David Yourdon is a Content Developer at Knewton. He specializes in finding the underlying structure of GMAT math problems.
A few weeks back, I wrote a post that discussed Dan Meyer's visionary talk about the future of math education. Halfway through the talk, Meyer mentions a classic problem in which students must determine how long it takes to fill a tank with water.
Usually, textbooks give students all (or most of) the necessary pieces and then ask them to construct the puzzle -- that is, plug the numbers into a formula. Meyer, however, advocates doing away with all the information and simply posing the question: "How long will it take to fill the tank with water?" Students then have to figure out what they need in order to answer the question. This approach forces them to think patiently and creatively.
It should be no coincidence, then, that we at Knewton teach the exact same question in the Data Sufficiency portion of our GMAT course:
An empty rectangular tank has uniform depth. How long will it take to fill the tank with water?
- Water will be pumped at the rate of 480 gallons per hour (1 cubic foot = 7.5 gallons).
- The tank is 100 feet deep and 30 feet wide.
- (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- (D) EACH statement ALONE is sufficient.
- (E) Statements (1) and (2) TOGETHER are NOT sufficient.
Okay, so the question isn't exactly the same. Meyer's tank is an octagonal prism; ours is a rectangular solid. Details aside, both questions require the same level of critical reasoning.
Imagine that the prompt was pared down to simply, "How long will it take to fill the tank with water?" Give it a moment's thought. What do you need to know to answer this question? This is, at its core, what the Data Sufficiency section tests. You never have to compute the final answer on a DS question; you just have to know what it would take to find the answer.
Here, you need to know how quickly the water is flowing into the tank -- that is, you need a rate. Next, you need to know how big the tank is -- you need a volume. The prompt tells you that the tank is rectangular, with uniform depth, so its volume will be a product of its length, its width, and its depth. In total, then, you need four quantities to answer the question: (1) rate, (2) length, (3) width, (4) depth.
Statement 1 gives you quantity (1), the rate, but nothing more; it can't be sufficient. Statement 2 gives you quantities (3) and (4), the width and depth, but since it doesn't give you the length or the rate, it can't be sufficient, either. When you put the statements together, you've got quantities (1), (3), and (4), but you still don't have quantity (2), the length of the tank. Even together, the statements are not sufficient.
If you approach this question critically, you can polish it off in a matter of seconds. "I need four quantities. I see only three quantities. Answer choice E is correct." If you approach it passively, though, you get lost in the numbers and are far more likely to get it wrong. Nearly 25% of students do.
Surely, questions like this are rare. In general, standardized math tests and creative problem solving do not go well together. As we change the way we teach students by drawing them further into the conversation, we will also need to change the way we assess them. The sooner, the better.