Yes, they’ve done it! Three stupid DC-CAS questions out of three!

I will swear on a dictionary, or a copy of Moby-Dick, that, so far, I have only looked at three of the released DC-CAS items for the 8th grade in math. And, to be quite honest, each one sucked. Here is number three. Do you see what I see?

The question is really asking students to compare four 4-place decimals to see which one is the smallest. However, the company that wrote this felt it was necessary to make this into a “story problem” with a context.

The problem is that in the real world, this problem makes almost no sense. Especially if you have ever cut down or measured a tree.

It so happens that I have done both.

The first difficulty is that trees don’t have the same diameter or circumference at all locations. If you measure at, say, 2 inches off the ground, well, you are going to have a very difficult time measuring at all because there are lots of roots and rocks and plants in the way, and you will find it nearly impossible to hold your tape measure level unless you have a whole bunch of helpers. Plus: where exactly is “ground level”? It varies, especially on sloping ground. In general, the higher you go on a tree trunk to measure the diameter, the narrower the tree will be. For legal purposes, one measures the distance around the tree at “breast height.” But whose breast, and which part thereof? It would depend on the age of the person, among other things. Allow me to quote wikipedia:

“Diameter at breast height, or DBH, is a standard method of expressing the diameter of the trunk or bole of a standing tree. DBH is one of the most common dendrometric measurements. Tree trunks are measured at the height of an adult’s breast, which is defined differently in different countries and situations. In continental Europe, Australia, the UK, and Canada the diameter is measured at 1.3 meters above ground…. In the US, New Zealand, Burma, India, Malaysia, and South Africa, breast height diameter is measured at a height of 1.4 meters. Previously 4.5 ft (1.37 m) was used…. Ornamental trees are usually measured at 1.5 meters above ground. On sloping ground, the “above ground” reference point is usually taken as the highest point on the ground touching the trunk, but some use the average between the highest and lowest points of ground. If the DBH point falls on a swelling in the trunk it is customary to measure the girth below the swelling at the point where the diameter is smallest. The two most common instruments used to measure DBH are a girthing (or diameter) tape and calipers.

“A girthing tape actually measures the girth (circumference) of the tree; the girthing tape is calibrated in divisions of π centimetres (3.14159 cm), thus giving a directly converted reading of the diameter. This assumes the trunk has a circular cross-section, which is typically accurate for most plantation trees.

“Calipers consist of two parallel arms one of which is fixed and the other able to slide along a scale. Calipers are held at right-angles to the trunk with the arms on either side of the trunk. Precision can be improved on non-circular stems by averaging two caliper measurements taken at right-angles.”

I submit that the results allegedly obtained by Eva are ridiculous, and should all be rounded off to 2 decimal places. What’s more, in the real world, they would probably be expressed in centimeters (i.e., whole numbers) rather than decimal fractions of a meter, unless she has enormous old-growth sequoias and redwoods in her yard, which I strongly doubt. For one thing, despite what Wikipedia says, trees NEVER have cross-sections that are exact circles. Secondly, anybody who says they can get the circumference of a tree to an accuracy better than one centimeter is deluding himself/herself. But even if we pretend that Eva did so, and then divided by pi to get the diameters, watch what happens when we work backwards.

Tree #1 supposedly has a width, which I guess means diameter, of 0.5091 meters. If we multiply this by the approximation 3.14 for pi, we get a circumference of 1.59874 m. Now, that is obviously impossible, because nobody can measure anything that accurately with just a flexible tape measure. So, shall we assume that she measured the circumference to the nearest centimeter? That would be 1.60 meters, or 160 centimeters, or 1 meter 60 cm. Now, let’s go forwards and divide that by 3.14. If I do, I get 0.50955414, which Eva should have rounded off to 0.5096, not 0.5091. If I use a calculator’s built-in approximation for pi, something lik3 3.141592653589 instead, I will get slightly different answers, but the bottom line remains the same: her answer are probably impossible.

Perhaps Eva really got a girth (circumference) of 1.59 cm instead? In that case, her reported thickness (diameter) should have been 0.506369… which would round off to 0.5064, not what she allegedly wrote according to the writers of this stupid question.

Yeah, I know that my nit-picking won’t change what the correct answer is (which I shall leave as an exercise for you, dear reader), but my point is this: when the writers of tests make up contexts for these problems, shouldn’t the contexts actually make sense?

