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Is this what you think math looks like? A book with many sections, each that has example problems followed by homework?
If that is your mental image of math, you probably use math to decide how to spend money, but you probably do not use math to earn lots of money.
What does the math for making much money look like?
Is it hard?
Could you do it?
Would you like any of those jobs?
Let's find out.
We start with a quick overview of the kinds of math people use to make lots of money.
You will probably notice that very few of them have every appeared in a math textbook like the one pictured above.
You may also notice that these kinds of math tasks seldom require more math than you know. They do often require a huge about of careful thinking.
Are you a boss? An employee? Either way if you have a job that pays well you often will need to use math to make good decisions.
Imagine you are trying to pick the right parking spot, apartment, or spouse. You need to look at a bunch to establish a baseline to make a good decision, but each that you look at without picking will be taken by someone else. How many do you use to explore the options? What is the balance between looking and leaping?
How to do these: Make a table listing the number of options (starting at 2) and the chance you get the best one if you leap at the obvious right time (which starts with leaping at the second of 2, which is the best 50% of the time). As the rows go down the percentages will grow closer and closer to the answer: about 37%.
But what happens if we know some information that reduces the number of options we need to look at before establishing our threshold for leaping? Or if there is some chance we will be allowed to pick an option we passed over earlier? A whole family of optimal stopping problems exist and are very important as businesses navigate endless options and decide which to take.
Imagine you are trying to sell your house. Each day we get one new offer from a buyer. We know how much it costs us to delay the sale a day. We know tomorrow's offer will be within one of seven dollar amount ranges, and we know the likelihood it will fall in each range. How many days should we keep the house on the market? This is a different kind of balance between looking and leaping.
How to do these: Make a table to find the expected value of a new offer. Divide it by the cost of waiting one day.
But what if we have less information about tomorrow's offer? Or the cost to us to keep the house on the market is not constant? A whole family of cost-benefit analysis problems exist and are very important as businesses decide "this or that?" or "now or later?".
Imagine you live in a city with many restaurants. When it is time to visit one, do you try a new one hoping it will become a new favorite, or return to one you already know you love? Or imagine a company wants to balance funding the research of new inventions and innovations with paying for its normal operations. How do we allocate our time or money when we do not know the chance that a new item is amazing or a waste of time?
How to do these: Start by giving each option a high value to represent that it might be amazing and you would regret not trying it once. Each time an option is tried, adjust its value (probably down) to show how well it lived up to your optimistic hope. As you keep trying the next option among the highest values, the entire spread of values slowly sinks like an air mattress unevenly deflating. Eventually the best options are revealed by their values that stop decreasing, and during the entire process you have minimized the time and money spent on options you are likely to regret.
But how do we adjust the values each time we try an option? This clearly depends upon how much we learn about an option by only trying it once. A whole family of regret minimization problems help businesses decide how much to explore new options based on the consistency and informativeness of exploration.
A similar situation happens when shopping websites simultaneously show different prices to different customers to actively explore what people are willing to spend (including at different times of day, and based on where the shopper lives). This is called A/B testing and is used not only to test prices but to determine the effectiveness of ad placements, fundraising efforts, website designs, and many other tools of business that are using math to shape society.
Are you a boss? A team lead? An office manager? A planner? Jobs that pay well involve using math to make good plans.
Slightly similar to regret minimization are scheduling tasks, such as a college trying to decide which classes to offer each term, and when during the week to schedule them. The key insight is again to try to measure and minimize the regret felt by students if a class they want is either not offered that term or offered at a time they cannot attend, but the math used will be quite different.
Similar tasks involve a store picking how much shelf space to give each item this month, a restaurant deciding what daily specials will minimize food spoilage, or a project lead planning which tasks to work on when the team cannot hope to finish them all that week.
Businesses have many flavors of scheduling tasks, most of which have mathematical guidelines for optimization.
How to do these: There is no "typical" scheduling task. We need to describe each carefully and then decide how to make a series of regret minimization decisions.
