Wednesday, December 16, 2009

On the Necessity of Ability-based Grouping in Mixed-Ability Classrooms

I have always believed that ability-based grouping is beneficial, but I never thought that failing to do so could be tremendously detrimental for students. As readers of this blog know, I have been helping my daughter's second grade by teaching math once a week to a small group of gifted kids. A few weeks ago, the homeroom teacher and I decided to try the supplementary material with the entire class. It is our opinion that the experiment was a complete failure despite every effort to choose problems carefully. I believe the reasons are the same reason that cause large, mixed ability classrooms to fail the majority of students.

The normal bell curve, or some variation thereof, has a funny way of manifesting itself despite loud proclamations by skeptics that ability-based grouping hurts education in general. My observations so far this year in my daughter's classroom have matched academic theory so closely as to be shocking! There are a few very smart kids, a few disengaged due to boredom, many of average ability, and a few clearly requiring special attention. Math is best learned by doing it. Hence, I always design my lessons as sets of carefully chosen problems. The idea behind my approach is to explain a few key concepts at the beginning of each lesson and then let the students learn the material by discovering the math while working through the problems. Unfortunately, this approach -- which is optimal for teaching mathematics -- fails miserably in a mixed-ability setting because the distribution of skill sets quickly interferes with everybody's learning process. The failure arises because learning speed and comprehension varies widely. Some kids cruise through the materials, while others work on it at the expected pace. Finally, a few struggle to the point where it becomes clear they do not even understand one-digit addition. My method calls for students to work independently, raising their hands when they get stuck. Unfortunately, many get stuck simultaneously in completely different places of the problem set. This puts such high, simultaneous demand on the teacher's attention, that no kid really benefits much from the session. Is my teaching method to blame, or is the organization of the classroom the real problem? I believe the latter is.

Today, my daughter's home room teacher and I decided to cluster kids according to ability starting in January. We will split the 24 students into three groups: one requiring additional help, one capable of average achievement, and one whose members are "gifted" learners. I write "gifted" in quotation marks, but my daughter is the only one in the class who has been for IQ or subject-specific aptitude. All we know is that the "gifted" group seems to learn faster, finish problems earlier, and work farther into the problems than the rest of the class. The key observation here is that we are grouping after assessing the kids. We did spend a ton of money getting everyone tested. I will report back in a few months on how this new experiment turns out. I am optimistic because things worked out very well earlier this trimester when I only taught the "gifted" group. Hence, I expect few problems when we come back from the holidays because I also have experience and have been successful with other levels of achievement.

One thing bothers me more than anything else. Why don't public schools group based on ability? There is no incremental cost over a mixed-ability setting. Take my daughter's school as a example. There are six second grade classrooms. This means that each could hold 16.7% of the second grade student body. Why is it so hard to dedicate 1 classroom for students with 130+ IQ and one for pupils requiring additional attention? Wouldn't these two populations perform better in classrooms designed to meet their intellectual needs. Bear in mind that this is a problem affecting not only gifted children but also those with modest intellect. I am not advocating a room for highly gifted kids (i.e. 145+). I am merely asking that we group together students of similar intellectual capabilities. Why is it so hard to realize that kids requiring additional help should be taught using appropriate methods and curriculum? The same goes for gifted kids. One argument against my idea is that it costs a lot of money to test for aptitude or IQ. Recall what I wrote above? We are grouping according to ability in my daughter's class without spending a small fortune on testing. The problem with the cost argument is that students are much better behaved and require less supervision then they are properly matched to the curriculum and the teaching methods (i.e. larger classrooms) and they are properly challenged. Hence, re-organizing schools around ability-based grouping could save money. In fact, small schools could be merged into larger campuses with five or six classrooms per grade, eliminating redundancies and providing a population big enough to exhibit clear distribution of abilities along a significant portion of the spectrum. It makes little sense -- other than political -- to ignore mountains of sound academic research as well as the clear evidence in our children's own classrooms.

Wednesday, November 11, 2009

Don't Think Your School Is Good Enough, Get Involved

Many parents believe their public schools are not good enough. They complaint incessantly about the deteriorating state of public eduction. The rants are warranted in many cases, but it is my experience that few parents do anything about it. By this I mean that very few donate money and time. My daughter's public elementary is one of the best in California. It ranks consistently in the top 5%. It has earned the designations of school of advanced studies and distinguished school, but it still is too slow to meet my daughter' s intellectual needs. There are many potential solutions to this problem, but most involve after-school, enrichment activities. Other potential solutions center on advocacy, but this tends to be confrontational and ineffective. As a result, homeschooling is growing in popularity. While I believe that homeschooling is ideal for some children, others like my daughter are extremely social and crave close, daily interaction with other children. This leaves parents like myself with two options: fight the system or make it work for you. I have opted for the second option, by getting involved in my daughter's classroom, and the results have been surprisingly positive.

LAUSD is crumbling due to California's budget crisis. Teaching assistants have been eliminated in many grades, and class sizes have increased markedly. Teachers are more overworked than ever and even have to clean their own classrooms because LAUSD has cut janitorial staff. I offered my daughter's homeroom teacher to teach the weekly computer class. She accepted gladly because my technical background is extensive, and kids like working with me. Computer lab worked out well. I taught the kids about graphs, data analysis, logic puzzles, simulations, and other fun topics. Things well so well that the teacher asked me to pull out a group of gifted learners for a weekly math class. The experience has been extremely rewarding. Over the past three weeks, my group has learned binary arithmetic and how it relates and compares to decimal arithmetic, various topics involving the platonic solids -- including Euler's formula --, and the concept of measure in 1D, 2D, and 3D. I am planning to introduce them to graphs (i.e. the discrete mathematical kind) and their relationship to the wire frames of the platonic solids. This will lead into a discussion of how to represent various problems in terms of graphs. I am excited because the kids have been enjoying our sessions and are always ready to work on problems and discover things on their own.

The moral of this little story is that you can always make a direct difference in your child's education by getting involved. Most parents complain, but they rarely get invest enough energy to make a difference. Getting involved means giving as much money and time as reasonable. Failing to do so is perpetuating the very problems you complain about. I am not certain what the next few years hold, but this one is turning out much better than expected.

