Category: physics

The Ups and Downs of Interpreting Graphs

Here’s a graph showing some guy’s position as he’s out for a walk:

This graph shows the position of a guy out for a walk. Can you tell what he's doing?
This graph shows the position of some guy out for a walk. Can you tell what he’s doing?

Take a moment and describe in your own words what he’s doing. If you said, “He went up a hill and down again,” I’m sorry, you’re incorrect. But don’t feel bad – that’s a common answer when you ask this kind of question in a first-year physics class.

Andrew Elby calls it WYSIWYG graph interpretation. Robert Beichner investigates these particular “kinematic graphs” that show distance, velocity and acceleration versus time while this terrific paper by Priti Shah and James Hoeffner reviews this graph-as-cartoon misconception and many others, with implications for instruction.

Almost every instructor in a science, technology, engineering or math (STEM) field,  and many in the Humanities, too, lament their students’ inability to “use graphs”. I sympathize with them. But also with their students: graph interpretation is one of those areas, I believe, where expert-blindness, also called unconscious competence by Sprague and Stewart (2000), is most visible: experts don’t even realize what they’re doing anymore. By the time they’re  standing at the front of the classroom, instructors may have looked at hundreds, even thousands, of graphs. We look at a graph and BAM! we see it’s key idea. I don’t even know how I know. I just…do.

Well, because of the line of work I’m in, I’m forcing myself to slow down and try to reconstruct what’s going on in my head. Task analysis, they call it. When did I read the axis labels? When did I notice their range and scaling? Was that before or after I’d read the title and (especially in a journal) the caption? When you finally get to looking at the data, how do you recognize the key feature – an outlier, the slope of the line, the difference between 2 bars of a histogram – that support the claim?

The ease with which we interpret graphs makes it difficult for us to teach it:

What do you mean it’s a guy going up a hill and down again?! Obviously he’s standing still for the first second – slope equals zero! D’uh!

I’ve been wrestling with this problem for a while. Every time it comes up, like it did this week, I dig out a piece I wrote in 2010 when I was part of the Carl Wieman Science Education Initiative (CWSEI) at the University of British Columbia. It was for an internal website so the version reproduced has been updated and some names have been removed.

Interpreting and Creating Graphs

I was at a 3-day meeting called Applying Cognitive Psychology to University Science Education which brought together science education researchers from the CWSEI in Vancouver and CU-SEI in Boulder and the Applying Cognitive Psychology to Enhance Educational Practice (ACPEEP) Consortium (or “star-studded” consortium, as CU-Boulder’s Stephanie Chasteen describes it.)

The skill of interpreting graphs came up a number of times. On the last day of the meeting, a group of us sat down to think about what it means to use a graph. One of us brought up the “up a hill and down again” interpretation of graphs in physics. An oceanographer in the group said she’d like to be able to give her students a complex graph  like this one and ask them to tell her what’s going on:

Vostok Petit data
Graph of CO2 (Green graph), temperature (Blue graph), and dust concentration (Red graph) measured from the Vostok, Antarctica ice core as reported by Petit et al., 1999. (Image and caption via Wikimedia Commons)
(Psst – how long did it take you to spot the 100,000-year cycle in the C02 levels? Not very long? How did you do that?) After thinking about the skills we ask our students for, a colleague sketched out a brilliant flow chart that eventually evolved into this concept map about graphing:


"Using graphs" means creating a graph (red arrows) and extracting information from a graph (green arrows).
“Using graphs” can mean drawing a graph (green arrows) or getting information from a graph (red arrows).

We see the information flowing inwards to create a graph and information flowing outwards to interpret a graph.

