Tutor New York City

Very Challenging Math Problems

noreply@blogger.com (Tutor New York City) — Thu, 16 Sep 2010 11:44:00 +0000

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

noreply@blogger.com (Tutor New York City) — Sun, 11 Jul 2010 11:42:00 +0000

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?

noreply@blogger.com (Tutor New York City) — Mon, 05 Apr 2010 15:02:00 +0000

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

noreply@blogger.com (Tutor New York City) — Mon, 05 Apr 2010 14:52:00 +0000

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

noreply@blogger.com (Tutor New York City) — Wed, 23 Dec 2009 15:42:00 +0000

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

noreply@blogger.com (Tutor New York City) — Tue, 22 Dec 2009 23:08:00 +0000

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.

Conceptual Physics

noreply@blogger.com (Tutor New York City) — Fri, 11 Dec 2009 17:44:00 +0000

Traditionally, physics has been considered the most difficult science, which should only be attempted by mature students with a strong mathematical background. There is some truth to this- being comfortable with advanced math can certainly open up some areas of physics and there are topics in a solid, college-preparatory physics class that require some pretty serious number-crunching. However, there is another side to this story and a growing feeling among educators (myself included) that even elementary school students can and should study physics.

One of the strongest arguments for introducing physics at a young age is that it is the most “basic” of the sciences. In other words, it’s really not possible to have a good understanding of photosynthesis if you don’t know anything about light. Learning about light is, of course, a branch of physics.

There’s currently a “physics first” movement that is trying to upend the traditional order of high school science classes so that the sequence becomes physics in freshman year, then chemistry, and finally biology. In this model, physics is taught as a more conceptual and less calculation-based subject because younger students do not have the mathematical knowledge of older ones.

Physics first could be the best thing that ever happened to physics education, if it is done correctly. Beyond the obvious perk for physics-enthusiasts everywhere that physics would studied by far more people than ever before, it could potentially be taught much more effectively. One of the flaws in traditional physics education is that it assumes that students will learn the concepts that the mathematics illustrate by applying the math to problems. Unfortunately, this doesn’t work well most of the time. Most people need to approach physics from a conceptual point of view before they can understand the mathematics in a meaningful way. This is true even for smart, mathematically inclined people, and it is at least part of the reason that so many people learn so little in high school physics classes. In essence, the goal of physics first education is to help all students achieve a solid qualitative understanding of the physical world. Interested students can then refine that pre-existing qualitative knowledge into a quantitative understanding in a later, more advanced class.

The philosophy behind physics first doesn’t have to be restricted to educating high school freshmen. It can effectively be applied to much younger children as well. For example, I teach science to a group of nine and ten year old homeschoolers, and we have done quite a bit of physics, with very little math beyond basic arithmetic. For example, last year we explored levers and turning effect. This year we will be looking into pulleys and simple circuits. I don’t shy away from using age-appropriate math, including simple formulas, but neither do I emphasize formulas over a verbal understanding of events. These kids are hardly advanced physics students, but they have achieved significant understanding while retaining a strong interest in learning more.

Critics of a conceptual approach to physics education worry that it is merely a way to lower standards. I share their desire for the highest standards possible, and I acknowledge that a poorly executed conceptual physics program could lack rigor. As always, details matter. A good conceptual physics program should involve hands-on (and “brain-on”) experiments, extensive writing, and some math. It should not be easy. On the other hand, it shouldn’t leave conscientious, hard working students feeling like they are studying an impossible subject- and traditional physics education leaves far too many students with that feeling.

Review of Dragon Genetics

noreply@blogger.com (Tutor New York City) — Fri, 04 Dec 2009 12:50:00 +0000

Piles and piles of research show that students learn better when they are fully and actively engaged in what they are learning, rather than attempting to passively absorb information. That is why good teachers in all subjects and at all levels find ways to get students fully involved. In science classes, one of the most important tools to achieve this goal is hands-on lab work.

Unfortunately, some topics don’t lend themselves very well to classroom experimentation. Genetics is a good example. Most studies of genetics require breeding large numbers of organisms through several generations. Some teachers manage to do fruit fly, bacteria, or Arabadopisis (a species of quick growing mustard plant) breeding experiments, but the logistics are daunting. Fortunately, there are alternatives. The very best alternative that I am aware of is Dragon Genetics.

Dragon Genetics is a free, downloadable program made by Pedogicica. As the name suggests, it allows users to explore the genetics of dragons in great depth. By happy coincidence, dragons have many genetic parallels with more familiar organisms, including humans.

Dragon Genetics is admirably clear and thorough. It starts out at the very beginning, and it goes all the way through the topics covered in AP Biology. From the beginning, it explicitly links the principles of genetics with the physical structure of chromosomes, rather than following the historical path of first discussing phenotypes and patterns of inheritance and only later connecting those facts to the physical structure of DNA.

Dragon Genetics teaches about monohybrids, dihybrids, mutations, and x-linked traits, among other topics. It does not discuss epigenetics, mitochondrial inheritance, or other similarly advanced subjects.

One of the characteristics that make Dragon Genetics effective is its sense of whimsy. The graphics are not cutting-edge, but the cartoon dragons are appealing and it is fun to figure out how to breed fire-breathing animals. At least as important is the structure of Dragon Genetics. The student must actively engage with the program, and do so successfully before being allowed to move on. In essence, the student learns about genetics by performing “experiments” and interpreting the results. Carefully structured questions guide the student through the process of interpreting “experimental” data. Of course the experiments are just computer models, with pre-determined results, but they are very good models and they allow for a genuine process of inquiry to take place.

As a tutor, I have used Dragon Genetics as an independent-study tool with considerable success. Because time is often quite limited when I’m helping students prepare for exams such as the Living Environment Regents, the SAT subject test in biology, and AP Biology, I will have students use Dragon Genetics to review genetics and related topics such as meiosis on their own, thereby buying more time to spend on everything else. In my experience, students who have completed Dragon Genetics almost always demonstrate a very good understanding of the material covered.

The program could also be used in a classroom context, as long as sufficient computers are available. Dragon Genetics is broken down into eight core lessons and four optional lessons. Each of the core lessons are designed to take about 45-50 minutes, so they will fit into most class periods. The optional lessons are shorter.

As a tutor who needs to get “results”, in the form of higher test scores, I appreciate the way that Dragon Genetics teaches content. As a teacher who genuinely loves science and cares about teaching logical thinking skills and the thought processes that underlie all science, I am enthusiastic about the way that Dragon Genetics teaches critical thinking and scientific method.

Review of Brain Maths, Volume 1

noreply@blogger.com (Tutor New York City) — Mon, 07 Sep 2009 17:39:00 +0000

Brain Maths, by Tan Thoo Liang, is a supplemental math book associated with Singapore Math. It is a two volume series, and this review is about Volume 1. It is aimed at students 8 years old and older.

On the back cover, it states that “Brain Maths is a series of two volumes which have been written to help you increase your IQ score and also develop your mental flexibility.” I admit, I’m quite dubious about this book actually making a person smarter, but the problems are genuinely challenging and varied and they require flexibility in mathematical thinking. In the credits, it lists a number of previously published books of puzzles and brainteasers- of the books listed; I am most familiar with several by Martin Gardener. If you are also familiar with Martin Gardener, you have some idea of the flavor of Brain Maths. However, Brain Maths is far easier than anything written by Martin Gardener, as befits a book written for grade school aged children.

