On of my favorite quotes I tell my students is

We could solve most of the world’s problems if everyone could just read a graph.

We could start with this pretty famous graph. The would would be in a much better place if we could understand it and its implications.

The Keeling curve. A graph everyone should be able to read.

Or this graph that tells us that next year’s digital camera is going to be way better than today’s (if you equate pixels with better, which you shouldn’t). Check those axes—this is a graph you’d never see in a newspaper, because almost no one could understand the logarithmic axis.

Why you should wait to buy a digital camera

And my personal favorite is this graph, from Dan Meyer, plotting the checkout time vs the number of items in your shopping cart. (I’ll give Dan a free pass on having naked numbers with no units in his graph, with wild exaggerated precision in the fit, and using $y$ and $x$ instead of $T$ for checkout time, and $n$ for number of items).

Dan’s graph is so profound it made national news. Why? The answer lies in that wonderful vertical intercept, which says shoppers spend 41 seconds on average paying for their items, and the slope, which says it takes only 3 seconds on average to scan an item. The bottleneck in lines isn’t how many items you have, it’s how many people are in front of you in line. So right there, if the world could read graphs, we would have eliminated express lines everywhere, saving shoppers everywhere a few minutes, that when aggregated across our population could be put to so much better use. But express lines still exist, wasting the time of the masses, and I conclude most retail management can’t read a graph. Sigh.

Of course there’s the old saw, “A picture is worth a thousand words,” but I wonder, if this is the case, why do graphs prove to be so hard to read? Newspapers know this, and it’s why you’ll almost never see anything other than simple pict-o-graphics and charts plotted with time on the horizontal axis (holla—USA Today).

I think I’ve found part of an answer in my own mixing of neuroscience, psychology and Seth Godin.

Let me explain, starting with internet god, Seth Godin, who, as far as I know, coined the term “lizard brain” to describe our reactions to certain stimuli. Seth rightly points out that deep inside your brain is the amygdala, which is very similar in size and function to the brain of a lizard. The amygdala drives our stress and fear responses—it responds to all primal emotions.

Here’s how it works. You see a picture of a bear.

Instantly your highly evolved visual cortex has evolved to actually react faster to a threat, and so it kicks your lizard brain amygdala into action and works to preserve your life with the stress response—your heart races, digestion slows down, adrenaline pumps through your body, and every part of you takes clear direction from the lizard brain—”forget whatever else you are doing, it’s time to run like hell.”

This is awesome when you see a bear, but the problem is, most us don’t live in a world with bears lurking around every corner. Instead, many of my students live in a world of continual stress and anxiety, anxiety over the piles of homework they face every night, anxiety over appearing dumb if they ask a question in class, and anxiety almost every highly charged social interaction that defines the teenage life few of us would want to relive.

So what do bears look like in the teenage world? I think they look something like this, on a test:

The "bear" for many of my students

And just like that, the lizard brain rushes into action, makes you “run like hell” and put down whatever comes to mind first (“that looks like constant velocity to me, since slope is velocity, right?)”. The problem is the lizard brain can’t read graphs. This isn’t a picture that you just respond to, it’s a picture loaded with deep and subtle symbolic meaning. The only chance a student has of properly understanding this picture relies on the pre-frontal cortex, the slow, lumbering part of the brain responsible for higher order thinking and reasoning that separates us from the rest of the animal kingdom. It’s the only thing that is going to be able to discern the very subtle difference in meaning between the $v$ on the vertical axis in a velocity graph, and the $x$ on the vertical axis in a position graph.

Problem is, once the lizard brain has taken hold and started to run, it isn’t easy for the pre-frontal cortex to re establish order. By the time it arrives on the scene, the stressed lizard brain has already dispatched the bear, and there’s nothing for it to do.

What happens if, however, you aren’t stressed and the prefrontal cortex, with its powerful reasoning abilities, has enough time arrive at the scene? It first begins to see something like this:

Your prefrontal cortex can distinguish the tiny difference in these symbols to deduce a huge difference in meaning.

Your prefrontal cortex looks at these symbols and realizes that even though they look almost the same, and are almost neighbors in the alphabet, there’s a huge difference in their meaning. Velocity and position! OMG! And it begins to see another graph, the position of that velocity graph:

The position graphs for these two object have almost nothing in common.

