If mathematicians addressed traffic problems

How would mathematicians solve traffic? Here are the suggestions from a 2020 book:

Photo by Life Of Pix on Pexels.com

All drivers need to be on the same navigation system. Cars can only be efficiently rerouted if instructions come from one center hub. One navigation system rerouting some drivers does not solve traffic jams.

Parking bans. Many urban roads are too narrow and cannot be physically widened. Traffic-flow models can indicate where parking spots should be turned into lanes.

Green lanes. For cities that want to increase electric car use, special lanes should be created for electric cars, providing an incentive for their use.

Digital twins. Traffic demands and available infrastructure can only be balanced with digital modeling that creates an entire “twin” of existing roadways. The software will be “an extremely useful thought tool in the hands of transport engineers.”

I have not read the book and this description is not long but it seems to depend on both understanding current and possible traffic flows through modeling. Often, Americans typically get more lanes added to roads – which then tend to fill up because there is more capacity and/or populations continue to grow.

I wonder how modeling would fit with other values underlying road and traffic decisions. A few examples:

  1. It might be better to have a centralized traffic and navigation hub. Is this technically feasible, would all car makers want to participate, and would there be privacy concerns?
  2. The politics of providing special lanes, whether for electric cars or high occupancy vehicles or bicycles, can get interesting. Americans often think the roadway should be for all users as opposed to particular users.
  3. The road system we have is the result of not just prioritizing efficiency but a whole host of actors and forces that includes privileging single-family homes (and generally keeping them away from busy roads) and highways in and out of major cities.

Fighting math-phobia in America

The president of Barnard College offers three suggestions for making math more enticing and relevant for Americans:

First, we can work to bring math to those who might shy away from it. Requiring that all students take courses that push them to think empirically with data, regardless of major, is one such approach. At Barnard — a college long known for its writers and dancers — empirical reasoning requirements are built into our core curriculum. And, for those who struggle to meet the demands of data-heavy classes, we provide access (via help rooms) to tutors who focus on diminishing a student’s belief that they “just aren’t good at math.”

Second, employers should encourage applications from and be open to having students with diverse educational interests in their STEM-related internships. Don’t only seek out the computer science majors. This means potentially taking a student who doesn’t come with all the computation chops in hand but does have a good attitude and a willingness to learn. More often than not, such opportunities will surprise both intern and employee. When bright students are given opportunities to tackle problems head on and learn how to work with and manipulate data to address them, even those anxious about math tend to find meaning in what they are doing and succeed. STEM internships also allow students to connect with senior leaders who might have had to overcome a similar experience of questioning their mathematical or computational skills…

Finally, we need to reject the social acceptability of being bad at math. Think about it: You don’t hear highly intelligent people proclaiming that they can’t read, but you do hear many of these same individuals talking about “not being a math person.” When we echo negative sentiments like that to ourselves and each other, we perpetuate a myth that increases overall levels of math phobia. When students reject math, they pigeonhole themselves into certain jobs and career paths, foregoing others only because they can’t imagine doing more computational work. Many people think math ability is an immutable trait, but evidence clearly shows this is a subject in which we can all learn and succeed.

Fighting innumeracy – an inability to use or understand numbers – is a worthwhile goal. I like the efforts suggested above though I worry a bit if they are tied too heavily to jobs and national competitiveness. These goals can veer toward efficiency and utilitarianism rather than more tangible results like better understanding of and interaction society and self. Fighting stigma is going to be hard by invoking more pressure – the US is falling behind! your future career is on the line! – rather than showing how numbers can help people.

This is why I would be in favor of more statistics training for students at all levels. The math required to do statistics can be tailored to different levels, statistical tests, and subjects. The basic knowledge can be helpful in all sorts of areas citizens run into: interpreting reports on surveys and polls, calculating odds and risks (including in finances and sports), and understanding research results. The math does not have to be complicated and instruction can address understanding where statistics come from and how they can be used.

I wonder how much of this might also be connected to the complicated relationship Americans have with expertise and advanced degrees. Think of the typical Hollywood scene of a genius at work: do they look crazy or unusual? Think about presidential candidates: do Americans want people with experience and knowledge or someone they can identify with and have dinner with? Math, in being unknowable to people of average intelligence, may be connected to those smart eccentrics who are necessary for helping society progress but not necessarily the people you would want to be or hang out with.