Monday, October 12, 2009

The Power of Math Circles

I recently discovered the wonderful world of math circles, and the activity has quickly become a favorite of my daughter. I approached the UCLA math circle with the idea of having Paulina do interesting math with other kids equally interested in the subject. I was also looking for a chance to step back from the role of parent/teacher and let her experience collaborative math without having my looking over her shoulder. I got the schedule wrong for the first meeting, but she managed to spend 30 mins working on cool introductory problems. I wanted Paulina to experience the joy of solving problems and working with other people, but I promised myself that I would not force her to return if she did not want to. To my surprise, Paulina said that the circle was fun and that she wanted to keep going. She now has attended five sessions, and she even fell asleep holding her book of problems the night of the second session. It appears that the math circle may provide Paulina with a source of intellectual challenge, excitement, as well as a place to enjoy math with other like-minded kids.

Why do I like the math circle? The answer is that the it approaches the subject very differently from the boring, tradition-bound, mind-numbing torture imposed on kids all across the US. The last few sessions have centered around binary numbers (i.e. base 2). This is a very simple concept for middle and high school kids, but few six, seven, and eight year olds have ever imagined that one could do arithmetic using anything but the decimals. The circle's leader -- a UCLA professor with kids in the circle -- started the afternoon by asking if anybody knew a way to represented numbers in any way other than using decimal notation. A few kids offered interesting, albeit impractical solutions, and one of them said that you could just use as many sticks as needed, counting each once. The instructor then suggested Roman numerals. The kids worked on writing Roman numbers and ended the discussion by figuring out how to correct the following equation by moving just one stick:

II + III = VI.

Clearly, the point of Roman numerals was to teach that numbers can be represented in different ways. As such, binary numbers were introduced, and this is where the teaching got clever. Instead of using powers of two, the instructor wrote the following sequence on the blackboard: 1, 2, 4, 8, 16, 32, 64... Every kid understood that you get the next number by doubling the preceding one. The instructor then asked the kids to imagine that each number represents a weight and to figure out how to balance an object using only those weights, assuming that each weight could be used only once. One way to think about it is to say that you are trying to balance a given weight with counterweights of 1, 2, 4, 8, ... It actually helps to draw a scale and ask the student to draw the counter weights required to balance both sides.

The beautiful thing about the balance / weight metaphor for binary is that it allows very young kids to understand binary without resorting to exponents and other more advanced material, and binary is, in my opinion, a great way to teach the basic principles of arithmetic.

Binary is just one of the topics taught at the UCLA Math Circle. Challenging word problems, principles of algebra, and other "advanced" material are introduced weekly to kids as young as six. The sad part of this is that most of the participants only get to enjoy fun math once a week. If it was up to me, math class would be abolished, and math circles would become the norm.

Sunday, September 13, 2009

Using Adaptive Programs to Teach Math

It is my position that math education must be as individualized as possible. However, it is impossible to do this in classrooms with 15 to 30 kids. This is where adaptive computer software fits in. My daughter has been using Stanford University's Education Program for Gifted Youth (EPGY) since December 1, 2008. Things went well with math , so we enrolled her in the English track this past May. The results have been outstanding, and I believe many other children could benefit from the system. In fact, I am convinced that a slightly slower pace would make EPGY suitable for kids at many different levels of cognitive development.

EPGY's math program -- at least the highly gifted version -- is probably a bit too fast for normal kids. It aims to cover K through 6th at a recommended rate of 2 grade levels per 2.5 quarters (approximately 7 to 8 months or nearly a full school year). My daughter has moved much faster because numbers and logic are one of her strengths. There are three elementary school courses:
  • K through 2nd grade
  • 3rd / 4th grade
  • 5th / 6th grade
However, EPGY continues through advanced undergraduate mathematics. This could allow an accelerated learner to go through the bulk of an undergraduate math degree before entering college.

Acceleration, adaptability, and early introduction of "advanced" concepts are key features of EPGY. The strong emphasis on acceleration is supported by decades of research showing that gifted children learn faster and make deeper abstract connections than the population within two standard deviations from the mean IQ. Adopting an accelerated curriculum need not lead to early college admission, but it provides a way out of monotonous concepts such as arithmetic so more abstract topics can be tackled early and in great depth. In practice, acceleration works well with gifted children they require fewer drills than normal children. This opens access to increasingly sophisticated material earlier than would be possible otherwise.

Adaptability is single, biggest reason why I love EPGY and why systems like it should be become part of the mainstream education system. The online platform tracks progress across six different strands:
  • Number Sense: Integers
  • Number Sense: Decimals and Fractions
  • Geometry
  • Logic and Reasoning
  • Measurement
  • Data/Statistics/Probability
A student could be at different levels in each strand. However, all strands must "graduate"to the next grade simultaneously. For instance, if a student completes the statistics strand one month before the others, the system increases the number of non-statistical questions until all strands make it to the next grade. The system adapts the difficulty and number of questions in a given topic according to the student's progress. Hence, every single student moves at a different speed and through a unique set of problems. The system's goal is to ensure comprehension and proficiency in as little time as possible. I find that the system works best when the student works with an adult to review difficult material. Additional explanations, exercises, and discussion can reinforce the learning process so the student moves as quickly as possible through the material. The most beautiful part of the EPGY system is that it requires zero homework. Because teaching happens through short lectures and the material is learned via exercises, there is no need for homework. Learning happens by doing. EPGY offers virutal classrooms as well as access to and feedback/guidance from very capable instructors. If EPGY were integrated into traditional classrooms, very little homework would be required. This would free up valuable time for reading, playing, and just relaxing.

I love EPGY's early introduction of "advanced" concepts. Ideas such as variables, equations, positive/negative numbers, proper survey design methodologies (i.e. avoiding leading questions, etc.), and statistical concepts surface as early as first and second grade. Early introduction could eliminate the shock suffered by many middle school students when confronted by these topics. Proper teaching techniques allow young children to understand what these things mean and how to use them. By the time second grade ends, variables and simple linear equations are second nature to EPGy students. I don't believe that the concept of variables is any more abstract that multiplication itself, but I have very little data to support my hypothesis (i.e. my daughter is my entire population). However, I believe that many children could handle at least some of the concepts if taught using appropriate techniques. Regardless of one's position on the early introduction of advanced concepts, some should be presented as early as possible. Some students will not understand what is going on, but many are likely to benefit greatly.

