Wikipedia:Reference desk/Archives/Mathematics/2009 January 5

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

January 5

Discussion for On-Line Algebra

How do you feel about the format of mathematics test; should they be multiple choice, open response, or a combination of both methods? Provide evidence to support your claims. —Preceding unsigned comment added by Johnny51R (talk • contribs) 11:47, 5 January 2009 (UTC)

Do you think this homework question is multiple choice,open response or a combination of both? Oh and answer in approved english (or what pases as that as education people keep inventing weird terms) as that seems to be more important than getting the maths right. And by the way I'd like to see more collaboration in answering which isn't tested in exams. Dmcq (talk) 13:28, 5 January 2009 (UTC)

Since this happens to be a topic which interests me, I'll include some arguments each way:

1) Pro-multiple choice:

a) Easy and quick to grade, especially if you use ScanTron forms (what, no article ?). Quick feedback to students is important in the learning process.

b) Objective. That means the teacher can't give better grades to students they like, or be so accused.

c) Allows students to solve the problem in any way they see fit, not just the way they were taught in class.

d) Reduces the likelihood of having a correct answer marked wrong. I once had an entire division test marked wrong because I'd answered correctly in decimals, rather than fractions. I suppose a particularly sadistic teacher could put both 10 1/2 and 10.5 in the answer list and only accept one, but that would at least make the students aware that they were being graded on the format of the answer, not just it's correctness.

2) Pro-open ended questions:

a) Teachers seem more likely to review this type of test afterwards, and show how the correct answer was determined. With some ScanTron tests I've just been told my grade, with no opportunity to challenge mistakes the teacher may have made or to learn from my own mistakes.

b) This allows the teacher to review the student's work to determine where they went wrong and let them know. This may be helpful to find each student's weaknesses and work on them. It might also help to identify learning disabilities, such as if the student is constantly transposing digits.

c) Partial credit can be given for answers that are almost right, like if they just got the sign of the answer wrong.

d) It's harder to cheat if you must show your work.

e) If a student gets a question right, and their work supports this, you can be sure they learned the material. On a multiple choice Q they may just have guessed right.

f) This doesn't allow students to work backwards. That is, they can't just plug in each of the possible answers to see which works. While this is a valuable skill for checking their work, students must also learn to solve problems on their own.

g) This is the only way to do some type of math tests, like proofs. You could list 4 different proofs and ask the students to select the correct one, but that's not the same thing as being able to write your own proof.

h) Tests often contain too much material for students to do a good job and double-check their work. This encourages them to learn to do things "quick and dirty", as opposed to "slow and carefully". The latter is what we should be teaching, at least in math, if we want to avoid spaceships crashing because of basic math errors. If the teacher must spend hours grading open-ended tests, they are less likely to make them too long for students to do properly in the time alloted. StuRat (talk) 17:05, 5 January 2009 (UTC)

It's Scantron, not ScanTron... but I made a redirect. -- Coneslayer (talk) 17:34, 5 January 2009 (UTC)

Thanks. Without the caps that could be taken as "Scant Ron", a web site concerned with the lack of Reaganomics in the current US Republican Party ? StuRat (talk) 17:49, 5 January 2009 (UTC)

3) Con-multiple choice

a) A monkey i.e. a total idiot can get on average a score of the reciprocal of the average number of choices in multiple questions just by picking at random. That makes low exam scores increasingly meaningless.

b) Some multiple choice questions invite multiple answers (e.g. "Tick every number that is a prime.") That raises a quandary about how to score when an answer contains both right and wrong "ticks".

I am not convinced by 2 f) above. Nothing stops a student investigating a problem "from both ends". Cuddlyable3 (talk) 11:57, 6 January 2009 (UTC)

I believe your 3a is similar to my 2e. For 3b, you have the same problem for such a question on an open ended exam, although if using an automated method for grading the multiple choice exams, they may not have any way to handle partial credit. In such a case that portion would need to be graded manually. As for working problems backwards, this is appropriate for some problems, like proofs, but not for others, like finding the solutions to a polynomial equation. If 4 answers are given to choose from, such a problem becomes trivial to work backwards, while solving it without multiple choice answers is not trivial. StuRat (talk) 20:46, 6 January 2009 (UTC)

I agree. My 3a develops the consequences of your 2e. You have developed 2f convincingly.

c) A problem that arose with automatic scoring of multiple choice math answers at a junior school where I taught was that students could use 3.16 or 22/7 as approximations to pi. They had not yet been taught about correct rounding of calculations involving pi. The problem was to accomodate the many different but acceptable answers which should include rounded and non-rounded versions using these or any better approximations (and also accept exact answers where pi is kept unevaluated as a symbol). Other irrational numbers could cause similar difficulty. (strikeouts by Cuddlyable3)

I've had a similar problem with multiple choice tests, where the number of decimal places wasn't listed in the question, and at least one answer was correct, but to a low number of decimal places, and there was also a "none of the above" option. For example: "What's the value of pi ? A) 22/7 B) 3.14 C) 3.1416 D) 3.1415926 E)None of the above". Of course, this isn't an inherent problem with all multiple choice exams, just with poorly written ones. Similarly, open-ended questions can have a problem where the answer is irrational and yet the number of decimal places is not specified. In such cases, I'd give as many decimal places as I, or my calculator, could manage. StuRat (talk) 19:16, 7 January 2009 (UTC)

I struck out some words in my post above because it described a problem with automatic scoring in general, not only multiple choices. One might even see an advantage in having multiple choices that "nudge" the examinee towards a particular answer form or resolution. StuRat, I would tick your "E" unless there are ellipses that I can't see after the digits in B, C or D. (What fun to lure students into the trap of believing 3.1415926 is closer to pi than 3.14... .) As for getting many decimal places from a calculator that's fine except for the last place. Calculators can't be relied upon to round that digit. Just push $10\div 3X3$ in that order on a cheap calculator and see whether it gets back to 10 or 9.9999999. Cuddlyable3 (talk) 19:59, 7 January 2009 (UTC)

c) Another problem with multiple choice questions is that they don't allow for multiple interpretations of a question. One I once had was "What's the perimeter of a 100 meter square yard ?". Not knowing if they meant 100 meters on a side, and thus a perimeter of 400 meters, or 100 square meters in area, hence 10 meters on a side and 40 meters in perimeter, I gave both answers. Had this been multiple choice, I might have gotten lucky if only one of the possible answers was listed and there wasn't a "none of the above", but I might not. StuRat (talk) 19:31, 7 January 2009 (UTC)

The heading says On-Line Algebra, so multiple choice seems the only choice, unless there is a live human teacher at the other end of the line. Cuddlyable3 (talk) 20:12, 7 January 2009 (UTC)