## Three Claims Function Carnival Makes About Online Math Education

Today Desmos is releasing Function Carnival, an online math happytime we spent several months developing in collaboration with Christopher Danielson. Christopher and I drafted an announcement over at Desmos which summarizes some research on function misconceptions and details our efforts at addressing them. I hope you’ll read it but I don’t want to recap it here.

Instead, I’d like to be explicit about three claims we’re making about online math education with Function Carnival.

1. We can ask students to do lots more than fill in blanks and select from multiple choices.

Currently, students select from a very limited buffet line of experiences when they try to learn math online. They watch videos. They answer questions about what they watched in the videos. If the answer is a real number, they’re asked to fill in a blank. If the answer is less structured than a real number, we often turn to multiple choice items. If the answer is something even less structured, something like an argument or a conjecture … well … students don’t really do those kinds of things when they learn math online, do they?

With Function Carnival, we ask students to graph something they see, to draw a graph by clicking with their mouse or tapping with their finger.

We also ask students to make arguments about incorrect graphs.

I’d like to know another online math curriculum that assigns students the tasks of drawing graphs and arguing about them. I’m sure it exists. I’m sure it isn’t common.

2. We can give students more useful feedback than “right/wrong” with structured hints.

Currently, students submit an answer and they’re told if it’s right or wrong. If it’s wrong, they’re given an algorithmically generated hint (the computer recognizes you probably got your answer by multiplying by a fraction instead of by its reciprocal and suggests you check that) or they’re shown one step at a time of a worked example (“Here’s the first step for solving a proportion. Do you want another?”).

This is fine to a certain extent. The answers to many mathematical questions are either right or wrong and worked examples can be helpful. But a lot of math questions have many correct answers with many ways to find those answers and many better ways to help students with wrong answers than by showing them steps from a worked example.

For example, with Function Carnival, when students draw an incorrect graph, we don’t tell them they’re right or wrong, though that’d be pretty simple. Instead, we echo their graph back at them. We bring in a second cannon man that floats along with their graph and they watch the difference between their cannon man and the target cannon man. Echoing. (Or “recursive feedback” to use Okita and Schwartz’s term.)

When I taught with Function Carnival in two San Jose classrooms, the result was students who would iterate and refine their graphs and often experience useful realizations along the way that made future graphs easier to draw.

3. We can give teachers better feedback than columns filled with percentages and colors.

Our goal here isn’t to distill student learning into percentages and colors but to empower teachers with good data that help them remediate student misconceptions during class and orchestrate productive mathematical discussions at the end of class. So we take in all these student graphs and instead of calculating a best-fit score and allowing teachers to sort it, we built filters for common misconceptions. We can quickly show a teacher which students evoke those misconceptions about function graphs and then suggest conversation starters.

A bonus claim to play us out:

4. This stuff is really hard to do well.

Maybe capturing 50% the quality of our best brick-and-mortar classrooms at 25% the cost and offering it to 10,000% more people will win the day. Before we reach that point, though, let’s put together some existence proofs of online math activities that capture more quality, if also at greater cost. Let’s run hard and bury a shoulder in the mushy boundary of what we call online math education, then back up a few feet and explore the territory we just revealed. Function Carnival is our contribution today.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Chris

January 27, 2014 - 12:29 pm -

Cool stuff.
1. Nice combination of graphs and open answer features. Some of it I know is in DME but not in the way it is combined here. There seems to be more overlap with the -not online- features that modelling software like Coach have. Will Desmos be looking at other examples than a man and cannon, because there are some examples with trajectories and quadratics of course.
2. Like the take on feedback. “though that’d be pretty simple.”, no it would be hard, I think, because WHEN is a graph correct or incorrect. Maybe even harder than mirroring. Which isn’t to say that both aren’t useful. Having played with it, I do wonder whether having multiple men when there are more function values for a given x really reveals enough. It will reveal that a student has done something wrong because, well, I have one man and the feedback shows several men, but of course the important question is why.
3. This is very powerful. How are the misconceptions assigned?
4. Yes, totally agree. Good design costs a lot of time and money.
Cheers

2. #### James

January 27, 2014 - 12:50 pm -

You’re my hero. My one thought is that with exercises like this physics and math are getting harder and harder to keep separate (which might be a good thing).

3. #### Joe

January 27, 2014 - 1:50 pm -

Two related design decisions that I really like:

* Height in the graph corresponds directly (and is right next to) height in the animation.

* Time in the graph corresponds to the moving time bar on the bottom.

An excellent way to scaffold out the task of scaling.

4. #### Mark James

January 27, 2014 - 3:38 pm -

Really awesome – and by that I mean the module itself as well as the well-articulated and justified claims. After doing a walkthrough the Carnival myself today, I immediately emailed the link to my entire HS math dept (14 teachers). Then, after dismissal I went and physically dragged 3 colleagues into my room so I could show them how amazing it is (as I had done with Penny Circle, et al – I’m still chipping away at the “I’m too busy to surf the MTBoS” mindset in my building). Looking forward to doing Function Carnival this Friday (if the Chromebooks arrive before then as promised) with my classes. I’ll let you know how it goes. Thanks Dan, Christopher, and Desmos!