It turns out that I am not the only one who thinks that adaptive software has a place in "traditional" classrooms. EPGY conducted studies in California (click here for PDF of study) to determine the effectiveness of EPGY's variables as a predictor of performance on CST (i.e. California Standards Test). Clearly, the purpose of the study was to determine if there was a statistically significant correlation between performance in EPGY and CST. However, it was very instructive to see the impact the program had on Title I students. The bottom line is that the overall population sample benefited greatly. Furthermore, because EPGY maps into California standards, there would few, if any, legal repercussions if a school adopted EPGY. Finally, EPGY offers a school-wide option to use EPGY, as well as grants and financial aid for students with modest resources. This all means that there is little reason to avoid using computerized, adaptive systems, and EPGY is an excellent option.

I would advice parents of gifted children to look into EPGY (click here for the program's website). If money is an issue, apply for financial aid. Some homeschooling charters like the Sky Mountain Academy give you as much as $3,000 toward curriculum materials, and EPGY is one of the approved curriculum providers.

As always, I hope you find this useful.

Saturday, September 12, 2009

Having the Talk at Age 6!

We just had THE TALK with Paulina. She is only six, and this definitely happened with her earlier than it did with me. I remember getting a great book from my parents when I was eight. I read it intently. I asked a lot of questions, but I was two years older than her when it happened to me. Paulina has been asking questions about babies for a long time. I think she was about three when she asked the first serious questions. We were very factual in describing the various systems of the human body. She especially loved the cardio-pulmonary system and, as she termed it, the "baby system." The questions got more and more probing until we simply could not avoid answering them precisely. Hence, we got an age-appropriate book and had the talk.

"The talk" went better than expected. There were a few of the giggles triggered by talking about sex with children and teenagers. There was very little interest on the intercourse section -- which will almost certainly change in a few years. Paulina was fascinated by the fact that the sperm and egg carry the DNA that defines a baby. She has known about DNA for quite some time and that half of the material comes from each parent, but it was a revelation for her that the sperm and the egg are the vessels.

I always wondered when we would have the talk with Paulina. It happened sooner than I expected, but things rarely happen when you want them to. It went better than I imagined it would, but I think things may get a bit more interesting when other kids in her class start talking about sex. In any case, she is the youngest second grader in her school, and it makes sense to help her prepare to handle the situation by teaching her facts and helping her understand what they all mean.

In case somebody is looking for a book appropriate for young kids, we bought

What's the Big Secret by Laurie Krasny Brown and Marc Brown.

Friday, September 11, 2009

Want to Skip Grades in LAUSD, Be Nice to Your Principal and Homeroom Teacher!

Our local public school agreed to skip my daughter from K to 2nd grade. I did not push for 3rd grade because she needs a transition year for her writing to catch up to her radical acceleration in other subjects. I estimate that it would take her approximately five or six months to catch up to third grade spelling expectations. Given her perfectionist tendencies, skipping one more year now could prove counterproductive. On the other hand, I am convinced that by the end of this school year she will be ready to skip to fourth or even fifth grade. Skipping grades in one-year vs. multi-year increments gives my daughter enough time to adapt, and I get plenty of time to figure out how to navigate my district's impenetrable bureaucracy.

I met with my daughter's homeroom teacher this afternoon to discuss grade the recent grade skipping decision. To my surprise, she was very well informed about gifted education and for many years has handled clusters of gifted kids, sprinkled with the rare highly and exceptionally gifted. We reviewed my daughter's test scores, academic record from EPGY, list of books read since last year, as well as her own assessment academic readiness. I was left speechless when she argued that my daughter could be skipped to third grade and that it could be arranged if I requested it. Say what?????? Did I hear the teacher argue for the radical acceleration of my child? Is this possible in the LAUSD? I explained to the teacher that I thought it best to allow one year for my daughter's writing to rise to third grade standards. We agreed that the best course of action would be to skip over third grade next year provided that the writing proficiency goal is accomplished.

The surprises continued this afternoon after I got to my house. I received an email from the homeroom teacher following a meeting with the principal. She informed me that my daughter will be accelerated to third grade math. Logistically, this means that Paulina will leave her homeroom every day to take math in one of the third grade classrooms and then return to second grade for the remainder of the day. Moreover, my wife and I will be allowed to come to class to help proctor Paulina while she spends part of her English and math classes working on EPGY. In exchange, we have offered to help the teacher since budget cuts mean she has no teaching assistant this year.

Here is a bullet point summary of what I learned today:
  • radical acceleration is possible in the LAUSD
  • homeroom teachers and school principals make the final decision to accelerate
  • it is possible to do single-subject acceleration simultaneously with grade skipping
  • this seems to work best when the teachers and principal are well-informed
  • offer to help when the school accommodates your child
The basic fact is that my family got lucky. Our experience is uncommon. Our public school is one of the best in California. The principal is well-informed. The school has a high concentration of high-achieving students, and many of the teachers have some training in gifted education.

I am having a bit of trouble coming up with a prescription for success. I did some things right. Good luck played a big role. However, I also believe that "Chance favors the prepared mind." This implies that you can best advocate for your child by being ready:
  • Talk to parents of current students to find out how the school has handled acceleration and grade skipping in previous years
  • Learn the rules and regulations governing grade skipping and acceleration in your district
  • Document your child's talents by collecting IQ test scores, grades from prior courses, scores from standardized exams, transcripts from gifted and talented programs (i.e. Stanford's Education Program for Gifted Youth, John Hopkins' Center for Talented Youth, etc.), evaluations from former teachers, etc.
  • Enroll the help of gifted education advocates. You may want to contact the Davidson Institute. The Davidson Institute's Davidson Young Scholars offers guidance, free consulting services, and may help you communicate with local school officials.
  • Read as much research as possible on the benefits of grade skipping, acceleration, ability-based grouping, etc. Become an expert. Knowledge is the most powerful tool at your disposal.
  • Become a relentless advocate for your child's rights.
There is no magic bullet here. Prepare yourself. Look out for opportunities and seize them when they present themselves.

