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Embarrassment
When Words Confuse Our Logic
Solving mathematically isomorphic problems gives the same result — or does it?
Posted October 8, 2020 Reviewed by Matt Huston
Take any two people and ask them to solve the equation “1 + x = 2”; chances are, both will understand more or less the same problem, and therefore, will arrive at more or less the same solution. In real life, however, problems are rarely presented as mathematical formulas. We have to solve them by understanding complex sentences, by reading between the lines—and this translation process often misleads us.
In real life, we don’t say things like “If A is true, then the opposite of B is also true.” Instead, we say things like “Small babies aren’t good at chess”—and take the risk that the listener might misunderstand some of the message.
Even outside math and science, in fields like journalism and politics, we’ve used formal logic to persuade and reason since the time of the ancient Greeks. However, expressing everything in mathematical formulas would be way too much extra work, and only in the rarest settings is it the most useful thing to do. This means that most of what we say, reason with and use to persuade others lives in text.
This is why it’s important to be good at solving word problems—and why it’s such a shame that most of us are not very good at it.
How old is the captain, really?
In a classroom setting, pupils often learn word problems as merely arithmetic tasks. We now know that many linguistic components impact a problem’s difficulty.
Standard word problems, the ones that have one clear solution, are the easiest ones to solve. But the problems we encounter in the real world aren’t usually standard word problems. Our world includes contradictions and often gives us problems with multiple solutions, or puzzles with no solution at all.
Do you remember the question about the captain’s age? In one study, 97 elementary students were asked a version of it that read: “There are 26 sheep and 10 goats on a ship. How old is the captain?” You might be surprised to learn that 76 of them attempted to give an actual solution to the problem.
How grownups get questions wrong
There’s a lot at play when we try to understand puzzles. Sometimes we make mistakes by incorrectly “translating” the text, and sometimes the calculation gets us. And even if we get both of those right, we still have to overcome our own biases.
It’s one thing when elementary students get calculations wrong in their classroom. It’s a much bigger problem when grownups make mistakes, especially if they work in a field where errors are expensive. Fraud detection, company valuations and business forecasts can go wrong because of similar misunderstandings and biases.
Investors often find themselves choosing between options such as these:
- Option 1: A sure profit of $5,000, or
- Option 2: An 80% chance of a $7,000 profit (with a 20% chance of receiving nothing)
Which one would you choose?
In an investment portfolio, the second option is the better choice. It has the higher expected value of $5,600 (for those interested, it’s “0.8 * 7,000 + 0.2 * 0”) compared to the first option’s $5,000.
Yet, many investors would still select the first option. It just sounds like a “safer bet”—and according to Nobel Prize winner Daniel Kahneman, people tend to be risk-averse when they think about prospective financial gains.
It’s not a simple translation
Research shows that formalizing word problems into equations is not a simple translation process. Turning sentences into numerical problems is done without a “translation” step between our “language brain” and our “math brain." Yet, we know that verbal cues can help the mathematical interpretation, and even small differences in phrasing can lead to significant changes in performance.
Compare the following two sentences from an experiment, also with elementary students:
- “Ben had eight marbles. Then he gave five marbles to Tom. How many marbles does Ben have now?”
- “Ben had three marbles. Then Tom gave him some more marbles. Now Ben has eight marbles. How many marbles did Tom give him?”
Almost all the children who were presented with the first sentence used a subtraction strategy: they deducted 5 from 8. For the second sentence, children counted up from 3 until they got to 8. Since most people can multiply quicker than they can divide, and add numbers faster than subtract them, text changes like this can have a big impact on the solution time.
The key to solving puzzles is understanding our own limitations. Real-life math problems are not standardized: they come with irrelevant information, missing or multiple solutions, and everything in between.
The next time you see a question, rephrase it to understand it better. Be suspicious if solving the new problem gives you a different result. Especially if the change you made was to calculate with marbles instead of dollar amounts.
