Why Email the files to a public folder? You could just register a Photobucket account and give everyone the username/password and we can directly upload it.

I still think cubic ft works better because a cubic foot is a cubic foot. End of story. We don’t use such units often enough to make it completely intuitive, but we use it far more than the weight of a cubic foot of dirt I’m sure.

Googling “density of soil table” gets a nice link to
http://web.ead.anl.gov/resrad/datacoll/soildens.htm
which has
http://web.ead.anl.gov/resrad/datacoll/img3.gif

You’re probably looking at sandy loam, which is 1.44 g/cm^3
but any number from 1.25 to 1.5 g/cm^3 is reasonable.

Incidentally, this sort of problem makes it clear why metric is easier: I can remember that water is 1 g/cm^3 or 1 pound/pint, but the conversion between cubic feet and gallons always escapes me. I have no such trouble with cm^3, liters, and m^3.

I’d do the whole problem in metric units and just convert kg to pounds at the end.

Cool idea. Gonna have to try it out, I’ll let you know how it goes.

“What app will let people email files to a public folder?”

I think that http://drop.io should work nicely for that task.

There is another factor to be considered here, energy expended. The weight of the dirt and the energy expended to lift it is the easy end of the process, breaking the dirt loose into a shovelable form requires more energy than simply lifting the loose dirt. Factor in rocks, dry compact soil at depth, dust and heat, and the size of the hole becomes more of a concern than weight of or amount of dirt. Digging holes in the desert isn’t easy and here real life experience plays a role coming up with the best solution to the problem.

Great math problem. Thanks for sharing.

As for file management, how about have students set up their own blog/website at the beginning of the year and they can upload their files, write comments, discuss via posts there.

Then you can just use your favorite feed reader to create a list of RSS feeds to all their sites. Puts all the work on them to maintain their pages and you get automatically updated on their assignments in real time (including date/time posted if that’s important to you). And it’s hosted on their site instead of clogging up your email.

Plus, they end up with a portfolio of work at the end that can be saved and used for variety of reflections, assessments, resources, etc.

I love weebly.com for this kind of stuff. Blogger would be simple to set up too. Weebly also has educational sites (education.weebly.com) where you can create one page that has a forum and allows for a file management system where students can upload files that are directly sent to you.

Great stuff…gonna try this Holes one w/ my middle schoolers this year.

I was just looking at the four submitted answers. Everyone assumed a cylindrical hole. I don’t remember from the book (or the flick) if a hemispherical hole was a possibility. I’ll turn one of my children on the task today as they wind up their summer vacation this week.

A challenge question: What is the difference in the amount of dirt between the two type of holes?

BTW: The twist on why the kids had to dig the holes is great. If anyone has not read this book, you should! The movie just doesn’t cut it. It is a three – four hour read max!

Nice post mortem. Ima gonna post my own additions in a bit.

>It remains to be seen if I’m crazy enough to drive down to the Mojave Desert with a scale and weigh a cubic foot of dirt

If you’re willing to wait until the end of October, I can do that for you.

Maybe this is obvious, but the actual, real world difference is that, for the whole day while you’re digging, you think “Hey, at least I have the short shovel”. That’s why it would be worth fighting over. The calculation, if it comes out favorably, is how you console yourself when you don’t get it (“Well, it only means this many extra shovelfuls. It’s not as big a deal as they make it out to be.”).

Pursuant to that version of things, the correct units are shovelfuls and the time period is probably per day, since you do your time a day at a time, and try not to think about the long haul.

Here are some thoughts I’ve had about the problem and the session over the intervening days.

With regard to the units used for the final answer, I think “holes” might be a natural and compelling unit to use. I’m not sure how to phrase the question, but something like, “By using the shorter shovel, X-Ray avoids digging the equivalent of how many full-sized holes over the course of a year?” With a different tack, maybe the question, “How much less work does X-Ray do in a year?” would allow for “holes” to be suggested as a unit, or perhaps different students would then actually opt for a variety of different units. The latter situation would be interesting because then there would have to be conversions made in order to compare hard-fought-for-and-invested-in answers.

