The Usefulness of Math

Catching up on items in my Instapaper queue we find a writer who says we should stop trying to sell math for it’s usefulness.1

One of the fall outs of children not understanding mathematics and the associated failure which often follows at some point in their 500 hour tour of the salt mines of mathematics?—?aka math education?—?is that teachers, through no fault of their own, start to sell/hawk mathematics like its some discontinued K-Tel kitchen product at a Saturday Flea Market.

Kids struggle with the number and symbolic manipulation we present as math for a variety of reasons, but the lack of context that necessitates that selling process is probably right up there. They are not dumb. Students understand that adults sometimes do use some math in their lives. But they also realize that there is likely an app for that.

Dozens of calculators to do the basic work, laser pointers that measure more accurately than a simple ruler, even software to produce each step of a process just as the teacher asked for. There may be a lot of trig involved with kite flying but learning from the mistakes of throwing it into the wind is more fun.

However, marketing math based on “its sometimes messy and intricate fun” and the intrinsic mystery of “symmetrical curvature” is also a dead end. And it won’t be especially beneficial to students in the long run.

Certainly playing with math can be both entertaining and lead to interesting discoveries. But even traveling that path, we will still arrive at the inevitable question: “when are we ever going use this?”, not to mention “is this going to be on the test?”.

I think there’s a middle ground between trying to sell kids on the need for our current formal mathematics program, most of which they will never use, and encouraging students to embrace the beauty and poetry of math.

How about using math to solve actual problems, other than those in books with mathematical titles? Like validating a survey in social studies. Gathering and analyzing data in science. Applying Geometric patterns to make art.

Maybe it’s time to eliminate the subject area silo called “mathematics” altogether in K12 (except for those few students in high school interested in that field), and instead incorporate those tools into the overall problem solving process. It would be a wonderful first step to tearing down all the artificial walls between subject areas.

I’m pretty sure someone can tell me why this is idea is impractical, unrealistic, or just plain looney. But there’s got to be something between “useful” math that really isn’t and “beautiful” math that few can appreciate (or even want to).

School Math is Void of Common Sense

A research mathematician turned teacher has a word problem for you: “There are 125 sheep and 5 dogs in a flock. How old is the shepherd?”

Most adults would quickly come to the conclusion that there is not enough information to find an answer. Not most students.

Now consider that, according to researchers, three quarters of schoolchildren offer a numerical answer to the shepherd problem. In Kurt Reusser’s 1986 study, he describes the typical student response:

125 + 5 = 130 …this is too big, and 125–5 = 120 is still too big … while … 125/5 = 25 … that works … I think the shepherd is 25 years old.

Remarkable. In their itch to combine the numbers presented to them, students negotiate three solutions. They show some awareness of context in dismissing the first two candidates. But a 25-year-old farmer is plausible enough for students to offer it up as their answer. The calculations are correct, but they are also irrelevant. Common sense has deserted these students in their pursuit of a definitive answer.

He says those findings are the direct result of the type of problems we ask students to solve during their travels through school math.

Students believe that all math problems are well-defined, usually with a single right answer. They strongly associate mathematics with numbers, to the extent that they will instinctively derive numerical answers to problems regardless of the context. They are subservient to computational procedure and trust that accurate calculations will always lead them to relevant truths. They accept that confusion and ambiguity is a staple fixture of mathematics, willingly offering up solutions that are void of context, meaning or even common sense.

So, what’s the alternative to our current standard math curriculum, featuring repetitive pages of calculations and ambiguous word problems with one right answer? The writer has some excellent ideas. However, they would require completely reimagining the way we teach math.

At the core of those changes is an emphasis on understanding process rather than finding answers.

Mathematics is a journey; it is defined by process, not rigid outcomes. That process can not be reduced to a series of discrete computation steps. It is governed by a flow of reasoning that guides the thinker towards a solution. Problem-solving often an unstructured, messy affair that requires several iterations of developing and testing assumptions. Error and ill-judgement are the most natural components of problem solving; they should be embraced. All mathematicians need pause to reflect on their problem solving strategies; to step back and retain full view of the big picture. Students must be afforded the same opportunities; their development as mathematical thinkers depends on this sense-making.

3-2-1 For 10-9-16

Three readings worth your time this week.

Following visits to elementary schools in Finland, the 2016 Kentucky teacher of the year wonders “What If High School Were More Like Kindergarten?”. The absolute best idea is the observation of a Finish business owner: “education is important, but learning matters more.” So why can’t we apply the “playful curiosity” approach to learning inherent in most young children to high school? (about 6 minutes)

I’ve playing with and watching the concept of virtual reality over the past few years and see a lot of potential for learning in the technology. However, there is also a lot of hype (some of which is on display in the Google announcements from this week). This article from the BBC offers some good examples of how VR might be used to help people understand places and experiences foreign to them, and maybe tell stories in new ways. (about 16 minutes)

A writer, comedian and “former Googler”2 asks Do You Take Yourself Seriously? Read this piece; then turn it around and apply the concept to your students. How many of them take themselves (and their ideas) seriously? What are you doing to help them change that attitude? Or possibly, maybe unintentionally, to reinforce it? (about 4 minutes)

Two audio tracks for your commute.

