I was making a test the other day and it got me thinking about question types. Which is fairer for students: a multiple-choice question, or a matching question? Of course, if a student knows the correct answer, then either question is equally fair since they will get full marks regardless of the question type. Instead, I was imagining a student who is uncertain of the correct answer and might need to make a guess. For example, consider the following 4 multiple choice questions:

Which of the following is a red fruit? a) Blueberry b) Banana c) Strawberry d) Orange Which of the following is a yellow fruit? a) Blueberry b) Banana c) Strawberry d) Orange Which of the following is a blue fruit? a) Blueberry b) Banana c) Strawberry d) Orange Which of the following is an orange fruit? a) Blueberry b) Banana c) Strawberry d) Orange

Instead of having a ridiculous number of multiple choice questions, we could condense the above question set into a single matching question:

Match the following fruits to their colours:

A. Blueberry B. Banana C. Strawberry D. Orange 1. Red 2. Orange 3. Blue 4. Yellow

Personally, I prefer the matching question; it feels clean and quick. Since most of you know your fruit colours, this isn’t a particularly difficult task. However, imagine if you had to take a guess on the colour of a blueberry. The problem is, if you incorrectly identify the blueberry as red, then no matter what you pick for the strawberry, your response will be incorrect.

Further, if you guess that strawberry matches with yellow, now the banana cannot pair with the correct colour. This cascading failure made me think that the matching question type might be setting students up for failure. Instead of going with my gut, I decided to investigate using math.

In the follow analysis, we consider the worst case scenario: a student who knows nothing about fruit colours and guesses at random. First, we consider the multiple choice question set. Since each question has 4 options, the student has a probability of getting a question correct. Given that there are 4 questions, this means that the student on average will get one question correct. Thus, they will earn 1 mark. This makes intuitive sense to those of us who have guessed on multiple choice tests before.

Second, we need to analyze our matching question. Unfortunately, it is unclear how to proceed. The student has a probability of matching the blueberry with the colour blue. But what if the student matches the blueberry with the colour red? This will affect the match on the strawberry, and so on. To make our analysis easier, we will consider a visual representation of the matching problem:

A possible student answer is:

In this example, the student scored 2 points because they had 2 correct matches. Now all we have to do is draw all possible matches and determine how many points each match gives. How many possible answers do you think there will be? We can start with answers that earn 4 points:

Not surprisingly, there is only one way to get everything correct.

Next, how many answers earn 3 points? If you think about it for a bit, you will realize that there are none. It is impossible for a student to score 3 points. How about earning 2 points?

We find 6 possible ways of earning 2 points. I will leave it as an exercise for the reader to confirm that there are 8 possible ways of earning 1 point. There are 4 ways of choosing A, then 3 remaining ways of choosing B (since A has already been matched to something), 2 ways of choosing C, and 1 way of choosing D, which gives possible ways of answering the matching question. Since we have found ways so far, this means there are possible ways of answering and earning 0 points. Summarizing in a table:

We calculate the average score by multiplying each number of points by the number of ways and then dividing by the total number of ways:



Wow, we get the same result as the multiple choice set! According to the above calculations, if you guess randomly on a matching question, you will get an average of one correct answer.

So which setup is fairer? Apparently, they are equal if we are measuring equality by the average score by guessing randomly. As a teacher, this gives me confidence that including matching questions on my tests is not harsh or unfair. However, the fact that it is impossible for a student to score a 3 on a matching question does raise further questions…