer a lot of extracurriculars), but I did have the 'nerds' in my class and large Asian populations, representative of Bergen County where I live. My thoughts were that someone from a high-performing high school, not even a specialized school, might react to your book by thinking "What makes him feel that SHS is so special. We've got AP kids here, International Baccalaureate programs in this school, we have some of the top students in the state." Is it that Stuyvesant is more one-dimensional compared to these other high schools? What made you feel that Stuyvesant stood apart?

K: The first thing to say about this is that I think that there are gifted students throughout school systems, not just at SHS. In fact I interviewed a lot of kids who did not get into SHS, you know, who took the entrance exam and fell short. I came away from those interviews convinced, without a doubt, that those students were just as capable, just as gifted, in many cases, as the kids who did get into SHS. The difference in many cases was that students who got into SHS spent a great deal of time and resources to prepare for that test, not everyone, but a number of students. In fact, I met many students who had spent years studying for this one test, literally years. In many cases they attended private academies or went to tutoring to gear up for this one test. There's a lot of pressure to take this all-or-nothing test, but, in fact, these students did do that and, in many cases, the students interviewed who did not get into SHS, by contrast had not studied at all or they'd studied a day or a week and naturally in many cases they didn't score as well. To me the entrance exam was more a function of whether students had actually prepared for the test. It's also a function of whether the middle schools prepared the students for the test with the kind of math and English that it requires. I think that's a variable that has to be looked at because there are clearly some schools that prepare students better than others.

MN: I agree, I completely agree and the irony for me, Alec, is that I have taught in one of these academies where students go after school until late in the evening and all day Saturday, for example, to prepare to get into the Bergen Academies as well as for SATs. And I think it's remarkable that these students are willing to give up the hours to do this. At the same time, it's expensive, so there is an economic factor here.

K: Yes, I think that's true. In fact, one of the things that intrigued me about the story was the kind of undercurrent of this question about elitism in public education because SHS is a public school, it's a $150M building, one of the most expensive schools ever built, it's a privileged place to go, yet it's funded by taxpayer dollars and 3% of the students who take the test get in. So it's incredibly exclusive in that sense and so there's naturally a good question about, "Is it fair to teach students in this manner, to separate so-called gifted and talented students from the rest of the school population?" When you talk to educators and policy-makers about this, there's a good deal of debate and controversy because many of them say that by separating these high achievers from their regular schools they are depriving these school of the kinds of students that help to raise the performance of the whole school through peer pressure or peer role models. Further, what are you telling students who don't get into those schools like SHS. You're telling them "you're not good enough" when, in fact, that's not the case. And are you sending the wrong message at a time when these kids are so young and their potential is just beginning to emerge. I was interested in that question and it's explored in the book and it's also kind of pervasive in the story line. In coming up with a title for the book, A Class Apart, I was kind of playing off a couple of ideas. The idea that these students are considered the cream of the crop, that they're considered the best and the brightest and thus they've been separated into this one school. But, the title, A Class Apart, also touches on this other idea that they've been separated from other students and it creates this kind of different class if you will.

MN: Define class there, Alec. Are you talking about some socioeconomic or some psychological distinction?

K: Well, it is partly economic in the sense that those who have the resources to send their kids to these tutoring and private academies have an advantage. But class in the sense that they're also creating separate classes, they're literally creating a different track for these students who get into SHS. Having said all that, there's no question that if you spend any time at SHS, you realize very quickly that what makes the school so special is in fact those students. That they are incredibly bright, incredibly driven, gifted and talented in so many ways. It's a special place in that sense. So I think it's the sort of question I struggle with. What's the best way to educate the gifted and talented. I think of students like Milo who is profiled in the book. I spent time with him, he was 10 years old, and he was already beginning to master precalculus and I didn't take precalculus until I think I was in my senior year at the age of 17 or 18. What is the best way to educate students like Milo? If you don't give him a different track to learn the kind of math that he's prepared for and you put him in his 5th grade class, which is where he was before, he would literally cry ever day he had to go to that 5th grade class where he was not being challenged. But he was thrilled though when he went to SHS and could begin to understand and learn precalculus and I think there has to be some way to address the needs for the gifted and talented to learn. You know they are largely ignored when it comes to policy debates about education in America today. Most of the emphasis focuses on students who are struggling and I think that's a good thing. We should be focused a lot on kids who need the most but I think there should be more attention paid to the kids who are on the fast track, educationally speaking, and who can really make a difference in the future when they'll be in positions of leadership.

