What Does the UK Brexit Math Tell Us?

The recent (and in many respects ongoing) Brexit vote in the United Kingdom provides a superb example of poor use of mathematics. Regardless of your views on the desirability or otherwise of the UK remaining a member of the European Community (an issue on which this column is appropriately agnostic), for a democracy to base its decision on a complex issue based on a single number displays a woeful misunderstanding of numbers and mathematics.

Whenever there is an issue that divides a population more or less equally, making a decision based on a popular vote certainly provides an easy decision, but in terms of accurately reflecting “the will of the people”, you might just as well save the effort and money it costs to run a national referendum and decide the issue by tossing a coin—or by means of a penalty shootout if you want to provide an illusion of human causality.

Politicians typically turn to popular votes to avoid the challenge of wrestling with complex issues and having to accept responsibility for the decision they make, or because they believe the vote will turn out in a way that supports a decision they have already made. Unfortunately, with a modicum of number sense, and a little more effort, it’s possible to take advantage of the power that numbers and mathematics offer, and arrive at a decision that actually can be said to “reflect the will of the people”.

The problem with reducing any vaguely complex situation to a single number is that you end up with some version of what my Stanford colleague Sam Savage has referred to as the Flaw of Averages. At the risk of over-simplifying a complex issue (and in this of all articles I am aware of the irony here), the problem is perhaps best illustrated by the old joke about the statistician whose head is in a hot oven and whose feet are in a bucket of ice who, when asked how she felt, replies, “On average I am fine.”

Savage takes this ludicrous, but in actuality all-too- common, absurdity as the stepping-off point for using the power of modern computers to run large numbers of simulations to better understand a situation and see what the best options may be. (This kind of approach is used by the US Armed Forces, who run computer simulations of conflicts and possible future battles all the time.)

A simpler way to avoid the Flaw of Averages that is very common in the business world is the well-known SWOT analysis, where instead of relying on a single number, a team faced with making a decision lists issues in four categories: strengths, weaknesses, opportunities, and threats. To make sense of the resulting table, it is not uncommon to assign numbers to each category, which opens the door to the Flaw of Averages again, but with four numbers rather than just one, you get some insight into what the issues are.

Notice I said “insight” there; not “answer”. For insight is what numbers can give you. Applications of mathematics in the natural sciences and engineering can give outsiders a false sense of the power of numbers to decide issues. In science (particularly physics and astronomy) and engineering, (correctly computed) numbers essentially have to be obeyed. But that is almost never the case in the human or social realm.

When it comes to making human decisions, including political decisions, the power of numbers is even less reliable than the expensively computed numbers that go into producing the daily weather forecast. And surely, no politician would regard the weather forecast as being anything more than a guide—information to help make a decision.

One of mathematicians’ favorite examples of how single numbers can mislead is known as Simpson’s Paradox, in which an average can indicate the exact opposite of what the data actually says.

The paradox gets its name from the British statistician and civil servant Edward Simpson, who described it in a technical paper in 1951, though the issue had been observed earlier by the pioneering British statistician Karl Pearson in 1899. (Another irony in this story is that the British actually led the way in understanding how to make good use of statistics, obtaining insights the current UK government seems to have no knowledge of.)

A famous illustration of Simpson’s Paradox arose in the 1970s, when there was an allegation of gender bias in graduate school admissions at the University of California at Berkeley. The fall 1973 figures showed that of the 9,442 men and 4,321 women who applied, 44% of men were admitted but only 35% of women. That difference is certainly too great to be due to chance. But was there gender bias? On the face of it, the answer is a clear “Yes”.

In reality, however, when you drill down just one level into the data, from the school as a whole to the individual departments, you discover that, not only was there no gender bias in favor of men, there was in actuality a statistically significant bias in favor of women. The School was going out of its way to correct for an overall male bias in the student population. Here are the figures.

                       

In Departments A, B, D, and F, a higher proportion of women applicants was admitted, in Department A significantly so.

There was certainly a gender bias at play, but not on the part of University Admissions. Rather, as a result of problems in society as a whole, women tended to apply to very competitive departments with low rates of admission (such as English), whereas men tended to apply to less-competitive departments with high rates of admission (such as Engineering). We see a similar phenomenon in the recent UK Brexit vote, though there the situation is much more complicated. British Citizens, politicians, and journalists who say that the recent referendum shows the “will of the people” are, either though numerically informed malice or basic innumeracy, plain wrong. Just as the UC Berekeley figures did not show an admissions bias against women (indeed, there was a bias in favor of women), so too the Brexit referendum does not show a national will for the UK to leave the EU.

Britain leaving the EU may or may not be their best option, but in making that decision the government would do well to drill down at least one level, as did the authorities at UC Berkeley. When you do that, you immediately find yourself with some much more meaningful numbers. Numbers that tell more of the real story. Numbers on which elected representatives of the people can base an informed discussion as how best to proceed—which is, after all, what democracies elect governments to do.

