### The Story So Far

In October of last year I posted about a lesson adapted from a textbook:

A common ant absorbs oxygen at a rate of about 6.2 milliliters per second per square centimeter of exoskeleton. It needs about 24 milliliters of oxygen per second per cubic centimeter of its body. An ant is basically cylindrical in shape, so its surface area S and volume V can be approximated by the formulas for the surface area and volume of a cylinder:

a. Approximate the surface area and volume of an ant that is 8 millimeters long and has a radius of 1.5 millimeters. Would this ant have a surface area large enough to meet its oxygen needs?

b. Consider a “giant” ant that is 8 meters long and has a radius of 1.5 meters. Would this ant have a surface area large enough to meet its oxygen needs?

Instead of the straight textbook version I approached it with an introduction video

and a simple question:

Suppose you had a super-enlarger ray. How large could you make the ant so it still survives?

However interesting the question hook might be, the important part is in the middle section where the students work things out, and in this case I was met with mute bafflement and had to lead them by the hand through the entire process.

I’ve been pondering since how to fix things, and part of the issue is a complicated discovery process is often nonlinear. Students may not realize things in the order expected, and when certain key insights need to be connected to form new insights the connections may not form at all even when the pieces are laid out.

Dan Meyer’s analogy of teaching to storytelling led me to thinking of interactive media, and how it’s possible to map a “story tree”. Could the same methods apply to teaching?

### An example circa 1979

The objective of *Adventure* for the Atari 2600 is to retrieve a chalice and take it back to the golden castle. Along the way there are locked doors (which must be opened with keys) and three dragons (which can be slain with swords) and several other obstacles depending on difficulty level. In the screenshot below our hero is holding the chalice but being chased by a yellow dragon through a maze:

Here’s the story tree of the first difficulty level:

Note the only required elements are “get yellow key, open yellow castle, get black key, open black castle, get chalice, put chalice in yellow castle”. The dragons don’t need to be killed since they can be evaded.

(If you’re keen on playing *Adventure* for yourself, there’s a nice clone over at Newgrounds.)

### Applying nonlinear design to the Giant Ants lesson

In my new attack on the Giant Ants lesson I first listed all the individual elements that might possibly go into solving the problem. I presume they have the question above, the two pieces of scientific data (the 6.2 and the 24), and that the ratio of the ant’s height to its radius is 16/3.

The ant can be represented by a cylinder

The height of the ant is 16/3 times the radius

The surface area is

The surface area is

The surface area is

The surface area is

The surface area is

The surface area is

The volume is

The volume is

The volume is

The 6.2 ml/sec per cm^{2}relates to the surface area formula

The 24 mm/sec per cm^{3}relates to the volume formula

The surface area is multiplied by 6.2 to get the total ml/sec of oxygen the ant gets

The volume is multiplied by 24 to get the total ml/sec of oxygen the ant needs

If the oxygen the ant needs > the oxygen the ant gets, the ant dies

Where the oxygen the ant needs = the oxygen the ant gets is the largest ant we can get

We need to solve for the radius

We need to solve

We need to solve

The ant can only be enlarged to a radius of about 0.6 cm

A radius of about 0.6 cm gives a height of about 3.2 cm

Including the fact a table approach may also be used, here’s the elements in tree form:

I’m unhappy with this: a student could go directly from the table to the answer, but it isn’t included; there ought to be an “OR” connective there. Wrong turns and bends are also omitted.

Still, the diagram gives some analytical perspective. It shows, essentially, three distinct realizations: the formulas and the substitution, how the values 6.2 and 24 are used, and the fact the ant must take in more oxygen than it needs to survive. When I first tried this lesson I narrowly expected discovery to initially reach the first realization, yet the other two are not only crucial but easier to attain.

The tree diagram is a possible in-class guide for the teacher, so that when giving hints either quite straightforwardly through hint tokens or more subtly through Socratic methods, the teacher knows what insights to target.

Filed under: Education, Mathematics |

Dan Meyer, on June 6, 2010 at 6:10 pm said:Brutal exemplar here, Jason. I found the problem tough to get my head around when you first posted it, but I like how you clarify it with a series of “insights the teacher can target.”

