My keen grade 8 students just took the last 15 minutes to run some numbers.
1) The ratio should be 20,822, as confirmed above.
2) Area of the small circle was measured at 0.070685834 cm cubed
3) Area of large circle should be: 1471.852 cm cubed.
4) Actual area of large circle: 246.7401 cm cubed
5) It is off by a factor of 5.965062
Using a black background may not be best in classroom use where lights are often up, but it is the best practice at TED where the lighting is set up appropriately. But I changed the small circle to a white-gray gradient to make it more visible.
(Haven’t really done geometry for years, so let me know if I need to make any corrections.)
On the other hand, one must consider the rhetoric of the thing. The end result was off by an area factor of about 20 — how many pixels would that be?
And it’s much better rhetorically to understate the difference by even a large factor than to overstate it by even a little. If you do the former, you get called out on math ed blogs. If you do the latter, it’s the New York Times.
I think a nice solution is to use isometric cubes. A small green cube, representing a pile of $3 million, and a large green cube, 27.5 unit cubes in each dimension, representing the 27.5^3-times bigger liability. Make the large cube look like it’s built out of the smaller $3 million cubes.
This could be done with a small cube roughly 11 pixels high, and the large cube roughly 303 pixels high. It doesn’t distort the data, and it’s still easy to read.
I did the Obama SOTU address problem with my class. We were discussing communication and it seemed relevant to point out that what you show could be wrongly interpreted by the listener/viewer.
In the subsequent discussion folks in about equal parts argued: 1) that diameter of a circle is an OK way to show the data and 2) that area should have been used accurately.
I have to say that, as the instructor, I disagree with point 1 in favor of point two. This, leads me to my question:
If you see a circle, what do you think of first? It’s area? Diameter? Equivalent spherical volume?
I think human nature needs to be understood when viewing something like this. Only then can we pick the right representation.
What about LINES for linear comparisons? Rectangles for area comparisons? And shoeboxes for volumetric comparisons? The “anisotropy” of the dimensions may force us to see the relevant metric (volume, area, length).
Great point, Dan, about the anisotropy (where did you ever find that word?!). I think your respective use of lines, rectangles, boxes, and Matt’s suggestion to use cubes, is right on.
I would argue that at first look, circles would represent area. Unless the design of it hints towards a sphere. My example above had a shaded gradient on the circle which might make it look more like a sphere, so that is something I would remove to make it more accurate.
There is another graph in his talk that caused my kids fits. It was the rectangular graph early in the talk. This led me to the same questions you posed. I wondered if the data presented in the rectangular graph was presented in a pie chart instead, would the kids still misread it?
I noticed the rectangular graphs that compares Available Resources ($76 Billion) to Budget Shortfall ($25 Billion), a ratio of 3.04:1.
I measured the rectangles of having respective areas of 561,267 pixels and 140,450 pixels, which gives a ratio of 4:1. At first glance it appears the ratios are incorrect.
However, I because the rectangles are overlayed, I wondered if I was looking at it wrong. If $76 Billion + $25 Billion is the total resources needed, then each rectangle represents their respective portion of those resources (those available and those lacking). In this case, the comparison would be ($76 Billion + $25 Billion) / $25 Billion which does give a ratio of 4:1.
It is great to see so much deep thinking about these graphs. I happened to watch the TED talk last week. I did think a bit about the graphical representation of all the money. However, I also think there is another message here. We are all so concerned about teaching math, about reaching all the kids, and presenting problems that are real and have a hook. And I think we have all gained a great deal by this forum. But I think it almost ironic (well maybe totally ironic) we put all this energy into dissecting the graphs in this talk.
We cut nearly three weeks from the school year because of budget shortfall. The curriculum did not get cut. Class sizes skyrocketed. When(if) things turn around, we will not be compensated for the heaps of work and stress we have endured.
Maybe I am in the wrong profession (is it a profession? I feel like a sparring partner).
OK, wouldn’t you know that 10 minutes after reading this blog entry, I was looking at an article that committed the same crime to geometry? The only saving grace is that the article scales the x-axis. This leaves the viewer with visual dissonance, but, an easily quantifiable comparison.