MN: Now why did Milo choose SHS as opposed to just accelerating in another high-performing high school? Could it be that he might have been more accepted at a place like SHS where the kids all feel unique anyway?

K: Well, Milo does fit in well at SHS because there are other prodigies like him, maybe not quite like him, but who are very advanced in many ways academically. So he clearly fits in and I don't think that would have been true at a lot of other high schools where students don't necessarily put a premium on academic achievement. But part of Milo's story is a matter of serendipity. He had a neighbor who was a babysitter when he was a little boy and that babysitter happened to be a math teacher at SHS later. That teacher suggested that Milo try taking some math classes at SHS given his incredible intellect and it just so happened that the math chairman at that time at SHS, Danny Jaye, was the kind of educator who didn't really care about the rules. You're not supposed to bring in a 10 year old who hasn't even taken the entrance exam. Danny Jaye, the math chairman, didn't care about that. He cared about the fact that there was a 10 year old who was ready for that kind of math.

MN: Oh, I read all about that and I was very impressed that Mr. Jaye is a very special person.

K: He is. He's one of the great educators and he's also one of the secrets that made SHS so successful, precisely because he didn't follow the rules. Instead, he found ways to break the rules to get the students the kind of education they need.

To be continued...

Posted by Dave Marain at 7:55 AM 12 comments

Labels: Alec Klein, gifted education, interview, Stuyvesant HS

## Sunday, October 14, 2007

### Last minute PSAT Prep... A Challenge for the rest of us!

NOTE: In addition to the PSAT/SAT practice in this post, I strongly recommend that interested readers go to the topic index in the sidebar, scroll down to SAT-Type Problems, click on this and you will pull up about 50 different posts of practice SAT questions!

While I'm working on the Alec Klein interview, here are a few problems for you or your students to work on as the PSAT fast approaches. These paired questions are generally similar to the test but some are on a higher level. The (b) part of each question should be somewhat more difficult. It generally helps, performance-wise, to practice with more challenging problems. Encourage students to visit the MATHCOUNTS home page (and many other math contest sites) to find similar challenges!

1) (a)How many 2-digit positive integers have exactly one digit equal to 4?

1) (b) How many 3-digit positive integers have exactly one digit equal to 4?

2) (a) How many 3-digit positive integers have a sum of digits equal to 3?

2) (b) How many 3-digit positive integers have a sum of digits equal to 6?

3) (a) When expanded, 10^{20} - 1 contains how many digits equal to 9?

3) (b) If k is a positive integer, 10^{2k+1} - 1 contains how many digits equal to 9? Express your answer in terms of k.

4) (a) In terms of k, what is the sum of all values of x for which (x-k)^{2} = x-k?

4) (b) If f(x) = x^{2} - 2007x, for how many integer values is f(x) less than zero?

Comments:

(i) 3(b) and 4(a) would of course be multiple-choice type questions.

(ii) A problem like 4(b) would probably not appear on the PSAT as it is more advanced. Something like it (but simpler) could definitely appear on the SAT.

Posted by Dave Marain at 10:51 AM 10 comments

Labels: PSAT, SAT-type problems

## Friday, October 12, 2007

### Stay Tuned... Interview with Alec Klein to appear in a few days

Alec Klein, author of A Class Apart, was interviewed by MathNotations at 2 PM today, 10-12-07. I am very grateful to Alec for staying on the phone with me for over an hour, answering questions about his book and commenting on some of the issues of gifted education. It will take me a few days to sort out all of Alec's responses and to have Alec read it over for accuracy.

Alec is a wonderful writer who has penned a compelling story about the students and teachers at Stuyvesant HS in NYC, a specialized school for students who have exceptional talents in math and science. Only 3% of the students pass the entrance exam, making it as competitive as any Ivy League school. Alec accurately and thoroughly presented both sides of the picture -- the positives of an extraordinary school that develops the talents of these unique students and the negatives of a pressurized environment in which some parents want to know why their children only scored a 97 on an exam. Of course, these talented students are also experiencing all of the joys of adolescence that make us look upon those years with such fondness and for which we yearn to relive each precious moment!