As a math tutor, I use this book in several contexts. It is useful for students who are preparing for entrance exams to gifted and talented schools, such as Hunter College High School or The Anderson School. Those exams expect children to be able to solve non-routine math problems using a wide variety of techniques, and the problems in this book promote exactly those skills. For that type of advanced student, this book is relatively easy- they can do the problems quickly, and they make good warm-ups that I then string together with more complex problems of a similar nature.

I also use this book with homeschoolers. No matter how good a math curriculum is, it is almost always flawed because the problems in the curriculum have a particular “flavor”. Students who use a curriculum can get very good at solving the flavor of problems in that curriculum, but still become flummoxed when presented with problems from a different curriculum, and therefore with a different flavor. For that reason, I always use supplemental materials including Brain Maths, books by Edward Zaccaro, Math Olympiad problems, and so on.

For students who are not particularly advanced, many of the problems in Brain Maths are challenging. I often present them as puzzles, and sometimes have appropriate manipulatives ready for the student to use.

I rarely teach math to groups, but if I did, I think that many of the problems in this book would be ideal for elementary school students to work on in small groups. They are complex enough so that they can be fruitfully approached from multiple perspectives, giving opportunities for interesting group discussion.

Although I like this book a lot, there are a few minor caveats to my recommendation. First, the book is from Singapore, so a few conventions may be slightly confusing. For example, they refer to Order of Operations by the acronym BODMAS (instead of the American PEMDAS). I’m still not quite sure what the B and the O stand for. A bigger flaw is that all of the cartoon people are either white or Asian. Finally, I want to make clear that this is not in any way a text book- rather, it is a book of supplemental math problems.

Reflections on a Reluctant Reader

noreply@blogger.com (Tutor New York City) — Fri, 04 Sep 2009 17:36:00 +0000

Of all of the work I have done as a tutor, I am perhaps most proud of the work I did with Cassandra (of course, not her real name). Cassandra was in the seventh grade when we first met, and she had always scored either a one or a two on her New York State language arts tests. (A one is the lowest possible score and a four is the highest possible score; both ones and twos indicate that a student is performing poorly.) Her parents were frustrated by the lack of progress she was making in school and decided that private English tutoring was the way to go.

This was near the beginning of my tutoring career, and at the time I had never worked with a student like Cassandra before. She was a lovely girl; behavior problems were clearly not at the root of her academic struggles. She had never been diagnosed with a learning disability, and while she may have had one, she lacked obvious symptoms such as atrocious spelling and reversed letters. Her family appeared to be warm and supportive, and they clearly valued education. I had no idea what was the root of her problem, so I didn’t have a clear idea of what to do to help her. The only think I could think of was to simply make sure she got a lot of practice reading and writing, so with some trepidation, I set about trying to make that happen. (Although we worked on both reading and writing, this essay is about the reading instruction that I gave Cassandra.)

When I asked Cassandra what she liked to read, she was unable to give me any answer. (Not surprising, since she didn’t like to read!) Therefore, I decided to just pick books for her. We started with easy books from my personal collection. I wanted to make sure that we started out with books that were actually a little bit below her reading level in hope that this would break the cycle of her having difficulty reading, feeling bad about that, and then trying to avoid reading in the future. The Fantastic Mr. Fox by Roald Dahl and The Toothpaste Millionaire by Jean Merrill are examples of early books that we read together.

I instituted a modified form of shared reading. When we were together, we would read the books aloud. I would start, read for a few pages, and then hand the book to Cassandra. Then she would read for as long as she felt comfortable, and when she was done, she would hand the book back to me. When I felt it was necessary, we would stop and discuss the book. For homework, I initially assigned her about 10 pages of reading per day.

As her reading became more proficient (and as I ran out of reasonably interesting, very simple books in my personal collection), I gradually introduced more challenging material and increased the number of pages that I expected her to read each day. I also began offering her limited choice. For example, after about three months, I would show up at a tutoring session with two books such as The Story of Madam Curie by Alice Thorne and Hatchet by Gary Paulsen that she could choose between. At this stage, I was assigning about 15 pages of reading per day.

After about six months of gradually increasing the volume and complexity of the reading that I required from Cassandra, she was up to about 30 pages a day and beginning to approach a grade-appropriate level of reading. Books that she selected from the choices I gave her included The Watsons Go to Birmingham- 1963 by Christopher Paul Curtis and several titles from “A Series of Unfortunate Events” by Lemony Snicket.

After about eight months, Cassandra really was reading proficiently and she was also becoming interested in choosing her own books, without guidance. This was when we found out how wildly our tastes diverged! She kind of liked the books I picked for her, but it turns out she loved books such as A Child Called “It”: One Child’s Courage to Survive by Dave Pelzer. I wasn’t familiar with this book, but started out eager to read it with her. That came to a swift end after I nearly threw up while reading a particularly gruesome passage. After that, we agreed to go back to reading books that she picked from a limited list that I gave her, but I searched for somewhat grittier tales. The Color of Water by James McBride is one of the books that we shared during this period. That was probably the most challenging book that we read together.

Ten months after we first met, Cassandra took her eighth grade New York State language arts test. She scored a fairly high three, and for the first time in her life was officially considered a proficient reader. Shortly after she got her test score back, her parents and I decided that my work was done, and Cassandra graduated from English tutoring. In ten months (about 40 hours of tutoring, plus many more hours of reading on her own) her reading level went up approximately 3 grade levels! I have rarely been so proud and impressed by a student’s improvement.

Since Cassandra, I have worked with a number of students with low reading levels. Although I have not always had such dramatic results, I have found that this method of gradually and consistently increasing the complexity and volume of reading while also increasing the amount of choice the student has works well. I recommend it highly.

Response to “Students Get New Assignment: Pick Books You Like”

noreply@blogger.com (Tutor New York City) — Mon, 31 Aug 2009 16:34:00 +0000

The New York Times published an article called “Students Get New Assignment: Pick Books You Like” by Motoko Rich (8/29/09) about an approach to reading instruction that reduces or even eliminates class books and replaces them with books that the students choose for themselves. In essence, this method works by allowing students class time to read each day and a relatively free rein to chose material (although teachers very well might prohibit “junk” such as “Gossip Girl”). To increase accountability, teachers monitor the number of pages that students read and require them to write about the books. Both teachers and students give reviews and recommendations to the class as a whole.

This is a thought-provoking article, which I would recommend to classroom teachers, English tutors, and parents. Ironically, I may even assign it to some of my students.

My personal feelings about the roles that free and assigned reading should play in education are complex. Part of me agrees wholeheartedly with letting students select their own books, if not from the entire domain of published works, then at least from a substantial and varied list provided by the teacher. As a tutor, it is often my job to get children to read more, and I’ve learned just how hard it is to select books for other people. Simply put, predicting what another person will like, even if you know that person intimately, is no easy task. At the same time, the only way children will read as much as they need to become truly proficient is for them to read work that they enjoy. (Have you ever noticed how excruciatingly slowly a kid will read if he or she does not like the book in question?)

As much as I like the idea of choice in reading, I have several quibbles with this approach. For one thing, I have found that students who are very reluctant readers are often even reluctant to select books. For these students, my personal experience suggests that reading teacher-selected books is an important intermediate stage for them to go through before they are ready to pick their own books.

A more fundamental qualm I have with this approach comes from me looking back at my own education. I loved reading books with my classes, and I would hate to have missed out on that experience. I would never have read most of the teacher-selected books on my own, and while I loved some and hated others, the class discussions taught me an awful lot. It is a great thing to look at a book from a wide variety of view points. A group discussion about a book the whole class has read provides this, but reading a book alone can not.

