Long before having children, I began formulating ideas about how I thought education should work. One of my initial ideas was that the way that we divide the math curriculum is arbitrary and causes unnecessary pain when it comes to understanding calculus. At that time, I stated a belief that calculus should be taught to third graders. My eldest son is eight years old and will be considered a third grader this coming Fall. So, where do we stand?
I started early on taking a split approach to mathematics. We did “normal” math, or what I now think of as calculation, using a fairly standard curriculum (Singapore and then Life of Fred starting this year). In addition to that, I spent time introducing additional concepts that were out of order according to the standards set by the school system. In kindergarten, I introduced him to variables and equations in a very simple, intuitive manner, not making a big deal out of anything. One book plus two books equal three books. One x plus 2x equals 3x. Just a simple extension of the way that normal arithmetic is taught. Then we worked on assuming x was some number and making it a puzzle to discover which number.
So, I’d give him 3x + 2 = 8 and leave him to try and figure it out. We began playing a game where I’d give him a puzzle like that before leaving for work and he’d present the answer when I got home. Then, we moved to multi-variable equations and systems of equations. Every now and then I’d teach him a new trick.
I think that was really key, not presenting things as “Now you need to learn this…”, but as “Here’s a clue to help solve the latest puzzle…”. In first grade, he was solving systems of three equations with three variables, limited of course to using the arithmetic operators he was familiar with. The end result of this is that when he sees equations now, he is not confused or afraid of them. Even if he can’t figure out how to solve them, he has an intuitive sense of what they represent. I attribute this mostly to familiarity and having worked with extremely simple equations for a long time. When we get to real algebra, it will be a straight-forward extension of things he’s been doing for years.
Similarly, we started on some geometry. Not focusing on proof techniques at this point, but again setting up puzzles. After a brief introduction to angles, parallel lines, etc, all of which are intuitively simple, we began solving puzzles trying to find the unknown angle. Again, I would periodically introduce a new trick to help him solve a harder type of problem. He’d still usually only work one or two problems a day, although he sometimes asked for more. By early in his second grade year, he was able to solve some of the example problems from the SAT (this was before we introduced long division, so I had to make sure they worked out simply).
Is this working because he has a talent for math? I don’t think so. Actually, he’s not very good at calculation. He frequently makes mistakes, loses interest quickly (we rarely work more than 15-20 minutes at a time), and often gets confused or distracted mid-problem. I never excelled in math at school either. It was always one of my worst subjects (admittedly still above average, but toward the bottom of the advanced class). Math was my lowest score on the ACT in high school. Among my engineering friends in college, I was always the best at Computer Science (I’ll talk about why I think this was the case in a later post) and the worst at math. I still have many of the same problems that he does, struggling to do mental math, getting confused mid-stream in complex problems, etc.
This year, we started on a few new things. The first is that we started watching The Joy of Mathematics together. In this series, Arthur Benjamin of Harvey Mudd College presents an extremely upbeat, infectious course about a wide range of math topics. We’ve only watched the first 10 episodes so far, and I don’t plan to get into some of the later episodes this year. Professor Benjamin is really excited about his subject and does a good job of making math seem fun. This is intended to be college level material, so don’t be surprised if you or your child don’t get everything, or have to back up and re-watch parts. After watching episodes, we frequently spent a while talking about what he covered, and I certainly didn’t worry about testing any material. I was excited when, after watching the episode on proofs, my son came up to me excitedly and showed me a visual proof of a multiplication problem. Not exactly ground-breaking, but he came up with it himself and had obviously been thinking about the idea of how to prove things on his own.
