Maths is about facing ambiguity, not avoiding it

Every so often, an expression like this goes viral on Twitter:

6 ÷ 2 (1 + 2)

What’s the result? It’s ambiguous.

There are two interpretations, depending on which order of operations you use. Whenever these expressions goes viral, there are heated arguments about which interpretation is correct.

Seeing these arguments reminds me of the school stereotype “maths always has one right answer!” – and how unhelpful it is. People want to find the singular answer, but there isn’t always one to be found. Like everything else, maths has plenty of ambiguous problems, and learning to deal with this sort of ambiguity is a key part of a mathematical, problem-solving skillset.

When faced with an ambiguous problem, you can do one of two things:

  1. Guess between the available options, or
  2. Try to find more information to aid your decision.

Both can be appropriate responses, depending on the context, and learning to choose between them (and how to go about them) is a skill.

For example, when I’m writing software, there are often aspects of the design which haven’t been explicitly defined by the person who wrote the spec. I have to decide how to how to approach those gaps in the spec, and how that affects the software I’m writing. I might guess if it’s a cosmetic detail, but ask for more information if it’s something security-critical.

There are lots of things that might affect whether I guess or ask for more detail, including:

If I guess, I should be able to explain and justify whatever option I end up choosing. If I try to find more information, I should be able to ask the right questions to resolve the ambiguity, and make sure I’m getting the details I need. These are both skills in their own right.

If we only ever look for a singular answer, we’re just avoiding ambiugity – we’re not addressing it. Problem solving often involves working with incomplete information, and we’re better off practicing this skill, not pretending problems will always be neatly specified. “There’s always one right answer” might be true in the artifical problems of maths textbooks, but it breaks down in the real world.

This post was originally a thread on Twitter.