But here’s the problem for students who are under too much stress and are too hurried to allow time for the prefrontal cortex to reason to the solution. Once the lizard brain leaps into action, and selects the wrong answer, the prefrontal cortex is shoved aside and there is no going back. The student gets to feel the wonderful emotion of failure, which usually adds to stress, and puts the lizard brain at an even higher state of alert. Research has shown that long term exposure to chronic stress can reduce the size of the hippocampus, the very brain structure responsible for helping us to consolidate what we learn. So it’s a vicious cycle that leads to our lizard brains running our lives by constantly seeking safety, minimizing pain and avoiding opportunities for learning and triumph. No wonder Seth Godin gets to be so famous trying to help adults overcome this and launch the next great marketing campaign.

## How we can help our students read graphs and tame their lizard brains

I think reading graphs is a skill that is so important, it is the perfect place to help our students pitch a battle against the shallow shortcut thinking of the lizard brain and feel the real power of their ability to reason with their prefrontal cortex. Here are some strategies I’ve developed to try to help this.

1. Help students develop a routine for interrogating graphs
• What is being plotted along the vertical axis?
• What is being plotted along the horizontal axis?
• What trend do I see in this graph—is the quantity increasing or decreasing?
• What is the meaning of slope in this graph? (Can I calculate it by starting with $\frac{rise}{run}$?)
• How is the slope changing?
• What is the meaning of area in this grpah?
2. The lizard brain loves easily memorized procedures. “Run from fire.” “Eat only the red berries.” It’s what pushes our students to memorize equations with no appreciation of the meaning of the symbols, and to tackle graphs by making endless stacks of little rules “if the slope in the v graph is positive that means the acceleration is positive.”

Instead of simple rules, I want my kids to learn a series of questions they turn to with every graph.

By design, each of these steps is a question, aimed to quell the lizard brain and fear, and to allow students’ prefrontal cortices the time they need to reason and answer these questions, which will lead to understanding.

3. Give students hooks in order to engage graphs.
4. The graphs above are hard to read partly because the only difference is a tiny symbol they must interpret to find meaning ( $v$ vs. $x$). I almost never do this to my students, and instead try to give my axes meaning labels, “velocity,” “time”, “position”, and I always add units as one more clue aid them. Unless you’re trying to test you students ability to discern tiny differences under duress, I think you will find you can get a far better measure of their understanding of graphs if you give them a handhold or two.

5. Learn the ways of Tufte.
6. In the world of information design, Edward Tufte has no equal. Buy his books, read his treatise on the woes of powerpoint and the role it played in the Challenger Disaster. Start to see graphs as “visual explanations“, find the value in the “smallest effective difference,” and devlop a filter for “chartjunk” which distracts focus.

Here’s a great summary of Tufte’s Ten Tips for Designing Visual Explanations.

If we’re going to get our students reading graphs effectively, we need to be sure that we are showing them graphs that are well crafted visual explanations, rather leaving them to find meaning in the world of bad information design on their own. This likely means abandoning most of the chart junk that Excel enables as the default option (one more reason I love Omnigraph Sketcher).

7. Look for ways to reduce student stress overall.
8. This is just one of many strategies I try to employ to help reduce student stress, including Standards Based Grading, incorporating metacognition into my curriculum, and larger outside of class initiatives to change the way students talk mindlessly about grades. I don’t do these things to be a “soft” or “warm and fuzzy” teacher, I do them because I genuinely believe that by making these changes to my classroom, and taking time to talk about how we learn, I help my students to be able to learn more deeply.

Despite all of these efforts, my students still make errors when reading kinematics graphs, or leap to wrong conclusion about the meaning of slope in a graph. But they are making progress, and one of the most gratifying things I saw last year was when a student would write “lizard brain mistake” when correcting her own work on a graph, perhaps building her confidence that her problem reading this graph has nothing to do with her ability, which is very strong. And the solution lies not in “becoming a physics person” but instead in reducing stress, slowing down, and giving your prefrontal cortex time to warm up.

1. January 22, 2011 2:00 pm

Love this post! Lots of awesome advice.

I admit I’m lazy and don’t write out the words on the axis on the graphs I give the kids. This is something I will start doing.

I also wanted to share a tip along the same lines that is working well for me: color coding graphs and diagrams. I use the color scheme from Knight’s text. I force the students to use it on whiteboards, exams, and notebooks. (I provide colored pens and pencils for them to use.) I use the same scheme in my “lecture notes.”

Here’s an example: http://www.webassign.net/knight/p2-31.gif

I find that asking students to use proper colors forces them to automatically start the interrogation process.

January 23, 2011 9:02 pm

Keep in mind the color-blind boys in your class. I have found that generally I have at least one in every class.

• January 23, 2011 9:22 pm

This is a good point. But I think the color in these graphs is just one more hook to try to help students properly interpret the graphs. Physics students should be able to interpret them just fine with no color, but the color might be one more handhold to get their pre-frontal cortex thinking.

