How my math skills improved dramatically

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.

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, and gave some hints of the re­la­tion­ship be­tween physics and calcu­lus.

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.

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.