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.
Monday, July 6, 2009
Monday, June 29, 2009
Number Sense
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.
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.
Monday, June 22, 2009
A Tutor's Perspective on the High School Application Process in New York City: Part II, the ISEE
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.
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.
Friday, May 15, 2009
A Favorite Math Project
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.
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.
Monday, May 11, 2009
A Tutor’s Perspective on the High School
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.
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.
Monday, May 4, 2009
What Makes Word Problems So Hard?
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.
Thursday, April 30, 2009
Welcoming a New Spanish Tutor
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!
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