Similarly, the teacher’s input was sometimes vital to my understanding of texts, especially ones from distant times or places. For example, I read Plato’s Symposium on my own as a teenager, and I’m mildly embarrassed to admit that although I loved it, I thought it was a light, comic novel. On the other hand, with my teacher’s guidance my classmates and I were able to come to a reasonably sophisticated understanding of Antigone.

In my ideal world, English classes would share books and all students would read independently for pleasure, following their own whimsy, during free time. In reality, I realize that many, probably most, students are not choosing to fill their free time with books. With that in mind, I’m not about to criticize an approach to reading instruction which focuses on student choice.

As with so many things, I suspect the best way is the middle path. To me, it seems fairly obvious that assigned books and free-reading choices serve different purposes. Assigned reading is a chance for students to carefully analyze literature with their class, for teachers to try to stretch students’ abilities and tastes, and for children to have the experience of sharing books with their peers. Free-choice reading allows students to explore their personal interests and exercise some control over how they spend their days. It is also important to realize that if students don’t read on their own, it is very unlikely that they will become good or habitual readers. Hopefully, by requiring students to choose their own books in the classroom, teachers will be encouraging them to pick up books outside the classroom, too.

Review of Available SHSAT Practice Tests

noreply@blogger.com (Tutor New York City) — Tue, 18 Aug 2009 14:23:00 +0000

The most essential tool any student needs to study for the SHSAT is high quality practice tests. There are quite a few sample SHSATs available, but they were not all created equal. Some are easier than the actual exam and a few are riddled with errors. Others are very high quality. There are four brands of SHSAT practice book readily available in most large New York City bookstores and several brands of test available online. The gold standard for practice tests are the two tests that are written by New York City and given to every student who registers for the SHSAT. In this article, I will review each one of these resources.

Kaplan has an SHSAT book that is published under at least two separate names. One is Kaplan New York City Specialized High School Admissions Test and another is Kaplan SHSAT Advanced Prep. A spot check suggested that these two books are identical. The Kaplan book has two full length practice tests and is written at an appropriate level of difficulty. Unfortunately, it also has quite a few errors. In particular, many of the scrambled paragraphs are poorly constructed and don’t necessarily have a single correct answer. If you use this book, I would recommend skipping the scrambled paragraphs and only using it to study for the other portions of the test.

The Peterson’s/Arco book is also published under more than one cover. It is even more confusing than the Kaplan book because it sometimes seems to be published by Peterson’s and other times seems to be published by Arco. Although the covers are quite different, the interior of the books appears to be identical. The title is Master the New York City Specialized High School Admissions Test and it is a clean book, with few errors. Unfortunately, it is not as challenging as it should be. Also, it doesn’t have a single full-length test. It does have 10 half-length tests, which make good supplemental study materials, but I do not recommend using them as a primary study tool. (As an aside, the Peterson’s/Arco book also contains practice material for the Hunter College High School admissions test. These tests are of very low quality- they do not accurately reflect the content of the Hunter test.)

The Princeton Review book, called Cracking the New York City Specialized High School Admissions Test, has only one full length test. The test is fairly good, but a student who seriously wants to raise his or her score will certainly need more materials. A large portion of this book consists of test taking tips and techniques. While the techniques themselves are solid (and in line with the techniques suggested in other publications) I find the tone unnecessarily snarky.

Of all the books available in bookstores, my favorite is the Barron’s, called Barron’s New York City SHSAT: Specialized High School Admissions Test. It has four mini-tests (half the length of the actual exam) and three full length tests. It is also the only book with a ninth grade mathematics supplement, which is a valuable tool for students seeking admission to the 10th grade. I like the Barron’s book because it is substantial, challenging, and contains few errors.

In addition to the tests available in bookstores, there are materials available online. Peterson’s has made three mini-tests available on line, but they are also included in the print book. Unless you can’t get to a bookstore and are in a big hurry, buying the physical book makes much more sense.

New York Academics, a tutoring agency which I own, has SHST tests available online. As of the fall of 2009, we are offering two full length, downloadable tests, with more planned for the future. These tests have been painstakingly written and edited to make sure that they accurately reflect the actual SHSAT in content and difficulty. I may be biased, but I don’t think any commercially available test surpasses the ones that New York Academics offers.

Another company that sells practice tests online is called PrepMate. They offer a package of 10 tests, which they deliver by mail. Since I am not familiar with this particular company, I was hesitant to give them my credit card information, so I sent them an email inquiring about the possibility of arranging a face-to-face purchase. Unfortunately, they did not respond. For this reason, I can not comment directly on the quality of their materials.

I hesitate to include the publication SHSAT Exam Secrets in this review, because it does not contain any practice tests. No full length tests, no half length tests, nothing! Please, don’t bother with this publication.

Of course, the gold standard for SHSAT practice tests is the tests that the New York City Department of Education gives to students who have registered for the exam. Since these tests are made by the people who make the real test, they are the most accurate representation of what the test is actually like. Better yet, they are free. (As far as I know, these are the only free tests available.) All students who plan to take the SHSAT should take these practice tests.

Review of The Cryptoclub

noreply@blogger.com (Tutor New York City) — Tue, 11 Aug 2009 14:03:00 +0000

One of the holy grails of education is making math fun, without reducing its rigor. Many people attempt this task, but few succeed. I believe that Jannet Beissinger and Vera Pless are among the rare success stories with their book The Cryptoclub, Using Mathematics to Make and Break Secret Codes. This book follows a group of fictional middle-school children who form a code-breaking club. They start out with historically ancient and rather simple codes, and work their way through progressively more modern, difficult, and mathematically rigorous codes.

The Cryptoclub is not a full math curriculum that could be used on its own - there are many math concepts that children should learn in middle school that it does not touch on. Rather, it is a fabulous supplement to more traditional curriculums. Pattern recognition, probability, prime numbers, exponents, modular arithmetic, factoring, organizing data, and inverses are among the topics that the book explores in a meaningful and interesting way. Even more important, it gives young (or maybe not so young!) students an opportunity to practice rigorous critical thinking in a meaningful context. Code cracking is a great example of open-ended problems that require tenacity, organization, and creative thinking to solve.

The skills that this book teaches are similar to the skills on certain admissions tests. In particular, I use this book with my students who are preparing to take the Hunter College High School and Anderson School entrance exams. These tests may or may not contain actual code-cracking problems, but they always require students to tackle long, non-routine math problems. Another reason I like to use this book with Hunter College High School and Anderson School prep students is that both schools are astronomically difficult to get into (they are literally more selective than Harvard). With that in mind, it is important to me that my students get something valuable from their tutoring sessions whether they are admitted to the school of their choice or not. I believe that this book helps me meet that goal.

No book is perfect, and as much as I like The Cryptoclub, it does have its flaws. The story that links the characters various adventures in decoding is not very strong and a bit cheesy. It works well with 6th graders, but if I were using it with 8th graders or high school students I can imagine them being alienated by the book’s tone. I think it could still work, if a teacher acknowledged the kitsch, but it would require skill on the teacher’s part.

Another difficulty with the book is finding a place to use it. Although the book itself is relatively slim (just under 200 pages), working through the whole thing would be enormously time consuming. Even working through just a few chapters would take more time than most math teachers could spare. I can imagine it working well as the backbone of an after-school club or an elective class. One day, I hope to use it with a group of homeschoolers in the context of a supplemental math class. It is also possible for a motivated, intelligent student to use it on his or her own, although I suspect that few kids would get far with that approach.

The Cryptoclub is a textbook with extensive problems for kids to work through. There is also an accompanying workbook, which I don’t own. The workbook is a convenience rather than a necessity, as it simply repeats the questions from the textbook in a format that is more convenient for students to work with.

