Dear Students Who Think Graphing is Stupid,
Right on! Graphs are stupid. Cosmically stupid. Deliberately stupid. In fact – and I hate to pull this rhetorical trick on you, but you leave me with no choice – that’s kind of the point.
A graph is not an end product. It’s more like a map – a simplified picture of something big and complex, a schematic diagram that shows important features and omits distracting details.
Take the function f(x) = x2. This relationship includes a dizzying number of input-output pairs, more than you could ever hope to list. Even with every atom in the cosmos at your disposal, you could never create a “complete” graph of it: to fill in the details, you’d need particles smaller than quarks, and to reach the extremities, you’d have to extend far past the most distant galaxies.
So when you graph f(x) = x2, you’re not depicting the whole function. You’re drawing a simplified version, a pocket-sized map. “Function for Dummies,” you might call it, or “Idiot’s Guide to This Function.”
A world map shows the key features of the planet – oceans, continents, the more formidable mountain ranges – and leaves out the overwhelming mass of negligible details. No world map includes the chocolate shop near my apartment, even though it is emphatically – and deliciously – a part of the world.
So it is with graphing a function. A graph of f(x) = x2 does not usually illustrate the fact that when x = 3.2, f(x) = 10.24, even though any mathematician will tell you it’s true. (Actually, most mathematicians hate arithmetic, so if you ask them for the square of 3.2, they’ll grimace and refer you to the nearest engineer. In fact, the original version of this post had the number wrong. Bad at drawing and arithmetic!) Instead, a graph will illuminate a few key truths about a relationship, such as the following:
- The square of a real number can never be negative.
- When you square opposite numbers (like 2 and -2), you get the same result.
- When you square a number larger than 1, it grows.
- When you square a number between 0 and 1, it shrinks.
- As a number grows bigger (say, from 2 to 3), its square grows even faster (in this case, from 4 to 9).
I’m not trying to sell anyone on the charms of f(x) = x2 – although it is a rather enchanting little function. My only point is that a graph should be useful. It’s a tool of clarification, a lubricant for problem-solving, a map of truths.
Our math classes aren’t always great at conveying this. (Mine included.) Too often, we teach you guys how to graph, without teaching you why to graph. Fight against this! The last thing you want to be is an expert in mapmaking but an illiterate in map-reading. That’s like being able to walk into any room and sketch the floor plan, but when pressed, being unable even to find the door.
Anyway, you have my support: Graphs are stupid. Just as they should be.
Best,
Ben
Okay here’s my stupid question. What is that function used for? Just to show how numbers are cool? What is the purpose in graphing the function? Who uses this? (Do astronomers use fuctions to figure out the oribits of stars? Metorologists to figure out possible cloud patterns?) If it is a tool for clarifiction — what is it clarifying (real world application please) ?
Great question.
To some extent… yeah, it’s just to show how numbers are cool. What a graph is clarifying is an abstract relationship, which may or may not have real-world utility.
This particular graph helps us understand what it means to square something. As it turns out, that is a useful understanding. To take a random example (which you actually hit on), the force of gravity (between a star and a planet, for example) depends on the square of the distance. The bigger the distance, the smaller the pull of gravity – except that it’s not a linear relationship. You have to square the distance. So as the distance increases, the force of gravity gets a LOT weaker. A good understanding of this graph can help explain, intuitively, that the sun’s pull on Pluto is MUCH weaker than its pull on Mercury.
To be fair, you can live a happy, full life without ever pondering the gravitational attraction between the sun and Pluto. But it’s fun to think about! And the more we think about questions and connections like that, the better we get at solving problems and understanding the world.
Not sure if that’s a satisfying answer, but it’s the best I’ve got! Maybe somebody else can chime in if they have ideas.
Frankly, I’m somewhat puzzled by people who can make it through a post-secondary math course without graphing every little function just to see what it does (I was always fascinated by the people who could look at a complicated function in mathematical notation and immediately comprehend how it behaves in response to larger or smaller inputs or whatever), but I might just be a visual learner. So I graphed everything.
And somewhere back in those undergrad years I dabbled in meteorology, and I can confirm that meteorology is a very function-heavy field. Google the Clausius–Clapeyron relation for an example of the types of functions meteorologists use. And astronomers do use functions to predict orbits. Stars probably not so much, but math comes in handy when you want to predict whether a particular asteroid is going to fly by harmlessly or give us a good walloping. (The orbits of bodies in space are generally predicted by the mathematics that describe ellipses–the ovoid shapes, not the punctuation.)
These days I work with population health data, and graphing is one way that we analyse trends over time and try to predict future trends. It’s kind of like the reverse of graphing a function to see how the function works: we graph the data we have and try to infer the underlying functions from the shapes of the graphs. Of course, diseases are complicated things that are influenced by multiple processes so the graphs are never as elegant as, say, the squaring of x, but an understanding of how functions appear when graphed is fundamental to this sort of inference. For instance, some colleagues of mine analyse trends in doctor and hospital visits for influenza and colds to try to identify outbreaks of pernicious strains. To do this, they have to consider that one, normal, expected process that influences such trends is seasonality: humans tend to get colds and flus in increasing numbers over the winter, and less so in the summer. So there’s a normal, expected sinusoidal shape to such graphs covering periods of a year or more. To detect potential outbreaks, one must then look for deviations from this regular pattern, such as a very short term (on the scale of days, or weeks) exponential increase in visits. Of course, a particular analyst’s experience and knowledge of the details of a particular disease and the health system in which it occurs will influence their interpretation, but it’s still all founded on the knowledge gained by graphing mathematical functions.
