I once saw a bumper sticker that claimed lotteries were “a tax on the innumerate”, meaning that most of the people who gamble on such do so because they don’t really understand the mathematics of basic probability (chance). It does seem to be alarmingly true that a great number of people don’t have a good understanding of odds. Sure, some people simply gamble for the gaming aspect, but casinos aren’t getting rich off folks like my grandma who got together with friends at each other’s homes once a month to chat and play penny-ante poker — they’re in business to make money off those who keep thinking that they’ve figured out some kind of “system” or that they’ve some kind of special “luck” or who are addicted to gambling.
There are some really odd ways the human brain works against reality, especially when it comes to understanding probabilities. The brain likes to find patterns, even when they aren’t there. People think that if there’s a pattern even if it’s only there for a short period of time, that things are not really random. Actually, “random” events do include patterns — it’s just that the patterns don’t repeat. (One of the ways we can tell that someone is creating false data is the lack of these pseudo-patterns — the data is too evenly random, which is an unlikely event.) When people see patterns that don’t really exist as true patterns, they think that they have found some kind of underlying truth or system, and assume that will enable them to win long-term. Unfortunately, that almost never happens.
People also engage in “magical thinking”, such as when they believe that the dice “remembers” and therefore assume that it will either roll the same way again, or that a certain number combination “has to” happen. Sorry, it doesn’t. When you roll a die again, the odds of a certain side coming up are still 1/6 (on a 6-sided die). Odds are also multiplicative; the odds of getting two 5’s is 1/6 * 1/6 or 1/36. In our schools, probability is a textbook chapter that happens during pre-algebra mathematics, after the students have learned how to do calculations with fractions. Unfortunately, we invariably have some students who although they can perform the calculations correctly, still refuse to believe in the true randomness of the universe.
There are other mathematical concepts that some people struggle with, such as negative integers. Apparently some people continue to struggle with these concepts through adulthood, as indicated by a recent story from the UK about lottery scratch cards:
Tina Farrell, from Levenshulme, called Camelot after failing to win with several cards.
The 23-year-old, who said she had left school without a maths GCSE, said: “On one of my cards it said I had to find temperatures lower than -8. The numbers I uncovered were -6 and -7 so I thought I had won, and so did the woman in the shop. But when she scanned the card the machine said I hadn’t.
“I phoned Camelot and they fobbed me off with some story that -6 is higher – not lower – than -8 but I’m not having it.”
Like Mark C.C., I find the line “but I’m not having it” really tickles me — as thought mere obstinacy can change the whole of mathematics! To be sure, to some people it’s not intuitive that -6 is lower, smaller number that is further from zero than is -5. Perhaps the wording of the lottery cards was part of the difficulty; the frustrated lottery player also explained,
“I think Camelot are giving people the wrong impression – the card doesn’t say to look for a colder or warmer temperature, it says to look for a higher or lower number. Six is a lower number than 8. Imagine how many people have been misled.”
When I have worked with students on this topic, we have used two different visual examples to help clarify things. One of those is using distance above or below sea level, and the other — perhaps surprisingly, in light of this news piece — is using the thermometer. Both the distance and temperature examples make use of vertical number lines, which seem to help disengage the usual secondary-school student’s distaste for ordinary horizontal number lines that are associated with little kids’ maths. It’s also no surprise that -3 is 3 below 0, or that -7 is two degrees lower than -5, or that if you are 5 feet above sea level and the bottom of the ocean dockside is 12 feet below sea level, that your anchor or fishing line would need to go 17 feet down to touch bottom.
Another thing that most students don’t see the utility of is “absolute value”; the absolute value of +10 is 10 and so is the absolute value of -10. What’s the point? ranted one student. What indeed! Absolute value is simply the distance from zero in any direction, which makes more sense when graphing and looking at X, Y and Z coördinates, as opposed to two directions on a number line.
Closely related to these are the difficulties people have with time zones. Greenwich is the 0 point of our circular number line, and the date line is where the two ends of the time zones meet. (Personally, I’ve always found these sorts of calculations to be much easier using the 24-hour military or train clock.) If you go west, the time gets earlier (subtract the number of time zones) and if you go east, the time gets later (add the number of time zones). If it’s 9:09 PM or 21:09 in New York City (Eastern US time zone), then it’s 2:09 AM in London (five hours later), and 7:09 PM or 19:09 in Denver (Mountain US time zone, two hours earlier). So how do time zones relate to absolute value? It’s always five hours later in London than in New York, and it’s always two hours earlier in Denver than in New York. It doesn’t matter what time of day is on the clock, whether it’s morning, afternoon or evening.
And no, despite the joke, crossing the International Date Line doesn’t affect when your time-release medication is going to wear off. The stuff lasts four or twelve (or whatever) elapsed, real-time hours since you took it. When I was given my hormone-replacement therapy medication (post-op from a hysterectomy), I was to put on a fresh patch twice a week. “Oh,” I said, after thinking for a few seconds, “that would mean Wednesday evening and Sunday morning.” My doctor beamed at me in delight — I had correctly halved the seven-day week!
What’s the use of learning maths? Oh, lots of little everyday things … like how to not call your overseas pals at 3 AM, or why lottery tickets are not a good way to make money.