Actually Understanding Physics

noreply@blogger.com (Tutor New York City) — Sat, 01 Aug 2009 18:57:00 +0000

My scientific background is primarily in biology, with a good dollop of chemistry and a fair bit of geology thrown in. Physics was never really part of the mix for me, so when I had the opportunity to take a couple of conceptual physics classes in graduate school as part of my teacher training, I was pretty excited.

One of those classes was particularly rewarding. It was primarily a study of naked-eye astronomy (i.e., the Sun and Moon). In this class, we did some just plain awesome things - most notably, we taught ourselves to tell time by looking at the moon. Most of my classmates were like me - science teachers (or future science teachers), but not physics teachers. Few of us had any significant knowledge of physics going into the course. One student, however, was different. He was a graduate student in the physics department, not the education department.

This fellow - I’ll call him John - gave me some insights into certain problems with traditional science education in general, and physics education in particular. I expected him to have a real edge in solving problems and understanding scenarios, but he did not. Without formulas to use and calculations to perform, he seemed lost. John did not have a better grasp of the big ideas in physics than the rest of the class. The only area where he showed superior performance was in mathematical ability. I don’t want to underestimate the importance of math, but being able to solve equations while being unable to explain the ideas in words seems a little like being a dancer who is physically fit enough to perform breathtaking stunts, but who lacks musicality and expressiveness.

Now, I am aware that there is an excellent chance that John had been assigned to that particular introductory-level class because his professors realized that he had an unusually poor understanding of conceptual physics. Nevertheless, I think he is symptomatic of a larger problem. Too often, science classes are structured in a way that emphasizes memorization over reasoning. These classes may appear to be rigorous, but instead of promoting high achievement, they often quash interest in science and most of their victims do not retain significant knowledge after the final exam is over. Because of the focus on details of problem solving, it is possible to do well in such a class without actually seeing the big picture.

In physics, there is currently a movement to correct this problem by teaching the subject first in a conceptual way, with relatively little mathematics. The goal of conceptual physics is not to eliminate math but rather to use it as a tool in the service of ideas. In this model, all students should leave high school with the ability to discuss Newtonian physics verbally and make accurate, qualitative predictions about the physical world. Students with a desire to deepen their knowledge of physics can then go on to take more math intensive, quantitative physics courses.

Given my strongly positive experience with conceptual physics, I am obviously a proponent of this approach. Indeed, I would like to see other sciences, especially chemistry, take note of this trend. If you would like to learn more about conceptual physics, I recommend Conceptual Physics for Everyone by Paul Hewitt and Physics by Inquiry by Lillian McDermott.

Oh, and John? By the end of that class, he was more proficient at solving equations, and able to discuss physics in words, too.

A Hard-Headed Argument for More Art Education

noreply@blogger.com (Tutor New York City) — Tue, 28 Jul 2009 13:53:00 +0000

As a math tutor, I often ask my students to sketch out problems. There’s nothing unusual about this. It is a common problem solving technique. Unfortunately, when students have absolutely no drawing skills, they often become so bogged down in making what should be a simple sketch of a cube, for example, that they lose sight of the bigger picture. The effect is similar to when a student becomes so absorbed by arithmetic calculations that should be automatic that he or she forgets the original purpose of those calculations.

It’s quite simple to solve this problem. Even a one semester drawing course that covers perspective would give them enough skills to make adequate math sketches. I don’t mind covering basic techniques in perspective, but it is a sad irony when schooling is so focused on math and English that students’ only art education comes from their math tutor.

Biology Regents Exam

noreply@blogger.com (Tutor New York City) — Wed, 22 Jul 2009 13:40:00 +0000

The Biology Regents Exam, officially known as the Living Environment Regents Exam, is one of four science regents available to New York State students who are pursuing a regents diploma. The test itself consists of a multiple choice portion and a free-response portion. Although there isn’t a lab component, students must have completed at least 20 hours of lab work to be eligible to sit for the exam.

Students normally take the Biology Regents after one year of high school biology. Not surprisingly, the test covers the topics one would expect to find in an introductory, survey course in biology. Cell structure, genetics, evolution, ecology, and classification all make an appearance. Experimental design and data analysis are also important parts of the exam.

Merely passing the biology regents requires a grade of 65 and isn’t very difficult. In fact, it can be done with shockingly little actual knowledge of biology because a significant portion of the test consists of reading comprehension questions and questions that test students’ ability to create and understand graphs and charts. However, because of the comprehensive nature of the exam, doing very well does indicate that a student has attained a meaningful level of proficiency.

If a student fails the Biology Regents (or appears to be in danger of failure), there is usually an underlying problem that needs to be addressed, above and beyond that student’s knowledge of biology. Study habits, English language proficiency, basic literacy, learning disabilities, and commitment to school are possible problems to be considered. Possibly, the teacher’s performance should be looked into, as well.

One often-cited reason for not passing the biology regents is that the student was not allowed to take the test because he or she lacked sufficient lab hours. Unless the student was seriously ill during the school year, this almost certainly indicates a chronic problem with cutting class. If you are a parent and your child tells you that he or she wasn’t allowed to take the Biology Regents, you should look into the situation very carefully.

If your child is struggling in his or her biology class, it is important to find and address the cause sooner, rather than later. Parents should start by talking to their child about the class. Talking with the teacher, either in person or by phone, is usually the best second step. If the problem has behavioral roots (i.e., not paying attention in class, cutting class, or not doing homework) there is a lot that parents can do to help. Asking your child about the class every day, looking at his or her notebook and homework regularly, and staying in touch with the teacher are all useful. Setting clear goals and consequences is also a good idea.

An example would be something like, “If you want to go out with your friends on Saturday, I need to see you spend at least 30 minutes every weekday evening on this class.” I prefer goals that relate to effort rather than results because a student who is used to failure may feel that good results are impossible to achieve, and therefore not try.

If the root of the problem is academic, then it is important to get appropriate help for the student. For example, if your child is having trouble in biology class because of a low reading level, you might want to look into after-school tutoring. Ideally, that tutoring should address literacy and biology together. In New York City, public schools should have at least some help available after school. Private tutoring can also be very helpful.

Sometimes, the problem is not with the child, it is with the class. Perhaps the teacher is not very competent or other students are so troubled that even the best teacher would be hard-pressed to do much more than classroom management. In either case, it is a sad situation, but a motivated student with reasonable skills could still look after his or her own interests and do well on the regents (and maybe even learn some biology).

If a class is useless, I recommend either a course of self-directed study, or study overseen by a tutor. As an aside, let me say that it is my opinion that self-directed study is sometimes needlessly dismissed. It is not always realistic, but it can work- I know this from personal experience. As an eighth grader, I recognized that my teacher was not competent and took charge of preparing myself for the Earth Science Regents. I passed with a score in the high 80s (with no adult help). For students who wish to study on their own, I recommend they start by first reading the textbook and answering the study questions included in it. Next, if they can talk about what they are learning with their parents, that would be very helpful. Finding relevant documentaries to watch and perhaps popular science books to read is also a nice supplement. In the month or two leading up to the exam, studying old regents exams is a very good idea.

Of course, this type of self-study is appropriate only for highly motivated students who are starting out with very solid academic skills. For other students, private tutoring can make all the difference. If you are seeking a tutor for this exam, you should look for someone with excellent references and a strong background in biology. After one or two lessons, your child should feel like he or she is really starting to learn a lot.

When studying for the Biology Regents Exam, it is important to remember that passing the test (while important) is not the main point. The main point should be to acquire a basic knowledge of life science, and the test provides a framework for people working towards this goal.

