Posted by: JonahSinick

## How my math skills improved dramatically - 03/05/14 11:33 AM

In Gifted children could learn math much earlier I suggested that children of IQ ~145 can learn math 5+ years ahead of schedule. Those of you who have children in this IQ range but who are not currently working grade level may wonder how one goes about doing this. It’s not necessarily a matter of trying to plough through the material more rapidly.

My own experience here may be illuminating. When I was a freshman in high school, I was a mediocre math student: I earned a D in second semester geometry and had to repeat the course. By the time I was a senior in high school, I was one of the strongest few math students in my class of ~600 students at an academic magnet high school. I went on to earn a PhD in math. Most people wouldn't have guessed that I could have improved so much, and the shift that occurred was very surreal to me. It’s all the more striking in that the bulk of the shift occurred in a single year.

I thought I’d share what strategies facilitated the change. Because I was at the high school level, the particulars of my experience may not be relevant to your own child’s situation, but some of the general principles will be.

I became motivated to learn more:

I took a course in chemistry my sophomore year, and loved it so much that I thought that I would pursue a career in the physical sciences. I knew that understanding math is essential for a career in the physical sciences, and so I became determined to learn it well.

I immersed myself in math:

At the start of my junior year I started learning calculus on my own. I didn’t have the “official” prerequisites for calculus, for example, I didn’t know trigonometry. But I didn’t need to learn trigonometry to get started: I just skipped over the parts of calculus books involving trigonometric functions. Because I was behind a semester, I didn’t have the “official” prerequisite for analytic geometry during my junior year, but I gained permission to sit in on a course (not for official academic credit) while taking trigonometry at the same time. I also took a course in honors physics that used a lot of algebra at the same time.

I learned these subjects better simultaneously than I would have had I learned them sequentially. A lot of times students don’t spend enough time learning math per day to imprint the material in their long-term memories. They end up forgetting the techniques that they learn in short order, and have to relearn them repeatedly as a result. Learning them thoroughly the first time around would save them a lot of time later on. Because there was substantial overlap in the algebraic techniques utilized in the different subjects I was studying, my exposure to them per day was higher, so that when I learned them, they stuck in my long-term memory.

I learned from multiple expositions:

This is related to the above point, but is worth highlighting on its own: I read textbooks on the subjects that I was studying aside from the assigned textbooks. Often a given textbook won’t explain all of the topics as well as possible, and when one has difficulty understanding a given textbook’s exposition of a topic, one can find a better one if one consults other references.

I learned basic techniques in the context of interesting problems:

I distinctly remember hearing about how it was possible to find the graph of a rotated conic section from its defining equation. I found it amazing that it was possible to do this. Similarly, I found some of the applications of calculus to be amazing. This amazement motivated me to learn how to implement the various techniques needed, and they became more memorable when placed in the context of larger problems.

Art of Problem Solving materials supply a rich source of interesting problems. (Disclosure: I teach for Art of Problem Solving.)

I found a friend who was also learning math in a serious way:

It was really helpful to have someone who was both deeply involved and responsive, who I could consult when I got stuck, and with whom I could work through problems. This was helpful both from a motivational point of view (learning with someone else can be more fun than learning in isolation) and also from the point of view of having easier access to knowledge.

My own experience here may be illuminating. When I was a freshman in high school, I was a mediocre math student: I earned a D in second semester geometry and had to repeat the course. By the time I was a senior in high school, I was one of the strongest few math students in my class of ~600 students at an academic magnet high school. I went on to earn a PhD in math. Most people wouldn't have guessed that I could have improved so much, and the shift that occurred was very surreal to me. It’s all the more striking in that the bulk of the shift occurred in a single year.

I thought I’d share what strategies facilitated the change. Because I was at the high school level, the particulars of my experience may not be relevant to your own child’s situation, but some of the general principles will be.

I became motivated to learn more:

I took a course in chemistry my sophomore year, and loved it so much that I thought that I would pursue a career in the physical sciences. I knew that understanding math is essential for a career in the physical sciences, and so I became determined to learn it well.

I immersed myself in math:

At the start of my junior year I started learning calculus on my own. I didn’t have the “official” prerequisites for calculus, for example, I didn’t know trigonometry. But I didn’t need to learn trigonometry to get started: I just skipped over the parts of calculus books involving trigonometric functions. Because I was behind a semester, I didn’t have the “official” prerequisite for analytic geometry during my junior year, but I gained permission to sit in on a course (not for official academic credit) while taking trigonometry at the same time. I also took a course in honors physics that used a lot of algebra at the same time.

I learned these subjects better simultaneously than I would have had I learned them sequentially. A lot of times students don’t spend enough time learning math per day to imprint the material in their long-term memories. They end up forgetting the techniques that they learn in short order, and have to relearn them repeatedly as a result. Learning them thoroughly the first time around would save them a lot of time later on. Because there was substantial overlap in the algebraic techniques utilized in the different subjects I was studying, my exposure to them per day was higher, so that when I learned them, they stuck in my long-term memory.

I learned from multiple expositions:

This is related to the above point, but is worth highlighting on its own: I read textbooks on the subjects that I was studying aside from the assigned textbooks. Often a given textbook won’t explain all of the topics as well as possible, and when one has difficulty understanding a given textbook’s exposition of a topic, one can find a better one if one consults other references.

I learned basic techniques in the context of interesting problems:

I distinctly remember hearing about how it was possible to find the graph of a rotated conic section from its defining equation. I found it amazing that it was possible to do this. Similarly, I found some of the applications of calculus to be amazing. This amazement motivated me to learn how to implement the various techniques needed, and they became more memorable when placed in the context of larger problems.

Art of Problem Solving materials supply a rich source of interesting problems. (Disclosure: I teach for Art of Problem Solving.)

I found a friend who was also learning math in a serious way:

It was really helpful to have someone who was both deeply involved and responsive, who I could consult when I got stuck, and with whom I could work through problems. This was helpful both from a motivational point of view (learning with someone else can be more fun than learning in isolation) and also from the point of view of having easier access to knowledge.