In an earlier post, I posted some word problems from a sample exam for Pennsylvania's new Common Core-inspired tests (the Keystone Tests). I found these problems vaguely troubling, but hadn't taken the time to figure out what was bothering me.

As Auntie Ann pointed out:

1) They're wordy.

2) They give you far more information than necessary. If kids really are supposed to understand this stuff, why does almost every problem write out the equation for the student? Why not have them generate it themselves? Shows a lack of faith that the kids *have* actually learned what to do.

Indeed, all cases, the expressions are set up for you. Furthermore, as I noticed upon going through these problems and doing them myself, no algebraic manipulations are necessary in order to solve any of them.

Here, again, are the problems, with my commentary below:

All that’s involved here, obviously, is a passive identification of the correct setup. Actively translating words into algebra is significantly more challenging.

------------

Here, not only is the equation already set up, but the variables are explicitly defined. All you have to do is map the equation to word problem to determine what the coefficients stand for, and then apply your knowledge of the conventional x, y ordering of ordered pairs.

------------

This one is not really a word problem: you can answer the question without even reading through the scenario. Since the four choices for x are all easily plugged into each equation with the resulting values for y easily calculated (from the second equation) and checked (via the first equation), the correct value of x can be determined by guess and check alone—i.e., by simple arithmetic—along with simple realization that the second equation is easier to start with.

------------

Here, you’re given the equations and told what the variables stand for. You can determine the answer by seeing which one of a handful of obvious whole number pairs works for the first equation. What works for the first equation, (4,3), also works for the second equation.

------------

Here, there’s just one variable, and what it stands for is obvious from the given scenario. Choices b-d look suspicious (it would be a coincidence if 185 is also the value of b; more likely, the 185s here are red herrings). Choice a is easily confirmed by plugging in 204 and doing some simple arithmetic.

------------

Here again, not only is the problem set up for you, but the variables are also identified for you. All you have to do is plug in the pairs of values in the different choices until you find one that works. Again, the problem boils down to simple arithmetic.

------------

Problem 18 involves passively picking out the correct expression; choices c and d can be eliminated instantly because the intervals are obviously too large, and choice b can be eliminated because the constant term (75 times 4453) is obviously too large. In problem 19, the equation is set up, the variables are defined, and the correct answer is readily determined by a quick inspection of the only plausible choices: c and d.

------------

Assuming an understanding of similar triangles and simple ratios, the answer to this problem is obvious (and the triangle diagram, as with the scenario in problem 12, is a pointless distraction).

------------

Here again the equation is already set up and the variables defined; one simply recognizes that x equals 0 when the machine is full, and that one can therefore eliminate x and solve for y.

Facility with algebraic manipulation is crucial for calculus. The idea that one can get sail through the Keystone algebra test by passively interpreting someone else's algebraic expression, and by plugging numbers into them, is deeply disturbing. What incentive is there left for algebra teachers to prepare students for higher-level math?