Creating a graph

Students should be able to use words and stories, mathematical models and equations, and numbers/data to create a graph. All of this information should be used to select the graph type – time series, histogram, scatter plot, y vs x, etc. – based on what we want to use the graph for, the type of data and what best tells the story we want to tell. Once selected, a useful graph should have

  • axes (for given variables, for combinations of variables that produce linear relations) with scale, range, labels
  • uncertainty, if applicable
  • visible and accurate data
  • title, legend if necessary
  • for graphs of functions, in particular, the graph includes (and is built from) characteristics of the function like asymptotes, intercepts, extreme points, inflection points

An instructor could assess a student’s graph with a graphing rubric with criteria like

  1. Does the graph have appropriate axes?
  2. Are the data accurately plotted?
  3. Does the graph match the characteristics of the function f(x)?
  4. and so on

The paper by Priti Shah and James Hoeffner reviews research into what people see when they look at a graph. It provides evidence for what does (and doesn’t) work. For example, if a graph shows the amount of some quantity, the amount should be the vertical axis because people see that as the height of the stack. On the other hand, if the graph is about distance traveled, distance should be the horizontal axis because that’s how people travel. One of my favourite snippets from Shah and Hoeffner: “When two discrete data points are plotted in a line graph, viewers sometimes describe the data as continuous. For example, a graph reader may interpret a line that connects two data points representing male and female height as saying, ‘The more male a person is, the taller he/she is’.” (p. 52) Their finding, as they say, have “implications for instruction.”

Interpreting a graph

More often in our Science classes, we give students a graph and ask them to interpret it. This is a critical step in figuring out and describing the underlying (that is, responsible) process. Just what is it we want students to do with a graph?

describe Describe in words what the graph is showing:

Given two distance vs time graphs, which person is walking faster?

What is happening here?

How have the CO2 levels changed over the last 400 000 years? [And we’ll save “why has it been doing that?” for the next question.]

interpolate and predict Use the mathematical model or equation to extract values not explicitly in the data:

Give the graph of a linear function and ask for the expected value of another (the next) measurement.

Give the graph, ask for the function y=f(x)

Find the slope of the graph

read off data Extract numbers already present in the data:

What is the value of y for a given x?

In what years did the CO2 levels reach 280 ppmv?

When is the man farthest from the starting point?

Join the discussion

I’m always looking to collect examples of graphs—the ones students in your discipline have trouble with. It’s very likely we’re having similar issues. Perhaps these issues could someday be addressed with a graphing concept inventory test that expand’s on Beichner’s Test of Understanding Graphs in Kinematics (TUG-K).

[Update: Just prior to publishing this piece, I looked more closely at the “guy out for a walk” graph. He travels 40 m in 2 seconds – that’s 20 metres per second or 20 x 3600 = 72 000 m per hour. Seventy-two km/h? He’s definitely not walking. Perhaps I should have said, “Here’s a graph showing some guy out for a drive.” I’ll stick with the original, though. Yeah, maybe I did it on purpose, just to make you put up your hand and explain your answer…]


  1. Elby, A. (2000). What students’ learning of representations tells us about constructivism. Journal of Mathematical Behavior 19, 4, 481-502.
  2. Beichner, R.J. (1994). Testing student interpretation of kinematics graphs. Am. J. Phys. 62, 8, 750-762.
  3. Shah, P. & Hoeffner, J. (2002). Review of Graph Comprehension Research: Implications for Instruction. Educational Psychology Review 14, 1, 47-69.
  4. Sprague, J., Stuart, D. & Bodery, D. (2013). The Speaker’s Handbook (10/e). Boston: Wadsworth, Cengage Learning.

Is going over the answers negative reinforcement?

My wife works with people with developmental delays, like autism and fetal alcohol spectrum disorder. Her niche is sexual health.  Imagine the hormones of a teenaged boy with the impulse-control of a 5-year-old. She often gets called in when some Grade 6’er starts whippin’ it out – either for the reaction he gets or because he doesn’t realize that’s not what typical Grade 6ers do.