Imagine you are planning your wedding. How many people should you invite? One factor is time. Consider...
If you double the number of wedding guests, the time required for everyone to get through the buffet line will double. This is called linear growth. It happens because each new guest only interacts with a fixed thing (the buffet).
But if every guest will want to hug every other guest, now they must all interact with each other. Time time required for these hugs increases as quadratic growth. (This is similar to how doubling a pizza's diameter will increase its area by four, because the length and width interact with each other.) If you had enough guests, this hug time would be much more significant than the buffet line time.
What if your family had some strange tradition where a game was played at each wedding: someone would start by saying a one-word compliment about the bride, then the next guest would say a two-word compliment, then the next guest would say a four-word compliment, and so on? This game takes exponential time because each step is double the previous step. Your family does not do this, because even if each word only took one second, 30 guests would need over 400 years to play the game!
In the business world there are lots of trends. Learning to quickly classify trends and identify which are linear, quadratic, exponential, or one of the few other common types of trends is important for many jobs. Metaphorically, if you have a thousand customers the buffet line becomes a non-issue if there is any hugging. And if you accidentally create an exponential time process, you should notice and fix it before your boss does!
How to do these: Exposure! Similar to learning to identify the ethnicity of a meal by the smell of its spices, once you have spent time dealing with different types of trends you develop a reliable intuition.
A related issue is sorting. You could sort books on a shelf by one at a time starting at the left, comparing each with the ones already sorted, and if needed taking it of the shelf and placing it where it goes in that "sorted so far" bunch to its left. Or you could sort the books by taking all of them off the shelf and placing them back on one at a time among the already reshelved books. Which method is usually faster? Is there a third method that would be expected to be even quicker?
Another related issue is searching. How should your department organize its file cabinets, bookshelves, and shared computer drive so that employees can quickly find what they need? Hm...this seems tied to sorting if the employee must put back the item when they are done.
How to do these: Exposure again! It turns out sorting and searching tasks fit into certain categories. Both of the methods of bookshelf sorting described above are the same category, for example. Once you have been taught the categories, and have spent time dealing with examples of each, matching new sorting and searching tasks with these categories is straightforward.
All the above issues were "people versus nature" types of problems. The world is this way. So what shall we do about it?
When we are instead striving against other people to get the best result, mathematicians use the term strategies. A few situations and dynamics are so common in the business world that their names should be familiar: zero-sum games, information cascades, the Prisoner's Dilemna, the Tragedy of the Commons, etc. As with trends, jobs that use math to make money tend to involve noticing when a business situation resembles one of these, modifying the situation if that's bad news, and using the result to adjust priorities and plans.
How to do these: Exposure again! There are only a few situations and dynamics whose strategies keep popping up in the business world. Learn to recognize them and you are set.
Business health requires knowing what stuff is worth. Unfortunately, price tags and salaries involve guesswork and finagling. Any job that pays well involves trying to decide how much time or money different tasks or items are worth.
Would you rather be given $100 now or $200 a year from now? How much does the value of money change when it is inaccessible for a while: invested in a bond, loaned to a bank customer, or spent to construct a new factory?
Most people realize that inflation is a part of this issue. What other factors are always involved? What factors only apply to certain businesses, situations, or commodities?
How to do these: Different types of businesses have quite different tools to determine future values. After all, jewelry, tractors, handguns, and cars all change in value quite differently. An acre of strawberries has limited times for harvesting and replanting, but an acre of lumber forest can be harvested next year as easily as today.
Today's price of a share of a company's stock is a guess. No one really knows how much the entire company is worth. (We do know how many shares are out there, so if we knew that entire company worth we could divide.)
How reliable is this guess? What do we really know by seeing the company's sales, costs, and profits? By counting the land, buildings, and equipment it owns? By looking at historical trends?
Hopefully we know a lot. The company has an intrinsic value about which knowledgable investors would have close agreement. The small disagreements explain why the stock price wiggles a bit, but seldom dramatically. The financial tools used to create a meaningful measure of intrinsic value are called fundamental analysis.
