Grade 3 Math Multiple Choice Questions
Jennifer Downey
Now that the project to upgrade my old multiple choice applet to a more modern and collaborative format is underway (see this server-side demo and this javascript/wiki demo, as well as the discussion here), I thought it would be a good time to collect my own personal opinions and thoughts regarding how multiple choice quizzes are currently used in teaching mathematics, and on the potential ways they could be used in the future. The short version of my opinions is that multiple choice quizzes have significant limitations when used in the traditional classroom setting, but have a lot of interesting and underexplored potential when used as a self-assessment tool.
In principle, it would seem that the unambiguous and precise nature of mathematical statements would lend itself well to the multiple choice format; in contrast to some other disciplines of knowledge, many questions in mathematics do have a single and objective correct answer, with all other answers being agreed upon as being incorrect. With a multiple choice quiz, a student can be tested on such questions in an objective manner; indeed, the grading for such quizzes can even be automated to be done by a computer or scanning machine. As long as the question was phrased unambiguously (and that the solution key is correct), the grading of such quizzes is less subject to dispute than other means of examination. As a final positive, the multiple choice format is extremely familiar to virtually all college students (who have probably had to have taken standardised multiple choice tests as part of the university admission process) and so the rules of the quiz require very little explanation.
On the other hand, the multiple choice format, as it is currently used in maths exams, has a number of serious weaknesses which, in my opinion, render it inferior to other examination options for most upper-division maths courses, although there are ways to remove the most glaring defects of the format. Perhaps the most obvious problem is the zero-tolerance approach to mistakes, which can distort the relationship between aptitude and credit: a student who had the right approach to a question, but made a single sign error or misunderstood the question slightly, could lose all points for a question, whereas a student who had no clue whatsoever what to do, and is simply guessing randomly, could manage to earn credit for a multiple choice question by pure luck, which is much harder to achieve in other examination formats. (Admittedly, one can mitigate this problem by keeping the questions simple and unambiguous, and ensuring that the incorrect answers obtainable from sign errors and the like are not given as one of the alternatives.) Another issue is that multiple choice quizzes are more susceptible to certain types of cheating and corruption than other examination formats, since the answer key is easy to copy and use, even by students who do not actually understand the material. (This particular problem can be guarded against to some extent by shuffling the questions separately for each student, though this of course makes it more difficult to grade the quiz, or to provide a solution key afterwards.) A third problem is that if the student arrives at an answer that is not among the options listed, this often encourages a rather tortured and not particularly logical fudging of the computation in order to arrive at one of the listed answers, which is not a good habit to instill in a mathematician.
I have discussed my reservations about the use of multiple choice quizzes in classroom examinations, particularly in upper-division mathematics courses. On the other hand, I do feel that such quizzes can play a very useful supporting role in self-examination for such courses, particularly with regards to foundational material (e.g. definitions or basic rules of calculation). I will illustrate this with a hypothetical course in high school algebra, though the point is certainly applicable to more advanced mathematical courses.
Such questions can address quite directly (and with immediate feedback) whether one has any misunderstanding on this specific point, without needing the live intervention of a lecturer or teaching assistant. (Ideally, an automated quiz should not only respond immediately as to whether the selected answer was true or false, but also to explain what the error was in the latter case.)
Another interesting possibility is to use multiple choice quizzes to explore specific mathematical problem solving tactics, which is an issue which is only indirectly addressed by most examination methods. For instance, in a single-variable calculus course, one could focus on integration tactics, using questions such as these:
As a European currently applying for graduate school in the US I was surprised to find out I needed to take multiple choice tests like the GRE subject math exam at this stage in my education. Do you think the benefit of having a standardised test to asses all candidates, which is also easy to administer, outweighs the various drawbacks mentioned in your post?
I suppose algebra lends itself more easily to this sort of feedback more readily than other areas of mathematics, blurring the line between quizzes and tutorial software. For what seems to be a good example of the latter, check out Theory Y Algebra.
In addition there are the types of meta-thoughts the student could have: For example in maths olympiads you could ofthen get an idea how to approach the question by using the fact that this was an olympiad question, i.e. it had to have a unique answer and usually there is a cute way of finding it. This often ruled out many dull things one would have tried otherwise.
Also, often the possible answers give something away: In question 1, anybody who actually reads all the possible answers and has only heard the waring one will remember that -sqrt(y) has to be considered as well once he sees this is a possible answer. If there were a problem where at some point x^2=y had to be solved for x and the possible answers would not suggest that one has to take care of the sign many more people would forget about -sqrt I would believe.
Similarly, question 2 is a no brainer since obviously only one answer can be correct and that one is obvious. One does not have to think about signs of squar roots to find x=y^2 as wrong. It would be better if it were not known in advance that only one answer is correct and some answers need several answers to be ticked.
My most vivid memory is solving all the statistical questions in a chemistry exam using calculus only to be asked by the professor afterward how I came to arrive at non-statistical solutions. Wondering what kinds of idiots I was being taught by led me to change majors.
One student can get the correct answer by guessing without even looking at the question and a student who studied hard for the exam who had a solution amounting to short bond paper but failed to correctly answer the question since he had the wrong sign for the final answer.
Typically they are going for the fact that these are all squares and the next number is 49. However this has always struck me as a poorly thought through question (despite the fact that variations of it are frequently recurring). f(x)=x^2 is an arbitrary polynomial, and each of the other answers will corespond to evaluation of some polynomial at consecutive integers.
The first thing that jumps out at me when i see sequence (*) is that they are all squares, but then again I spend a fair amount of time thinking about number theory. Who am I (or rather the guys at the college board) to say that this is the best answer? Is this really what college admissions should be based on?
I challenge my disadvantaged high school Science students, and before them my university and high school Math students, by giving them both short quizzes (5-10 questions) which they can master, and longer (also tricky) multiple choice tests with 20-30 questions that almost none of them can complete in time. The latter are to force them to develop better strategic test-taking skills, such as triage.
When will mathematics be viewed as it should: a very cognitively nourishing spectrum of opportunities and a way to overcome many life puzzles? This cannot be the view if one is bound by perhaps faulty imaginations of another party in the name of the examiner. Mathematics should be a liberating rather than an enslaving and thought restricting discipline. It should be a bedrock of scientific disciplines as well as arts by setting the mind free to manipulate ideas and prove results. This encourages innovation and adventure and is a recipe for industrial revolution in any economy.
hi! I am MORWO lIKINJIYE FROM TANZANIA. I AM INTRESTED WITH THE DESCRIPTION ABOUT MULTIPLE CHOICE IN MATHEMATICS GIVEN ON THIS PAGE. PLEASE CAN I GET A MORE ACADEMIC ARTICLE CONTAINING THE EXPLANATIONS ABOVE? KINDLY HELP A MY RESEARCH ON MULTIPLE CHOICE IN MATHEMATICS.
MY EMAIL IS grm...@outlook.com
I'm creating a math quiz and can get equations and math symbols in the body of the question, but I can't figure out how to put equations or math symbols in the answer choices, using fill in the blank question. I was trying to put in the infinity symbol and can't figure it out.
Unfortunately, as far as I know, you cant directly type math symbols like pi or infinity into the fill in the blank questions. The only way I have been able to achieve this is to type it up on something that can type these symbols, such as Word, and then copy/paste into the answer choices.
It's true that multiple choice questions support the Rich Content Editor, but that is not supported for fill in the blank questions, and some math people are not okay with converting everything into multiple choice.
c80f0f1006