The other day, we were talking about how to change people’s behaviours and she gave me an example of positive, no wait, negative, erm, reinforcement. I’m out of my depth when it comes to psychology so let me remind me (and you) about the difference, in overly-simplified terms I can get my head around. Oh, and when I’ve mentioned I’m writing this post, everyone I’ve spoken to gives a different definition of negative reinforcement, so it’s possible the one below is different than yours…

Positive reinforcement is something that’s added, typically by the person in authority – a parent, teacher, boss – after a person does something good. Like a high-5 by the coach after a good play, for example. That action strengthens the person’s motivation to repeat the behaviour.

Negative reinforcement strengths the unwanted behaviour. Your kid is having a fit because she doesn’t want to clean her room. Suppose you say, “Okay, I understand you don’t want to do it. Why don’t you watch TV for half an hour, calm down, and then clean your room….” It reinforces the undesired behaviour.

Every source I googled made sure to point out negative reinforcement is not the same as punishment. Getting grounded because you haven’t cleaned your room is not negative reinforcement.

(Geez, this is subtle. I can imagine some amazing clicker questions about positive reinforcement, negative reinforcement and punishment. [Update March 19, 2012: A couple of days after I wrote this post, Derek Bruff wrote about a clicker workshop he gave, including some pos/neg reinforcement clicker questions created by one of the participants.]  Okay, back to the conversation with my wife.)

Scene 1: Grade 6 classroom

There’s this boy, let’s call him John. John like to strip his clothes off at school. Like in the middle of class. His teacher intervenes. Frustrated with John’s continual stripping, the school decides they have no choice but to send John home when he strips, punishing him for his behaviour. But here’s the thing – John might have a developmental delay but he knows what’s what: he doesn’t like school. He strips so he can get sent home. In fact, John has started stripping on the school bus on the way to school so he doesn’t even have to go through the charade of going to class. Sending John home, which the staff feel is punishment for his behaviour, is, in fact, a reward for John. What they think is punishment is, in fact, negative reinforcement for John.

“So what are they supposed to do?” I asked her.

They shouldn’t send John home. And they shouldn’t praise him for keeping his clothes on. Instead, throughout the days when John is at school, the teachers should say, “We’re so glad you’re here with us today, John!” That’s positive reinforcement, something added to John’s school day that strengthens the good behaviour of keeping his clothes on.

What I’ve left out is what to do during the difficult transition time between he continually rips off his clothes and when he keeps them on. The teacher needs to intervene somehow. Calling my wife is a good start!

Scene 2: University physics lecture hall

The physics instructor has realized that his traditional, “all lecture, all the time” style of teaching does not promote learning like he wants.  He’s decided to make the class more student-centered. He gives 10-15 minute mini-lectures and then hands out worksheets which are supposed to guide and scaffold the students through the next stage of the development of the concept. The problem is, the students don’t do the worksheets. They just sit there, staring at the empty spaces on the page or desperately scribbling down formulas like I described here, biding their time, because they know he’ll be going over the answers in a few minutes. Sure enough, after a while, he goes over the answer to Question 1. The students madly scribble down his solution or, increasingly, grab their phones and start snapping pictures.

He’s not punishing them for not doing the worksheets (“Why have you not answered the questions!? You will all Remain. In. This. Classroom. Until I see some work!”) Rather, he’s reinforcing their behaviour of not doing the worksheet. They get what they want (the answers) and he thinks he’s helping. This seems to be an example of negative reinforcement, at least according to the definition I posited earlier.

“So what is he supposed to do?”

Good question.

Let’s look at this top down: What do the students need to get out of the activity? They need feedback on their answers in a timely manner. “Timely” because feedback a month later when they fail the exam is too late. One way to give them feedback is to go over the answers so they can check. That’s not the model used by the significant portion of the astronomy education community who use the Lecture Tutorials worksheets. Instructors do not go over the answers. Instead, the worksheets have built-in feedback and most instructors follow the worksheets with a sequence of peer instruction questions. If you get those questions correct, you know you’re okay on the worksheet. If you don’t get the questions correct, your peers will straighten you out. At the very least, you’ll know which concepts you didn’t get and can talk to the prof or TAs about them. More positive reinforcement comes when you ace those identical or “identical except some parameters changed” questions on the exam.