Percentile Rank: A Case of Unfortunate Naming?

noreply@blogger.com (Tutor New York City) — Mon, 06 Jul 2009 15:28:00 +0000

On a fairly regular basis, I’m faced with a parent who wants help interpreting some standardized test scores. Often, the test scores are in the form of a percentile rank. Many people know what percentile rank means, but in case you’re not entirely sure, it is a measure of how well someone has done in comparison to the rest of their group. For instance, if a child achieves a percentile rank of 60 on a certain portion of the ERB or ISEE, it means that he or she has done better than 60% of his or her peers. It does not indicate that the child got 60% of the questions correct. In fact, it doesn’t say anything at all about the percent of questions that the child answered correctly.

Most of the parents I work with know quite well what percentile rank means. But, they often act on the numbers as if they are percentages on a classroom test. By this I mean that they will look at a ranking of 75 percentile, as compared to independent school students (generally, an elite group) and act as if that’s an inferior score.

I suspect many parents make a subconscious connection between percentile rank and percent and therefore don’t see a 75 for what it is because that number would normally indicate something unsatisfactory.

In a way, this phenomenon is good for me as a tutor- when parents worry, I get new clients. Nevertheless, I’m not happy about it. It results in unrealistic expectations that cause both students and parents great anxiety. After all, it’s not possible for very many people to be in the 95th percentile, nor is it a measure of failure to be below it.

I believe that appropriate changes in the way scores are reported could go a long way towards alleviating unnecessary anxiety (without lowering standards). One approach might be to eliminate percentile rank and replace it with a more fortuitously named ranking system such as stanines. Stanines are a method of breaking down scores into nine groups, with the top group having a rank of nine and the bottom group having a rank of one. Stanines give roughly the same information as percentile rank and are much harder to confuse with percentage. One disadvantage is that few people understand stanines, so although the ERBs and ISEEs already report scores as stanines (in addition to reporting them as percentiles), most people overlook them. Another disadvantage of stanines is that because the groups are relatively large and all stanines are integers, two scores in a single stanine are sometimes farther apart than two scores in adjacent stanines.

Alternatively, actual percentages could be added to score reports. This would have the effect of shifting the focus to the individual student’s performance and away from comparison with the group. Of course, when it comes to school admissions, comparison with the group is most important from a practical point of view, so a percentage would not be a sensible replacement for percentile rank, merely an addition to add context.
Neither one of the suggestions I make is fully satisfactory, although I think they move in the right direction. I’m not a statistician and therefore don’t really have the tools to come up with the best way to report scores. My goal here is to point out a problem and hopefully open a discussion.

Number Sense

noreply@blogger.com (Tutor New York City) — Mon, 29 Jun 2009 18:21:00 +0000

Some people are just better with numbers than other people. The people who are better with numbers have something that math teachers sometimes refer to as number sense. It’s like common sense, but for math.

Your friend who can reliably estimate which brand of peanut butter is the least expensive even though they come in differently sized jars probably has good number sense. Likewise, the fifth grader who makes a calculation error on a test but then notices that the answer doesn’t look right. Number sense is useful throughout life and generally makes all things mathematical much easier. The question is, how can we help children (or teenagers, or adults) develop number sense?

In the most basic terms, people who think more deeply about math are more likely to develop and intuitive sense of how numbers act. But that just begs the question of how we can encourage children (who are often reluctant) to think deeply about math? One method is to expose them to a wide variety of problems, especially problems that are genuinely interesting.

Here’s an example. Pick a number from 1 to 9. Now multiply that number by 3. Finally, multiply it by 37,037,037. What happens? Why? (Can you figure it out? Hint: what is 37,373,737 times 3?) Thinking about problems like this, and especially talking about them can be a great way to develop number sense. You can find more interesting math problems like this by doing an internet search for terms such as “math magic” and “surprising math”.

Certain games are good ways to develop number sense, too. “24” comes to mind. “24” is really a family of games designed specifically to hone mathematical thinking, and they’re well worth checking out. I’ve had good luck getting even reluctant learners to enthusiastically participate. Other games might not have been designed with math in mind, but they rely heavily on probability and therefore can provide great exposure. Examples of this type of game include tabletop strategy games made by the Games Workshop or Wizards of the Coast. An older, better known game that can also serve this purpose is Risk.

Math problems that have a serious bearing on real life rather than the silly, unrealistic problems that most students are usually exposed to can be stimulating too, especially for older kids. Studying compounding interest and the way credit cards, bank accounts and other financial instruments work come to mind as an example. Edward Zaccaro’s book 25 Real Life Math Investigations is another rich source for this type of problem.

A related strategy is to study the mathematics of interesting physical phenomenon. For instance, if you have the opportunity to build and launch rockets with kids, it’s possible to use all sorts of math in exciting ways (combined with physics, of course). For example, you can use algebraic formulas to calculate how high you expect the rocket to fly and trigonometry to actually measure the path it takes. The topic you investigate doesn’t need to be as spectacular as rocketry- there are a wide variety of amenable subjects (primarily from physics). Using the density formula while designing model boats is one of my favorite projects using math.

Developing a strong number sense isn’t quick or easy, but it also isn’t magic. Anyone can do it with practice and it is certainly worthwhile. Done right, it might even be fun.

A Tutor's Perspective on the High School Application Process in New York City: Part II, the ISEE

noreply@blogger.com (Tutor New York City) — Mon, 22 Jun 2009 15:14:00 +0000

I’ve been a tutor in New York City since 2004, and since I specialize in working with middle school age children, I have become quite familiar with the high school application process. There are many categories of high school and the application process is by no means the same for each category- because of this, the whole process can sometimes feel even more complex and draining than applying to college. This essay will be about applying to New York City Independent Schools, with particular focus on standardized testing. Other essays will address applying to Specialized Public High Schools and Selective, Non-Specialized Public Schools.

Applying to private school at the middle or high school levels is a lot like applying to college. Students submit transcripts, teacher recommendations, and an essay. They list their extracurricular activities and go on interviews. Instead of taking the SAT or ACT, they take the SSAT (or, much more commonly) the ISEE. Every portion of the application process counts and different schools grant different weights to the various aspects of an application. Perhaps the most difficult part of the process is recognizing that in addition to all of the elements that families can control, there is an element of randomness and luck as well.

It’s a stressful process, for sure. Try to remember that just as children need schools, schools need students. Do your best, but try not to let it take over your life. Also, remember that going to the right school is important, but there is no one right school, and what a student makes of his or her education is at least as important as the school he or she attends.

As a tutor, I am far more familiar with the ISEE portion of middle and high school applications than I am with any other part of the process. Fortunately, many of the long-term strategies that lead to a good score on the ISEE also promote good grades, strong teacher recommendations, and well-written essays. (Hint: If you’re reading this essay, there’s a pretty good chance that you want to know what to do about the ISEE now. Skip to the end for short-term suggestions.)

Long-Term Strategies

The ISEE rewards students who are avid readers. Most obviously, kids who read a lot do much better on the reading comprehension part of the test. No matter what any testing company or private tutor may tell you, there is simply no way to compensate for the hundreds of books reluctant readers haven’t read compared to their enthusiastically reading counterparts. Of course, certain test taking techniques can help book-avoiding students, but they can never fully bridge the gap. In addition to reading comprehension, the test contains straight-ahead vocabulary questions, many of which are quite sophisticated. Reading is certainly not the only way to develop vocabulary, but it is probably the single most consistantly effective way.

