On 1/22/2011 at 10:15 pm, Jonathan Crabtree wrote:
> Thank you all for your input. I really appreciate you > putting your minds to this unusual question. > > I'm curious about naiive math or intuitive math. > > Twice as many involves counting or subitising at its > simplest. Twice as long, twice as high and twice as > many may be more left-brain measurement. > > I remember learning to draw by a technique in which > you copy a detailed image. Yet the detailed image is > upside-down. The idea (which seems to work) is that > the non-numeric right brain then asserts control on > the visual processing. > > Then the artist turns their drawing the right way up > and is delighted to see art! > > The alternative is adults will draw the image via > measurement and estimation of angle, which results in > a bad outsome. > > Perhaps twice as big or twice as large relates to > size before the left brain and symbols get in the > way. > > So just as with the square, if I asked adults to > draw a circle and then draw a circle twice as > big/large they still have no idea what the result > looks like, let alone how to draw it. > > I believe children think of scale and ratio by > comparing relative sizes well before they learn > counting. My hypothesis is that you shouldn't need a > formula to draw circles or squares approximately > twice as big... > > So rather than number, I'm wondering about ratios > between consistent forms. > > A farmer who can't count, may intuit the 'twice as > big/large' shape by wondering about how much crop > could be sown inside the shape. > > A painter might intuit the twice as big/large shape > by contemplating how much paint might be need to > paint the shape. > > A cook might contemplate how much extra dough might > be needed to make a pizza twice as big/large. > > To me, visual mathematics may be better taught the > way people naturally observe the world without > symbols and formulae interfering too soon, which is > why I'm interested in this area of learning. So while > my approaches may not be for everyone, I believe they > may help those with either math anxiety or > dyscalculia. > > As for MAB, it's an acronym for Multi-base Arithmetic > Blocks. Perhaps you know them as Cuisenaire or Dienes > in the Northern hemisphere. > > As it appears there may be no research on this > question, I will start surveying adults and children > to see if young children are better at estimating > 'twice as big/large' an area than adults. > > Thank you all again for your interest. > > Jonathan Crabtree > Geelong, Australia
I have seen that as a common error: the assumption that doubling the sides of a square or rectangle doubles the area or doubling the length, width, and height of a box doubles the volume.
Though I agree with the others who have said that this question might be vaguely worded. At first I did not because I had assumed "twice as big" for squares means "twice the area." This is probably the most common meaning of what we mean by "twice as big" for two- dimensional shapes. Graphs that do take advantage (or abuse) of two-dimensional shapes use areas to their advantage (or abuse). That is, our eyes tend to respond to the sizes of the areas rather than the sizes of the dimensions. But one can reasonably argue that even this common meaning is vaguely worded.
But there is no doubt in my mind that it is a common error to assume that doubling the dimensions of a geometric figure doubles the area or volume. This misconception is probably one reason for why many students make the error of converting square inches to square feet or cubic inches to cubic feet by dividing by 12 instead of by 12^2 or 12^3 (of course, some make the error simply by misreading or by misunderstanding that feet and square feet and cubic feet are all different). Whatever the reasons for this error in any one particular student, it does remind me of the general error that multiplying the dimensions of a figure by k multiplies the area or volume by k as well rather than by k^2 or by k^3, respectively.
A lot of graphs will use this error--perhaps intentionally--to distort proportions. For instance, one graph I had seen in a math book that discusses various ways that graphs are distorted had drawn a graph of the spending value of the dollar at the time the book was written (which was within the last few years) in terms of the dollar from about 1980 or so. Today's dollar is equivalent to about 43 or so cents back then, which is roughly half. The graph draws dollar bills in a way that the dollar bill for today had length and width about half of the length and width of the dollar bill representing a dollar from the previous generation. That is, the graph visually suggests that today's dollar has about only a quarter of the spending value of last generation's dollar rather than half. I will have to track down that book of mine if I am to get the title and author. My first guess a few minutes ago was incorrect. So were my next several guesses.
I don't know if it is the same book or not, but I remember another example involving either prices or uses of oil over time, and the graph drew oil drums in such a way so that twice the amount being measured was represented by an oil drum 8 times as large.
Such examples might not answer your original question, but they do illustrate this same error involving areas and volumes.