kitchen table math, the sequel: Advanced Math for Young Students

Monday, December 15, 2014

Advanced Math for Young Students

Phil Keller sent me a copy of his beautiful new book Advanced Math for Young Students.

(Here's his blog. And here's an early review!)

I told Phil, as soon as I laid eyes on the book, that it has the look and feel of an authored work, nothing like the many-authored splendor of a typical commercial textbook.

Plus (and I have to say this) the cover and pages are creamy and smooth, a throwback to the physical beauty of books pre-crash (although of late I've had the sense that decent paper may be making a comeback).

The book is so compelling I may have to buy a second copy to work through myself so the other one can stay on the coffee table.

From the Introduction:
For 26 years, I have been a high school physics teacher. I work in an excellent, well-regarded high school and I have been fortunate to have many talented students who soak up all the physics I can teach them, and more. But every year, I also teach students who struggle to master the topic, despite their great efforts and mine. And I know from discussions with colleagues, both within my school and arose the country, that we are not the only ones struggling. There is something getting in our way. Maybe this will seem obvious to anyone who has struggled in physics, but here's what I think: I think it's the math.

Physics applies math. It's all about finding relationships and solving the puzzles that the laws of physics present. For the most part, this work is done in the language of mathematics, and more specifically, the language of algebra. So to be comfortable learning physics, a student has to be fluent in that language. Algebra cannot just be a memorized set of procedures for finding 'x'. It has to be a symbolic way of representing ideas. But for many students, that level of fluency is not attained unjust one year of algebra--which is all that many students have had when they start studying physics. It's no wonder that some struggle.

It is not only physic students who struggle. For even more than 26 years, I have been teaching students how to prepare for the math portion of the SAT. What I have seen over the years is that most students are not fluent enough in algebra to successfully apply algebra on the SAT. One goal of my SAT course is to teach alternative, non-algebraic approaches to SAT problems. It is also a major theme of my math SAT book, The New Math SAT Game Plan. And I will tell you something you may find surprising (or even distressing): on the SAT, these non-algebraic methods work very nicely. They won't get you to an 800, but they will take you pretty far. And even my top scorers report that they like to mix in the non-algebraic methods along with the standard approaches (which, as top students, they also know how to use).

The non-algebraic methods, however, won't get you very far in physics. In fact, a student who does not really learn the language of algebra is going to struggle in all later math and science classes: physics, statistics, computer science and beyond. That STEM door is swinging closed because one year of algebra class did not lead to sufficient fluency. So why spend only one year? Why not start earlier?

I am not saying every 7th grader should be in a high-school version of Algebra I. But I am saying that every middle school student should, over the course of the middle school years, start learning about and thinking about the ideas of algebra (even some ideas that won't reappear until Algebra II or Pre-calculus). These are ideas that take some time to ponder.


SteveH said...

My son liked PWN the SAT for math which has pushed the idea that SAT-Math is not a math test for quite some time. Doing well on the SAT might get you into the admissions bucket of a better college, but it says little about STEM preparation.

Advanced math in K-12 is more associated these days with the Art Of Problem Solving (AOPS) and the AMC series of tests. I can't say that I'm a big fan of that (competition versus exploring advanced material), but it's the name of the game these days. Some colleges ask for your AMC scores.

Glen said...

So why spend only one year [on algebra]? Why not start earlier?

To be fair to our local administrators who are trying to eliminate as much "advanced" math as possible in the name of Common Core, they repeatedly claim that algebra is so fundamental to future studies that they want to make sure it is well-learned and not skimmed over.

I strongly agree with them that getting algebra right is the key to success in all subsequent STEM studies. You can forget most of geometry and be embarrassingly shaky with your calculus and still become a successful practitioner, even professor, in many STEM fields (not physics), but if your algebra is shaky, you won't understand the explanations in most of your STEM textbooks. Algebra is less about solving equations than it is a language for thinking and communicating about how things are related, how they work, what has been discovered, what is being hypothesized.... It is a critical component of critical thinking.

Unfortunately, instead of "why not start earlier?", the position of our administrators is to delay everything else in order to continue pre-algebra and algebra longer. Even worse, the proposed changes implicitly suggest that if taking longer to reach and finish algebra is helpful to some, no one else should be allowed to move ahead, no matter how well prepared.

All should be delayed until everyone is ready.

Froggiemama said...

This post says pretty much the same thing as the letter from the Middlebury College professor I posted earlier. In that letter, the professor blamed the rush to AP Calculus for her students "shockingly weak algebra skills". There was quite a bit of disagreement from some of the regulars here.

The current post mentions "fluency" a lot, which on this blog tends to be conflated with the ability to quickly solve problems. But the poster also says "Algebra cannot just be a memorized set of procedures for finding 'x'.". I think he is arguing for something more. I think he is arguing for students to have the chance to fully internalize the language of algebra, so they really understand what they are doing.

The poster seems to be arguing that algebra should be started in 7th grade. That, however, assumes that students have a high level of mastery of topics such as fractions. My kids could have done it, but many kids even in this high performing school district could not. I do not think kids should be pushed into algebra until they have demonstrated mastery - not just the ability to quickly solve problems, but real understanding.

I like Common Core's emphasis on mastery of a smaller set of key topics, especially fractions, which appears to be a topic where many kids go astray.