Rare events may happen multiple times due to the law of truly large numbers plus the law of combinations

Rare events don’t happen all the time but they may still happen multiple times if there are lots of chances for their occurrence:

Improbability Principle tells us that we should not be surprised by coincidences. In fact, we should expect coincidences to happen. One of the key strands of the principle is the law of truly large numbers. This law says that given enough opportunities, we should expect a specified event to happen, no matter how unlikely it may be at each opportunity. Sometimes, though, when there are really many opportunities, it can look as if there are only relatively few. This misperception leads us to grossly underestimate the probability of an event: we think something is incredibly unlikely, when it’s actually very likely, perhaps almost certain…

For another example of how a seemingly improbable event is actually quite probable, let’s look at lotteries. On September 6, 2009, the Bulgarian lottery randomly selected as the winning numbers 4, 15, 23, 24, 35, 42. There is nothing surprising about these numbers. The digits that make up the numbers are all low values—1, 2, 3, 4 or 5—but that is not so unusual. Also, there is a consecutive pair of values, 23 and 24, although this happens far more often than is generally appreciated (if you ask people to randomly choose six numbers from 1 to 49, for example, they choose consecutive pairs less often than pure chance would).

What was surprising was what happened four days later: on September 10, the Bulgarian lottery randomly selected as the winning numbers 4, 15, 23, 24, 35, 42—exactly the same numbers as the previous week. The event caused something of a media storm at the time. “This is happening for the first time in the 52-year history of the lottery. We are absolutely stunned to see such a freak coincidence, but it did happen,” a spokeswoman was quoted as saying in a September 18 Reuters article. Bulgaria’s then sports minister Svilen Neikov ordered an investigation. Could a massive fraud have been perpetrated? Had the previous numbers somehow been copied?

In fact, this rather stunning coincidence was simply another example of the Improbability Principle, in the form of the law of truly large numbers amplified by the law of combinations. First, many lotteries are conducted around the world. Second, they occur time after time, year in and year out. This rapidly adds up to a large number of opportunities for lottery numbers to repeat. And third, the law of combinations comes into effect: each time a lottery result is drawn, it could contain the same numbers as produced in any of the previous draws. In general, as with the birthday situation, if you run a lottery n times, there are n × (n ? 1)/2 pairs of lottery draws that could have a matching string of numbers.

Rare events happening multiple times within a short time also tends to provoke another issue in human reasoning: we tend to develop causal explanations for having multiple rare events. These multiple occurrences can still be random but we want to know a clear reason why they occurred. Having truly random outcomes doesn’t mean outcomes can’t be repeated, just that there is not a pattern to their occurrence.

The Simpsons slips in a math lesson now and then – and it probably doesn’t change much

Check out this listing of The Simpsons episodes that feature math lessons. Having seen some of these episodes, I remember a few of these moments quite clearly. Yet, while it is clever that the writers dropped these in, how many of the viewers noticed? We could use these examples as a sign that The Simpsons is an erudite show, one often derided as an animated comedy with little redeeming value that typically punches above its weight in terms of cultural references and ideas. But, if the viewers don’t notice or care, what is the point (beyond making some critics happy)?

It sounds like we need an experiment where viewers are asked to watch these episodes and see if they spot these math moments. Or, we might set something up to see whether viewers of these episodes, compared to viewers of other episodes, learned something more. The answer, I suspect, is that including the math doesn’t change most viewers.


Argument: The Myth of ‘I’m Bad at Math’

Two professors argue being good at math is about hard work, not about genetics:

We hear it all the time. And we’ve had enough. Because we believe that the idea of “math people” is the most self-destructive idea in America today. The truth is, you probably are a math person, and by thinking otherwise, you are possibly hamstringing your own career. Worse, you may be helping to perpetuate a pernicious myth that is harming underprivileged children—the myth of inborn genetic math ability…

Again and again, we have seen the following pattern repeat itself:

  1. Different kids with different levels of preparation come into a math class. Some of these kids have parents who have drilled them on math from a young age, while others never had that kind of parental input.
  2. On the first few tests, the well-prepared kids get perfect scores, while the unprepared kids get only what they could figure out by winging it—maybe 80 or 85%, a solid B.
  3. The unprepared kids, not realizing that the top scorers were well-prepared, assume that genetic ability was what determined the performance differences. Deciding that they “just aren’t math people,” they don’t try hard in future classes, and fall further behind.
  4. The well-prepared kids, not realizing that the B students were simply unprepared, assume that they are “math people,” and work hard in the future, cementing their advantage.

Thus, people’s belief that math ability can’t change becomes a self-fulfilling prophecy.

Interesting argument: if you believe you can’t do well at a subject, you probably won’t. The authors then go on to hint at broader social beliefs: Americans tend to believe in talent, other countries tend to emphasize the value of hard work.