My sense with these problems is that forestalling the math is rarely the worst course of action. In this case, even before “The ant can be represented by a cylinder.” I think I’d need a lot of discussion about physiology and anatomy. “For what possible reason

wouldn’ta huge ant survive?” Indeed, that’s the part I find most difficult to wrap my head around.Past that, I’m convinced there’s a certain structural sweet spot, where the paralysis and intimidation of

too much structuremeets the uselessness oftoo little structureand the teacher has everything she needs and no more. My instincts tell me the list of elements is the ideal structure whereas the story tree is too much. Just a guess, though.Jason Dyer, on June 6, 2010 at 7:34 pm said:I agree this is a gnarly question. The science aspect likely could form an entirely different tree before this one.

I also agree the tree structure is probably overkill, although I think the list might need a little more organization; maybe three or four topic categories.

I could still see the tree being used by a curriculum planner / textbook writer, someone who then distills the parts in a more teacher-readable format.

Mr K, on June 7, 2010 at 9:38 am said:This is indeed a rough problem. I think the toughest part is that the question you want to ask isn’t immediately obvious – there are plenty of fictional examples in stories and movies that counter they actual physics problem.

I’m more interested in the notation for mapping a lesson like this. Creating an open ended lesson requires that the kids be allowed to follow their own paths, rather than the well paved highway Dan mentions in his textbook speech. To be able to guide them, we need to understand the landscape they might travel through. We need to understand not only the shortcuts or the scenic routes (your black and red lines), but the dead ends. On top of that, some dead ends have value in that following them will break assumptions they have made, and force them to develop new concepts, other dead ends are silly and a waste of time.

This sort of planning is antithetical to the linear nature of traditional lesson planning. I think your diagram is a good first step (more than I’ve done, at least) to developing a detailed language for developing and sharing open ended lessons.

Jason Dyer, on June 7, 2010 at 5:30 pm said:I would like to see this with the wrong turns, but I’d think it would be best for that to use a simpler example.

The problems with Lemov’s Teach Like a Champion « The Number Warrior, on June 8, 2010 at 9:00 pm said:[…] Nonlinear lesson design (Giant Ants of Doom) […]

Keith Sherwood, on December 29, 2010 at 2:00 pm said:Jason,

This is just awesome. Fantastic to see someone doing this kind of creative stuff. Bravo!

Love the story tree idea. If you want to pursue that I have something that might be useful to you. It’s called Causal Loop Modeling (CLM), and is a system dynamics tool for modeling complex systems. It is simple and might be the perfect tool for this lesson.

I have tried using using mind maps, trees, cause and effect diagrams, and CLM, I find that kids eyes glaze over. I think the reason is that it takes time for the understanding to take shape, so they get bored or lost in the interim, while you are building the diagram. I have had good results using it backwards though. What I mean is I work backward from the desired result to the solution…similar in some ways to working through an algebra problem. So take a game for example. Start the diagram with a desired outcome that represents a win. Then add predecessor components, continuing to work backward until the strategies and tactics become apparent. I have actually taught my daughter to win games this way. Happy to provide more info if you think this is useful to you.

Ben Morris, on October 31, 2013 at 10:11 am said:I know this is an old post, but I’d love more information.

TI-Inspire « Andrew's Math Site, on January 3, 2011 at 1:53 am said:[…] Keith Sherwood on Nonlinear lesson design (Giant Ants of Doom) […]

Nonlinear lesson design: on definitions « The Number Warrior, on May 24, 2011 at 10:06 am said:[…] lesson design: on definitions Posted on May 24, 2011 by Jason Dyer Last year I wrote about nonlinear lesson design as a way of fixing issues I had getting students to understand a difficult […]

Outline from my talk on Lessons from Game Design for the Classroom « The Number Warrior, on January 21, 2012 at 6:05 am said:[…] Example from Adventure (Nonlinear lesson design) […]

Will Giant Ants ever invade the planet?!? | Teaching Mathematics, on February 18, 2012 at 1:31 pm said:[…] did a quick search on the internet and found that (fortunately) someone has done it before me. See Jason Dyer’s blog for the worked solution. Share this:TwitterFacebookLike this:LikeBe the first to like this […]