I’d love to create a sequence of clicker questions to follow the worksheets in this physics class but that’s not the simplest alternative because it requires the instructor to be agile with worksheets AND with peer instruction. One thing at a time…

What about this? The instructor watches the students doing the worksheet questions, monitoring their progress. If everyone is getting along just fine, don’t stop them. When it looks like students are stuck, and individual attention by the instructor or TA can’t handle the widespread confusion, intervene with a class-wide discussion. Don’t begin with, “I’m so happy you answered Questions 1 and 2 by yourselves!” (“John, I’m so glad you kept your pants on today!”) Instead, work together to get past the sticking point. Get the students to contribute to the solution, using the work they’ve already done to chip away at the problem. A pat on the back or a high-5 for a good tidbit of problem solving. The students are praised and rewarded for the work they’ve done, even if it’s not complete. That’s positive reinforcement for good behaviour, right?

(Unless that’s an example of “intermittent” negative reinforcement which, according to my wife, is even stronger than continuous.)

Yes, there will be difficult transition period, when students are not solving the problems and the instructor is not going over all the answers. Sorry, tough it out.

What if the students were never allowed to get into the habit of not doing the questions? What if, from Worksheet 1 on Day 1, this collaborative solution approach was the way it’s done. Ahh, now that would be something, wouldn’t it?

Alright, I’m not exactly sure where I’m at. I know the current method of going over the answers isn’t working. And that if we make changes, there will be a difficult period of transition. I like the collaborative problem solving approach — I’ve seen it happen in a physics class of about 30, where the agile instructor knew everyone’s name and kept track (in his head) of who hadn’t contributed yet, calling on them for input.

One other thing I know:  I should learn some more psychology.

Image: RaaksBeton2 by Dan Kamminga on flickr CC. In my mind, it shows people working together to reinforce what they’re building.

What does open communication mean to you?

I’m struggling with an issue. I can’t decide, or maybe I’m afraid to admit, if I’m being naive. Or perhaps so inexperienced, I’m blinded by imposter syndrome, the feeling that you really don’t belong in the group of experts you find yourself in. I’m hoping that by the time I get to the end of this post, I’ll at least have a better understanding of my confusion.

In a few months, there will be a Gordon Research Conference (GRC)  that I’d like to go to. It’s called Astronomy’s Discoveries and Physics Education. The theme is finding ways to use the latest discoveries in astronomy (and astronomy education, knowing the invited speakers) to motivate and enhance undergraduate physics education.

I haven’t been to that many conferences – maybe a dozen over my academic career, often by the same organizations. With my limited experience, there are 2 aspects of the GRC that are new to me.

1. Attendance by application and selection

You have to apply and then be accepted to attend. Not the usual,  accepted to present a paper or hang a poster, but accepted to be there. Kind of like TED talks, I hear. I guess that ensures that the people attending are motivated to be there and, more importantly, are sufficiently knowledgeable about the subject that they can make meaningful contributions to the conference.

2. All communication is treated as private.

This is the one that’s got me confused. By accepting the invitation to attend the GRC, you agree to their “Disclaiming Statements” which, because you can’t link directly to them, I’ll reproduce here:

