TAKS Item 8 leads TEA to flawed reasoning

TAKS Item 8 Leads TEA to Flawed Reasoning

The Texas Education Agency (TEA), on August 6, 2003, issued a press release regarding a problem in the Math section of the TAKS exam, a state exam for 10th graders in Texas. The press release was disturbing for reasons the TEA apparently did not consider. In fairness to the panel that reviewed this question, it is possible that there was deep and meaningful discussion of the true nature of the error that was committed, as well as the consequences of making various decisions. If so, the press release revealed none of it.

The article states, “It was discovered … that item eight … could have been read in such a way that the question had more than one correct answer.” The Houston Chronicle (August 7), in turn, removed the qualifying phrase and simply reported that the problem “had more than one correct answer.” Here is the question:

This is a straightforward question that cannot be interpreted in more than one way; neither is it possible for an octagon to have more than one perimeter. We can definitely rule out any possibility that there is more than one correct answer.

However, it appears that we are not to worry about that. What is important, it turns out, is how students approach the problem, rather than what is correct. According to the Houston Chronicle, “the test-makers thought students would use the Pythagorean Theorem to find the answer, which the agency originally said was 36 centimeters…. After reviewing the question, the agency realized that had a student used trigonometry, the answer could be 27 centimeters, which was another choice.” So, the TEA decided the fair thing to do was give full credit for any answer.

Now excuse me, but, while different methods can result in slightly different answers due to rounding, this is a glaring inconsistency. Math educators rightly place a lot of emphasis on developing “number sense,” and the math educators in charge of writing a statewide exam should have enough “number sense” of their own to realize that there is something very wrong here. These "perimeters" show a 25% deviation (or 33% depending on which you use as a base). Different methods and rounding errors don’t produce variations like that.

What is really disturbing here is not that there was an error on the test, nor that a faulty question was eliminated. Mistakes happen. The TEA says there have been only two errors in 23 years on the TAKS; if true, that is an exemplary record. What is disturbing is the educators’ apparent indifference to the mathematical issue, i.e., What Really Went Wrong Here? A cavalier acceptance of two widely divergent answers on a simple perimeter problem, justified by the fact that two "correct" approaches to the problem produced these answers, just illustrates the kind of “math illiteracy” that so many educators complain about. Do we really want our future engineers and architects to believe that the perimeter of some octagon could be either 27 or 36, and it doesn’t matter which you use, since both are derived through "correct" methods? And, taking this to its logical conclusion, we’ll just accept 41 or 18 as well! It really doesn’t matter; after all, if "correct" methods give different answers, you may as well just guess! Take any answer you like!

Another concern is the expectation that students will use specific methods to solve the problem. In fact, because this is a multiple choice problem, it is not necessary to calculate a precise answer. It turns out that a simple estimation procedure that requires neither the Pythagorean Theorem nor trigonometry could have been used. An elementary student could do it. The perimeter of an octagon can be approximated by the circumference of a circle. An inscribed circle with radius 4 will slightly underestimate the perimeter. Taking a diameter of 8, times 3 for pi (rounding down just makes the estimate lower), gives 24. Thus the perimeter must be a little more than 24. The only reasonable answer on the list is 27. Done in five seconds, no calculator required. Still not sure? Let’s try a circumscribed circle, using a radius of 4.6. This will overestimate the answer, so let’s just round up to 5. Then the diameter is 10, which, multiplied by pi, gives approximately 31.4. There can be no doubt that the perimeter must be between 24 and 31.4, so the only possible answer is 27.* (This agrees with the trigonometric approach cited.)

But wait, using the Pythagorean Theorem gives approximately 36! How can this be? We just found that the answer has to be less than 31.4! To diagnose the problem requires a little trigonometry and a closer look at the diagram given. If a right triangle has a leg of 4 and a hypotenuse of 4.6, the other leg is approximately 2.27, which, multiplied by 16, gives the answer of 36. But what is the angle between the first leg and the hypotenuse? Using the cosine inverse function, we get approximately 29.6 degrees. If this were a regular octagon, the angle would be exactly 22.5 degrees. (These dimensions would be about right for a hexagon, which requires the angle to be 30 degrees.) So, the angle is over 30% too large, which accounts for the fact that the perimeter is inflated by over 30%. If the leg was intended to be 4, the hypotenuse could have been given as 4.33 instead of 4.6 (yielding answers between 26.509 and 26.528 using different methods--now that's the sort of variation you expect). Interestingly, the press release never suggested that there might be something wrong with the drawing.

Since the given figure is geometrically impossible, there is really no solution to the problem. Unfortunately, "None of the above" was not a choice. Given the answers available, the only thing a reasonable person could do is mark 27. But wait, the TEA spokesperson says, “It was just a coincidence that 27 was an answer on the test. We always would attempt to just have one answer.” Hmmm. Math illiteracy carries the day.

Dr. Dwight Galster
Asst. Professor of Statistics
South Dakota State University

References:
The Press Release
The Test
The Houston Chronicle "TAKS plays Solomon, rules all answers right," August 7, 2003

*Alternatively, the perimeter can be estimated by using a square. The perimeter of the octagon must be less than that of the square pictured here, which is (apparently) 32.

On the other hand, the perimeter must be greater than that of the square shown here. This time we won't escape using the Pythagorean Theorem, which gives a side of approximately 6.5, and a perimeter of about 26.

So, we have a "proof" that, if this octagon exists, its perimeter must be between 26 and 32.

Another approach using circles, rather than employing a "squeeze theorem," would be to suggest that the perimeter must be close to that of a circle with radius about half-way between 4 and 4.6. The circumference of a circle with radius 4.3 is about 27.018!

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I retract much of what I said above--because Mark Loewe is right. I made the crucial error of assuming a point that appeared to be in the center was in fact the center, when it is not labeled as such.