OUR MATH STORY by Brook Randal
My son was homeschooled for 12 years. He then graduated from MIT with a double major in math and computer science, plus a master’s degree. He was always extremely talented at math. I’ve frequently been asked about our journey. This was written in response to one of those requests.
We've never worried about doing anything sequentially. We always used the Tetris model of homeschooling with the random pieces falling from the sky, never the traditional beads-on-a-string approach that schools use. Here's what we did for math. It might not work for less-mathy kids, but it sure worked for us. It is sort of like learning a foreign language by moving to a new country and having to survive, rather than learning a foreign language by studying 10 new vocabulary words a day and introducing 2 verb tenses per semester. I think S would have done much less if we had taken a traditional approach. "Real math" is not sequential little increments. "The usual order" is something totally arbitrary that pedagogues came up with for the convenience of teachers.
I used to say we had a "game-based curriculum". We played a lot of dice, card, computer, and board games that involved math. The best ones are the ones where you need to use math to play the game. I avoided the ones where you are drilled for a few problems, and then "rewarded" by shooting a few spaceships or something. We talked a lot about mathy stuff in the car or waiting for food in restaurants. There was no agenda to it. We would just talk about anything off the top of my head to keep him entertained. We did not use a traditional math textbook with formal homework assignments until he was beyond AP calculus. But we used a lot of children's math books from the library. (eg., "Number Devils", "A Gebra Named Al" by Wendy Isdell, "Math for Smart Pants", puzzle books by Theoni Pappas and others). I also had all of those "What Your Nth Grader Should Know" books, and I went thru the math sections like a check list to make sure he had all of it. We bought all the Doug Downing "Easy Way" books (algebra, trig, calculus) and I read them to him like story books. Whenever he got bogged down in the explanations, I just skipped ahead to where the story picked up. A year or 2 later I would go through it again, with the explanations, a year or two after that we would make the final pass and include all the footnotes and end-of chapter problems. These all overlapped. We did the first pass on the algebra one in 3th grade, and the first pass on the trig and calc ones in 4th grade. We did all the teaching company math videos. But I never asked him to do any problems with them. He would just watch them. Also the old Square One on PBS. He never memorized the multiplication tables (neither did I). I taught him the little tricks I figured out as a kid to compute them on the spot, and he invented some new ones. The tricks involve a much higher level of mathematical understanding than memorizing the table, plus they also work for bigger numbers than 12X12.
MIDDLE SCHOOL 6-8
At the beginning of 6th grade we discovered Mathcounts, and our lives changed. The math was an exact match for him. The problems are extremely challenging, varied, interesting, and combine multiple areas of math in one problem. They sometimes require stuff like combinatorics and number theory that I didn't see until college. Competition math is totally non-sequential. When newcomers start on Mathcounts, they just jump in the stream wherever everyone else is working. At first most of the kids can't do it. They need to work backwards from the solution, or have someone walk them through a bunch of problems before they start to get the hang of it. Then, the young first years can maybe get 20% of the problems without help. The second year, the problems don't get any harder, but the kids are better at it. Maybe they can get 50%. Third year maybe 70%. There is a tremendous amount of preparation material out there for contests. We had a huge supply of problem books with solutions, for contests at various levels. S learned by doing problems, with all the math mixed together. In 7th grade we started hiring math grad students to tutor him. Initially it was just general enrichment, unrelated to any curriculum. Later it became focused on math competitions.
the summer after 8th grade S started attending USA-Canada Mathcamp. 5 weeks of math-nerd paradise! A huge smorgasbord of offerings at all levels. Kids pick and choose what they feel like doing every day. Some classes are one-lecture long, others might be 2 weeks, or the entire 5 weeks long. Topology, Field Theory, Optics, Problem Solving, Cryptography. A total mish-mash of subjects that the staff is interested in (often has PhDs in) and makes available to the kids. It is totally non-linear. No pre-reqs, homework sets, exams, grades. The kids just jump in it and play around. Everyone is thrilled to be there and looks forward to it all year. My son went for 5 summers in a row, and still sometimes gets together with other alums in the Boston or NY area.
9th Grade -
S had already had 3 passes thru Calculus the Easy Way spread over 5 years, plus watched some calculus videos. I got a bunch of AP review books and old AP problems. He worked through those, then took the AP Calculus BC exam at the end of 9th grade. I bought a calculus textbook, but he never used it.
10th grade -
he did distance learning courses for Multivariable Calculus and Linear Algebra. This was his first experience actually working through a math textbook and dealing with homework sets and exams. He HATED that aspect of it (too much drill-and-kill even at this level). I had to force him to sit down and do it while he bitched bitterly, particularly on the multivariable which didn't involve as much new material as we had assumed. But he got through, and made As. He also read an AP Statistics review book and took the AP exam.
11th grade -
started auditing grad level math courses at UT (Abstract Algebra, Algebraic Topology).
12th grade -
Audited 4 grad courses. Because of conflicts with college visits, he was able to do varying amounts of the work, including attending. But he got something significant out of each of them. Also self studied differential equations using lectures and materials from MIT's OpenCourseWare site (free),
We ended up with 10 AP scores (all self-study), 4 scores on SAT II subject tests (in addition to the regular SAT I scores), 3 grades from UT distance learning courses, 2 letters from UT profs stating what his grade would have been in their grad course if he had been allowed to register. I put everything together on one big master transcript. I included the things he studied at home, sort of arbitrarily bundled into "courses". I did not include any parent assigned grades. The transcript is 2 pages. There is an additional 5 page document with course descriptions (textbooks used, etc. Max few lines per course). There is also a one-page "school profile" describing our general educational philosophy. It makes it clear that we were homeschooling in order to attain the highest possible level of academic success, not for any religious reason. It also makes clear that the student had plenty of opportunity for social interactions. We got a rec letter from one of his math profs at UT, from one of the coaches of the USA Computing Olympiad, and from a homeschool parent who taught several classes that included my son. He also had an extensive list of national and some international awards in math, physics, computer science.
He got automatic credit for his score on the AP Calculus exam (plus some non-math stuff). We submitted a transcript and copy of the syllabus for the two UT distance learning courses he took, but he did not get automatic credit for that. He was able to get credit for Multivariable Calculus and Differential Equations by taking MIT exams. There are other courses that he covered in high school that he could have gotten credit for by taking MIT's exam (linear algebra, topology). But he decided not to bother since they were not required for anything, and he was not trying to rush through. MIT is fairly loose about prereqs, so he did not have to repeat anything he had already covered. He started right in on graduate level math and CS courses as a freshman, with about a semester worth of credit.
ADULT LIFE Alex currently works for a finance firm in Manhatten. His job is very technical and involves trying to make markets more efficient. He describes it like designing Battle-bots, only instead of robots in an arena throwing ping pong balls at each other, they exist inside of giant computer systems and fight with wads of cash. He still runs into people in his professional life that he met, or knew of, because of contests in middle and high school.