Very Challenging Math Problems

Often, when people mention difficult math, they are referring to advanced math. I would agree that there is some correlation between those two things, but they are not the same. It is possible to find examples of relatively easy math problems that rely on relatively difficult concepts, and vice versa. For example, the current New York State Integrated Algebra Regents Exam is easier than a list of the topics that it covers would suggest, because of the way that many of the multiple choice questions are structured (more on this topic in a future blog posting). On the other hand, New York Academics’ Very Challenging Math Worksheets (available at shop.tutornewyorkcity.com) are much harder than you would probably expect if you only knew that they are a collection of problems involving fractions, decimals, proportion, simple number theory and similar topics.

Doing math is more than simply learning and applying a set of algorithms. It can also involve creative thinking, estimating, organization, and persistence. A fairly simple example is this word problem: A dog has six puppies. Each dog (young or old) has 15 fleas. Each flea has six legs. How many legs are there? (*The answer is below). In theory, everyone who can multiply and add whole numbers should be able to easily solve this problem. Nevertheless, many students who are competent in the required arithmetic still get this problem wrong because they lose track of where they are in the middle, or they forget to include the mother dog, both of which are examples of how organizational skills are important in math. Sometimes less confident students panic when they see a problem like this one and resist making any attempt to solve it (an illustration of how important persistence is).

The problems included in the Very Challenging Math Worksheets are more difficult (some of them are much more difficult) than the above example. Nevertheless, the principle remains the same. For example, the next problem is a moderately difficult problem that is found on one of the Very Challenging Math Worksheets. (**The answer is below.)

The Acme Doodad Factory employs 500 people. 80% of them are factory workers who make $55,000 per year. 12% are office support staff who make $45,000 per year. % are management who make $95,000 per year. Everyone else makes $250,000 per year. What is the mean salary at Acme Doodad Factory?

Finding the solution to this problem requires an understanding of percents and mean (specifically, weighted mean). It also requires that students have the ability to stay organized and calm through a problem that has many steps and some unattractive numbers (i.e., a fractional percent). While all of these topics are theoretically covered in standard middle school math curriculums, only a relatively small number of middle school students are able to correctly solve this problem the first time they see it. (To be fair to middle schoolers, many high school students and adults aren’t much better off.)

In my experience, the main things most kids need to solve challenging, multi-step problems like this one is more guided practice. Mathematically gifted students are more likely than other children to be given problems like this, and I certainly approve of these kids being challenged, but average students also get very real benefits from exposure to challenging, non-routine math problems, too (as long as they get enough support and encouragement along the way!) Doing this type of math problem helps develop the ability to think in a logical but flexible way and organize information. It also helps to build the persistence necessary to work through situations where the path to a solution might not be immediately obvious. As a bonus, the ability to do these kinds of problems can help children get into gifted and talented programs (such as Hunter College High School, the Anderson Program, Specialized Public High Schools, and the Center for Talented Youth at Johns Hopkins) and can help adults have a wide range of career choices open to them.

*There are 388 legs in all.
** The mean salary = $57,310

Review of Barron’s SAT II: Biology E/M

When I tutor students for the SAT Subject Test in Biology (formerly the Biology SAT II), I make use of a variety of preparation books, including Barron’s How to Prepare for the SAT II: Biology E/M by Maurice Bleifeld. (This is an older book which has been replaced by Barron’s SAT Subject Test Biology E/M by Deborah Goldberg, but I still find it useful. I will review the more current book separately at a later date.) This book attempts to include a thorough review of the biology topics covered on the SAT Subject Test in Biology as well as four full-length practice tests and one mini-test. That’s a lot to pack into a single book, and while I find this book quite useful, it doesn’t quite live up to the promises it makes.

The book has specific subject reviews on topics in biology ranging from “How we Fight Disease” to “Green Plants- Basis of All Life”. These reviews are relatively short (in the range of 10-25 pages each, including review questions and answers). I’d say that these reviews contain good information, but they should in no way be considered a substitute for a real text book or a thorough biology class (despite the framing of the book, which suggests that this book might be all you need to prepare for the test). Some of the review questions at the end of each section tend to be overly specific, for instance, “Germs that enter the body in milk may cause (a) pneumonia (b) tetanus (c) mumps (d) tuberculosis (e) influenza”. A question like this seems to be more aimed at checking a student’s comprehension of the chapter, rather than actually targeting information likely to appear on the SAT Subject in Biology. On the other hand, because of the brevity of the review sections, it is inevitable that many topics are not covered thoroughly.