Hope this helps,


Monday, September 7, 2009

Skipping Grades in the LAUSD (i.e. Los Angeles)

Is it possible to skip grades in the Los Angeles Unified School District (LAUSD)? The answer is yes but only within very limited constraints. It is possible because my daughter was allowed to go straight from K to 2nd grade. I am fairly certain that she will have to skip grades a few more times over the next few years. Hence, I have set out to learn how LAUSD and the California Department of Education handle radical acceleration -- defined as three or more years. My findings have been discouraging so far. The LAUSD's Board of Education enforces minimum age limits for admission to various grade levels. I summarize my findings below and will update the information when I learn more.

I have to admit that my experience so far has been much more pleasant than I expected. My daughter was recently allowed to skip first grade, but her school is one of the best in California. I believe that our school's willingness to accommodate my daughter is the result, among other things, of its administration's focus on academics and the large concentration of high-achieving students. A thorough review of my daughter's academic record, including her EPGY scores, convinced the principal and first grade teacher that skipping to second grade was optimal. Unfortunately, I fear that my daughter will only be able to skip one more year before violating LAUSD's minimum age requirements. I believe my daughter will be allowed to skip from second to fourth grade because the principal is open minded and LAUSD's rules allow one more year for somebody in my daughter's situation. Unfortunately, her school is a K through 5th grade elementary, and I fear I will not find a public middle school that will allow her to skip from 4th to 6th or 7th.

Here is a summary of the pertinent age limitations for grade skipping within LAUSD:
  • Age of Admission to K - For admission to kindergarten, during the first school month of the school year, the fifth birthday of the child must be on or before December 1 of that calendar year. (Education Code, Section 48000) For good cause, a child of proper age may be admitted to a class after the first school month.
  • Age of Admission into 1st Grade - A student who is at least five years of age and who has been lawfully admitted to a kindergarten class in the Los Angeles Unified School District may be placed in the first grade, in accordance with regulations established by the Superintendent of Schools, when the administration determines the child is ready for first grade work. For admission to first grade during the first school month of the school year, the sixth birthday of the student must be on or before December 1 of that calendar year. Verification of age shall be required as provided in Board Rule 2001. For good cause, a student of proper age may be admitted to a class after the first school month.
  • Minimum Age to Enter Middle School - The minimum age for students entering middle high school who have been accelerated because of superior mental ability is 10-9 years of age on September 1 of the school year.
  • Minimum 6th Grade Attendance Before Middle School - Students transferring from regular Los Angeles Unified School District elementary classes to middle high school must have been enrolled in grade 6 for a minimum of one semester.
  • Exceptions to Rules for Entering Middle School - If exceptions to this policy become necessary for the overage student, the elementary and the middle high school principals involved must confer prior to the transfer of the student. It is understood that the final decision relative to exceptions shall rest with the elementary school principal.
  • Minimum Age for Senior High School - The minimum age for students entering senior high school who have been accelerated because of superior mental ability is 13-9
The single biggest implication of the above rules is that It makes no difference to LAUSD if your child is the smartest, most accomplished person in the planet. Your second grader already knows pre-algebra and reads seventh-grade books. Who cares? The poor child will be forced to study addition, subtraction, and how to write the simplest of sentences.

The following list summarizes the required ages by December 2nd of each grade level:

1st grade, 6
2nd grade, 7
3rd grade, 8
4th grade, 9
5th grade, 10
6th grade, 11

Middle School
7th grade, 12
8th grade, 13
9th grade, 14

High School
10th grade, 15
11th grade, 16
12th grade, 17

LAUSD's minimum, allowed age for seventh grade is eleven, provided the birthday happens on or before December 2. This is because the student must be 10 years and 9 months old by September 1st (i.e. the start of the 7th grade school year). Under this rule, the smartest kid in the world could only accelerate one year ahead of the expected, seventh grader. As you can see from the age rules above, high school admission is regulated more or less the same way.

It is very important to understand that the California Department of Education is powerless to enforce grade skipping. In fact, the California Department of Education takes the following position.

-----------START OF CITATION ---------------------
A child who was not age-eligible for kindergarten (that is, the child turned five after December 2 in the school year) and who attended a California private school kindergarten for a year is viewed by the CDE as not legally enrolled in kindergarten, pursuant to EC Section 48000 requirements. Therefore, this child, upon enrollment in public school, is enrolled in kindergarten, assessed, and may (but is not required to) be immediately promoted to first grade if the child meets the following State Board of Education criteria, pursuant to Title 5, Section 200:
  • The child is at least five years of age.
  • The child has attended a public school kindergarten for a long enough time to enable school personnel to evaluate the child's ability.
  • The child is in the upper 5 percent of the child's age group in terms of general mental ability.
  • The physical development and social maturity of the child are consistent with the child's advanced mental ability.
  • The parent or guardian has filed a written statement with the district that approves placement in first grade.

A statement, signed by the district and parent/guardian, is placed in the official school records for these five-year-olds who have been advanced to first grade (EC Section 48011). This action prevents a subsequent audit exception for first grade placement of an age-ineligible student.


-----------END OF CITATION-------------------

The above verbiage means that it does not matter if your child has already completed K. It is entirely at the discretion of your school's administration to let your kid into first grade. The State of California does not even recognize private K as a real K for legal purposes. The California Department of Education recommends alternatives when grade skipping is not approved. For instance, individualized instruction, multi-classroom settings (i.e. moving to a higher grade for a single subject), etc. However, my research into the subject shows these alternatives are seldom if ever implemented in LAUSD.

-----------START OF CITATION----------------------------
Local districts have discretion regarding promotion and retention. According to California Code EC48070, "each school district and each county superintendent of schools shall adopt policies regarding pupil promotion and retention. A pupil shall be promoted or retained only as provided in the policies adopted pursuant to this article." EC48070 Promotion and Retention EC48070 is fairly precise about retention policies, but it is glaringly vague on acceleration.