For more background, read the preview of this interview I posted a few days ago.

Posted by Dave Marain at 7:14 PM 2 comments

Labels: Alec Klein, gifted education, interview

## Monday, October 8, 2007

### The 25,000th Positive Odd Integer to Celebrate!

[There's a wonderful discussion in the comments regarding the challenge problem at the bottom of this post. Read tc's and mathmom's astute explanations that generalize to the combinatorial problem of placing k indistinguishable objects into n containers.]

Just a quiet acknowledgment to my readers, an expression of gratitude for helping a math blogger who was unknown before 1-2-07 to reach the 25,000th visit on October 7th. Thank you...

And for our middle schoolers and on up, here's a simple A.P. (that's arithmetic progression, not advanced placement!) problem that is designed to help students see the variety of problem-solving techniques one can employ before they reach for the calculator or plug into a formula.

Could a 6th or 7th grade student or group find a way to determine the 25,000th positive odd integer? How would the instructor guide the process?

Well, let's see...

1st.....2nd.....3rd.....4th.....5th.....

1.......3.......5.......7.......9.......

If students have tackled similar problems and are accustomed to making and analyzing tables, looking for patterns, making conjectures (forming hypotheses) and testing their ideas, perhaps some would arrive at the result. They may even surprise you with their ingenuity!

I'll share my favorite approach but don't expect students to think the same way:

Position...1st...2nd...3rd...4th...5th...25,000th...nth

Even.......2.....4.....6.....8.....10....50,000.....???

Odd........1.....3.....5.....7......9....?????......???

Now, what is the formula for the nth positive even integer? the nth positive odd integer?

Today's problem may not be sophisticated but the issue of pedagogy is never trivial, is it?

Oh, ok, I know my readers want more of a challenge to sink their teeth into. So, I'm adding the following:

The answer to the above problem is 49,999. The sum of the digits of this 5-digit positive integer is 40. Determine the number of 5-digit positive integers with this property. This combinatorial problem should keep you busy for at least a few nanoseconds!

Posted by Dave Marain at 6:01 AM 16 comments

Labels: arithmetic sequence, combinatorial math, discovery learning, investigations, middle school math, patterns

## Sunday, October 7, 2007

### Preview of Upcoming Interview with Alec Klein, Author of A Class Apart

You may recall from about six weeks ago that I reviewed a new book, A Class Apart, by Alec Klein. You may want to read that post first for some background. Here's an excerpt from that post:

...this is an honest and well-written view of one of the highest-rated high schools in the country, Stuyvesant HS, with its long tradition of excellence and famous alumnae. Stuyvesant is a selective (based on an entrance exam) school for the gifted, particularly in math and science.

Alec has agreed to an interview for MathNotations to take place sometime in the next few days. Unlike my recent interview with Prof. Lynn Steen, I plan to have an opportunity to ask some follow-up questions. Alec and I are still working on the format which might take the form of a phone conversation (and my transcribing my notes).

You may also want to read the review of the book I wrote on Amazon. The issue of gifted education is a hot topic and was recently discussed by mathmom in an excellent post referred to in Jonathan's latest edition of the CoM. Selective schools (aka magnet schools, academies) like Stuyvesant HS in NYC (the focus of the book), which require an entrance exam, have been the target of many critics who view them as elitist and overly pressurized for adolescents. I personally have mixed feelings about this and this ambiguity will emerge in my questions. Alec, an alumnus of Stuyvesant, gives an even-handed objective view of the school and the issues of specialized schools in general. Although, he doesn't claim to be an expert on education, he is an expert journalist who knows how to eloquently present issues fairly and thoroughly. Even more significant is his humanism and his compassion for students which resounds in A Class Apart.