The main value of the Barron’s SAT II Biology preparation book is the practice tests. My students report that the actual Biology SAT II is a bit more challenging than the tests in this book, but the topics are accurate. Unlike some other practice books, this book is well-edited and the practice tests are composed of reasonable questions with appropriate answer choices and correct answer keys. (This may seem like something that should go without saying, but unfortunately, some test prep books are written very quickly and are insufficiently edited.)

Going through the practice tests in this book should be sufficient test prep for a student who is doing well in his or her biology class and will be satisfied with a good (but not fabulous) score on the SAT II Biology exam. My recommendation for very serious students who want to do extremely well on the SAT II Biology is to use this book as one of several supplements to their biology class. Specifically, I would suggest going through the tests in this book (and other SAT II Biology prep books) relatively early in your study period and then moving on to AP Biology prep books for tougher practice in the weeks leading up to the exam.

Is a Specialized HS Right For Me?

As an SHSAT tutor, my job is to help students get the best score they can on the specialized high school entrance exam; it’s not to help them decide which school is the best fit for them. Nevertheless, parents often ask me what I think about Stuyvesant, Bronx Science, and other specialized high schools. I have not done any in-depth investigations- my knowledge about these schools is limited to anecdotal feedback that I get from kids I know who attend them. But, for whatever it is worth, I have some thoughts which may be useful.

First, I think that all students and parents should know that the New York City specialized high schools are demanding and competitive. This may seem obvious, but I am often approached by parents who would like their child to take the SHSAT, even though their child is struggling with academics in middle school. This simply does not make sense. Parents, if your child is ready, willing, and able to take on a workload that is significantly heavier and harder than what most middle schools offer, a specialized public high school may be an excellent choice. Otherwise, it is not.

Second, I think that families should be aware that any large public high school is going to be significantly less personal than the average small private school (or small public school, for that matter). It’s simply a matter of scale. No matter how warm and caring an individual teacher may be, personal relationships will be more difficult to form in a school with thousands of students than in a school with two or three hundred students.

On the other hand, the very size of the larger specialized high schools means that they can offer a diversity of social opportunities and extra curriculars that smaller schools simply cannot match. For this reason, schools such as Stuyvesant, Bronx Science, and Brooklyn Tech can be especially good places for students who crave a broad array of social experiences or who feel like they have had trouble fitting in in the past.

As far as the actual quality of the education goes, I think the experience of a girl I know may be illustrative. Last year, I helped her prepare for the SHSAT and I was able to observe that she is not only smart, but also hardworking. To call her driven would not be an overstatement- she is ambitious to a degree that is unusual at any age. She got into Stuyvesant. In her freshman year, she got an excellent geometry teacher and is thriving in that class. She is also having a good experience in English (although her parents wish she did more writing), Spanish, and art. Unfortunately, her social studies class is appalling- the students are essentially asked to remember long lists of disjointed facts. Worse, her physics class has been almost nightmarish. The workload is intense, but it is not educational. Neither she nor any of her classmates are able to follow the teacher’s lessons. After a rough start, she is now getting excellent grades in physics and understanding the material, but only because she is receiving tutoring from another New York Academics tutor. Those of her classmates who can get private tutoring are doing well; unfortunately, the rest of the class is failing.

Based on my anecdotal observations, many students who attend New York City specialized public high schools have similar experiences to this girl. They get some fabulous classes, quite a few very good classes, and a smaller, but still significant, number of very bad classes. Taken as a whole, the quality of the education is still very good, but it is not without problems.

Homeschool Group Classes

The dominant image of homeschooling involves one child, or several siblings, studying with their mother in a socially isolated household. While those homeschoolers exist, they appear to be a definite minority, at least among New York City homeschoolers. One reason that most homeschoolers are far less isolated than many people assume is the popularity of small group classes.

Group classes take many forms. Some are taught by parents, others are taught by professionals; some are highly academic in nature, others are primarily intended to be social; some are traditional in their format, others are highly innovative. As a tutor who serves many homeschool families, I have come to strongly support group classes, since they provide an opportunity for a dynamic exchange of ideas among students and an affordable way for parents to ensure that their children are well educated, even in areas where the parents may not be comfortable teaching the material.