This lines up with what I was recently reading about athletes in The Sports Gene. The author reviews a lot of research that suggests training and genetics both matter. But, genetics may not matter in the way people typically think they do – more often, it matters less that people are “naturally gifted” and more that some learn quick than others. So, the 10,000 hours to become an expert, an idea popularized by Malcolm Gladwell, is the average time it takes one to become an expert. However, some people can do it much more quickly, some much more slowly due to their different rates of learning.

How the Facebook equation 6÷2(1+2)= reveals the social construction of the order of operations

An equation on Facebook that has generated a lot of debate actually illustrates where the mathematical order of operations comes from:

Some of you are already insisting in your head that 6 ÷ 2(1+2) has only one right answer, but hear me out. The problem isn’t the mathematical operations. It’s knowing what operations the author of the problem wants you to do, and in what order. Simple, right? We use an “order of operations” rule we memorized in childhood: “Please excuse my dear Aunt Sally,” or PEMDAS, which stands for Parentheses Exponents Multiplication Division Addition Subtraction.* This handy acronym should settle any debate—except it doesn’t, because it’s not a rule at all. It’s a convention, a customary way of doing things we’ve developed only recently, and like other customs, it has evolved over time. (And even math teachers argue over order of operations.)

“In earlier times, the conventions didn’t seem as rigid and people were supposed to just figure it out if they were mathematically competent,” says Judy Grabiner, a historian of mathematics at Pitzer College in Claremont, Calif. Mathematicians generally began their written work with a list of the conventions they were using, but the rise of mass math education and the textbook industry, as well as the subsequent development of computer programming languages, required something more codified. That codification occurred somewhere around the turn of the last century. The first reference to PEMDAS is hard to pin down. Even a short list of what different early algebra texts taught reveals how inconsistently the order of operations was applied…

The bottom line is that “order of operations” conventions are not universal truths in the same way that the sum of 2 and 2 is always 4. Conventions evolve throughout history in response to cultural and technological shifts. Meanwhile, those ranting online about gaps in U.S. math education and about the “right” answer to these intentionally ambiguous math problems might be, ironically, missing a bigger point.

“To my mind,” says Grabiner, “the major deficit in U.S. math education is that people think math is about calculation and formulas and getting the one right answer, rather than being about exciting ideas that cut across all sorts of intellectual categories, clear and logical thinking, the power of abstraction and a language that lets you solve problems you’ve never seen before.” Even if that language, like any other, can be a bit ambiguous sometimes.

Another way to restate this conclusion from Grabiner is that math is more about problem-solving than calculations.

This reminds me of well-known areas of sociology that deal with the norms of everyday interactions. In order to interpret the actions of others, we need to know about agreed-upon assumptions. When those assumptions are blurry or are not followed, people get nervous. Hence, as this article suggests, many people get anxious when the rules/norms of math are seemingly violated. If these sorts of basic equations can’t be easily figured out, what hope is there to understand the rest of math? But, norms are not always cut and dry and that can be okay…as long as the people participating are aware of this.

Researchers develop an equation to help predict the next hit song

A team of researchers says they have developed an equation that helps predict which songs will become hit singles. Here is how the equation works:

We represent each song using a set of 23 different features that characterize the audio. Some are very simple features — such as how fast it is, how long the song is — and some are more complex features, such as how energetic the song is, how loud it is, how danceable and how stable the beat is throughout the song. We also took into account the highest rank that songs ever achieved on the chart.

The computer can combine a song’s features in an equation that can be used to score any given song.

We can then evaluate how accurately the computer scored it by seeing how well the song actually did.

Every single week now we’re updating our equation based on how recent releases have done on the chart. So the equation will continue to evolve, because music tastes will evolve as well.

As the researchers note, this equation is based mainly on the musical content and doesn’t factor in the content of the lyrics or budgeting for the song and music group. The equation seems mainly to be based on whatever musical styles and changes are already popular so I wonder how they account for changes in musical periods.

If this equation works well (and the interview doesn’t really say how accurate this formula is for new songs), this could be a big boon for the culture industries. The movie, music, and book industry all struggle with this: it is very difficult to predict which works will become popular. There are ways in which companies try to hedge their bets either by working with established stars/performers/authors, working with established stories and characters (more sequels, anyone?), and trying to read the cultural zeitgeist (more vampires!). But, in the end, the industries can survive because enough of the works become blockbusters and help subsidize the rest.

At the same time, haven’t people claimed they have cracked this code before? For example, you can quickly find people (like this and this) who claim they have it figured out. And yet, revenues and ticket sales were down in 2011. There is a disconnect here…

The “value of estimating”

Here is another way to help students develop their mathematical skills: learn how to estimate.