Here's a short bio on Alec:

Alec Klein is an award-winning journalist, playwright and author. His first book, *Stealing Time: Steve Case, Jerry Levin, and the Collapse of AOL Time Warner*, was an acclaimed national bestseller that was translated into Japanese and Chinese and excerpted in Great Britain. *Stealing Time*, required reading in several college courses across the nation, was selected as one of the “Best Business Books” of 2003 by *Library Journal* and *Strategy + Business* and hailed by *The New York Times* as “vivid and harrowing ... a compelling parable of greed and power and hubris.” His second book of nonfiction, *A Class Apart: Prodigies, Pressure, and Passion Inside One of America’s Best High Schools*, was just published by Simon & Schuster, and it is being nominated for the National Book Award and the Pulitzer Prize.

I hope my readers will be patient as I work through this process. I may still post a math blog here and there of course! That's one habit I can't seem to kick!

Posted by Dave Marain at 6:28 AM 0 comments

Labels: A Class Apart, book review, gifted education, interview

## Saturday, October 6, 2007

### CoM 18th Edition -Sooo big, Sooo good!

Enjoy the 18th CoM at jd2718. Jonathan has put together a delectable sampling of over 50 outstanding posts from a wide variety of math blogs:

Research Level - 9; Other Campus (politics, teaching) - 6; HS level - 6; Recreational (any level) - 8; Elementary - 6; Math Ed in general - 5; Modeling/economics - 3; Computer Science - 4; Humor - 3; History - 2. They come from: Greater NY - 5; Midwest - 7; California - 7; New England - 4; Mid-Atlantic - 3; Unknown US - 9; Canada - 1; Britain - 5; Other Europe - 3; Australia/NZ - 2; No idea - 3

Congratulations, Jonathan, for a Marathon of Math!

Look ahead to the next edition over at Good Math, Bad Math.

Posted by Dave Marain at 3:33 PM 3 comments

Labels: carnival of mathematics

## Tuesday, October 2, 2007

### But it's not in the Standards: Finding imaginary roots, completing the square, factoring and other 'obsolete' topics...

Remember the good old days when students solved 'quadratic-type' equations? Of course, many are still doing this but it is fast becoming a lost art (and some of you may feel it should be!). It is not required in any state math standards or Achieve's, so there's no reason to mention it, right?

Below you will find a 4th-degree (quartic) polynomial equation. The rational root theorem won't help because there are no rational roots. The graphing calculator won't help because there are no real roots! Ok, maybe Mathematica and other Symbolic algebra software could do this, but who exactly programmed this?

Using substitution to rewrite certain 4th degree equations as quadratics (so-called 'biquadratic' equations) used to be covered in some Algebra 2 or advanced classes. Some of you may feel nostalgic about this. However, our challenge today is to solve this by at least TWO 'radically' different methods and then show the solutions are equivalent!

Here's your equation:

x^{4} + 3x^{2} + 4 = 0

(a) Explain, without solving, why this equation has no real roots. Should ALL students in Algebra 2 and beyond be able to answer this one?

(b) Solve, by substituting y for x^{2} and using the quadratic formula. You should eventually arrive at 4 imaginary solutions. This is the way I was taught to solve it, eons ago.

(c) Solve by completing the square and factoring. [Definitely not the first method I would have thought of way back when...]

(d) Show your results are equivalent. This may be annoying! So, which method is easier in your opinion?

(e) Any other method for finding imaginary solutions?

QUICK OPINION POLL

(1) Completing the square (not to mention factoring) is no longer an important topic and should be deemphasized in our curriculum (or omitted).

By the way, is it explicitly mentioned in your state's math standards for Gr 8-12?

(YES NO)

(2) The equation in this post has little relevance to the 21st century and Dave should be ashamed for publishing such trash. Besides, this topic is not included in the Algebra 2 Standards developed by Achieve and ADP.

(YES NO)

You've perhaps assumed that since I've been discussing and complimenting Achieve's standards and the new Algebra 2 End of Course Exam, that I would no longer advocate exposing students to this kind of traditional mechanical 'exercise.'

Well, I taught from the AP Calculus syllabus and I still made time to discuss some ideas and methods that were not 'required'! Further, who exactly will be the ones left on this planet who know how to find imaginary roots for this type of polynomial equation that has no real roots! In case you're wondering, this kind of question has traditionally been taught in Asian countries and still is! (Dave, can you document that? Sure...)

Posted by Dave Marain at 7:02 AM 20 comments

Labels: advanced algebra, algebra 2, completing the square, complex numbers, theory of equations