-----------END OF CITATION--------------------------

The age limits for grade acceleration can be found at

I hope this information helps you advocate for your child. Inform yourself. It is the best tool at your disposal.

Saturday, September 5, 2009

Using Graph Theory With Young Kids

I recently spoke about the challenges of teaching math with a good friend from college who is now a computer science professor at Yale University. I have fond memories of our freshman year sitting at a pizza joint doing math on napkins. Our lives took completely different directions, although we both got advanced, math-related degrees. He stayed in academia, where he has been very successful, and I went to work in Wall Street. We kept in touch throughout the years, and we now find ourselves with young daughters at similar stages of development. Our conversation focused on how to to spark interest in math and develop abstraction using topics accessible, yet absent from the traditional, elementary school curriculum. He said he had been trying a bit of graph theory with his daughter. This sparked my interest. While I never liked discrete mathematics, I now realize that it is probably the most accessible area in the discipline. Graphs are simple to explain, and, most importantly, many opened problems in the subject could be explained to a smart six or seven year old.

Let me give you an example. Show your kid the following star. Ask your kid to figure out if it is possible to draw the star without lifting the pencil or going over the same line twice.

Obviously, it is possible to draw a five-point start in the manner described above. That's not the point. Look at the following star-shaped graph.

Have you ever heard about the traveling salesman problem? The problem aims to find the shortest tour of a group of cities without repetition. Clearly, this is equivalent to draw the star without lifting the pencil or going over the same line twice. In other words, you are introducing your kid to a major area of graph theoretic research by asking how to solve a cute, little drawing problem.

Here are two additional graphs to try. Determine if it is possible to draw the following two figures without lifting your pencil or going over the same path twice.

If a solution is found, ask if there is a shorter way to draw the figure. Depending on the maturity of the child, you may draw little circles at the corners of the diagram and then explain that this is really a graph that can be used to represent lots of real-world problems such as how to travel to a bunch of cities along given roads in the least amount of time. Whether or not you choose to tell your kid that he is working with graphs and what they mean is irrelevant. The key here is to get the kid to think about optimal ways to draw these shapes. At the end of the day, you are teaching your child to think about optimal algorithms. Who cares if you achieve this by drawing silly figures or proper graphs?

Saturday, August 22, 2009

Sample Challenging Problems

I think it is important to challenge kids to think beyond mere formula recall. This is particularly the case with gifted learners because absorbing spoon-fed material is not a challenge. Acceleration merely increases the pace, but it does little to challenge the brain. I believe that most gifted educators without formal advanced training in mathematics fail to see the difference because most have no clue what real mathematicians do.

In keeping with last week's post about challenging vs. accelerating, here is an elementary problem requiring only simple mathematical facts while going significantly above simple formula recall.

1. Draw a rectangle. Draw a triangle such that its base is one side of the rectangle and the opposite vertex is chosen arbitrarily on the opposite side of the rectangle. Find a relationship between the areas of the rectangle and the triangle.

2. If your kid has a rudimentary understanding of probability, I would phrase the problem differently.

Repeat the rectangle / triangle construction above. Pick a point at random inside the rectangle. What is the probability that you picked a point inside the triangle?

Note that the triangle is picked before the random point. Hence, we asking for the probability of picking a point interior to the this particular triangle.

The answer to this problem is simple. The area of a triangle is 1/2 of its base times its height. Since the height and base are always the same regardless of the location of the triangle's apex, the area never changes. In fact, the area is 1/2 that of the rectangle. You will pick a point interior to the triangle with a 50% probability, and this independent of where the triangle's apex is located.

If your kid is advanced enough, you can ask him or her to generalize this problem to 3 dimensions. The simpliest, 3D generalization is to think of pyramids inside prisms. The answer is 1/3 probability of picking a point inside the pyramid. This is because the volume of a pyramid is 1/3 * area of base * height of pyramid.

Grade Skipping, Nation Deceived

This will be a very short post. Every educator and parent of gifted children should visit Nation Deceived.

Acceleration and grade skipping are not for every intelligent kid. However, when the conditions are right emotionally, intellectually, and socially, acceleration and grade skipping are extremely beneficial. In fact, for many gifted children, failing to accelerate or skip grades could be extremely detrimental.

The link above takes you to A Nation Deceived: How Schools Hold Back America's Brightest Students. Please, read the paper. Share it. Use it to advocate for gifted children.

As a side note, look in the left navigation bar at the Nation Deceived website. Download the PowerPoint. It is a good summary. You will be asked to registered before downloading it. It will ask for a data and topic of your talk. Just enter "TBD" for both. You will be able to download the PowerPooint.

Wednesday, August 19, 2009

What to Do When The Distributive Law Does Not Make Sense

My daughter has understood, for some time now, a few basic applications of the distributive law. However, it became clear yesterday that she did not really comprehend why the law is true and what it means. I have never seen a book properly tackle the concept -- for very young kids. The law is usually stated as such. Applications are introduced. An algebraic justification may be given. However, I have never seen good motivational examples. This was precisely the problem with my daughter. She saw no connection between the real world and the law.

Food is a big deal in our family. So are parties and friends. So, it occurred to me to use food and children to explain the distributive law.

Distributive Law for Division
- Get a bunch of M&Ms. Get 20 in total. Put them in a pile. Tell you child that you want to divide the 20 M&Ms evenly between 2 children. Clearly, the answer is 10. Now, ask you child to split the 20 M&Ms into two piles: one with 6 and one with 14. Ask your child if you get the same number of candies per child if you divide each of the two smaller piles first. Does each child get the same number of M&Ms? Try this with varying number of children and M&Ms.

The key here is to make you kid understand that you could divide the whole pile of candies or just divide any two smaller piles and then put everything together. You may have to explain this multiple times. Do it. Repetition works wonders. Have your child do this over and over with M&Ms until it becomes natural. Have the kid choose the sizes of the two piles.

It is time to work on paper. Set up another M&M pile. Break it into two smaller piles. Divide each pile evenly among the desired number of children. Now, write the equivalent arithmetic equation. Ask the child to match the various piles of M&Ms and the correct parts of the equation. Do this several times.

It is now time to use variables to see if your kid really understood the distributive law and its interpretation in terms of candy and children. Ask him or her to solve the following problem:

(2m + 4n) / 2 = ?