Much of that “one level down” data was collected by the BBC and published on its website. It makes for interesting reading.

For instance, it turned out that among 18-24 years old voters, a massive 73% voted to remain in the UK, as did over half of 25-49 years of age voters. (See Table.) So, given that the decision was about the future of the UK, the result seems to provide a strong argument to remain in the EU. Indeed, it is only among voters 65 or older that you see significant numbers (61%) in favor of leaving. (Their voice matters, of course, but few of them will be alive by the time any benefits from an exit may appear.)

Source: http://www.bbc.com/news/uk-politics- 36616028

You see a similar Simpson’s Paradox effect when you break up the vote by geographic regions, with London, Scotland, and Northern Ireland strongly in favor of remaining in the UK (Scotland particularly so).

It’s particularly interesting to scroll down through the long chart in the section headed “Full list of every voting area by Leave”, which is ordered in order of decreasing Leave vote, with the highest Leave vote at the top. I would think that range of numbers is extremely valuable to anyone in government.

There is no doubt that the British people have a complex decision to make, one that will have a major impact on the nation’s future for generations to come. Technically, I am one of the “British people,” but having made the US my home thirty years ago, I long ago lost my UK voting rights. My interest today is primarily that of a mathematician who has made something of a career arguing for improved public understanding of the sensible use of my subject, and railing against the misuse of numbers.

My emotional involvement today is in the upcoming US presidential election, where there is also an enormous amount of misuse of mathematics, and many lost opportunities where the citizenry could take advantage of the power numbers provide in order to make better decisions.

But for what it’s worth, I would urge the citizens of my birth nation to drill down one level in your referendum data. For what you have is a future-textbook example of Simpson’s Paradox (albeit with many more dimensions of variation). To my mathematician’s eye (trained as such in the UK, I might add), the referendum provides very clear numerical information that enables you to form a well-informed, reasoned decision as to how best to proceed.

Deciding between the “will of the older population” and the “will of the younger population” is a political decision. So too is deciding between “the will of London, Scotland, and Northern Ireland” and “the will of the remainder of the UK”. What would be mathematically irresponsible, and to my mind politically and morally irresponsible as well, would be to make a decision based on a single number. Single numbers rarely make decisions for us. Indeed, single numbers are usually as likely to mislead as to help. A range of numbers, in contrast, can provide valuable data that can help us to better understand the complexities of modern life, and make better decisions.

We humans invented numbers and mathematics to understand our world (initially physical and later social), and to improve our lives. But to make good use of that powerful, valuable gift from our forbearers, we need to remember that numbers are there to serve us, not the other way round. Numbers are just tools. We are the ones with the capacity to make decisions.

* A version of this blog post was also published on The Huffington Post.

How do you find good math learning apps?

There are approximately 20,000 math learning apps available on the App Store (classified as such by their creators). Google Play does not provide the corresponding figure for Android apps, but presumably there are a lot there as well.

Most of those apps do little more than provide repetitive practice of very basic skills, primarily about numbers. They are essentially just animated flash cards.

How can a parent, or a teacher, decide which apps are likely to benefit their child, or their students? I’ll come back to that later.

First, let me say that there is not necessarily anything wrong with an app that is essentially just an animated flash card – unless parents buy them (or just download them, as the majority are free) thinking that putting them on their children’s iPad or whatever is all they need to do to improve their performance in math.

In the days when the gateway to mathematics, and indeed much of everyday life, lay in mastering the multiplication tables and memorizing a few formulas for calculating areas and volumes, mastery of the basic number facts was indeed enough to start with. So it’s a pity those fun learning apps were not available back then. They would have made the acquisition of those fundamental facts and skills so much easier and far more enjoyable.

Unfortunately, the very digital technologies that have put those learning apps into eager young hands have also provided tools that have rendered procedural mastery of those basic skills all but irrelevant.

In today’s world, we use cheap, ubiquitous devices to do our calculations. It’s no longer important that all members of society have procedural mastery of basic arithmetic. What is required is the ability to make effective use of those digital devices, and what that depends upon is a good understanding of number – what is often referred to as number sense.

Roughly speaking, having number sense means being proficient with quantities and operations with numbers. A person with number sense is able to represent number concepts with models, words and diagrams, to communicate numerical ideas, and solve problems involving numbers. She or he can flexibly compose and decompose numbers for computation and solving problems. They can evaluate the reasonableness of solutions to numerical problems, and make connections between multiple solution methods. They can communicate their number sense verbally and in writing. They notice and explore number patterns, make connections and conjectures, and communicate their thinking to others. Number sense goes beyond solving word problems and memorizing basic facts and procedures. It involves engaging in numbers and operations in ways that develop a deep understanding of the content, which provides a firm foundation for mathematical success. In particular, a strong background in number sense sets the stage for later success in algebra and other parts of mathematics.

If that last paragraph sounds like something that emerged from a committee of mathematics education experts, it is because in essence it did. You find language like that in the National Research Council’s 2000 report Adding it Up, (which you can download for free from the National Academies Press) and in the preamble to the Common Core State Standards for Mathematics, which emphasize the development of number sense in young children.