5. #### l hodge

January 27, 2014 - 5:11 pm -

Very nice. Very clean and clear.

Interesting to note that multiple jumpers appear if you draw a graph that indicates the jumper needs to be in more than one place at a particular time. Relevant for functions, intersections of two functions, parallel lines, etc.

Would be great if it were possible to tweak some of the features or questions. In particular, I would like to be able to adjust the equation controlling the projectile(s).

6. #### Chris Painter

January 27, 2014 - 5:59 pm -

I am vastly behind in the technological realm in comparison; however, I wonder if there is a way to start pairing animations such as these along with short films as there are on the graphing stories webpage. I imagine an animate figure being imposed on the video footage itself. I am sure there is a way. Just imagine how much this would free up time during the construction of such problems (assuming of course that the animation overlay isn’t a nightmare to pull off).

7. #### Kathleen Cushman

January 28, 2014 - 5:13 am -

I can’t wait to try this myself. All the math education I ever got has come through following this blog for the last few years! May it go on forever!

8. #### Jason Dyer

January 28, 2014 - 6:43 am -

Have you seen the PARCC sample questions? It’s got some ingredients of what you’re doing in there. (I’m guessing Smarter Balanced is doing something similar, but I haven’t been keeping up to date.)

http://www.parcconline.org/computer-based-samples

It’s also got some super-frustrating parts of the interface, but that’s a different ballpark of worry.

9. #### Elaine Watson

January 28, 2014 - 8:44 am -

This is a great tool. I love the way that it shows the reality of what a student’s graph shows. The fact that both the green “reality of the story” and the blue “interpretation of the story” are both visible on the playback is a wonderful gift that helps the use to tweak their interpretation.

I found myself getting the general shape of the graph, but my blue item reached the max height sooner than the real green item. I had to continue to tweak my graph to get it as “correct” as I wanted to.

My frustration was with my own inadequacy in being able to physically manipulate the graph I drew. I assume that this would improve with practice. Perhaps if I had a touch screen, it would be easier. I was doing it using my MacBook Pro touch pad.

This site is a great contribution to moving math education into the 21st century. Thanks and kudos to you and Christopher for creating it and also for sharing it freely.

10. #### Howard Phillips

January 28, 2014 - 8:57 am -

Very nice !
I have one suggestion:
When using the line segment option, it would be easier for the user to click on the newly positioned dot and the line from the previous dot to be drawn automatically.
Keep it up!
Howard

11. #### Bryan Bazilauskas (@nobackswing)

January 28, 2014 - 10:41 am -

this is big time awesome. stuff like this is a huge part of the future of math education.

12. #### Dan Meyer

January 28, 2014 - 11:41 am -

@Jason, just checked them out on your recommendation. Not a lot that dazzles me. They’re duplicating existing hard copy tools online (the protractor, the ruler), they have students clicking coordinates on a grid and dragging items around (which I can’t get too worked up over), and then there’s a pile of fairly ill-considered interactions. Like making a kid click 48 separate boxes in a grid to indicate an answer. Or the dropdown menus to create the parts of an equation.

13. #### Jason Dyer

January 28, 2014 - 2:49 pm -

@Dan: The equation editor is especially painful. I challenge anyone to type in the geometry proof they are asking for without tearing their hair out and/or having the browser crash and lose all their data.

The interesting thing is, philosophically, it makes all the same points as your #1, yet somehow it comes off worse.

14. #### Dan Meyer

January 28, 2014 - 3:42 pm -

Jason:

The interesting thing is, philosophically, it makes all the same points as your #1, yet somehow it comes off worse.

I see how it aims in the same direction. It’s an instance where the tools get in the way of thought, where the same task (constructing a geometric argument) would be easier on paper than digitally. It doesn’t enhance thought. It’s like running an already-pretty-tough race with leg weights.

15. #### Elaine Watson

January 28, 2014 - 7:49 pm -

Jason, I tried them out too. Ouch! I would not agree that the tasks aim in the same direction as Dan’s and Christopher’s tasks. I answered all of the items and was told that 4 of the 6 did not have answers. This was because I evidently didn’t know the nuances, the correct format, of the system. It was not intuitive at all. It also did not include the feedback loop that the blue and green icons supplied in Function
Carnival. Granted that the PARCC tasks were not necessarily functions. Considering only functions with an independent and dependent variable allowed allowed Function Carnival to be more focused. Thanks, though, for sending the link. I am hoping that there is a major overhaul before this becomes the real assessment!