If your kid gets stuck, tell him or her that you have a total of 2m + 4n M&Ms. Suggest that he or she think of the big pile with 2m + 4n M&Ms as two smaller piles put together: one with 2m and one with 4n. If your child cannot do this on his own, it is time to start explaining things again. Give lots of written examples. Use simple examples. Always revert back to the interpretation using candy/children. Do not get discouraged. Keep trying because the distributive law is incredibly important.

Distributive Law for Multiplication - Use the same method as for division. For instance,

2 * 10 = 2 * (4 + 6) = 2*4 + 2*6 = 20

can be interpreted as two children, each getting 10 candies. This is the same as doubling each of the two piles of 4 and 6 M&Ms. It should not matter if you double the big pile of 10 or proceed to double the smaller piles first.

You may be tempted to explain the distributive law of multiplication by using the definition of multiplication as repeated addition. However, I advise against it because children like to relate things to concrete, familiar concepts. Candy and children are definitely concrete, so they ought to work well.

Tuesday, August 18, 2009

It's Good Enough!

Parents of highly gifted children know how hard it can be to deal with perfectionist tendencies. This is perhaps the single biggest challenge when dealing with intelligent kids. My daughter is very well adjusted. However, the perfectionist gene is a bit of an issue for her. She is not only an extreme perfectionist but also hyper-competitive. Unfortunately, the combination of these two personality traits can create serious difficulties, and dealing with them is the biggest challenge in our daily routine.

A basic strategy advocated by psychologists is the "good-enough" method. This basically means that parents tell an obsessed child that what he or she did is good enough and does not have to be perfect. This is easier said than done. While high intelligence certainly contributes to success, reasonable intelligence coupled with persistence and focus often leads to more optimal outcomes. Hence, it is extremely important for parents to learn when to encourage the perfectionist tendency and when to say "good enough."

We have been working on this for approximately one year. Our daughter is much better now when it comes to academics and the arts. She knows when to stop. She is beginning to understand when it's good enough. We cannot say the same thing about sports. Whether we play video games on the Wii, play tennis, race each other, or wrestle, she must always win. Losing is simply not an option. It does not matter to her if a kid twice her size beats her at basketball. She simply melts. She cries. She cannot help herself. This is clearly a problem. There are always going to be quite a few people better than her at sports. Hence, she will have to learn her to deal with it. There will be very few people better than her academically until she enters -- if she ever makes it -- an elite university. Hence, we need to help her deal with the reality that no matter what she tries, there is almost certainly somebody better than her.

It occurred to me during our latest summer vacation that golf is probably a good game to teach patience, hard work, and humility. My daughter likes the game. I plan to play with her a few times a month. I have explained to her that golf is possibly the most frustrating game in the world. She says she understands. We will see. Perhaps, the lessons of the game will carry into other aspects of her life. Plus, it is very nice to enjoy quality, father-daughter time every few weeks!

Accelerating vs. Challenging

I have been thinking a lot about the difference between accelerating and challenging in mathematics. My prior blog posts clearly show that I am a big believer in acceleration. However, there is a big difference between doing things fast and challenging the brain to think deeply. I don't mean to say that gifted children should be challenged instead of accelerated. I mean that the process of acceleration does not seem to include enough hard problems, and this may become an issue later in the child's educational career.

How should I balance acceleration and depth of thinking? My daughter is only six. Hence, I am very careful about the amount of time we spent working on traditional coursework. We do about one hour of English and math every weekday. That is plenty for her. She moves so fast that I worry quite a bit about retention. The key here is that I want her to learn the fundamentals -- math and language arts -- while having enough time for independent exploration. She likes to do art, write stories, read about the universe and the human body, etc. Hence, I need to think carefully about how to challenge her without imposing much formal study time beyond what we do with EPGY.

Here is one strategy I have been testing. Assign one tough problem every week. Come up with a challenging problem that can be solved using what the kid already knows. The problem does not have to have an exact answer. That's not the point. The point is to teach him or her to work on something hard over an extended period of time. This means that you give the child the problem, make sure it is understood, and then leave him or her alone to think. Every few days, you ask if he or she wants to talk about the problem or need a hint.

Here are a few examples:
  1. Compute the area of right triangle once it is understood how to compute the area of a rectangle.
  2. Compute the area of an arbitrary triangle once you know how to compute the area of a right triangle.
Ans to 1: Make a copy of the right triangle. Stack it on top of the original along the hypotenuse.
Alas, you got a rectangle. Divide by 2 the area of the rectangle. You want the kid to come up with a formula, algorithm, etc. Don't reveal the answer. Give only small hints if they get stuck.

Ans to 2: Set the triangle on one of the sides and call that side the base. Drop a perpendicular from the base to the opposite vertex. Alas, you you two right triangles now. Use the answer to question 1.

Yes to Acceleration

Why is it that it is okay for gifted athletes to go straight from high school into the NBA? Forget the NBA. Isn't it a NORM for gifted, young tennis players to go pro while in their teens? The same is true for soccer. How come we accept this form of acceleration, but we don't think it is okay for kids to skip grade levels? This is complete nonsense. Kids should be accelerated whenever they are ready to handle it.

I believe that acceleration in math and science is even more critical for girls than boys. I believe that profoundly, exceptionally, and highly gifted girls (145+ IQ on the Stanford Binet) are at a disadvantage over boys because of their tendency to "dumb themselves down." Yes, when boys come into the picture and being accepted by the crowd becomes important, girls tend to dumb themselves down. While I have little data to prove it, I believe it is quite hard to act intellectually average when working three or more levels above chronological peers. Moreover, I believe that working above grade level has an ego boosting effect that may help teenage girls to deal with the pressure to be normal.

Trying to get a school to allow grade skipping can be a major undertaking. It really does not matter whether the school is public or private. Administrators will use every strategy imaginable to keep your child with kids in the same age group. The reasons could include:
  • Your child is not emotionally ready to be with older kids
  • Your kid will miss crucial material and then fall behind
  • We have experience with gifted kids, and grade skipping is dangerous for the kid and his or her peers
  • What is the rush, let him or her be a "kid"
The last one is my favorite. "The rush?" What kind of a thoughtless question is that? This is not about rushing through school. It is about intellectual challenge, stimulation, developing a life-long love for learning, and striving to maximize potential in a psychologically healthy way.