One distinctive feature of the societies pictured in Star Trek and other science fiction is the lack of money. But some countries here on Earth in 2016 are moving quickly towards a cashless life. Freakonomics Radio takes an interesting look at some of these efforts and asks Why Are We Still Using Cash? Personally I love Apple Pay and think it would be great if every business would stop taking my money. (45:59)

Much of the political discussion about immigration is framed in very stark black and white. But there are many, many different pieces, including the issue of guest worker programs that shouldn’t be included at all. The DecodeDC podcast offers an interesting look at the problems US farmers are having in finding workers to pick their crops, and how Congress is getting in the way with their simplistic fights. (34:01)

One video to watch when you have a few minutes.

Why are most middle and high school students in US schools sent down a math path that begins with Algebra and aims straight towards Calculus? Especially since “[a]t most, 5 percent of people really use math, advanced math, in their work.”, according to the author of The Math Myth. In this segment from PBS Newshour he discusses why students need mathematical literacy far more than that the formal structure of our current curriculum. As a former math teacher and member of NCTM, I can’t support the president of that organization interviewed in the video. (7:03)

Questioning the Math

Weapons of Math Destruction

Any book with a title like that is worth looking into, right?

In a recent interview, the author, Cathy O’Neil, says she started looking into how data and math was being applied to a variety of human processes following the financial crisis of 2008, in which she had a “front-row seat”. Her research found that the “very worst manifestation” of those applications was “kind of a weaponized mathematical algorithm”.

One of the first of those algorithms she investigated was the “value-add” model for assessing teachers being used in New York City and other districts. This is the process where student test scores are mixed with other data to determine which teachers are given raises, and which should be fired.

However, O’Neil, a data scientist and “former Wall Street quant”, someone who might actually understand the mathematics (and explain it to the rest of us), was denied access.

It’s opaque, and it’s unaccountable. You cannot appeal it because it is opaque. Not only is it opaque, but I actually filed a Freedom of Information Act request to get the source code. And I was told I couldn’t get the source code and not only that, but I was told the reason why was that New York City had signed a contract with this place called VARK in Madison, Wisconsin. Which was an agreement that they wouldn’t get access to the source code either. The Department of Education, the city of New York City but nobody in the city, in other words, could truly explain the scores of the teachers.

It was like an alien had come down to earth and said, “ Here are some scores, we’re not gonna explain them to you, but you should trust them. And by the way you can’t appeal them and you will not be given explanations for how to get better.”

O’Neil says similar secret formulas are used by financial institutions to determine who can borrow money, by courts to decide who goes to jail and how long they will stay there, by Google in presenting the results of your last search, and many more. Some of this activity is benign and some is extremely manipulative.

But which is which? And what do we do about it?

The very first answer is that people need to stop trusting mathematics and they need to stop trusting black box algorithms. They need to start thinking to themselves. You know: Who owns this algorithm? What is their goal and is it aligned with mine? If they’re trying to profit off of me, probably the answer is no.

I’m doubt many people actually “trust” math, certainly not after the classes they took in school. But I’m pretty sure most of us don’t understand or even know about those “black box algorithms” being used by companies and governments to analyze our information.

However, just like the data manipulation behind polling and studies that drive public policy, its clear we all need to start asking questions.

Anyway, I’ve just started reading the book. Based on just the introduction and the first chapter, I’ll probably have much more to rant about later.

Why This Math?

A recent, very short post on the NPR Ed blog covers almost 500 years worth of math curriculum.

However, as a Harvard professor explains, the trip doesn’t really require all that many words since not much has changed in that time. After reviewing some very old textbooks, he says, we find “a curriculum that is so similar to the curriculum we have right now it might as well have been written by the good folks who wrote the Common Core”.

That professor is Houman Harouni, who became interested in this topic when his elementary students asked the question all of us who teach math have heard at one point: why do we have to learn this stuff?

But it isn’t just why we teach math that fascinates Harouni. He is particularly interested in why we teach math the way we do: “Why these topics? Why in this order? Why in this way?”

He says history offers the best answer.

Harouni has studied texts dating to ancient Babylonia, ancient Sumer and ancient Egypt, and, he says, he has found three main ways of teaching math, each associated with a different economic group.

The three types of math Harouni identified are “money math”, used by traders and merchants, “artisanal math”, for carpenters, masons and other craftsmen, and “philosophical math”, which was only studied by “elites”. The first two groups arranged for their math to be taught to their children, trainees, and apprentices, solely with the goal of extending their influence and wealth. Relatively few people outside of colleges studied philosophical math until very recently.

Today the math curriculum used in most schools is a mashup of all three, with elementary kids mostly working in money math, because “we live in a world where money matters”, with some artisanal math in the form of Geometry. That’s followed in middle and high schools with most students receiving a heavy dose of that philosophical math in the standard path from Algebra to Calculus.

The bottom line is, the school math we impose on students in most American schools is largely a legacy from centuries long past. Much of it needs to be thrown out (or drastically rewritten) and replaced with concepts and skills that better fit with the way math is applied in the 21st century rather than the 16th.

For most kids in K12 schools, math should be studied as it was 500 years ago, reflecting how it is used in today’s real world: as a tool for solving problems in many different aspects of life. And not as an independent, overemphasized and excessively tested, stand-alone subject.