But what if your kid isn’t a reader? There are several approaches you can try to encourage reading. First, of course, it helps to read to children extensively when they are little and have them read to you when they learn how. What many people don’t realize is that older children often enjoy reading with their parents almost as much as pre-schoolers do. Even if your child is in middle school, you may be able to start a habit of reading together as a family. It’s also very helpful for children to see their parents reading regularly. “Do as I say, not as I do” is rarely effective.

Nothing can replace reading, but other activities can support vocabulary development and the acquisition of background knowledge, both of which are important components to reading comprehension generally and success on the ISEE particularly. Activities to consider include watching more sophisticated films, listening to books on tape, and partaking of the many theatrical offerings we’re lucky enough to be surrounded by in New York City. I can’t prove it, but my instincts tell me that acting in school plays and being on a debate team are also beneficial activities.

The ISEE also rewards students who are thoroughly comfortable with math. Obviously, doing well in math class in school is critical. Students who pay attention in class, ask questions, study, and most especially think about what they are learning do far better on the ISEE than students who don’t. Some math curriculums give students insufficient practice working with thought provoking, non-routine problems. If you suspect your child’s school is using this type of curriculum, joining a math team is a really good idea, if it is at all possible. Other students don’t get enough practice with basic calculations. If you suspect this is the case, I would suggest putting the calculator away for a good long while, no matter what the teacher allows.

Outside of school, it’s a good idea to involve children in the math that you do on a daily basis. If you go to a restaurant, tell your kid what percent you want to tip, and have him or her do the calculations. If you go shopping, look for sales and have your child calculate the actual price of items. If you’re comfortable with it, get your child to balance your checkbook for you.

One important aspect of doing well on the ISEE, and in academics more generally, is to deal with small problems before they become big ones. If you sense that your child is struggling in a particular subject, take action before the situation becomes dire. Talk to your child’s teacher- maybe you’ll find out that extra help is available. If extra help isn’t available or doesn’t seem to be doing the trick, consider private tutoring. Sometimes, even a short course of tutoring can get a student back on track.

Short-Term Strategies

There are three basic ways to prepare specifically for the ISEE: independent study, group classes, or private tutoring. Some families combine one or more of these methods while other families choose not to do specific test preparation at all. (Not doing any specific test prep is actually a perfectly valid strategy for students who are very strong academically and proficient test takers. I would not recommend it for any other type of student.) Each method of preparation has strengths and weaknesses.

When a student prepares through independent study, he or she simply gets one or more of the commercially available ISEE test prep books and works through it, perhaps with a bit of help from a parent or older sibling. One great advantage of this method is the cost, which is extremely minimal. Another advantage is that if the student succeeds in getting a good score, he or she can take full ownership of that accomplishment. Obviously, independent study only works for highly motivated students. Perhaps less obviously, they also have to have a strong skill set to draw on. This method will not work for kids who don’t already have command of the fundamental academic skills the ISEE tests. If you want to try independent preparation, I suggest starting early so that you have time to move on to a plan B, if necessary. One final note on independent study- it should be relatively easy to evaluate how well your child is doing simply by examining their sample test results. However, in my experience the vocabulary sections in the commercial prep books are very difficult for almost all students. Therefore, I wouldn’t necessarily worry too much if your child is having trouble on that one section.

In general, group classes are my least favorite way to study for the ISEE. There are many places you can go to take a group class, and some are obviously better than others. Unfortunately, the big players don’t seem to get very good results and they do seem to take a big chunk out of their students’ quality of life. Sitting through wearisome 3-hour classes with a bunch of other kids, slogging through huge piles of homework, and getting little personal attention is drill and kill in the worst sense. It’s mind-numbing and not particularly educational. It’s cheaper than private tutoring, but I would call most group classes a false economy.

On the other hand, there are a few group classes that are actually quite good. You should look for small groups (no more than 8 students per teacher) and individual classes that are a reasonable length (perhaps 1.5 hours). The teachers should be experienced and should be able to produce excellent references. Homework and practice tests should obviously be part of the program, but you should not feel that a class takes over your child’s life or your family’s life.

For most students, private tutoring will be the most effective option. The individual attention makes a big difference when working on challenging and potentially tedious material. It’s more efficient, because a good tutor focuses on exactly what an individual student needs and it’s easier to remain motivated when accountability is provided by one-on-one lessons. Unfortunately, tutoring can be quite expensive. Rates vary widely, but $85-$150 per hour is the general range you can expect to pay for an experienced, effective tutor in New York City. Tutoring is expensive largely because tutors must spend a great deal of time traveling between appointments and you are paying for their travel and lesson planning time as well as the time they actually spend with your child.

If one-on-one tutoring is prohibitively expensive for you but would otherwise be your first choice, there are a few strategies you can try to lower your rate. If you know another family that lives very near to you who also wants tutoring, you might consider looking for a tutor together. If you can arrange to have lessons back-to-back, with only 5 minutes or so of transportation time in between, you’ll very likely be able to arrange a discount. Likewise, semi-private lessons (with two or perhaps three students and one tutor) can be much more affordable and still very effective. Most tutors don’t advertise semi-private lessons, but if you ask, you’ll find that many tutors are amenable.

A Favorite Math Project

noreply@blogger.com (Tutor New York City) — Fri, 15 May 2009 22:01:00 +0000

A number of years ago, I had a student who was assigned an algebra project called “Grid Co.” I don’t know if her teacher invented it or got it from someone else. I don’t even know the teacher’s name, so I am unfortunately unable to give proper credit. Regardless of where it comes from, it is one of my favorite math projects for students studying algebra.

The premise of Grid Co. is that you (the student) are a consultant hired by Grid Co., the nation’s leading supplier of fine grids for industrial and home use. (Kids always want to know what the grids are for- sometimes I have them come up with a fun use as a small creative writing extension to the project.) The people at Grid Co. want a rapid, accurate way for their sales staff to quote prices when customers call up to purchase custom grids, and it’s your job to make that possible.

Of course, what I am really looking for here is a student-made formula that can correctly spit out a price when you insert information on the dimensions of a grid. It’s a great way to help demystify formulas, which often seem like magic.

I break down Grid Co. into two portions: 2D grids and 3D grids. The 2D grids are considerably easier, so I always start with them. To give you a better idea of how I structure the project, I have included an example of the worksheet I give students at the end of this article.

Some students are pretty much able to take the assignment and run with it. However, most students need some guidance. One way I provide this is by giving students data collection tables. I find this provides focus and gets them thinking about what details are important and what details are not. (Sample data collection tables are also included at the end of this article.)

It may seem obvious, but it is critical that students actually build models and count the parts carefully. Some students run into trouble early by trying to cut corners when collecting data. For 2D grids, I usually have students just draw grids out on paper. Using dot paper (like graph paper, but with dots instead of lines) makes drawing grids easier, but it is a luxury, not a necessity. If kids want to, I’ll let them build actual 2D grids, but it does take more time than using drawings. For 3D grids, I always have students build physical models. A very wide variety of materials can be used for building grids- use what is convenient or what will appeal to your particular students. In the past, I’ve had particularly good luck with toothpicks and stale prunes (fresh prunes are a little too soft). A grid built with toothpicks and large gumdrops was a little less stable, but far prettier. If they’re available, commercial model building sets can be nice, and they won’t attract ants.