Quick, take a guess: how tall is an eight-story building? How many people can be transported per hour on a set of train tracks in France? How many barrels of oil does the U.S. import each year?

Maybe you gave these questions your best shot – or maybe you skimmed right over them, certain that such back-of-the-napkin conjecture wasn’t worth your time. If you fall into the second, just-Google-it group, you may want to reconsider, especially if you’re a parent. According to researchers who study the science of learning, estimation is the essential foundation for more advanced math skills. It’s also crucial for the kind of abstract thinking that children need to do to get good grades in school and, when they’re older, jobs in a knowledge-based economy.

Parents can foster their kids’ guessing acumen by getting them to make everyday predictions, like how much all the items in the grocery cart will cost. Schools, too, should be giving more attention to the ability to estimate. Too many math textbooks “teach how to solve exactly stated problems exactly, whereas life often hands us partly defined problems needing only moderately accurate solutions,” says Sanjoy Mahajan, an associate professor of applied science and engineering at Olin College…

Sharpen kids’ logic enough and maybe some day they’ll dazzle people at cocktail parties (or TED talks) the way Mahajan does with his ballpark calculations. His answers to the questions at the top of this story: 80 ft., 30,000 passengers and 4 billion barrels. To come up with these, he guessed at a lot of things. For instance, for the number of barrels of oil the U.S. imports, he made assumptions about the number of cars in the U.S., the number of miles driven per car per year and average gas mileage to arrive at the number of gallons used per year. Then he estimated how many gallons are in a barrel. He also assumed that imported oil is used for transportation and domestic for everything else. The official tally for U.S. imports in 2010 was 4,304,533,000 barrels. Mahajan’s 4 billion isn’t perfect, but it’s close enough to be useful – and most of the time, that’s what counts.

It sounds like estimation helps with problem solving skills and taking known or guessed at quantities to develop reasonable answers. I tried this question about the barrels of oil with my statistics class today and we had one guess of 4 billion barrels (among a wide range of other answers). This also suggests that there is some room for creativity within math; it isn’t all about formulas but rather takes some thinking.

This reminds me that Joel Best says something similar in one of his books: being able to quickly estimate some big figures is a useful skill in a society where statistics carry a lot of weight. But to do some of this, do people have to have some basic figures in mind such as the total population of the United States (US Census population clock: over 312 million)? Is this a commonly known figure?

The article also suggests ways to take big numbers and break them down into manageable and understandable figures. Take, for example, the national debt of the United States is over 15 trillion dollars, a figure that is perhaps impossible to comprehend. But you could break it down in a couple of ways. The debt is slightly over $48k per citizen, roughly $192k per family of four. Or you could compare the debt to the yearly GDP.

Finding the right model to predict crime in Santa Cruz

Science fiction stories are usually the setting when people talk about predicting crimes. But it appears that the police department in Santa Cruz is working with an academic in order to forecast where crimes will take place:

Santa Cruz police could be the first department in Northern California that will deploy officers based on forecasting.

Santa Clara University assistant math professor Dr. George Mohler said the same algorithms used to predict aftershocks from earthquakes work to predict crime.”We started with theories from sociological and criminological fields of research that says offenders are more likely to return to a place where they’ve been successful in the past,” Mohler said.

To test his theory, Mohler plugged in several years worth of old burglary data from Los Angeles. When a burglary is reported, Mohler’s model tells police where and when a so-called “after crime” is likely to occur.

The Santa Cruz Police Department has turned over 10 years of crime data to Mohler to run in the model.

I wonder if we will be able to read about the outcome of this trial, regardless of whether the outcome is good or bad. If the outcome is bad, perhaps the police department or the academic would not want to publicize the results.

On one hand, this simply seems to be a problem of getting enough data to make accurate enough predictions. On the other hand, there will always be some error in the predictions. For example, how could a model predict something like what happened in Arizona this past weekend? Of course, one could include some random noise into the model – but these random guesses could easily be wrong.

And knowing the location of where crime would happen doesn’t necessarily mean that the crime could be prevented.

Fantasy football in the classroom

Fantasy football is not just for adults or for recreation.  Some teachers are now using it in the classroom to help teach math:

Empirical data show that classroom fantasy-sports programs help improve grades and test scores.

In a 2009 survey of middle and high school students by the University of Mississippi, 56 percent of boys and 45 percent of girls said they learned math easier because they played fantasy sports in class. And 33 percent of boys and 28 percent of girls said their grades improved.

This sounds like a fun way to learn math. And the story suggests that whole families got involved with the process and helped the children decide whom to draft and how to score.

On another front: will everyone will be playing fantasy football in the future?