I hope this helps to answer samiJ’s not-at-all-stupid question at least somewhat.
Don’t know if that answers samiJ’s question, but that’s one of the coolest things I’ve read all day.
Next chance I get, I’m definitely going to relay to my Precalc students your description of public health data as a kind of functional forensics.
“Next chance I get, I’m definitely going to relay to my Precalc students your description of public health data as a kind of functional forensics.”
Wow, I’ll be pretty pleased if it helps a student! Glad you found it cool.
Im Ben Hirsch, I work a college readiness advocate in Austin, and also went to high school as Ben. A more relevant example to my high school seniors who are not graduation eligible may be that the square footage of a room increases much more quickly than its perimeter. Knowing something like this can prevent you from getting scammed when you are say renting an apartment.
Another graph I use with my students that has incredibly powerful real world application is a graph of compounding interest over time. A graph really gets the point across that they should pay off their debt before it really starts to increase rapidly. And it isn’t stupid, it is so smart it can communicate to people who really dont have the ‘background knowledge to understand the mathematical concept
Hey Ben, thanks so much for reading. I plan to pick your brain sometime about your college readiness work – I talk a lot with our 12th graders about their futures, and I’d love to hear your insights.
Great examples, by the way – square footage and compounding interest are both pretty potent examples of high school math’s real-world value.
Thank you – it is a satisfying answer. I struggled with math all throughout school and you hit it right on the head regarding how frustrating it is to hit that brick wall of not understanding. There is some comfort in knowing what the application or purpose is — otherwise I just feel like I’m memorizing recipes from a cookbook written in Greek using foods that no longer exist — I can recognize the patterns but never feel like I know what it is for, or what the finished result will look like, which is maddening.
“I feel like I’m just memorizing recipes from a cookbook written in Greek using foods that no longer exist”
Awesomely stated – I’m going to quote you on that, if you don’t mind.
(I tell my students sometimes, “Math shouldn’t just be following recipes. If that’s all we were learning, we’d be better off using actual recipes – at least then we’d wind up with cake at the end.” But you take the analogy to another level.)
Like Ben, I love your metaphor. There might even be ancient Greek words whose meaning we don’t know – for example, because all the Greek writers, and those after them who knew what that food was, took it so much for granted that they never left any clues to what it was (e.g. saying “you can’t have bacon without killing pigs”), until some point in history when that food passed out of fashion (perhaps in favour of some other food that used the same raw materials), so that we now have no way of telling what it was – and yet, there you are memorising the recipe that uses it. (Indeed, consider the English word “mutton” – it just means “lamb” that got to live a bit longer before being killed, and used to be common food; but you won’t see it in shops anything like a much as you did when I was a child. Maybe that’s on its way to being this problem for people in two thousand years time.)
Mathematics has quite a lot of sub-fields and some of them use words (or notations) alien to the others, or use the same words as each other to mean utterly different things. Mere rote learning of their formulae and definitions could lead to some baffling confusions or out-right disasters if recipes from different branches get used side-by-side, seemingly with the same ingredients, but actually meaning something completely different – as if, in cake-baking, “butter” meant that stuff made from milk (or some common substitute for it like margarine) but, in sweet-making, it meant cocoa butter and no-one bothers to qualify it as “cocoa”. Your sweets would work out dull if you used the cake-baking meaning in their recipes. Or if domestic cooking used the word “peck” to mean a pinch (e.g. of salt) while industrial cooking used it as the unit of dry volume of that name (equal to two gallons); using the latter in domestic cooking would be disastrous !
As to the function that squares its input, the graph, turned upside-down, also gives you the shape of the trajectory of a compact dense object thrown (moderately hard) upwards and a bit side-ways in still air near Earth’s surface. If you spin this shape, a parabola, about its central axis (that the shape is its own mirror image in) you get a surface that has some neat properties: there’s a point (called the “focus”) a little way up the axis where, if you put a light there, its rays, reflecting off the parabolic bowl, will all come out parallel to each other, and to the axis. That has been used in torches and light-houses to project a tight beam outwards; and, in the reverse direction, is used by antennas to collect incoming signals (e.g. radio or TV broadcasts) from one direction onto the focus. So it happens that the square function’s shape is remarkably useful. (The same can’t be said for many other graphs, but understanding their shapes can still help you understand the processes the function graphed describes.)
First of all, to answer the first commenter – Yes. Astronomers and meteorologists and lots of others who use numbers to do their work will use graphs to understand how those numbers behave pretty frequently.
Secondly, and much more fun – Ben, you have proven your own point about mathematicians and arithmetic. 3.2 squared is 10.24.
Ladies and gentlemen, the fruits of a B.A. in mathematics!
*takes a humiliated bow*
Anyway, thanks for catching that, Scott. Fixed it above.
I found this site from your post about being a math teacher who’s bad at math. This post did not disappoint me. The square of 3.2 is 10.24 That’s a particularly egregious mistake because 3.2=32/10, so therefore 3.2^2=32^2/10^2=1024(which every mathematician should know off the top of their head)/100=10.24
Other than that, I greatly enjoyed your post, and I look forward to reading and, if necessary, correcting many more in the future.
Thanks, Aaron. I have a feeling there may be ample need for your services in the future…
Excellent point about graphs!
Additional thought: All too often, most students don’t get into the stuff from discrete mathematics where graphs don’t even have to use axes…trees and networks and so on. Practicality is way too overemphasized nowadays.
i hate drawing graphs so much-