My primary personal experience with group homeschool classes is in the area of science. Science is particularly well suited to group classes, for two main reasons. First, it takes quite a bit of effort to get together the materials necessary for lab classes. As an illustration, I recently completed a unit on simple machines with a small group of upper elementary school age children. This relatively simple unit involved spring scales, three different kinds of pulley, two kinds of plastic tubing (to make Archimedes’ Screws), as well as copious amounts of odds and ends such as cardboard, rubber bands, and popsicle sticks. It took several hours of concentrated effort and about $45 to get all the requisite materials together. While this was hardly a heroic level of preparation, it is far more involved than the preparation that normally goes into preparing lessons in arithmetic. The cost/benefit analysis simply makes more sense when that effort is going into a lesson for several children rather than just one.

The more important reason why science is an ideal subject for group lessons is that many parents feel inadequately qualified to teach science. Even at the elementary school level, they may be intimidated by their lack of knowledge and their own perceived failures in the subject. Of course, as the material becomes more sophisticated, these perceived shortcomings only grow, and in fact often turn into real shortcomings. It is my belief that most educated adults could master enough science to teach their 4^th grader with the benefit of some good books, but the same can’t be said for a 10^th grader. It really does take specialized knowledge to effectively teach high school science classes well.

Just as a lack of real or perceived knowledge in science means that group classes are particularly valuable in science, other topics that many parents are uncomfortable with or lack knowledge in also make good candidates for homeschool group classes. Poetry, second languages, and more advanced mathematics are all fairly obvious candidates.

Since New York Academics has multiple teachers with different specialties, we are well-placed to offer group classes for homeschoolers in a variety of subjects including a wide range of topics and levels in science, most areas of math, Spanish, poetry, writing, and literature. We make it our policy to tailor the content of our small group homeschool classes to the needs and desires of each group that we work with (although all of our instruction is academically rigorous and secular).

How I Made My Peace with Math

I was always a dutiful student. I did those things kids are supposed to do: pay attention in class, complete homework assignments, study for tests, read. For the most part I did pretty well: I got into a gifted and talented public middle school program and went on to be accepted to Stuyvesant High School. I also got into both of the private schools I applied to, with 95% scholarships. Obviously, I was doing pretty well in all of my core subjects.

But, despite this evidence, I never felt like I was good at math. I also disliked it, largely because I never felt like I really understood what was going on, no matter what my test scores may have said.

Because I had been in a G&T program in middle school, I entered high school one year ahead of schedule in math. I struggled with geometry freshman year and fought to retain consciousness throughout sophomore algebra II and trigonometry. The thought of pre-calculus made me ill, so in my junior year I escaped to “Dr.Kolb’s math class for kids who hate math”. (Not the official course title!) It wasn’t a prestigious class, and the tests were pretty easy, but we actually did some deep mathematical thinking, at a relatively relaxed pace. It wasn’t too bad- and I even learned something about logical thinking. Nevertheless, I was hugely relieved to escape from math my senior year. I made a promise to myself that I’d never put myself through that (i.e., math) again. That’s how I ended up applying only to colleges that allowed students to major in biology without taking calculus or statistics.

My mathematical epiphany happened in the summer of 1999, just before my senior year of college. I was doing a summer scientific research internship at Emory University, and my research had gone pretty well. My advisor told me that if I could do a good statistical analysis of the research, I could probably get the work published. Wow! Did I want that! With a little help from a professor of statistics and an old textbook he recommended, I taught myself the necessary math. I understood what I was doing and why I was doing it. For the first time in my life, math felt like an ally rather than an enemy.

Since then, I have continued to have a positive relationship with math, but I still vividly remember my days of math avoidance and anxiety. This combination has proved to be an ideal background for tutoring.

When I sit down with a student who becomes desperately anxious at the thought of doing math, I’m able to be compassionate (which is critical for gaining the student’s trust) and also to see a way out of the negative feedback loop. Each student is different, but there are some repeating themes that come up more often than not when I help students who don’t like math. First, I find that it is important to acknowledge that not everyone has to love math or excel at math- but that a fundamental competency is possible for nearly everyone and will open doors in a wide variety of fields. Next, I think that it is very important to retreat to the last relevant topic where a student achieved mastery, and proceed from there (i.e., if a student is having trouble with algebraic fractions, it may be necessary to backtrack all the way to the idea of what a fraction actually is, review simple arithmetic fractions, then more complex arithmetic fractions, before finally returning to the topic of algebraic fractions). Finally, it is absolutely critical to underscore that math is not effortless- students who appear to “just know everything” almost always study more than their peers and their classmates assume they do.

It is ironic that I once vowed to avoid math at all costs, but I now make my living largely by teaching math. Not only that, but I enjoy it. I think my story is a useful one because it demonstrates that early discomfort with a subject does not have to be the end of things- there is always an opportunity for a second chance.

