New York State Math Exams Baffle Teachers and Students Facing the Tradeoff Between Breadth and Depth

Ambushed by the Exam
original artwork by Tavet Rubel, created for Dewey to Delpit

Last week, New York City public school students sat for the state math exams. The scores won’t be announced until the summer, but there is reason to expect a dip. State exams are not typically available for public viewing until a month or two after the exam is administered, but the board of ed took special precautions this year to ensure that every copy of the test-booklets were returned to their offices and that no information about the exam leaked. What I hear from teachers who administered the exam, however, is that the 5^th and 8^th grade tests were completely unprecedented, both in their content and in their difficulty.

The transition to the national Common Core Standards is not slated to begin until 2014, so educators I’ve spoken to are struggling to understand why this year's 5^th grade exam abandoned topics like decimals and percents that have traditionally been the meat and potatoes of that exam, and focused instead on difficult pattern and area problems, some of which lie outside the stated 5^th grade curriculum altogether.

The Sweet Spot: Balancing Conceptual and Procedural Instruction in Grade-School Mathematics

The purely procedural approach to long division.
I stole these colorful illustrations from this website.

A few years ago, I was working in a charter school where the sixth-grade math teacher was having a hard time teaching long division. (I myself, let it be noted, was having a hard time teaching anything at all.) I sat in on some classes, and it was easy to spot the problem. The teacher had reduced the process of long division to a series of five steps: divide, multiply, subtract, bring down, repeat, with a mnemonic, which I forget, to help remember them. This must have seemed like an easy-to-follow script when they were planning the lesson, but it’s deceptively complicated: you have to know which numbers to divide, multiply, subtract, and bring down and what to do with all those quotients, products, and differences once you get them. (See Appendix A, for an idea of just how complex this gets.) The students didn’t understand why they were doing any of these steps, so they found all that information extremely difficult to keep track of. Long division, taught this way, became a dull and intricate labyrinth, riddled with small procedural booby-traps to derail the unsuspecting scholar. Yet, pure procedural methods like this one seem to be prevalent in contemporary instruction—try Googling “long division” and see what comes up.

More recently, my girlfriend asked me to teach her long division, a skill she’d never gotten her brain around back in elementary school. Eager to show her that mathematics is logical and comprehensible, not arbitrary and byzantine, I dove into a thorough explanation of the inner-workings of the long-division algorithm, complete with diagrams and concrete examples. After ten minutes, she was frustrated

Concepts vs. Procedures – the Great Math Debate

In last week's post—read it before you read this one—I presented evidence of deep math concept deficits in four 5^th graders in inner-city Brooklyn. Math is America's weakest subject, according to the results of international exams,^[1] and it's traditionally the most feared and hated among students. There's a lot of debate as to why that is, and as with so many other issues, that debate tends to polarize around two camps. For years, I have found myself torn between those camps, but I believe the evidence that I presented last week offers a glimpse of an elusive middle-ground and a more nuanced approach to math education.

The Confusion Beneath Confusion: Brief Glimpses of Number and Quantity

For the past few months, I’ve been spending a couple hours each week teaching remedial math to four fifth-graders in inner-city Brooklyn. My assignment: help my students to develop a conceptual grasp of numbers and elementary mathematics.

Over the course of my first week or two with my charges, I gauged my students’ knowledge and understanding—or so I thought. In fact, I had discerned only the superficial: their addition tables were shaky, but they could perform two-digit column-addition; they could carry but were prone to mistakes; three of them could borrow, but again with errors; they could name the place values of two- and in some cases three-digit numbers; they could quickly add multiples of ten in their heads.

It took me several weeks to realize what should have been obvious: that that was a procedural assessment, not a conceptual one. Beneath those weak basic skills were fundamental gaps in my students’ understanding of number and quantity.