To encourage open communication, each member of a Conference agrees that any information presented at a Gordon Research Conference or Gordon Research Seminar, whether in a formal talk, poster session, or discussion, is a private communication from the individual making the contribution and is presented with the restriction that such information is not for public use. Prior to quoting or publishing any such information presented at a Conference in any publication, written or electronic, written approval of the contributing member must first be obtained. The audio or video recording of lectures by any means, the photography of slide or poster material, and printed or electronic quotes from papers, presentations and discussion at a Conference without written consent of the contributing member is prohibited. Scientific publications are not to be prepared as emanating from the Conferences. Authors are requested to omit references to the Conferences in any publication, written or electronic. These restrictions apply to each member of a Conference and are intended to cover social networks, blogs, tweets or any other publication, distribution, communication or sharing of information presented or discussed at the Conference. Guests are not permitted to attend the Conference lectures and discussion sessions. Each member of a Conference acknowledges and agrees to these restrictions when registration is accepted and as a condition of being permitted to attend a Conference. Although Gordon Research Conference staff will take reasonable steps to enforce the restrictions against recording and photographing Conference presentations, each member of a Conference assumes sole responsibility for the protection and preservation of any intellectual property rights in such member’s contributions to a Conference.

(Source: follow the Disclaiming Statements link on the right-side menu here.)

Buried in the middle of this statement is a restriction on communicating any information from the conference via “social networks, blogs, tweets or any other publication, distribution, communication or sharing of information.”

In other words, I will not be able to tweet from this conference. And that’s got me, well, disturbed.

It’s not that I’ll have to disconnect my iPhone from my hand and won’t be able to follow what @RealSomeFamousPerson had for #theirmeal. Fine, whatever. I can catch up with my followers and those I follow on Twitter each morning at breakfast or evening at the pub.

Rather, it’s that as I’ve attend more conference and benefited from people I follow who share their conference experiences, I’ve learned of 2 remarkable ways that Twitter enhances my conference experience and my professional development:

  1. Twitter creates a forum for people at the conference to share ideas and reactions to the speakers. This “back channel” connects people around the room and in different parallel sessions.
  2. Twitter invites the outside community, the people not at the conference, to be a part of what’s happening there. In fact, and this is the heart of my confusion with the GRC policy, I benefit so much from following colleagues who tweet and blog their conference experiences, I feel an obligation to share the inspiration, ideas and resources that I am privileged to gather in person.

I posed this dilemma on Twitter and received replies from John Burk (@occam98), Chris Goedde (@chrisgoedde), Brian Utter (@quantumtweep), Phillip Cook (@cookp) and Joss Ives (@jossives) that helped me begin to understand the policy. Both Chris and Joss suggested that policy allows people to speak more freely and more easily share their latest ideas and results, without the fear of being scooped. I think that’s what the opening line of the Disclaiming Statement is all about: “To encourage open communication…” I get that, especially if the GRC about breaking research, which many GRC’s are. If you’ve on the verge of discovering a better way to assay your samples or process your data or distill your protein, and want feedback from your peers, then you want to keep that communication private. Phillip suggests this is pretty common with pre-published research.

I’m having a hard time applying this model to education. I suppose I’ll come away from the conference a better science education practitioner, which should cascade to my colleagues and their students. But I don’t feel like I’m doing this for me. I don’t have that killer instinct that might be necessary for academics (see “imposter syndrome”.) In my heart, I do what I do for the students (see “naive”.) Obviously I’m benefiting from this job and salary and perks (like attending conferences) but I continually filter my activities through, “Who will benefit from this?” If the answer isn’t students or their instructors, I think twice. In my mind, I can think of no better way to pique the interest and boost the enthusiasm of science educators than to share the latest discoveries, approaches and practices from the experts in the field.

Hmm, all this writing has helped. I won’t not go to the GRC because of this policy. A colleague who has an important presentation at this GRC has offered to introduce me to the organizer, Charlie Holbrow, so we can talk about the origin of the policy and the breadth of the restrictions it imposes. In the end, perhaps I’ll just have to turn off my phone. But that doesn’t seem like “encouraging open communication” to me.

Have you attended a GRC? Maybe these restrictions are relaxed or ignored. What about other professional events where communication with the outside world is restricted – what have you done before, during or after those? Drop a comment below if you have any thoughts, thanks.

Image: Communications Artwork by thomasfrank09 on flickr CC