The next big change this year was the switch from Singapore Math, which is based on the national curriculum developed by Singapore and is one of the best traditional systems, to Life of Fred. Life of Fred still covers material in the traditional order, but using a very novel technique. Everything is presented as part of a narrative, the story of the life of a 6 year-old boy named Fred Gauss, who is a math professor. The story is silly and fun. Every math concept is presented in a context where it is used in some situation that Fred faces. The stories are also rich with information from other subjects as well – history, art, science. I knew that the series was a success when I had to tell our son to stop reading his math book and go to bed for the third time one night. He blew through the first 10 books in the elementary series and is now on Fractions. In the early books I didn’t make him work the problems, I just periodically quizzed him using some of the chapter questions. With Fractions, we’re making him do the end-of-section exams, although he is allowed to, and encouraged to, read ahead.
So, will we be ready for calculus this year? First, I need to clarify what I mean by “learning calculus”. I don’t expect a third grader to work all of the problems in a college calculus text. That would require trigonometry, advanced algebra, and a bunch of other concepts that have nothing to do with calculus. Those problems exist in the standard texts because of where calculus is placed in the curriculum.
On the other hand, it would be trivial to teach third graders how to solve basic derivatives without understanding. The transformations for simple derivatives and integrals are no more difficult than, say, long division. So it would be no great achievement to teach a child how to perform the basic transformations. It also wouldn’t be very useful to them.
So, what does it mean to understand calculus? To answer that, I think we need to look at the context in which calculus was developed and the problems it was invented to solve. To that end, we are watching a series on the Great Ideas in Classical Physics by Steven Pollock. It does a good job of setting up the stories of Gallileo, Kepler, and Newton in developing the ideas of motion, gravity, and acceleration. The course is a good one for younger students because it doesn’t focus on solving many hard problems, but on the concepts and ideas. As of lecture 8, he has only introduced one equation, and spends time talking about equations not as something to be solved but as representing relationships between concepts. Of course, since we have established a habit of looking at equations, this makes sense and is not intimidating, but interesting. We have also talked briefly about graphing equations as relationships.
I plan to have our son work through “Fractions” and “Decimals and Percentages” this summer, and we should start next year with pre-Algebra. In parallel, we will continue studying physics and begin formalizing an understanding of calculus as the relationship between position, velocity, and acceleration. I don’t plan to focus on solving difficult calculus problems involving lots of algebra, but rather on understanding the nature of the relationships involved. We will work through the idea that derivatives and integrals are inverse operations, just like multiplication and division, and try to gain an intuitive sense of why that is true.
The idea is not to be able to pass an AP calculus exam at the end of the year, but to have a strong foundation on which to build. Just as I introduced algebra from the beginning with simple equations, and we are still working towards understanding all the messy details, by the time we get to high school, calculus will be something very familiar that he’s been working with for years, and he can fill in the latter details as it makes sense. The idea is to gain a familiarity and comfort with the concepts.
We will probably start working through Calculus Without Tears, which I only recently discovered. Here is a quote from the author’s website:
Calculus has always been taught ‘theory first’, that is, before a student studies calculus, he/she spends years studying abstract and difficult mathematics including geometry, algebra, and trigonometry. Then the study of calculus is encumbered with the notion of mathematical proof, and the student is required to mathematically prove the simplest facts about calculus before using calculus to solve problems.
We could take the same approach to teaching arithmetic. We could start with a series of courses on symbolic logic. Then, as our arithmetic textbook we could use Whitehead and Russell’s Principia Mathematica, an important work that proves the basic properties of arithmetic. Never mind that it is two thousand pages long and comes in three volumes. The definition of number is on page 234, and the proof that ‘1+1=2’ is on page 362 (see the proof at http://www.idt.mdh.se/~icc/1+1=2.htm ). Using this approach, multiplication would be taught midway through college! Fortunately, it’s not the way arithmetic is taught. Unfortunately, it is the way calculus has been taught. CWT teaches calculus the way arithmetic is taught, by starting with the basic operations applied to easy examples. The result is that the student has a good intuitive grasp of calculus, something that often eludes students in college calculus classes.
I still think the example explanations provided on the website are too quick to jump into modern notation, but his philosophy is very similar to my goals. I’ll keep you posted on how it works out.