(I actually do have to measure things to 4 decimal places when measuring stuff for grinding and polishing telescope mirrors. And you know what? It’s extremely difficult, and takes very specialized and accurate tools, and a fair  amount of knowledge, skill, understanding of sources of error, and old-fashioned experience to measure accurately to 4 decimal places. A bit more understanding, knowledge, experience and skill than is demonstrated by the writers of this worthless test, on which the fate of so many DCPS staff members will ride. And a test which does virtually NOTHING to help teachers decide what needs to be taught, because the questions are so poorly written.)

Another remark: This question is a bit like a person saying that the sign that gives the age of the moon rocks on display at the local science museum is all wrong. The person reasons like this: ‘The sign says they are 4,000,000,000 years old (that’s four Billion years), but they were brought back to the Earth in 1971, and that was 39 years ago, so the sign should really say 4,000,000,039.’ They don’t realize that “four billion years” is merely an estimate made by some geologists based on the best evidence that they could find at the time, and is likely to be changed by many millions of years (but probably not by Billions) as better evidence comes in; what’s more, the first group of geologists probably gave their estimate of the age with a range of error. In other words, they probably said that their answer is ‘give or take 10%’, which would mean they would not be surprised at all if their date was off by 400,000,000 years either way. That’s plus or minus somewhere near FOUR HUNDRED MILLION YEARS!!! In comparison with that, 39 years is way less than the blink of an eye.

A foolish quest for spurious precision is the hobgoblin of little minds.

Published in: on August 5, 2010 at 8:17 pm  Comments (16)  

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16 CommentsLeave a comment

  1. I see your point. I agree that these questions do not have any real-life applicability. However, I think the most important thing to consider when reviewing a test question is whether the question aligns with the standard. What is the standard supposedly being tested here? Is it something related to ordering numbers with multiple decimal places? If so, then, even if the context is preposterous, I’d say it’s an “acceptable” (but not ideal) question.

    In other words, you ARE being too nitpicky.


    • If that’s all they want to test, then why put it into a fictitious context that doesn’t make sense
      And shouldn’t we be teaching students that mathematics actually makes sense, rather than the opposite?


  2. Meh. I don’t see the big deal. A reasonable DC student would just figure out that the question is asking what’s the smallest number.

    I think you’re 3 for 3 on whining about items that, while not perfect, seem pretty reasonable.


    • Smart!!! Keep up the good work! We need people with the energy and precision to keep up the pressure and expose this testing nonsense for what it is… the “haves” wreaking education for the “have not’s” …


  3. EG’s response reminds me of the MSPAP test given in Maryland back in the mid 90s to early 2000.
    The test set up “real life” situations. The one I “read ” for 8th grade science involved density, specific gravity, and mineral hardness.
    After several questions involving those three areas, the kids had to answer the question, if you found a rock, how could you determine if it contained gold.
    The kids were to answer that they would calculate density and specific gravity and compare those figures and the hardness number to the charts they were given.

    The best answer I read was “Take it to a jeweler and ask him, but don’t let him take it into the backroom.”


  4. Hey! New reader here. Love your blog 🙂

    At first glance, the critiques look a bit nitpicky, but if you consider that they’re supposed to be testing math here, then this might be pretty rough on a kid who’s good at math but has trouble reading. Imagine for a moment that you’re not a well-educated adult with the same cultural capital as the test-makers.
    -You’re reading the problem, chugging along until you get to the word ‘approximate’.
    -“Ugh,” you think, and you attempt to break it down. It may or may not mean anything to you when you’re done.
    -Now you’ve lost the meaning of the rest of the sentence. If you’re resilient, or if this is an early question and you’re not too burned out yet, you go back and re-read. If you just don’t give a crap, you might keep chugging through, though if you’re that kid, you probably aren’t so sure what “calculate” means either, and you may not care to expend enough mental energy to notice that it looks a lot like “calculator”, a word with which you’re familiar. Or maybe you don’t have enough facility with language to even consider looking for known words inside of unknown words.
    -Either way, by this point, you’re a bit tired. Hopefully, if you read on, you realize at the end that the other stuff was BS information that you didn’t need, and you answer the question correctly. But if this was the last question, or you’re bummed ’cause your friend’s moving, or you just noticed it’s snowing outside, or you’re hungry, you might just fill in a random bubble and move on so you can be done with this stupid thing and play with your pencil or take a nap.

    If that happens enough times, then you’ve failed. What looked like a math test is really testing 1) reading; 2) patience; 3) your skill at thinking like a test-maker (“What is this question *really* asking me?”).


