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.

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:

• Short Research Summary: http://www.davidsongifted.org/db/Articles_id_10349.aspx
• Long Review of Research: http://www.templeton.org/pdfs/funding_areas/10112_Final_Rpt_Bibliography.pdf

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.

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.