Math class is a special place. We've talked before about some of the special assumptions that are based into that context: teachers pose questions, students answer questions, all questions have answers, questions include all the necessary information, answers are usually "nice," problems can be answered with the tools students have (just been) taught, diagrams are indicative while the underlying true forms are perfect, etc.
Of course, not all math classes make these assumptions or leave them implicit, or are constant about which ones are in force, etc.
In this post, I want to pick up a thread related to the "one true answer" myth: problems that have multiple interpretations.
Example
You are driving from your house to a soccer tournament. The distance is 120 miles. For half of the trip, you drive 60 mph. For the other half, you drive 30 mph. What is your average speed over the whole drive?
Where's the ambiguity?
For the teacher who poses this problem, there is no confusion. Obviously, students are meant to calculate that it takes 1 hour to drive the first 60 miles and 2 hours for the second 60 miles. That means it took 3 hours for 120 miles, or 40 mph average speed.
The catch: what does "half of the trip" mean? As an alternative, it could mean half the time of the drive. If that feels contrived, consider the following natural statements about travel measured in time instead of distance:
- "The drive took 3 hours; we stopped for a snack half-way." In this case, time and distance are equally natural in normal conversation.
- "The flight took 6 hours; I read half the time and slept the rest." In this case, time is the more common metric, but it wouldn't be considered unusual for someone to talk about the distance they flew.
- "We were gone for 2 weeks, half at the beach, half visiting our cousins." Here, time is the natural metric, while it would seem strange to focus on distance. However, a vacation spent hiking the Appalachian trail or cycling across country would shift the balance back to distance.
Sources of ambiguity
I came up with four potential sources of ambiguity in math questions:
Things that
can be measured in multiple ways.
This extends the idea from “half a trip”
ambiguity about distance or time. J1 and I had a discussion a couple of weeks
ago where we measured chocolate bars and cookies using three different metrics:
mass, cost, utils. For example, which is more:
- 100 grams of chocolate that costs $2.00 and you value at 100 utils
- 80 grams of fresh baked sugar cookie that costs $2.50 and you value at 90 utils
In business, it is common to have to deal with
the ambiguity of whether “stuff” is measured in physical amounts or monetary
value.
Pronoun ambiguity
For example: Ellis had 10 strawberries. Ellis gave 4 to his father and he ate 2. How many does he have now?
Who is meant by each occurrence of "he"? In each case, it could mean either Ellis or the father which leads to 3 distinct answers: 6, 4, or 2.
I accept that this is an example of bad English, but we're in math class and never claimed to be masters of language (did we?)
Tense
ambiguity
In the prior story it could be that giving the strawberries away and
eating them happened before the state where Ellis had 10. Let's add some extra story context to make this alternative more clear:
Ellis still had 10 strawberries. He bought a pack of 16, but Ellis gave 4 to his father and he ate 2.
I think this alternative interpretation is more of a stretch, but I've seen cases where the uncertainty about when things were happening is more natural.
Assumption of scalability
Joe can bake 2
cookies in 20 minutes. How long does it take him to bake 4 cookies? 400
cookies? 4 million cookies? 4 quadrillion cookies?
I saw one math class question that involved writing books, a task which is very unlikely to
happen at a constant rate.
Your challenges
- Find other sources of ambiguity that can infect (or add spice to) math class problems.
- For N a positive integer, create a puzzle that has N distinct solutions based on (reasonable) alternative interpretations.
