Learning the Fundamentals in Algebra I
Anyone who has taught calculus to undergraduates will tell you that the main thing these students struggle with is not calculus itself, but basic algebra. Algebra I is where many students fall behind, and without a solid foundation to build on, they never manage to recover. I therefore feel an enormous sense of responsibility teaching this course: all the mathematics that my students will ever do, at Proof School and beyond, relies on what they are learning with me this block.
If the hardest part of calculus is algebra, then in some sense the hardest part of algebra (or at least Algebra I) is arithmetic. One has to understand how numbers work at a very deep level in order to manipulate algebraic expressions. For instance, in order to multiply x² + 3 by x³ + 4x² - 7, one has to understand the distributive law and the laws of exponents. In order to compute
one has to truly understand integer factorization and fractions. Of course, one can also learn how to do these things by memorizing a bunch of rules without understanding; this works in the short term, which is why so many students graduate from high school convinced that mathematics involves moving symbols around according to a set of arbitrary procedures and formulas. But in the long term, this strategy fails: by the time they get to calculus, these students can no longer manage the keep the increasingly complex sets of arbitrary rules in their heads. In contrast, I want my students to know -- to truly believe, in their heart of hearts -- that the symbols in mathematics always stand for something real (in Algebra I, it's actual, honest-to-goodness numbers) and that the rules for manipulating them always reflect this reality. If you want to know whether a certain manipulation is valid, you don’t need to consult your notes; you just need to consult your understanding of reality!
To drive the point home, I told the class a story about a friend of my family who, while traveling in a remote region of Russia in Soviet times, had a chance to observe a math class in a small village school. He was shocked by what he heard: the teacher was telling the students that to add fractions, you just add the numerators and add the denominators. After class, our friend politely explained to the teacher that he was a mathematician from Moscow University who happened to be visiting the area, and that the correct way to add fractions was, in fact, quite different. He sat in on the next class to see what would happen. The teacher told her students: "Children, there is an important man here from Moscow. From now on, we have been instructed to add fractions in the following way..."
It took a short historical aside to help the students understand the humor of the story, but they loved it! I hope this anecdote serves to remind them that mathematics possesses a truth that is not based on the authority of a teacher, or a visiting mathematician, or even the Communist Party of the Soviet Union. You never need to memorize arbitrary rules; you just need to understand what is really going on.
To a certain extent, such understanding has to come from experience, so a lot of Algebra I consists of practice, practice, practice. I tell the students that it's like learning to play a musical instrument: to get anywhere with it, you have to practice your scales. However, you can't learn to love music with scales alone, so I’ve made sure to intersperse fun activities in our class as well. We’ve had an "underwater scavenger hunt" (our class meets in the Fish Tank, which got decorated with origami sea creatures during the last Build Week) and sent each other secret messages encoded with linear equations. I often tell anecdotes from the history of mathematics to help make the subject come to life. But what the students seem to love most of all is when I give them a peek at more advanced mathematical concepts. (It turns out that Proof School students are always eager for more cool math!) For instance, while talking about Euclidean distance, we also discussed the taxicab metric, in which circles are squares. And as an application of the rules of exponents, we learned about the Ackerman hierarchy and got acquainted with some numbers that are so big that they make your brain want to explode.
Many of my students hadn't finished prealgebra in their old schools. This class has therefore been extremely challenging for them, as it amounted to leaping over a year or more of the middle-school math curriculum. However, being Proofniks, the students all rose to the challenge and never lost their good humor. Now that they've started their journey toward abstract mathematical thinking, there is no turning back. Of course, the journey does not end here: they will be practicing the skills they learned in this class for months and years to come. Still, I am honored to be the one to have given these kids their first glimpse of the great art of al-ğabr wa’l-muqābala.