But maybe that's hogwash? The psychological value theory says real life is way too vague and uncertain, so any attempt to calculate an intrinsic value is a waste of time. Instead focus on the beliefs and behavior of the crowd: today's stock price is valuable only as a group consensus about the company's worth, and the big question about tomorrow is how the crowd's consensus will change. The financial tools used to create a meaningful prediction of the crowd's future behavior are called technical analysis.
The same issues appear when we try to find the value of a car, antique painting, employee, or grocery shelf with fifty types of cereal.
How to do these: Both theories of value have some useful applications, but neither is complete by itself. The types of analysis involve a lot of bookkeeping and hunting for patterns in charts, but no math harder than the compound interest formula or an expected value table.
A related topic is when hindsight reveals a price bubble. The psychological value theory was clearly correct about Dutch tulips in the 1630s, the South Sea Company in the 1720s, the American stock market boom of the 1920s, and the internet bubble of the early 2000s. What allowed savvy people with increasingly better financial tools to get swept up into these disastrous crazes? Why did they believe certain justifications for the bubble's high prices--and how can we avoid being duped by similar justifications today?
Imagine that most people think the price of a collectable toy will go up, but you are convinced it will go down. How could you make money off that?
Your sibling owns 10 of the toys. You ask to borrow them for three months. Your sibling agrees, but wants to charge you $1 each. You pay her, and are currently down $10.
Then you immediately sell them for their current price of $30 each. Now you are up $290.
Three months later you buy them back, and as you predicted the price has dropped down to $20 each. So you spend $200 to get ten, hand those to your sibling, and end up with a $90 profit.
That type of borrowing the thing to sell it as a reverse investment is called a short sale. What would have happened if your prediction was wrong, and the price went up to $35? Do short sales have more or less risk than a normal investment?
How to do these: Basic arithmetic. When using the formula percent change = change ÷ original be careful if it is not obvious which values to plug in.
Next imagine you own 10 of those collectable toys, but are less certain than most people that their price will go up. You sell ten options to buy for $2 each, with a $30 price and one-month duration. These are like coupons that give the purchaser, anytime during the next month, the opportunity to buy one of your toys at today's price. Your hunch is correct and the price goes down for the next month. Hooray, you made $20 for selling coupons no one used.!
But what if the price went up to $40? All ten coupons would be used. You would miss out on a $100 profit, instead having only the $20 of coupon sales. And each coupon customer could immediately sell the toy for an overall $8 profit.
Wow, each coupon customer spent $2 to earn $8, which is a 400% profit! In contrast, a normal investor would have spent $30 at the start of the month to earn $10, which is only a 33% profit. Option coupons really mess with risk and reward!
What would an option to sell look like? Is it equivalent to a short sale?
When should a business sell or buy real investments, and when should they use options instead? Does the answer change if a rival business is making the other choice? (Maybe we are back to the Tragedy of the Commons here?)
Tangentially, is investing a zero-sum game if short sales and options allow people to make money off stuff that other people own? Is this why the stock market keeps going up over the long term?
Can you explain the formula that describes how many toothpicks it takes to make a row of boxes? (The three dots represent vagueness: we are letting the last row of the table represent a line of n boxes, which we cannot really draw unless we know what number is plugged into n at the moment.)
This problem is pretty easy. Did you get y = n + 3? Many jobs involve this same thinking but with more complicated patterns.
Often there is the extra complication of change being caused by more than one variable at the same time, so observations or experiments must be carefully arranged to identify the effect of each variable on its own.
For example, in 1990 the formulas for basal metabolic rate (BMR) were updated by Mark D. Mifflin and Sachiko T. St. Jeor. For men, the formula is:
Men's BMR = (weight × 4.55) + (height × 15.88) − (age × 5) − 161
Two scientists named J. Arthur Harris and Francis G. Benedict had earlier confirmed that a useful estimate for BMR only needed to include weight, height, and age. But how important the three variables were was iffy. Mifflin and Jeor looked at lots of patterns to find better values for the numbers multiplied by the weight, height, and age. Yet the basic idea was like the table of toothpick boxes: find the patterns and use them to get an equation.
