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.