How much is every other?

On a recent math test for Swedish 7th graders, one question asked: How many percent is every tenth? (Hur många procent är var tionde?) You’d think the answer is a straightforward 10%, but is it? Several students have had difficulties with that particular question. Why?

In order to arrive at 10% you have to assume infinity. The question requires you to imagine a situation where you can pick every 10th apple (for instance) out of an infinite amount of apples. Hence it requires a certain level of abstraction that isn’t necessarily intuitive, because in reality no one has an infinite amount of apples to chose from.

In fact, the expression every Nth can correspond to a whole number of varying percentages unless you have been taught to assume infinity. For instance, every 4th equals 25% only when the total amount is either four, eight, twelve, any other number divisible by four, or else if the total is (the hypothetical) infinity. If the total is, say, five, then picking every 4th apple may only give you 20%. That is, { 1, 2, 3, 4=pick, 1 }, leaving you with four unpicked apples, namely, the first three and the one remaining after the one you picked. If the total is seven, every 4th equals a mere 14.28%, i.e. { 1, 2, 3, 4=pick, 1, 2, 3 }, or one out of seven.

If the
total is:
1 2 3 4 5 6 7 8 9 10 11  … 
then
‘every
4th’
equals:
0% 0% 0% 25% 20% 16.7% 14.3% 25% 22% 20% 18.2%  … 

An important point here is that every Nth is not the same as a Nth. While a 4th always equals 25%, irrespective of the assumed total, every 4th does not, simply because every Nth is unit-based, meaning it counts only whole units. That is, you don’t cut up any "remainders" as you would if you were to chose a 4th of all apples.

I don’t have any statistics showing how often students experience difficulties with this seemingly uncomplicated question, but I know that at least some do. This is undoubtedly a tricky question, if not a trick question. Answering it in the expected way is in any case a cultural feat (i.e. you have to be taught to assume the abstract concept of infinity) just as much as it is a logical or mathematical one.

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4 Comments

  1. Chrono from Gallifrey Base said,

    Monday, October 5, 2009 at 19:41

    It’s interesting how the wording of exam questions can cause problems. If one imagines that the examiner is asking a “trick” question, one can get into a real tangle. Often, the required answer is clear (to the educators at least!) from the course specification and the level at which the students are studying. You would not expect younger students to think too deeply about this, but at an advanced level, such considerations might even be expected.
    On this specific example, I agree there is a difference between every 10th and a 10th.
    On the other hand, as the question asks specifically about per cent, then it could be argued that, as this refers to per 100, then every 10th is indeed 10 if one has exactly one hundred to count.

  2. jfmaho said,

    Monday, October 5, 2009 at 20:50

    Hey Chrono! I’ve just spent the evening watching The Chase, an old favourite with a bitter-sweet ending.

    In theory, you should be correct, and I think you are. When the correct (expected) answer is explained to the student who didn’t give an answer, or gave a faulty answer, s/he will invariably have an aha-experience.

    I was trying to analyse why some students have difficulties understanding the question. I did some substitute teaching during last spring, and had some math classes. One of those was during a test which included that same question. Some of the 7th graders struggled with it, and it was a bit frustrating to watch and not being able to help out (it being a test and all).

  3. Chrono from Gallifrey Base said,

    Tuesday, October 6, 2009 at 10:20

    I did a small scale study some (20?) years ago on the subject of the language used in physics exam questions. In particular, how the choice of, sometimes, a single word can make a big difference to the students’ understanding of the question, and subsequent score in the answer. We wanted to look at how students for whom English is not the first language are disadvantaged by the use of certain culturally-weighted or unusual words that have no relevance to the physics or mathematics being tested. We used their pure maths scores, their physics scores and their English language scores. We found that, for students with similar (good) maths scores, their physics score was reduced in a way that seemed to depend on their English score. We were assuming (and tried to arrange) that the maths questions and answers were minimally language dependent and gave a good indication of the students’ raw ability. The variation in the physics scores could then be correlated with the language score.
    It would have been nice to do some more work on this. Maybe someone has by now!

  4. jfmaho said,

    Tuesday, October 6, 2009 at 10:43

    That sounds very interesting.

    Educational disadvantages due to ethnic/linguistic background is a major field of study, especially in multilingual countries and countries with sizable immigrant populations. I’m only familiar with that research in broad outlines myself.

    I know there was a major debate in the 60s/70s following some controversial research by Basil Bernstein, who looked at how working class children in the UK were socialised differently than other children, mainly as a result, he argued, of them having less access to Standard English. This also affected their potential to properly take advantage of education, which is largely mediated through Standard English.


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