For sure, you cannot have number sense without being able to solve an arithmetic problem and get the right answer. What has changed is that it is no longer important to solve that problem by the fastest method, or by a standard method that leaves a paper audit trail that others can check. Our calculating devices do those for us.

Much more important in today’s world is to be able to reason about the numbers in a problem from first principles, in a way that embodies the internal structure of the numbers. For as humans, we need to be able to operate when and where that calculator cannot: namely, when we are faced with a novel problem the real world has thrown up at us.

It was a lack of recognition that the world has changed fundamentally that led the consequently-Internet-famous “Jack’s Dad” to pen his satirical “letter to his son” that went viral on social media earlier this year. (See the next link below.)

Actually, Jack’s dad is an electronics engineer, so he was certainly aware of how much today’s world was different from the days of his own childhood. Unfortunately, as someone outside the world of education, he had just not connected the dots to understand what changes in education were required in order to properly prepare today’s kids to live, not just in our present world, but in the world they will help shape from it.

One of the best summaries of the issues behind that social media firestorm that I came across was the April 6 response to Jack’s Dad written by the math education blogger Christopher Danielson.

Danielson’s observations about different kinds of expertise rang very true to me. Having devoted the first part of my mathematics career to mathematical research, it was my appointment to serve on the Mathematical Sciences Education Board in 2000, and the close contact with leading experts in mathematics teaching that resulted, that brought home to me just how little I knew about how people learn mathematics, and how (consequently) we should teach it.

Put plainly, having a PhD in mathematics and a string of published research is absolutely nothing like enough background to speak with authority about K-12 mathematics learning. People like me can provide good advice on mathematical content; but not on mathematics teaching. That requires different knowledge and expertise.

My own university, Stanford, famous for its very high standards in research, apparently recognizes this when it comes to hiring new faculty in Education. While I cannot speak with authority for the School of Education’s policies, I have observed that no one gets appointed to the faculty who has not spent several years in K-12 teaching. (In addition to having done and published first class research!) Whether or not K-12 experience is official hiring policy, it certainly plays out that way, and it seems to me to be a sensible criterion to demand.

Going back to the standard algorithms and Jack’s Dad, a few months after his first post, on October 8, Danielson posted another excellent blog on the degree to which the position occupied by the standard arithmetic algorithms (in actual fact, there are many variations, so there is no such thing as “the standard algorithms) has changed in the educational landscape – from being the main focus as a method for daily use, to an interesting and historically important example of a set of highly efficient paper-and-pencil algorithms that quite literally changed the world. Their significance was a consequence of the dominant information storage and communication technology of the time: flat, static writing surfaces such as parchment, blackboards, and paper. (I describe that story in my book The Man of Numbers.)

I will note, in passing, that Danielson’s October post indicates that some math learning apps may in fact do harm to a child’s mathematics learning, an observation that should be coupled with my earlier remarks about choosing basic skills educational apps.

What put these thoughts onto my front burner recently were some discussions I was having with members of the Scientific Advisory Board for my educational technology startup company BrainQuake.

If you check out our company’s Team page, you will find we have recruited a number of world renown experts in mathematics education. Now you may think they are just there for marketing purposes – website name dropping. But you would be wrong. Each one is there because they bring very valuable, very specific expertise to the table.

To someone not an expert in mathematics learning, the arithmetic puzzles in our launch app, Wuzzit Trouble, may look as though they are just a series of problems we generated in an essentially random fashion, following the simple rule that the numbers should get "harder" the further a player goes in the game. But that is not the case. In a mathematics learning game, the mathematics ramp is just as critical as the level design of the game, and both require a lot of expertise to get it right.

(Interestingly, another name on our website, John Romero, is a world expert in level design – the ramping in game-play – but he joined forces with us only after we had brought out Wuzzit Trouble, so you will only see the results of his genius in future products we bring out.)

Which brings me back to my promise to provide advice on how to select good learning apps. It’s probably not a foolproof method, but a quick and easy way is to check out the website of the creators, and see who they have advising them on the learning side.

There is always the danger that some of the names are there for little more than window dressing, but the majority of education experts (indeed, experts in any domain) are not likely to lend their name to an enterprise they do not believe in. So the presence of names of distinguished mathematics educators should give you a lot of confidence in the product.

More to the point, the absence of such names should be taken as a serious warning. Quite frankly, it is not possible to design and build an educationally sound and effective learning app without a lot of expert input.

And I mean a lot of expert input. I bring years of my own expertise to BrainQuake, but Wuzzit Trouble would not have been anything like as educationally successful as it has, if it had just been me on the mathematics side.

There is your quick-and-easy quality check. If you use it, you will find that list of 20,000 apps suddenly shrinks down to a significantly smaller number. Fortunately, that number is not zero. There are some great math learning apps out there. You just have to choose wisely.