## Sunday, August 9, 2015

### Equal? Ambiguous test questions

We have been looking at questions from a local exam contest. I thought two were worth discussing.

## Climbing a mountain

Kaew had been in the mountains climbing for the last four days and recorded her distances on this graph (the first one below). The other graph shows the climbing distances from Ploy.

Below the graphs are 4 statements by other people and we're asked how many of the statements are true. The statements are:
1. Weera claims: "Each day, Kaew climbed more than Ploy."
2. Piti claims: "Kaew climbed more than 10,000 meters, in total."
3. Mana says, "In total, Ploy climbed more than Kaew."
4. Choojai says, "Each day, Ploy climbed 500 meters more than Kaew."
How many statements are true?

I'm only really interested in statement 2, did Kaew climb more than 10,000 meters?
In math class, we often ask students to enter a world of perfect accuracy and exact logic. Usually, the questions in this world only involve "nice" numbers.
Entering that framework,  we see that Kaew climbed 1500m+3000m+2000m+3500m = 10,000m
Because she climbed exactly 10,000 meters, she did not climb more than 10,000. Piti's statement is false.

## Why is this unsatisfying?

We have two issues. First, how accurately did Kaew record her climbing distances? Second, how accurately are we reading the graph? Surely there is uncertainty here that prevents us from having any confidence in assessing Piti's comment.
My sense is that the data is accurate to the nearest 500 meters (so each day's distance is ±250m). Based on that, I feel confident that Kaew climbed between 9000m and 11,0000m over the four days.

## Taking our medicine

In another question, we are told that a pack of pills comes in a 3x3 grid, pictured below. Each day, the patient has to take 3 pills. We are told that once a pill has been taken, the patient can't take a pill from the same row or column. How many ways are there to take the medicine?

Interpretation
How did you interpret the question? I guess you thought that each day the patient can take only one pill from each row and column, that we are asked how many ways the patient can take this medicine over three days. Right?

Looking carefully at the question, I came up with five interpretations:
• no two pills back to back from either the same row or column on the same day vs no two pills in sequence from the same row or column (possibly over two different days) vs no two pills on the same day from either the same row or column (choice 1)
• how many ways to take the medicine on the first day vs how many ways to take all 9 pills over three days (choice 2)

## Putting it in perspective

I'm glad that we had ambiguous questions on the contest paper. It undermines the authority of the exam, the idea that there is only one right answer, and the importance of answering questions correctly. Instead, we are left with a collection of prompts that can lead us to interesting thoughts and interesting discussions.