16. #### Holly Brie Thomas

January 29, 2014 - 8:32 am -

Gorgeous! It’s wonderful to see someone actually designing lessons (and software) that adds value by using technology, instead of essentially duplicating a paper-and-pen experience digitally.
I checked out the PARCC sample items yesterday and had the exact same reaction as Dan’s responses to Jason — they are basically old-school tasks with digital data entry, and with a clunky interface at that — Who clicks 48 separate boxes adjacent boxes when they want to select them in a spreadsheet? Why must I structure my work as a paragraph proof because the entry box is text-based & linear? Why is there no opportunity to SKETCH anything anywhere? Why didn’t the interface CONNECT the plotted points in the “graph 3 vertices of a rectangle and find the fourth”, which would at least add _some_ value beyond pen-and-paper graphing. And don’t get me started on the quality of the tasks as actual assessment items capable of providing meaningful, nuanced feedback about what a student can actually do…
Dan, if you have ANY ability to get what you are doing in front of the PARCC/SBAC folks, please do! Tasks like Function Carnival beautifully illustrate a paradigm shift from using technology to duplicate “the way we’ve always done it”, to a using technology as a tool to give students a deeper experience that takes them further into the math. The teacher feedback with miniature versions of the graphs students created is wonderful — much more powerful than any kind of alpha-numeric data would be.
This reminds me of a methods course I took focused on teaching functions using graphing calculators. I’m just old enough that graphing functions in high school was all pencil & paper, and there was still resistence to letting students use standard calculators in lower-level courses like Algebra. Consequently, it took a while to graph a quadratic function because you had to calculate 5-7 points and then plot them, so exposure to parabolic functions was mostly limited to those that could be expressed as x^2 + bx + c or -x^2 + bx + c, and which were located relatively close to the origin. However, with graphing calculators, the focus could shift from calculating and plotting individual functions to examining permutations within a class of functions (i.e. viewing a quadratic function through the lens of standard form, vertex form or factored form, as well as looking at how changes to variables impact the graph), and easily comparing classes of functions. The calculators could have been used to look at old-school functions faster (HOW the material was presented), but the real power actually came from changing WHAT material was presented — the automaticity of graphing the calculators afforded allowed for deeper exploration of functions (and hopefully deeper understanding, as well).
Since so much of what is happening in education right now is being driven by high-stakes assessment, it would be amazing if the folks designing the high-stakes assessments were thinking along the lines of Function Carnival, rather than drag-and-drop, point-and-click, and type-the-number-in-the-box, as they currently (sadly) appear to have the ability to reach the widest audience.

17. #### Maryke

January 29, 2014 - 12:44 pm -

That WAS fun! I was mesmerized for a good ten minutes, and embarrassed by my first attempt at the cars activity. I should really know better ;). After fixing my mistake, it was very satisfying to see the “feedback” problem that I now knew how to fix! If go this excited over “being wrong and then right” and only imagine how my students would feel!

18. #### Matt Vaudrey

January 29, 2014 - 3:03 pm -

My 8th grade math intervention class just did this, and these thoughts need to get written while they’re fresh:

1.) The user interface is clean and easy-to-use, even on iPads (though it automatically selected blocks of text when students tapped-and-held too long).
2.) I gave easy corrections while the class was working, viewing individual graphs (projected on the wall) and offering feedback in real time. Many who were stuck would just watch the screen as I critiqued others, then apply my advice to their own.
3.) The Carnival supported students working at their own pace. Three finished the whole 6 steps (nearly perfectly) before 1/4 of the class finished cannon man.
4.) That class–and probably all my classes–need remediation on “the graph of a line is a series of points all close together”. I had the same conversation 15 times one-on-one.
5.) More students were engaged with this activity than during my lesson the hour previous.

Well done.
~Matt

19. #### Nyima Drayang

January 29, 2014 - 3:13 pm -

Spot on! This is actually VERY similar to how motion is controlled in 3D animation software (except in 3 separate axes: x,y and z – which kind of hurts your brain in a good way!)
The feedback tool is exceptionally well thought out – I like how you can move the play-head and place a dot at key locations etc. all the time seeing both the video and your own version.
As Chris Painter mentioned: using this tool in conjunction with other videos would add another level of challenge. Could pupils upload videos of their own and then use the tool to graph them? Maybe that’s a big ask? : )

20. #### Jacob Klein

February 3, 2014 - 6:15 pm -

Kudos! I like how the image of Cannon Man appears when you hover over the graph. Even before committing to any answer, I get instant feedback. Also love the diversity of drawing tools – line, dots, freehand.

How many different “filters for common misconceptions” did you put in for each puzzle? And what kind of AI or sorting algorithms are you using to detect misconceptions?

21. #### Zachary

February 24, 2014 - 8:54 pm -

Great activity! My students were pretty into how you can see the man or car described by your graph laid over the original. They were a little bummed because they couldn’t figure out how to save orreturn to their work (they started after they took a quiz, and had about 30 min in class to work). Is this possible or do they need to complete the exercise in one sitting?

22. #### Dan Meyer

February 26, 2014 - 8:11 pm -

Hi Zachary, great critique. We’re aware of the issue and will be resolving it in future updates. Thanks for taking the time.