None of the above reasons against acceleration is based on fact. They are myths. Despite what the professionals may say, acceleration and grade skipping are the right way to go. In fact, outcomes are far more favorable for highly gifted children who are radically accelerated (3+ years) than for those who are not. Furthermore, research shows that radically acceleration leads to a more fulfilling and successful adult life.

Here is some research in the subject of radical acceleration:
My best piece of advice is to arm yourself with knowledge, preferably of the scientific kid. It is one of the few tools at your disposal when advocating for the rights of your gifted child.

Friday, August 14, 2009

Strategies for Teaching Multi-digit Multiplication

Multi-digit multiplication can be challenging at first. It is quite easy to teach that multiplication means repeated addition of the same number. However, kids typically never understand the traditional method of multi-digit multiplication because their teachers themselves don't understand it. In fact, ask adults why the way they were taught to multiply actually works, and the vast majority won't know.

I first taught my daughter how to multiply using the traditional method. Unfortunately, it soon became clear that she did not understand why the method works despite the fact that she could carry out multiplications with an arbitrary number of digits.

Let me preface the following discussion by saying that multiplication with an arbitrary number of digits requires knowing the following:
  1. Regrouping into powers of 10: into ones, tens, hundreds, thousands, etc. For instance, 15 tens = 1 hundred plus 5 tens.
  2. Multiplication by powers of 10: multiplying by 10 adds a zero to the right. Multiplication by 100 adds to zeroes to the right. And so on, and so forth...
  3. The distributive law: should be understood at a basic level. It is far more important for kids to understand this than to blindly multiply an arbitrary number of digits.
A bit of advice on the distributive law for multiplication. If your kid does not really get why a (b + c) = ab + ac, try the following. Suppose you have two boxes full of candies. It is the same to:
  1. put all the candy together and then double the number
  2. separately double the number in each box and then put them together
If your kid does not get it at first, have him or her try it with real candy. M&M come in very handy here. Think about it. If you are going to double a pile of things, you might as well double two small sub-piles. It makes no difference how you do it. You always get the same answer. This is precisely what the distributive law means, and the way kids should understand it. There is no sense in forcing them to blindly memorize that a (b + c) = ab + ac.

I used several strategies to teach my daughter the reasons why multi-digit multiplication works.

1. Multiplication as Repeated Addition

I gave her examples of the distributive law --without telling her what I was doing -- using numbers, variables, and a combination of variables and numbers. For example:

c (a + b) = ca + cb
5 (4 + 6) = 5 * 4 + 5 * 6
5 (x + 2) = 5x + 5 * 2

I explained that 5 (x + y) = 5x + 5y because 5(x+y) means to add x+y to itself 5 times. Hence, you must have 5 x's and 5 y's. I actually wrote out

5(x+y) = (x+y) + (x+y) + (x+y) + (x+y) + (x+y) = 5x + 5y.

I then explained that 23 * 5 is the same thing as 23 * 5 = (20 + 3) * 5 = 20*5 + 3*5 because the definition of multiplication means I must have five 20's and five 3's. Hence, we can do the following:

23 * 5 = (20+ 3) * 5 = 20*5 + 3*5 = 100 + 15 = 115.

2. Alternate Method Using Distributive Property
We proceed similarly to the above method, but this may be easier to understand at first. Let's repeat the above example. 23 * 5 is the same as

(20 + 3) * 5 = 20 * 5 + 3 * 5 = 15 + 100 = 115

Let's expand this the product of two 2-digit numbers and introduce some useful record keeping method. Consider 23 * 25. This is the same as
The above method is much less confusing for young kids. The only drawback is that it uses many more rows that the traditional, grade school algorithm.

3. Alternate Method With Counting Followed by Regrouping

This is my preferred method. It is the closest to the method taught in school. Let's give an example. Suppose that you want to multiply 23 and 45. The first number is 2 tens and 3 ones. The second number is 4 tens and 5 ones. This means that the product will contain:
  1. 3 ones, five times = 15 ones
  2. 2 tens, five times = 10 tens
  3. 3 ones, 4 ten times = 12 tens
  4. 2 tens, 4 ten times = 8 hundreds
It should be clear that you kid needs to understand that ones x ones = ones, ones x tens = tens, tens x tens = hundreds, etc. Putting the above information into my notational framework, one gets:
The final step above is to regroup into standard numerical notation. For instance, the 15 ones is equal to 5 ones and 1 ten. Hence, the 22 tens plus the 1 ten from the 15 ones becomes 23 tens. This is equal to 3 tens and 2 hundreds. The, 8 hundreds plus the two hundreds from the tens add up to 0 hundreds and 1 thousand.

One advantage of this method is that it spells out every step done in the traditional method, while coming as close to it as possible without carrying and multiplying simultaneously. It is an easy jump to the traditional method once this method is understood.

I hope that the above strategies help your kid to learn multi-digit multiplication. At the end of the day, it is of the utmost importance to understand the process. The actual method is irrelevant.

Patience: This is the key to teaching highly gifted children. Do not assume that they need no repetition. They do. They just need fewer drills than normal kids. Do not assume they will get every concept right away. They will not. These kids learn amazingly fast, but they tend to forget things unless they are absolutely passionate about them. Hence, you may have to explain things a few different ways.

Consistency: This is extremely important. You need to set a routine. Kids like structure and routine. Do not do too much in any one day. If you spend 4 to 5 days a week doing 30 mins per day, your highly gifted child could finish elementary school math in less than two years. Don't push it. Take a week off here and there. Don't study during the weekend if they are very young. Do a little every day and don't let too many days go by without doing a little bit of mathematics. Consistency is the name of the game.

Praise Them When They Work Hard. Don't Praise Them for Being Smart: Kids need to be praised when they work hard or accomplish something that is difficult for them. Kids should not be praised for being smart. There is nothing they can do about the way they were born. You job is to teach them how to set goals and work to accomplish them. Being smart is not an accomplishment.

