## Sunday, April 6, 2014

### Word problems in a sample in Keystone Algebra exam--deconstructed

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.
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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.

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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.

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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.
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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.
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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.
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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.
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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).
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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?

Auntie Ann said...

To paraphrase J.K. Rowling:

“Surely the whole point of Algebra class is to practice algebra?” [said Hermione]

“Are you a NCTM-trained educational expert, Miss Granger?” asked Professor Umbridge in her falsely sweet voice.

“No, but—“

“Well then, I’m afraid you are not qualified to decide what the ‘whole point’ of any class is. Math-wizzes much older and cleverer than you have devised our new program of study. You will be learning about algebra in a secure, risk-free way—“

lgm said...

The test writers seem to be trying to isolate various skills, rather than pose problems that would derail students that are missing skills from preAlgebra and below.

I can solve the triangle problem much faster using the triangle than setting up ratios. It reminds me of an SAT math problem...if you understand, it's fast. If you resort to equations, its tedious and sucks up your time.

#16 you might want to rework. They are testing two things...the use of the calculator result vs the problem constraints, and the understanding of the inequality sign..both concepts from K-8 that should make this problem very easy, but one that ime some K-8 teachers don't make sure that their students learn.

Hainish said...

It's very likely that each of these questions was written to align to a particular standard. As a result, each on narrowly focuses on a very specific skill.

Katharine Beals said...

I'm intrigued by the idea that these problems are each measuring a particular skill rather than, say, the general skills of setting up word problems algebraically and manipulating the resulting expressions to solve find solutions. And it makes sense that the test-designers are attempting to align particular problems to particular Standards) to make sure that students who failed to master skills from pre-Algebra and below are able demonstrate what they can do.
However, it seems to me that there's a significant overlap in what these 10 problems measure, and that the skills they measure boil down to:
1. passively recognizing the correct set up of an equation
2. correctly plugging in numbers and doing the arithmetic
3. using estimation to rule out unlikely candidates
4. knowledge of ordinal pair conventions
5. understanding of the inequality sign
6. facility with arithmetic

This list of skills, unfortunately, includes few, if any, of the core skills of first year algebra.

Anonymous said...

These questions are too easy. Even if the equations were not given, I was pretty sure we did this type of questiosn as a 6th grader when I was a student in China more than 20 years ago.

lgm said...

Yes, anon, most of these questions are from PreAlgebra of that era in the US when students were placed by instructional ability, except back then they were expected to generate the expressions and equations from the word problem themselves.