Review of The Art of Problem Solving

I was recently introduced to The Art of Problem Solving, which is a series of math textbooks aimed at gifted middle and high school students, especially those who are interested in math competitions. I only looked through one of the books, Introduction to Algebra, by Richard Rusczyk , and this review will be about that text only. Other books in the series delve into topics such as geometry and probability.

The Art of Problem Solving bills itself as a book for 6th to 10th grades. This evaluation is quite ambitious! It is, however, in character with the rest of the book, as ambitious is the best word I can come up with to describe the overall tone. (According to the author’s biography, he was a high level math competition champion as a child, and I think it would be fair to suggest that he wrote this book with his younger self in mind.) I would warn parents and teachers to take the pre-test provided on the website very seriously. If your student(s) can not get a perfect score on the pre-test without your help, they are not ready for this book, regardless of their age.

As an adult who is comfortable with math, I loved this book. Both the text and the problems are thoughtfully written and very interesting. The explanations provided are lucid. If time was not a constraint, I would joyfully devote an hour or two a day to methodically working through this book- it would probably take me a year or so to finish, and I have no doubt that I would learn a great deal. However, while my endorsement of this book is strong, it is also very limited and specific. So that you can understand, let me tell you a little bit about myself.

As a child, I was an insanely conscientious student. Not surprisingly, I did well in school and was placed in an accelerated math program in middle school. Nevertheless, I found no joy in math (and always had the nagging feeling that my success on tests and report cards was due to some sort of cosmic mistake rather than real achievement on my part.) In high school, I stopped pursuing math as soon as I decently was able. I never took pre-calculus, never mind calculus. I chose my college partly based on where I would be able to major in biology without taking higher math classes. Fortunately, I experienced an epiphany at the age of 21.

My epiphany was the result of a research project that I was perusing- I was researching certain aspects of ancient salt marshes, and my advisor told me that if I could successfully do a statistical analysis of my data, it could most likely be published. With that enormous inducement, I began studying elementary statistics, and with almost no instruction except from a textbook, I soon understood statistics well enough to analyze my data. My paper was published and, much more importantly, my fear of math was conquered.

Years past, and I became a tutor. I teach test preparation and science as well as math, but I spend the largest portion of my time teaching math to 8-14 year olds. (I’ve hired other tutors to teach more advanced math.) I’m very good at what I do, and I think it is in large part because I have a very thorough understanding of math through high school algebra, a genuine affection for the subject and, simultaneously, a clear memory of a time when math was not my friend.

All of this history is a roundabout way of explaining why I feel like I have a lot to learn from this book- although it starts out with basic algebra, it ends up covering topics normally reserved for pre-calculus. Furthermore, when I look at The Art of Problem Solving, Introduction to Algebra through the lens of my remembered childhood feelings about math, I see a terrifying tome. It does not gently lead the student forward, first with easy problems and then with gradually more challenging ones. Instead, it dashes ahead and dives straight into hard problems. This approach is great for a motivated, interested person with a solid background in the pre-requisites, but it could easily prove miserable, frustrating, and ultimately counter productive for students who do not meet that description.

I intend to begin using The Art of Problem Solving, Introduction to Algebra, but only with a select group of students who are already robustly successful in math and who are coming to me for enrichment. For example, I will incorporate Art of Problem Solving problems into my work preparing students for the Hunter College High School and Anderson School entrance exams.

I wish to offer one further warning about The Art of Problem Solving, Introduction to Algebra, specifically to homeschool families. If your child is ready for this textbook and eager for the challenge it presents, then that is a wonderful thing. However, if you plan on integrating your child into a school environment, you should be aware that the book does not touch on topics that are important in both middle and high school curriculums (primarily geometry and probability) and you might therefore want to provide supplementation in these topics.

Suggested Non-Fiction Reading for Advanced Students

Last year, I worked with a particularly gifted student applying to Hunter College High School (she got in, by the way). She was only 11 years old, but she had a fully adult level of reading comprehension and a desire to tackle interesting, sophisticated, and challenging material. At the same time, she was still a child and books written about certain adult experiences and emotions simply held no interest for her. Working with this girl got me started thinking about books that might interest a child or teenager while being challenging for even the most advanced students.

It’s a crime to think of excellent literature in terms of test preparation. And yet, as a tutor, I sometimes find myself framing reading in those terms. Instead of doing that here, let me just say reading literature at this level makes the reading comprehension on a test such as the SHSAT or even the SAT seem like child’s play. It is also probably the only way for a child to have a real chance at a school like Hunter College High School or The Anderson School.