After World War II it looked like lots of people would have jobs in electrical engineering and rocketry. Those were the exciting and promising careers that appeared to show a new direction for math in society. Those jobs do involve trigonometry, calculus, linear algebra, and imaginary numbers, which became the core of "college level" mathematics.
Those "college level" math topics are indeed used frequently by many jobs in electrical and chemical engineering, aerospace, economic modeling, actuarial science, and astronomy. But it turns out that society did not need as many people in those careers as anticipated after WWII.
Many other STEM jobs also need to know those "college level" math topics, but only for occasional use. So it is certainly true that mastering traditional "college level" math can lead to a high paying career. But many, many more people are earning lots of money without "college level" math topics using the types of math topics listed above. (For example, America has as many marketing managers as it does electrical engineers, and more than twice as many market research analysts.)
America has lots of accountants, and they make decent money.
But if an accountant learned a bit more math, he or she could go into operations research, databases, or information research and earn more money using computer software!
So using a spreadsheet or using a similar data crunching program can indeed be the foundation of a nice career. But aim higher!
The English language is terrible about only having nice words for opposites and extremes. We say people are tall or short, even though most people are in between. We say test scores are high or low, even though most test scores are in between.
Similarly, a statistic can summarize a lot of information into a single number, which may or may not be useful for a particular decision. Two baseball players' batting averages tells you something about those players, but not necessarily which is widely regarded as the better player. Two cities' average incomes tells you something about those cities, but not necessarily which has better paying jobs for most people.
People who use math to make much money might say words like "high" or "low" when at a meeting, but in their head (and on a slide or handout for everyone to see) is a histogram or some other visualization of the entire spread of values.
A teacher trying to improve teaching cannot succeed by thinking about "high" and "low" test scores because most test scores are in between. A business trying to improve sales cannot succeed by thinking about "high volume" and "low volume" items because most items are in between. And so on.
Trends tend to be more important than the current values. The best mental images involve changes over time.
Here is an example where you can use a slider at the bottom to move from 1799 data to 2018 data.
People who use math to make much money are used to picturing moving charts when they think about data.
Thinking about averages can be just as unproductive as thinking about extremes. Averages do have uses. But for most decision making they are often an unhelpful way to hide the true spread of data behind a single number.
For example, why would anyone think the histogram above could be condensed in a meaningful way into a single number? We could find the average test score, but it is difficult to invent any practical decision for which know would help the teacher to improve the class, or the students to learn better.
Most statistics try to summarize several types of information, and to do this must decide which types to emphasize more or less than others. For example, the United Nations Human Development Index rates countries' prosperity by including averages for each country's income, life expectancy, and education in a way that gives equal weight to each of those three aspects of life.
Now imagine a health care business deciding which country to next expand into. This business might want a similar but different statistic that gives much more weight to life expectancy than income and education, perhaps doubling or tripling the life expectancy part of the calculation. Or that business might measure other aspects of health besides life expectancy (such as hospitalizations, deaths from certain causes, or money spent on health care) and build their own summary statistic. For this decision it is obvious that using math can help the business make a good choice, but it is not obvious how to get started.
People who use math to make much money are used to distrusting statistics understand the ways to handle them properly, and can judge when a certain statistic is useful for a particular decision.
When driving you care about where you are as well (your position) and also your car's speed and direction (your velocity). The business world is also in motion. People who use math to make much money are good at focusing on both position and velocity.
A company's managers might worry if sales are high but decreasing. A food cart's owner might rejoice when popularity of a new high-profit item on the menu is still low but increasing.
For some issues position dominates. For other issues velocity dominates. Sometimes both are equally important.