  5. Dc_CAS 2010 school scores are up:

    Over all, Sousa saw scores shift from adv to prof and from basic to below basic
    exception was prof Math went up 9 points
    Looking at each grade level, Sousa 7th did the best
    reading Prof 28.92% to 45.21%
    Math Prof 31.33% to 54.79%
    But when you examine 2009 grade 6 to 2010 grade7 (ie the same kids, basically) they basically stayed at the same level of performance


  6. New reader here. Excellent posts (all three). Keep up the critique – you are on point!


  7. You and your acolyte, efavorite should study your lesson on false precision. Maybe together, you can come up with better versions of you cavils over DCPS tests score and revise some of the presentations there: Clue: The placement of the decimal point is NOT the key to confidence intervals.

    Yes, astronomy buffs are supposed to know deeply, beyond creed, the difference between precision and accuracy and to not exaggerate either. But, then, you’d have to know quite a lot about testing and error in cognitive measurement to know the difference between signal and noise coming from tests and test reports.


  8. ELLs might also be thrown by the use of distance and width, when they are more familiar with circumference. I think the use of distance is a little weird, you don’t normally talk about the distance around a tree unless you are talking about measuring around the whole yard. I worked at a program for students with learning difficulties, many of them were amazing at math but poor at reading. They would come up with so many accurate reasons why many of the word math problems couldn’t possibly make sense or give other reasons precisely because their knowledge of math and random facts was so extraordinary. I was thrown so many times, like they actually knew things like the average circumference of a take-out pizza.


  9. Much as the question distracts the 8th grader with a challenge to vocabulary (for some), readers should know that a significant portion of extensive year-long test preparation* is given to directing the student to eliminate extraneous information and get to essentials: find the smallest or largest number of a set. That says a lot about American education: teach testmanship to extract from distractors the nub of the problem. But, yes, in brief, the question requires connection between synonyms “width” for diameter — the blogger used “cross-section”, and ‘distance around” for “circumference.

    For all the nattering about real deviations from the idealized cylindrical tree, the blogger hasn’t noted that not so many adult US readers here mentally pictured four trees all about 19 inches wide. That, too, is irrelevant to solving the problem.

    *That includes four simulated full-length pre-tests.


  10. I love your blog but I have to agree with some of the commenters on this one. Your criticism of this question is awfully nitpicky.

    If a question says “you have these things of that width” it’s totally irrelevant that trees can actually have different widths. In this hypothetical question, they do not.

    It would be like arguing in a question about two trains leaving the station, well, in real life, trains stop sometimes, how do we know whether or not they stop?

    In any test like this, that is not the point. The point is to come up with the answer based on what you’re told. There is no reason this question would confuse anyone.

    I’m with you that these questions generally are not well-written, but don’t try too hard to find problems if there really isn’t one. It’s just a standardized test, if it’s clear what the intent of the question is, even if it’s not perfectly written, I don’t think there’s a real problem.

    As for your last one (the question with >90 and 100), though, that’s awful. I can’t believe anyone would write a question like that…


  11. I’d much rather have seen a which number is smaller question — if that’s what they want to test. Otherwise, what they’ve created is a reading comprehension/math sense puzzle. “Measured distance” then “calculated width” takes some parsing. Once you decide it means she “measured circumference” then “calculated an approximate diameter,” your number sense kicks in and you start to wonder if that can be right. If true, Eva’s got a bunch of massive trees in her backyard (Under DC law, they’re all officially “special”!) And if she reported her data to me in this manner I’d make most of the same points Guy did — especially the ones related to false/misleading precision.

    Basically, by converting this into a story problem, the testmaker risks confusing not only kids who have problems with reading comprehension generally, but kids who think carefully about mathematical issues (terminology, appropriate measures, scale).


  12. I spend all year teaching students to read the problem. Then the test comes and it teaches them NOT to read the problem.

    In my mind, it doesn’t even matter if the scenario is perfectly reasonable. If there is a lot of writing that doesn’t actually offer any useful insight or information for the problem, it teaches students not to read.


  13. Guy:
    Will you be looking at how many schools made AYP this tiem around as well as analyzing DC-CAS results for us? I would love to hear your take on the newly released results.Thanks!

    Candi Peterson
    (AKA The Washington Teacher blogger)


  14. Teachies makes an important point — which is that tests aren’t just evaluative tools; whether we like it or not, they perform a pedagogical function as well. (“Will this be on the test?”)

    This question says don’t read and think about the problem — just try to figure out what answer the test-taker is looking for and give it. It also tells you that the metric system and precise measurement complicate rather than simplify basic tasks. And that no one testing you can think of a real world example of why you should know this stuff.


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