Using full-sized holes as a unit gives a different feel to the result–the immensity of pounds or cubic feet is stunning, but the number of holes ties the quantity more closely to the narrative context. A bunch of pounds means a lot of work, sure, but it’s clear from the start that everyone is digging a lot of dirt. The fact that the shorter shovel could save X-Ray from digging the equivalent of 35 holes in a year–more than a month’s work–makes for some compelling narrative.

Note that asking it in this way removes the need for finding out the density of dirt. For better and worse.

It also allows for a nice geometrical connection to be made, namely, to scale factor. From this perspective, even the shape and size of the original hole are irrelevant. The linear measurements of the smaller hole are scaled by 29/30 from the full-sized hole, and so its volume is scaled down by that cubed, which means that the smaller hole has about 90.3% of the volume of the original one. So X-Ray is digging only 90.3% as much, and that’s the same as digging only 90.3% of the days: 330 full days, 330 full-sized holes.

As far as my experience of some aspects of the session: I didn’t feel like I had enough time to really process others’ estimates. I think that asking for revised estimates (or a consensus estimate) once everyone has seen everyone else’s guesses could be a good strategy, both for drawing attention to the activity of estimation and provoking some critical thinking with regard to revision.

Having a smoother way to share our written work will be nice.

I agree that typing in your questions (in addition to saying them aloud) is a good idea. You might also be able to find a way to use the whiteboard space to help guide the flow of conversation and to highlight interesting points that arise. (Maybe stick the estimates up there?)

The whole experience felt a little rushed, but I think that was mostly just me getting used to the format and the technology. I felt like having time to watch the video twice might be important. It was nice that we had some time to debrief and toss some questions and opinions around at the end.

Thanks for hosting, Dan, and for demonstrating the potential of online PLNs going “live”!

@ Dan Lemay:

I think digging hemispherical holes is more common, although I’m not sure “typical” is the name of the game at Camp Green Lake (yes, I loved the book, too!). Good point there.

@ Justin:

I love the idea of “holes saved per year.” That makes it much more accessible (though not necessarily easier), which is important to people who get psyched out by math (me!).

Finally, maybe I’m a weirdo….But before I even thought of the size of the hole, I thought about whether it’s easier to dig with a shorter or longer shovel. That’s the thought that flitted into my mind when I read the book, with no mathematical context (i.e. me thinking “there’s a math question in this somewhere…”). That has to do with energy expended rather than amount of dirt. My brief encounters with high school physics don’t give me enough background knowledge to deal with that issue…but maybe it would put a good spin on it for a physics class?

For what it’s worth, my reflections wrt this model of online PD. It’s been a great learning process!

“You’re probably looking at sandy loam, which is 1.44 g/cm^3
but any number from 1.25 to 1.5 g/cm^3 is reasonable.

Incidentally, this sort of problem makes it clear why metric is easier: I can remember that water is 1 g/cm^3 or 1 pound/pint, but the conversion between cubic feet and gallons always escapes me. I have no such trouble with cm^3, liters, and m^3.

I’d do the whole problem in metric units and just convert kg to pounds at the end.”

– Funniest thing EVER. If the kids actually get this caught up in it, someone should make a movie.

I love this problem. I think that the lack of an “answer video” is a good opportunity to discuss the fact that, in real life, you can’t flip to the back of the book for the answers. The mathematician must think of ways to verify his/her answer – and one way to do that is to collaborate with others.

If you want to have a closing video, you could go to a place that sells dirt and take a video of yourself with a pile of dirt that is roughly equivalent to the amount of extra dirt that Stanley has to shovel. (It would be funny to video yourself shoveling the pile and then pan out to show how big the pile really is.)

I also like the idea of envisioning how deeply the extra dirt would fill the classroom and also bringing in a cubic foot of dirt in a container. I had no concept of how heavy that container would be.

(Recapping the problem, for the video-impaired)

I was confused about who’s who with the shorter shovel, where’s it shorter (shaft, blade, …?) and why does it matter.

To quote the video: “you dig a hole every day, five foot deep, five foot diameter, _your shovel is your measuring stick_”.

Then, Mr. Goldenberg says “Hence, a shorter shovel SAVES digging, not the other way around.” in response to the problem statement, “[Stanley takes another kid’s shovel which is slightly shorter … ] How many pounds of extra dirt is Stanley going to dig at the end of a full year?”

