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.

Digital Conversion

In the last few years, many districts in this area have been promoting a “digital transformation” in their schools, including Fairfax, the system that employed me for many years. It’s a nice phrase and one that is often linked to 1-1 programs. But what does the phrase really mean? What exactly is going to be transformed?

Dig into the plans – posted on websites, presented at conferences, explained in conversations – and you hear a lot of elements not related to learning. The discussion is about technology and support issues: What device should we buy? Do we have enough bandwidth? We need more power outlets. How do we pay for all this? What happens if a student does something wrong with the machine we’re handing them?

Almost completely missing is an explanation of the major changes that will come in curriculum, pedagogy, assessment, or pretty much anything else instructional, as a result of buying all the equipment, software, and infrastructure.

Ok, I know transformations like this take time, especially in a tradition-bound institution like American education. And I’m also sure this kind of external communication doesn’t cover all the pieces districts are considering in their planning.

So, at the risk of covering issues already being addressed, I have a few questions for districts and schools undergoing a digital transformation.

How are you planning to change the curriculum teachers and students will be working with?

Shouldn’t the concept of learning change when information is no longer scarce? When the process of “teaching” is no longer one way from teacher to student? Asking students to recreate the same research papers their parents wrote makes no sense. Plodding through sheets of problems that their phones could solve in seconds, and which add nothing to their understanding of mathematics, wastes everyone’s time.

Are you providing enough support and time for teachers to learn the pedagogy to accompany all the digital?

Managing computers in the classroom is important. Knowing how to work Google Classroom or Office 365 is certainly part of the mix. But using Google is not necessarily transformative. Shifting the standard assignments from paper to digital is not at all transformative. And it’s going to take a lot of time for teachers (measured in years, not semesters) to make the major alterations to their practice that takes complete advantage of the new opportunities available in their classroom.

How will evaluation change to match the transformed expectations for learning?

Certainly there is basic knowledge and fundamental skills that we should expect any educated person in our society to know. Beyond that, digital tools allow for exploring the personal interests and talents that all students bring to school. So how do we assess their learning of both the essential materials and their individual goals? It’s not through standardized tests and we need to figure it out if this transformation is ever going to happen.

And finally, where are the students in your transformational planning?

Educators talk all the time about how the kids are the most important part of school. However, we rarely include them in any of these discussions. Not with surveys. Not by asking their opinion about school rules. Not with a few focus groups once most of the plans are in place. Students need to be at the table when we are finding the answers to all of the questions above. It’s their education. They will benefit most from their work in school (or possibly benefit very little). They need to have an equal voice.

This is just a start. There are many, many other questions that need to be asked, all part of the process of creating real change.

Because if you are using technology to digitize the same old learning process, what you get is a digital conversion, not a transformation.

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.

School Math is the Wrong Subject

Conrad Wolfram, a mathematician and “director of what’s arguably the world’s ‘math company’” (that would be Wolfram Research), believes “today’s educational math is the wrong subject”. Meaning that what we present to students as “mathematics” is not anything like what it is in the real world.

In the real world we use computers for calculating, almost universally; in education we use people for calculating, almost universally.

This growing chasm is a key reason why math is so despised in education and yet so powerful and important in real life. We have confused rigor at hand-calculating with rigor for the wider problem-solving subject of math. We’ve confused the once-necessary hand mechanics of the past with the enduring essence of math.

At its heart, math is the world’s most successful system of problem-solving. The point is to take real things we want to work out and apply, or invent, math to get the answer. The process involves four steps: define the question, translate it to mathematical formulation, calculate or compute the answer in math-speak and then translate it back to answer your original question, verifying that it really does so.

Teachers and textbooks give lip service to math as a tool for problem-solving but do little to help students understand the process Wolfram describes. As a result, the work kids do in “math” class is dry, boring and largely useless. For the most part, students learn to step through algorithms that the real world turned over to computers and calculators many decades ago.

I love how he describes “word problems” (now often euphamistically called “applications”) “as toy problems and largely outside any context most students can relate to”.

But it’s not just about turning the computation part over to computers. Wolfram says we need to completely replace the current mathematics curriculum taught in most schools.

Instead of rote learning long-division procedures, let’s get students applying the power of calculus, picking holes in statistics, designing a traffic system or cracking secret codes. Such challenges train both creativity and conceptual understanding and have practical results. But they need computers to do most of the calculating — just like we do in the real world.

All of us who have taught math in K12 have heard one common question from our students: “when are we ever going to use this?”. The fact that the honest answer is “probably never” should be a clue that something needs to change.