When Possible, Gifted Children Should Skip Pre-Algebra

I have never understood the point of pre-algebra, and I have plunged Paulina straight into algebra as a result. I am guided here by my personal experience and because she has yet to exhibit any deficiency from missing the traditional course.

I recall finishing arithmetic in seventh grade and then jumping straight into algebra over the summer. Of course, I did this on my own because my school had no system in place to accelerate eager students. Well, this is not entirely true. There was a summer pre-algebra program to accelerate those deemed to posses mathematical talent. I did not make the cut despite being a “straight A student” because my seventh grade teacher did not think I had the mental capacity for higher-level, abstract thinking. So, I got really pissed. I bought myself an algebra book, and I solved every single problem. This was the first time in my life that I actually enjoyed math. By the end of the summer, I was ready for geometry and trigonometry. Isn’t it funny how that worked out? I went through algebra while the “smart” kids in my class barely had enough time to learn pre-algebra. Once again, I taught myself geometry and trigonometry. I did this while taking the standard 8th grade pre-algebra sequence at my school. Frustrated with the pace of instruction, my mother convinced the local Universidad Catolica de Puerto Rico to let me enroll in individual courses. From 10th grade through my senior graduation, I took pre-calculus, calculus I, number theory, physics, and microbiology. I tried to take calculus II, and calculus III, but there were not enough students at the university to justify offering the courses. So, I taught myself calculus II and calculus III before my freshman year at Yale. The point of this story is that I never took pre-algebra or trigonometry as standalone courses, and I turned out just fine. Yes, I did study trigonometry. However, I did not waste an entire year on the subject, and I certainly never wasted a second on pre-algebra. Based on my personal experience, I have been teaching Paulina algebra using The Art of Problem Solving while she completes 6th grade through EPGY. So far, things are working out just find.

I want to expand on some of the ideas I discussed above. First, I want to explain how I deal with those instances when Paulina does not know a concept needed to solve an algebraic problem. Second, I want to explain my theory on the early introduction of advanced material.

Occasionally, Paulina bumps into an algebra problem dependent on a concept that she has yet to learn. When this happens, I stop the algebra lesson, and I write  a custom problem set to teach her what she needs. I always start with a brief summary of the key ideas, and I then follow with a series of exercises to help her “discover” the mathematics on her own. Exponents are a good example. A problem set we worked on last month required knowledge of exponents. I designed a set of exercises to teach her exponential arithmetic. I also wrote a series of problems to explore the relationship between dimension, measurement, and boundary. The questions dealt largely with the impact on measurement from changes in the sizes of objects along one or more dimensions and the relationship between the dimension of the objects and the dimension of their boundaries. We went back to algebra after finishing the problem. Needless to say, Paulina cruised through the algebraic problems and now knows a lot more about exponents and their applications that she would if we had wasted a whole course on pre-algebra.

There is one aspect of my instructional approach that is a bit peculiar. I believe that one should introduce advanced concepts along side the rest of the material even if the student does not get it at first. While this may sounds like a waste of time, there is logic behind my philosophy. Paulina surprised me a few years ago when she automatically knew how to do a bunch of problems on symmetry after only hearing the definition. I was surprised because some of the problems were difficult in the sense that they required visual intuition. When I asked Paulina if somebody at school had taught her about symmetry, she replied that I had explained it to her when she was four years old (she was six at the time) and that she remembered what I said. The funny thing is that we only talked about symmetry a few times, and she was not able to solve at four what she could at six. This gave me the idea of introducing carefully selected, advanced topics simultaneously with her regular curriculum. This has worked out wonderfully. She may not get every concept at first, but her dormant knowledge always wakes up at the right time. It is almost magical.

Guided by the principle of dormant knowledge, I have started to teach Paulina about trigonometric functions and limits in the context of sequences of areas. I will soon post in this website a problem set about trigonometric, areas, and limits.

I want to stress that I am well aware that not every kid can handle skipping courses like pre-algebra. However, the point of this blog is to reflect on the issues affecting the mathematical education of gifted youth. I believe that mathematically talented children should attempt algebra after mastering arithmetic. It is my opinion that pre-algebra is an artificial step to remedy poor training.

In summary, I would encourage you to try two ideas. First, skip pre-algebra after mastering arithmetic. Second, introduce advanced concepts early. This should make it easier to learn them the second time around. The early introduction of advanced concepts may help your child develop connections it may otherwise not make until much later.