I think the problem is being inconsistently stated; if X-ray has the longer shovel, he’s the guy digging more. If Stanley has the longer shovel, the the set-up is wrong. Something inconsistent is going on here. But if you swap one of the variables, it all comes out right (or just let the answer be negative).

Also, one might also state the question more openly: “How _much_ extra dirt [is being dug by Mr. Röntgen or whomever drew the shorter straw or shovel or something]”, leaving the decision of unit up to the student and/or class discussion. Pros, cons?

My point was that if you only read/hear the question without seeing the video (or can’t see the video, not to bring up the fact of visually impaired students), there is ambiguity. It really does sound like Stanley has the short shovel and it’s not clear that he loses it before he gets to use it. And in fact, doesn’t he later wind up with that shovel as a gift from X-Ray?

This is, of course, a minor quibble, but my viewpoint was properly captured by Jonas despite my carelessly omitting the issue of being video- or visually-impaired. Ideally, our written problem statements are consistent with the available visual evidence and there is mutual support between them.

Like any other sort of problem-situation ambiguity, it’s possible to see strengths in not being consistent, clear, well-posed, etc., as well as weaknesses. I MIGHT have been the only person on the planet to wonder what was up (and I’ve actually seen the movie a couple of times, which could be what’s sticking in my craw a bit), but my experience is that if ANYONE sees something as unclear, s/he’s probably not alone.

All that said, I enjoyed lurking on Math 2.0 last night. Most of the ground seemed familiar, but it’s very fertile ground. Frankly, Dan, graduate school will be mostly a waste of time for you. You’re already so far ahead of the thinking of so many mathematics teachers and, dare I say it? mathematics teacher-educators that I wonder if what you’re going to be exposed to and expected to conform to in a doctoral program will improve or dull your mind. Maybe that’s unfair to Stanford, or merely reflective of my own ambivalent relationship with doctoral programs and academia. And perhaps also part of my fond wish that more folks with really great, original minds just forego the rigidity of traditional Ph.D programs if at all possible and carve out their own ground, establish legitimacy through the high quality of their work (as you are CLEARLY well on your way to doing), and let the paper chasers do what seems to pass for establishing their bona fides as insiders who alternately sneer at and quake from fear of originals and iconoclasts.

Hmm. Maybe another cup of coffee is long past due for me.

Dan, @Michael Paul Goldenberg,

WRT the penultimate paragraph of (Dan in grad school), I thought the same thing when I read that you were pursuing a PhD at Stanford. Let me put it this way: “Commanding a starship is your first, best destiny; anything else is a waste of material.” Well, not a waste, but if I could wave a magic wand I’d have you at Stanford and at the same time teaching high school (some place that doesn’t fire you every year).

For WCYDWT desk: and here I am looking for ideas because this bit of information has been on my mind since I read this (in Julia Witty’s piece “BP’s Deep Secrets”, Sept-Oct Mother Jones). Read, marvel, and think.
from the data stream recorded in the ocean near Hawaii…”The animations show spinner dolphins gathering in a circle of 16 or 28 animals, always an even number, each dolphin paired with another, the pairs arranged in an echelon formation: one animal slightly above and ahead of the next, while maintaining about three feet of separation. A perimeter of roughly 300 feet is precisely maintained as the dolphins swim in an undulating circle, trapping the fish inside the net of their swimming bodies.
“One after another, in fixed sequence, two dolphin pairs directly opposite each other dart into the ball of fish to feed. As they return to the circle, four more follow. And so on. The action is extremely fast, the dolphins darting in to feed at a rate of about 1.25 prey per minute, all while swimming and circling in their roller-coaster pattern. After five minutes below, each pair has engaged in four feeding dashes, and the dolphins simultaneously surface to breathe. They typically grab only one or two quick breaths before diving, repeating the underwater rodeo over and over throughout the night without rest.”
This occurs at the DSL (deep scattering layer, the twilight area between surface light and darkness) so no film. But Julia Witty describes a picture easy to visualize. Is mathematics involved if the dolphins can’t articulate what’s going on? Is the mathematics all in our heads? These are questions that mathematics has struggled with for hundreds of years.

I used this in my physics class today to talk about unit conversions and problem solving techniques? My favorite question that the students came up with is how long will it take until they run out of room to dig the holes? I printed them off a screen shot of holes at the begining of the clip to help them out. Thanks again!