Students can almost always come up with portions of the formula on their own. For example, it quickly becomes apparent that every 2D grid contains the same number of 2 hole connectors (i.e., four- one at each corner of the grid). From there, it’s easy to see that simply multiplying the price of a 2 hole connector by 4 will obtain the total cost of the 2 hole connectors. On the other hand, figuring out how to model the number of 4 hole connectors is significantly more difficult. I let students mull it over for quite a while- usually I let them toy with the problem over the course of several days. At first, I give virtually no clues beyond the data collection chart, but after a while, I will gradually start suggesting ways to look for patterns in the data (interesting, but they don’t usually get the algorithm from this). Then, I will start helping them look at the grid from a more functional point of view. In other words, I’ll ask questions such as “Why is the number of 3 hole connectors on a given side always fewer than the length and width as measured in beams?” and “How does the number of 4 hole connectors in a row relate to the number of beams in that row?” After a while, this type of leading question will lead to the breakthroughs that students need to finally crack the algorithm.

Finally, a note for classroom teachers. I am a tutor and I also teach small groups of homeschoolers. My student who originally clued me into this project went to a small, exclusive independent school. My point is that I don’t need to deal with classroom management issues, and neither did that teacher. If I were teaching in a traditional classroom, I would structure this project somewhat differently because in its current form, the students have to deal with a lot of frustration. In the environments in which I currently work, I can manage that frustration, and I think it is educationally valuable for students to sometimes really struggle (especially when they ultimately succeed and end up with a result that really wows the people they show it to). However, in the classrooms I used to teach in, that level of frustration could easily have led to a classroom management disaster. I would still tackle this sort of project, but to head off a crisis, I would probably break the project into smaller bites and announce in advance that clues would be given out at certain pre-determined times.

P.S.- I know I haven’t given the algorithm here- I’m confident that you can figure it out if you try!

Student Handouts

Grid Co.

The Situation

You have been hired as a consultant by Grid Co., the country’s leading manufacturer of fine grids. They have hired you because they are having a difficult time quickly and efficiently quoting prices when potential customers call. Your job is to construct an algorithm that will allow any employee at Grid Co. to quickly, easily, and accurately tell a customer how much a grid of any given dimension will cost.

The Grids

Grid Co. sells both 2- and 3-D grids. In the first part of the project, you will develop an algorithm for pricing 2-D grids and in the second part of the project you will expand the algorithm for 3-D grids.

A two dimensional grid consists of connectors which may have 2, 3, or 4 holes. Because it is expensive to drill the precise holes that Grid Co. prides itself on, the connectors with more holes are more expensive. Grids also have beams which are uniformly priced.

Price List

2-hole connector:………. $0.20
3-hole connector:………. $0.30
4-hole connector:………. $0.40
5-hole connector:………. $0.50
6-hole connector:………. $0.60
beam:……….....…………... $0.15

Requirements

1. An algorithm (in the form of an algebraic expression) that can be used to calculate the price of any 2-D grid. You may use as many variables as you feel is appropriate, but you must define all of your terms.

2. A data table showing how many of each type of grid component is required to build various grids. Your table should show a minimum of 10 different examples.

3. An instructional manual for employees of Grid Co. explaining simply and clearly how to use the algorithm to compute the cost of any 2-D grid.

A Tutor’s Perspective on the High School

noreply@blogger.com (Tutor New York City) — Mon, 11 May 2009 22:41:00 +0000

Application Process in New York City:
Part I, the SHSAT

I’ve been a tutor in New York City since 2004, and since I specialize in working with middle school age children, I have become quite familiar with the high school application process. There are many categories of high school and the application process is by no means the same for each category- because of this, the whole process can sometimes feel even more complex and draining than applying to college. This essay will be about applying to New York City Specialized Public Schools. Other essays will address applying to Independent Schools and Selective, Non-Specialized Public Schools.

The most straightforward of the selective schools to apply to are the Specialized Public High Schools. As I write this, there are nine Specialized High Schools that base admissions on the SHSAT. Three of them are the old, storied behemoths: Stuyvesant, Brooklyn Tech, and Bronx Science. Six more are newer and (mostly) smaller: The Brooklyn Latin School, The High School for Mathematics, Science, and Engineering at City College, The High School for American Studies at Lehman College, Queens High School for the Sciences at York College, and Staten Island Technical High School. There are only two considerations for admission to each of theses schools: New York City residency and SHSAT scores.

Applying is simple, but getting in is hard. The SHSAT is a 2 ½ hours long, multiple choice test with a math section and an English section. The math section bears a distinct resemblance to the math section of the SAT. Granted, it only assumes an introductory knowledge of algebra, but the “flavor” is the same. The English section of the test is more unusual. In addition to challenging (but run-of-the-mill) reading comprehension passages and questions, there are also logical reasoning questions and scrambled paragraphs that students must unscramble. As far as I know, the scrambled paragraphs are unique among standardized tests.

Not surprisingly, the best way to prepare for this test is to be a good student. Kids who get in are virtually always kids who pay attention in class, do their homework thoughtfully, and study. Reading far beyond school assignments is also a significant predictor of success. No matter what anyone may tell you, no prep course, no prep materials, and no tutor can fully compensate if these factors are not already in place. (Beyond which, a student who is not academically inclined and interested in working hard is unlikely to be happy in a specialized high school, even if he or she did manage to get accepted.) Unfortunately, being a good student is not enough. It’s an unfair, even tragic fact that many middle schools are not rigorous enough to give their students a fair shot at doing well on the SHSAT. I will address some long-term strategies for students going to sub-standard middle schools at the end of this article. If your child is already doing all of the right things, and going to a rigorous school, there are a number of steps you can take to further improve your child’s chances of doing well on the SHSAT.

Your child can:

•Study on his or her own, using commercially available prep books;
•Take a prep class;
•Study with a tutor;
•Or, a student can do some combination of the above.

Each one of these study methods has its pros and cons. I recommend that parents
and children look at the options together, and make decisions about how to prepare as a family.

Independent study is the cheapest way to prepare, by an enormous margin. All that is needed is a few test prep books- they’re not expensive and they can even be borrowed for free from a library. For highly motivated students who have a strong academic foundation, this can be an effective way to study. I’d recommend selecting one prep book (Barron’s is my favorite) and working through it, from beginning to end. You’ll be able to gauge your progress and decide whether you are on track to meet your goals. Ideally, you would start this process in the spring or early summer before the test so that you have plenty of time and can add in other study methods if they are indicated. When self-guided study is effective, it’s wonderful how students can really take full ownership of their success.

Taking a group class to prepare for the SHSAT is generally my least favorite option. There are many places you can go to take a group class, and some are obviously better than others. Unfortunately, the big players don’t seem to get very good results and they do seem to take a big chunk out of their students’ quality of life. Sitting through wearisome 3-hour classes with a bunch of other kids, slogging through huge piles of homework, and getting little personal attention is drill and kill in the worst sense. It’s mind-numbing and not particularly educational. It’s cheaper than private tutoring, but I would call most group classes a false economy.

On the other hand, there are a few group classes that are actually quite good. You should look for small groups (no more than 8 students per teacher) and individual classes that are a reasonable length (perhaps 1.5 hours). The teachers should be experienced and should be able to produce excellent references. Homework and practice tests should obviously be part of the program, but you should not feel that a class takes over your child’s life or your family’s life.

For most students, private tutoring will be the most effective option. The individual attention makes a big difference when working on challenging and potentially tedious material. It’s more efficient, because a good tutor focuses on exactly what an individual student needs and it’s easier to remain motivated when accountability is provided by one-on-one lessons. Unfortunately, tutoring can be quite expensive. Rates vary widely, but $85-$150 per hour is the general range you can expect to pay for an experienced, effective tutor. Tutoring is so expensive largely because tutors must spend a great deal of time traveling between appointments and you are paying for their travel and lesson planning time as well as the time they actually spend with your child.