My basic criterion for choosing these books is that they had to be interesting to me, contain rich vocabulary and complex thoughts, and not be sexually explicit or wantonly violent. Furthermore, they had to have something in them that I think would be interesting to many young people- for example, many of these books feature young protagonists. To be part of this particular list, the book had to be non-fiction- at a later date, I will write a similar list of recommended fiction books for advanced readers. Obviously, this list is highly idiosyncratic and far from comprehensive. The books range from moderately to extremely challenging.

Non-Fiction

Wild Swans by Jung Chang

In Wild Swans, the author tells the history of modern China through the history of her own family, in particular her grandmother, her mother, and herself. This book is emotionally difficult as well as technically difficult: it is long and complex, and it deals with some of the darkest aspects of human nature. On the other hand, it is richly informative and gripping. It also shows some people being the best that humans can be.

It seems to me that many classic novels that are often read in middle and high school (1984, Lord of the Flies, Deliverance) are popular at least in part because they encourage an exploration of society gone wrong. In my personal opinion, a flaw of many of these books (and all of the ones I just listed) is that it is too easy to dismiss the mistakes of the characters because they are not real and, ultimately, not fully convincing. Wild Swans on the other hand, paints portraits of people who do deeply evil as well as exceptionally heroic things, that are much harder to dismiss because they are actually real and because the author is able make her characters’ motivations understandable.

Uncle Tungsten: Tales of a Chemical Boyhood by Oliver Sacks

In this book, the author tells of his childhood love affair with chemistry. It takes place in England, around the time of World War II, and perhaps the most immediately astonishing part of the story is how much freer children were at that time. His parents are clearly loving and responsible, but he nevertheless conducted a wide variety of hair-raising chemistry experiments that would never be allowed today. Indeed, many of the materials that he so casually bought as a boy are now almost impossible for a private individual of any age to purchase legally.

The writing style of this book is relatively challenging, but most of the content should be easy for young people to identify with. While Oliver Sacks was no doubt an unusual boy, his excitement when he discovers something new to him or when he causes a particularly good explosion is contagious. Likewise, his emotions surrounding negative events caused by World War II are easy to relate to.

The Cannon by Natalie Angier

When I think about science books, I don’t usually think about books filled with deft word play, cultural references, and delicious prose that reads almost like poetry. And yet, that is exactly what The Cannon delivers. This book is written by a New York Times science reporter, and it is about the biggest, most important ideas in science, as defined by a wide range of working scientists. The science is clearly intended for a non-scientific, yet sophisticated, audience. I think that many students would get a real kick out of reading about what they are learning in science class, but in a far more sophisticated and artful way than any textbook has ever been written.

There is nothing simple about the writing style of this book; indeed I would go so far as to say that a big part of the tragedy of low reading comprehension levels is that people who have them will never be able to enjoy a book like this. On the other hand, this book is truly a reward for those skilled enough to enjoy it.

The Interesting Narrative by Olaudah Equiano

The title of this book does not lie; it is an exceptionally interesting narrative! Olaudah Equiano was born around 1745 in what is now Nigeria, was kidnapped when he was about ten years old, and was sold into slavery. As a slave, he served as an officer in the British Navy and then as a clerk, laborer, sailor, and even ship captain for a private owner. Eventually, he purchased his freedom. As a free man, he had an astonishingly wide range of experiences, which included working as an overseer of slaves, attempting to sail to the North Pole, having an intense religious conversion, marrying an Englishwoman, and working hard to end slavery.

The Interesting Narrative was written as part of Olaudah Equiano’s anti-slavery efforts. It is a complex book that is difficult to summarize. It is not an easy read; it is from a very different time and both the language and the concepts are somewhat different from what we are used to. However, it is also an extremely vivid, rewarding read that opens a window into a world that no longer exists.

The Last Algonquin by Theodore L. Kazimiroff

In 1924, a 12 year old boy who was studying for a Boy Scout merit badge encountered Joe Two Trees, a man who believed that he was the last living member of the Algonquin tribe. Joe Two Trees, who was living a traditional hunter-gather life and who had not regularly spoken to another human being for decades, decided to tell his life story to that boy. This did not happen in a remote, wild area- it happened in New York City. Specifically, it happened in Pelham Bay Park, a very large park in the Bronx.

The story that Joe Two Trees told is exciting, moving, and sometimes very sad. It is also a reminder that astonishingly drastic change can take place during one person’s life time. The boy he told it to was Theodore Kazimiroff, Sr and The Last Algonquin was written by his son.