2. January 22, 2011 2:21 pm

Here are some better examples:

Notice how velocity is always green: the velocity vectors on motion maps, the curve on a velocity-time graph, and the area under an acceleration graph. I love it!

• January 22, 2011 2:27 pm

Frank, this is awesome. You would make Tufte proud!

3. January 22, 2011 3:00 pm

I like this explanation of the difficulties with reading graphs. I also find Excel chartjunk very irritating, so I teach my students to use gnuplot (which has been free for a long time and is likely to remain available for a long time, unlike many of the ephemeral tools with graphical interfaces). I also recommend Tufte’s books to my students, though only a few have followed up to the extent of actually reading anything.

One other piece of advice a give students is that the only plot people interpret reasonably is a straight line—everything else gets interpreted as if it were a collection of straight lines. Thus it is very important to transform the axes or the data to a form in which straight-line interpretation makes sense. If you expect exponential growth or decay, use a log-linear plot. If you expect power-law behavior, use a log-log plot. If you have something like a diode characteristic, plot square-root of current vs. voltage, rather than current vs. voltage , …

• January 22, 2011 3:27 pm

Is gnuplot easy to use? Can you make piecewise functions fairly easily? I’d love to find a tool to share with some PC owning colleagues—there really is a lack of decent, easy to learn, affordable options out there.

I also totally agree with you on the linearization thing. This is the heart of the modeling curriculum. My often fall into the trap of seeing everything that curves as exponential (the most recent curvy function they learned). So we learn to to plot various combinations of the variables (squrare, log, etc), until we arrive at a linear graph that can then be interpreted to arrive at an equation of fit.

• January 22, 2011 5:11 pm

gnuplot is easy to use if you are used to command lines and scripts, rather than menus. My 14-year-old son finds it straightforward, but he is already an accomplished programmer, so I don’t know how much that means.

Maybe you should look at the tutorial at
http://www.duke.edu/~hpgavin/gnuplot.html

4. January 23, 2011 10:31 am

Math anxiety – or lizard brain – is exactly why I thought I could not do math after advanced 6th grade math because I’d panic every time I had to quickly recall math facts. It wasn’t until Freshman year of college, when I taught myself how to master basic probability to pass a course, that I realized that I’m not half bad at math… I’d just equated panic with inability. And, it took a number of years in the work force to teach myself to calm down when I saw a graph.

Your kids are lucky that you are teaching them about this so they don’t have to wait for six years to realize they can get past the barrier – you are saving them years of doubt/lack of progress and possibly completely changing their minds about what they can achieve.

I’m lucky because I can show my kids your blog — now they will have my story about math anxiety and yours about the lizard brain to help them move past panic and to trust in their abilities.

A very creative, insightful post. Thanks!

• January 23, 2011 11:52 am

Thanks for the kind words—I’d say lizard brain graph reading is slightly different from math phobia, though the two are probably related (math phobia probably excites the lizard brain even more, causing it to be more controlling). But lizard brain thinking a sort of stress induced shortcut taking that fails to grasp the deeper meaning of graphs in general—this graph looks like a threat (a bear), and I must dispatch it as quickly as I can.

I have written, a while ago, about how I try to take on math phobia (Dan Meyer calls it the angry wolverine—a great post), in Beating up on the Wolverine, and Beating up on the Wolverine, and Sometimes the Wolverine Bites Back (password protected, so email me if you want to see it).

Dan Meyer is one of the best at describing and taking on math phobia, and his site is filled with wonderful math lessons that really get students to see the power of math to describe the world around them. If you haven’t discovered it yet, be sure to check out his blog, and dig deep into his archives.

• January 24, 2011 10:58 am

I think I may have read Dan Meyer’s blog before but thanks for leading me back (it has been awhile) & to the Angry Wolverine links.

Love his line: “Throw it a stick and it brings two back. Throw it two sticks and it brings four back. Don’t be scared to speculate what’ll happen when you throw those four sticks. Whatever happens will happen and that’ll inform your next hypothesis. In the meantime, you won’t ever find math impatient or angry. It’s always eager to play.”

I think my older son understands this and it is why he enjoys higher math & problem solving. I am working with my younger son who struggles with anxiety & math is a trigger — he loves animals – maybe he will relate to the comparison of math to the wolverine.

LOL at: “if we’re graphing 3x + 2y = 12, I’ll ask a student for a solution. The student will reply “I don’t know,” because, well, I mean, look at the fangs on that thing.”

It will take more time (and more peaceful environment – the boys are playing their instruments right now) to digest the content in links to Beating up on the Wolverine & Sometimes the Wolverine Bites Back.

Thanks again for sharing! MKJ