If one-on-one tutoring is prohibitively expensive for you but would otherwise be your first choice, there are a few strategies you can try to lower your rate. If you know another family that lives very near to you who also wants tutoring, you might consider looking for a tutor together. If you can arrange to have lessons back-to-back, with only 5 minutes or so of transportation time in between, you’ll very likely be able to arrange a discount. Likewise, semi-private lessons (with two or perhaps three students and one tutor) and be much more affordable and still very effective. Most tutors don’t advertise semi-private lessons, but if you ask, you’ll find that many tutors are amenable.

A note on what to do if your middle schooler goes to an academically weak school:

If your child is stuck in a school that leaves a lot to be desired, you can do several things to ameliorate the situation. Of course, switching to a better school is an ideal option, but that is not always possible. Assuming that switching schools isn’t realistic, I strongly recommend that you make sure that your child is getting supplemental enrichment. Exactly what you do will obviously depending on your budget, time constraints, and interests, but you should start as soon after realizing that there is a problem at school as possible. The list that follows is not exhaustive, but it will give you a place to start.

•Read. No matter where your child goes to school, it’s important for him or her to read independently. This becomes extra-important if the school is poor. For middle schoolers, a book a week is a reasonable rule of thumb. If your child doesn’t like to read, read together. Let your child choose his or her own books, and don’t be judgmental about them (unless you feel a particular book is morally unacceptable).

•Do math. Do actual math, not just test prep materials. If school math is severely lacking, consider working through a curriculum or enrichment materials at home. I very much like all of Edward Zaccaro’s books- they are challenging and thought provoking, with good explanations for home study.

•Go to cultural events. Museums, theater, concerts, walking tours, poetry slams, and book readings can all be fun, cheap, and enriching. They are opportunities for exposure to literature, history, art, and science, all of which add to the store of background knowledge which is critically important for effective reading comprehension.

•Take up a hobby. There are a variety of hobbies that provide opportunities to use math and reading in meaningful, concrete ways. Consider robotics, model railroading, building radios, or working in a community garden. You may want to look into joining a club where you and your child can meet more experienced hobbyists and become part of a community.

•Take classes. It is sometimes possible to take classes that will help fill in the gaps that a weak formal education can leave. Be careful though, that these classes are thought provoking and useful rather than just a series of drills.

What Makes Word Problems So Hard?

noreply@blogger.com (Tutor New York City) — Mon, 04 May 2009 21:32:00 +0000

It’s an old story- kids have been having trouble with word problems for a long time, probably since word problems were invented. Elementary school children have trouble with word problems and high school seniors do, too. This problem plagues public schools, private schools, and parochial schools. It’s found in wealthy areas and poor areas. It seems to be practically universal. Why is this?

Sometimes it’s very easy to see why a particular student is having trouble with a word problem. Let’s invent a 4th grader named Bob. Bob likes getting good grades and praise from his parents and teacher, but he doesn’t much care for homework. He’s got better stuff to do! So when it’s time for him to sit down and do his homework, he rushes through it as fast as he can. He skims the word problems, looking for the key words that his teacher taught him. Let’s look over his shoulder for a moment and see the problem he’s working on:

Sally is making lunch for 4 people. She wants each person to get two sandwiches and 3 cookies. What is the difference between the number of slices of bread and the number of cookies she must have?

Ahah! He sees the word “difference” and knows there must be subtraction in the problem. Easy. 4-3=1. Moving on to the next one..

And so Bob gets the problem wrong.

It was probably easy for you to see that Bob got his problem wrong because he didn’t read the problem carefully enough. Of course, as any parent knows, just saying “Do your work more carefully!” is unlikely to get results. As a tutor, I have a variety of techniques to force students to be more careful. One of my favorites for elementary school children is to have them draw pictures of the problems that they get wrong or have trouble understanding. In this case, after drawing a picture of the situation, Bob would almost certainly understand his mistake and how to correct it.

Or maybe the situation is a little more complex. Let’s look at a fictional 10th grader, Lisa. Lisa is a conscientious student, but she never especially liked math and as the years have gone on, she’s become less and less confident in her skills. Her class is reviewing rates and she’s working on this problem:

If Curly can paint a fence in 8 hours and Moe can paint the same fence in 4 hours, how long would it take them to paint the fence if they worked together?

Lisa thinks for a minute, and then writes down 6 hours, since this seems kind of like a situation where an average would be appropriate. She’s a little bothered by this answer, but she’s used to not feeling very sure about math so she just shrugs off the feeling and moves on to the next problem.

As a tutor, when I see this type of mistake I speak to my students about the importance of estimating- and then paying attention their estimates. Lisa’s key error here (and it is a very common one) was that she didn’t trust her own judgment. If you think about this problem for a moment, it’s probably not hard for you to see that the Curly and Moe should finish painting the fence together more quickly than either of them could working alone. So immediately, it becomes clear that the answer must be less than 4 hours.
(If you’re having trouble solving this problem here’s a hint - What fraction of the fence could Curly paint in one hour? What fraction could Moe paint in one hour? What fraction of the fence could they paint together in one hour?)

There is no single solution when a student struggles with word problems. Undoubtedly, that’s a big part of what makes this such a challenging area to teach. It’s also the reason why tutoring can be particularly effective in this area- students really do benefit from individualized attention when there is no one-size-fits-all solution.

Welcoming a New Spanish Tutor

noreply@blogger.com (Tutor New York City) — Thu, 30 Apr 2009 21:21:00 +0000

Leaning a second language can be hard - I know this from my own personal struggles to become bilingual. The right teacher can make the task more approachable, enjoyable, and rewarding as well as making the student more successful. The wrong teacher can make learning another language feel downright impossible. The best teachers possess fluency in the language they are teaching, a solid understanding of grammar, detailed knowledge of educational techniques, good humor, and boundless patience. That’s a lot to ask for. People with all of this in abundance are rare - which is why I feel so lucky to have found one of them.

She’s an experienced teacher, a Fulbright Scholar, and incredibly enthusiastic about learning and teaching. Our new Spanish tutor is Katie Gordon, and we’re very excited to have her!

Review of "The Last Slice of Rainbow"

noreply@blogger.com (Tutor New York City) — Sun, 26 Apr 2009 17:59:00 +0000

As an English tutor, I am regularly faced with the problem of assessing a student’s reading skills when I am also meeting her or him for the first time. This is a difficult task, because I know little or nothing about the student, yet an effective assessment relies on using materials that are near the student’s actual reading level. A further complication is that I want to start out with something enjoyable, because it is important for me to make a good first impression on my students.

The short stories collected in The Last Slice of Rainbow by Joan Aiken have turned out to be some of the best assessment tools I’ve found, especially for middle schoolers (the age group I work with most frequently).

These stories have the excellent quality of being able to be read on multiple levels. Taken literally, the tales are simple and almost fairy-tale like. For the more sophisticated reader, however, the stories are filled with thought-provoking ideas that can be teased out and discussed. Likewise with the vocabulary: on the whole, the vocabulary is quite basic, but the stories are peppered with enough unusual words to allow me to assess a student.

None of this would work if the stories weren’t really good. I’ve had my copy of The Last Slice of Rainbow since I was a child, and it is one of the few books from my childhood that I have elected to hold on to through multiple moves and bouts of clearing out my apartment. The stories are charming, but not saccharine; sweet, but not gentle; fairy-tale like, but not predictable. The protagonists are boys and girls with problems, hopes, flaws, and successes that are easy to relate to, but magical enough to be an escape from everyday life. Also, they are short enough to fit comfortably into a one-hour lesson.

The Last Slice of Rainbow is a book that I heartily recommend to students and teachers alike. Frankly, if adults were to read it, most of them would enjoy it, too.