And we are not just talking about values. An items whose position seldom moves that starts moving erratically or quickly is very different from another item with the same current velocity that is always moving about. As with chart thinking, trends matter most.
In his book Noise, Daniel Kahneman begins by describing four targets at shooting competition in which two rifles had their sights adjusted badly.
His important point is that from the back of the papers the two left-hand targets look similar, as do the two right-hand targets. From the back of the papers we can only see how closely grouped the shooter's shots were, not whether the hits were centered or not because the rifle's sights were adjusted badly.
The math term for sloppy shooting (left versus right targets) is noise. You might have seen it represented as "error bars" when graphing points. Noise is not intentional, but comes from limitations when measuring (or measuring improperly). Often noise relates to the habits of mind already discussed. For example, a business that assumes all its customers have one average income will do a lousy job offering the products its customers want, and their sales data will look like numbers all over the place until someone realizes they shoud instead divide their customers into several income group and then look at the sales data for each group separately.
The math term for consistently aiming off-center (top versus bottom targets) is bias. This is not about prejudice, it is simply about focusing on the wrong place. A business that is consistently doing the same things, but those things are not working well, has a bias problem. Remember when Starbucks appeared and turned a cup of coffee from a 5¢ addition to diner food into a several dollar dessert disguised as a beverage? Many coffee shops tried to keep selling coffee in the traditional way and went out of business despite being good at their obsolete business model. But many other coffee shops loved the change, and also learned how to take advantage of customers willing to pay so much more for a cup of coffee.
There are techniques for dealing with data that help unravel noise and bias. Not all people who use math to make much money need these techniques. But they certainly need the habit of noticing when data has noise, bias, or both before making decisions and plans.
What is the difference in economic impact between a stimulus package that costs $1 billion and another that costs $1 trillion? How much farther away is Saturn than the moon?
How is increasing a likelihood from 0.1% to 0.11% both a ten percent change and a hundredth of a percent change? What actions are sensible precautions to avoid a 0.01% annual chance of fatal injury?
It can be tricky to think about very big and very small numbers. The types of errors we tend to make change, and the reasons for making those errors can surprise us. People who use math to make much money have tricks appropriate to their type of work for handling very big or small numbers accurately and meaningfully.
People have different levels of understanding for math subtopics. We can even have different levels of understanding for different example problems and situations in the same math subtopic. For example, many people understand basic probabilities with cards but the Monty Hall problem breaks their brain.
There are ways to talk about math to help people move to a deeper level of understanding. These communication skills are often different from those we use when talking about non-STEM topics.
The use and pace of communication when cooperating on mathematical group work is different from other types of group collaboration.
There are ways to take good notes for math ideas that can be different from other types of note-taking, especially if your notes will be shared with other people.
When we believe a statement is true, how do we convince someone else that it is true? In most parts of life we use persuasive language skills called rhetoric. Mathematical communication can include some rhetoric as icing on the cake, but mostly involves listing logical steps. (Sometimes, such as when finding the error in a calculation, it only involves steps.
Here is an example. I claim (perhaps facetiously) that this statement is true: American white women should get married between the age of 20 and 24.
You and I could certainly create lots of rhetorical arguments for or against that claim. It is the type of claim that could be intelligently discussed without any mathematical reasoning at all. (Think about this claim a bit now, please.)
Here is a chart. Notice the four underlined data points are the same set of women. They were in their early 20s in 1999, their late 20s in 2004, their early 30s in 2009, and their late 30s in 2014.
In 2004 someone from this group of women might have asked, "Where did all the good men go?" Well, roughly half of that group of women got married during the past five years, so we would expect they took at least half the good men. Probably more, since the women getting married first would be picky instead of marrying random guys.
We can criticise the steps used to think about this chart. The chart is out of date. What makes a "good" man anyway? Can those men even be identified when they are younger?
But notice how your thinking about the claim changed after you read about the chart. Once data and logical steps are on the table, they exert a magnetic pull on our reasoning and rhetoric.