Thursday, August 13, 2009

When should variables be introduced?

When should variables be introduced? I debated this for many days when I learned that my daughter was highly gifted. I asked myself if I should concentrate on basic arithmetic, reasoning, etc. I went back and forth between first teaching her to add two and three digit numbers and introducing variables simultaneously with those concepts. The key here is that my daughter could already add single digit numbers and some two digit numbers.

I tried various strategies to teach the concept and use of variables. First, I explained the idea of an equation. I made it simple. I simply said that an equation is when you have two things that are equal. I taught her how we write an equation using standard notation. I used simple things like 5 = 5, 1 + 2 = 3, etc. I then showed her examples of things that are not equations. I used inequalities like 4 <> 3, 4 <> 1 + 2, etc. The key was to present a ton of examples. The next step was a fun one. I taught my daughter how to solve simple linear equations using M&Ms. Yes, you read that correctly. I used candy. We typically give our daughter some sort of dessert after dinner. So, I figured it may be fun to use dessert to teach elementary math. I got a bag of M&Ms and a small bowl. I set 1 M&M on the table and 2 under the bowl -- making sure she did not know how many were under the bowl. I then put 3 M&Ms the other side of the table. I asked her to tell me how many M&Ms would have to be under the bowl so both sides of the table had the same number. It should not shock you to learn how quickly she figure it out since I told her she could only eat the M&Ms under the bowl if she got the right answer. M&M algebra became a daily favorite of my daughter after dinner. I just made sure she never had too many! We played this game for about a week. Once she was getting the answers quickly, I took out some paper. I set up an M&M problem and then wrote down the corresponding equation on a piece of paper. I put an empty box to represent the bowl. I set up a number of examples asking her to tell me which of the M&M piles corresponded to what number and what corresponded to the empty box. I then asked her to tell me the number needed in the box to make the equation true. The next step was easy. We replaced the empty box with letters like n, x, etc. I used different letters and moved back and forth between empty boxes and letters always asking her to set up an equivalent M&M algebra problem. A few days later, variables became second nature to her. She understood they were just meant to represent the number we don't know. Clearly, I let her eat the M&Ms when she got the right answers.

I know. Some parents are going to argue that I shouldn't have used candy. However, you can be careful with the amount of sugar and still make the learning process fun. After all, we all know kids love candy. This may not work for your kid for medical or other reasons, but there is always something you can use in place of candy. Just make sure your kid loves it.

Let me get back to the original question of this post. When should variables be introduced? I now believe they should be taught as early as possible. The concept is only marginally more abstract than that of a number. Why we wait until pre-algebra is beyond me. In fact, Stanford's Education Program for Gifted Youth first teaches the idea in K and 1st grade. It uses many different examples in many different situations. The upshot of this early introduction is that 2nd and 3rd grade EPGY students can solve an impressive array of equations and understand how to translate word problems into equations and vice versa. For instance, one of the standard questions in EPGY's 2nd grade final exam is to solve a system of equations like the following:

m + n = 5
m - n = 1

This is asking to find two numbers 1 apart that add to 5, and this is precisely how they are taught to solve the system. They do not look at it the same way that algebra students do. They learn the role of variables and what the problem means.

I hope this post gives you some creative ideas to teach about variables and equations. I will write about how to teach other concepts in future posts.

Can Your Kid Get a Truly Individualized Education?

Can your kid get a truly individualized, primary education? I believe this is next to impossible. Public and, to a larger extend, private schools argue that they educate the "whole" child, catering to individual needs. This philosophy sounds wonderful, but the reality of day-to-day instruction is quite different. This is not the result of apathy. It is a consequence of basic economics and simple arithmetic.

Teachers can and do attempt to individualize instruction. Unfortunately, this is typically limited to remedial help or additional worksheets. Gifted programs such as California's GATE use additional curriculum and ability-based grouping, but they do not really allow students to move at their own pace. Good private schools market themselves as capable of handling accelerated learners. However, a few well posed questions quickly reveal that the vast majority of schools are incapable of handling students two or more standard deviations from mean IQ. Simply put, your child will NOT get what he or she needs when faced with special needs or radical acceleration.

I visited a well-known, Los Angeles private school last year when my wife and I were searching for a challenging primary education. Our daughter has exhibited an affinity for math and science, so we approached the director of the elementary science program. After enduring a fifteen minute marketing pitch, we asked the director to explain how she would handle a first grader who already knows reading, multiplication, fractions, etc., and is years ahead of her chronological peers. Her answer shocked us. She said that every child has weaknesses and that she would simply "hold back" my daughter's mathematical development to help her on areas where she might be weaker. She also stressed that while kids enter first grade at varying developmental stages, most exit second grade at similar degrees of proficiency. My wife and I listened intently how the school would hold our baby back while we paid $20,000+ per year. This was ludicrous. There are real, easily identifiable, intellectual differences. Bear in mind that it is irrelevant for this discussion whether fat tails or a standard bell curve is the best description of the distribution of ability in a population. It is a well-established fact that the set of kids with IQs above 145+ is quite small relative to the overall population. It is also a fact that the range of subject-specific abilities follows a curve with a heavy mass density around the mean. In other words, there is a clear distinction between the rare cases of high ability and the rest of the student population at the beginning of first grade. What the science director above implied is that her school has a magical method to homogenize brains!!! Furthermore, she argued, the majority of students exit second grade on even ground. Isn't it amazing how the outliers predicted by ability distribution curves re-appear after this magical, first-through-second grade lobotomy? We visited a number of other well-known, private schools, but the same story repeated itself over and over. It was deja vu, over, and over, and over again. The bottom line is that traditional schools, however well-intentioned, are not equipped to deal with exceptional students. There are a handful such as The Mirman School for Gifted Children that offer a unique environment ideal for many highly gifted children. However, even Mirman may be wrong for your kid for reasons that have little to do with academics.

How should one educate a highly gifted child? This is a difficult question whose answer clearly depends on the child as well as the family's ability to provide a balanced, maturity-appropriate social environment. Homeschooling is clearly an option provided there is flexibility and time to embark in this serious commitment. However, I believe there are other options which I will explore in future blog posts.