Take game shows for example. I remember watching the first ever episode of "Who Wants to Be a Millionaire?" with a much younger and more vibrant Regis. The idea of a million dollars fascinated the public. How could such a large amount of money be given away? Back then one million was at the forefront of the world's consciousness. Everyone thought that their life would be easy if they were "millionaires". Not many understood the actual breadth of that wealth. Nowadays, we see game shows like "Deal or No Deal" have special $10,000,000 shows, "Wheel of Fortune" puts a million wedge on the wheel, and recording artists sing about how they want to be billionaires "so friggin bad". (Radio Edit Included). The world's recognition of large quantities has diluted, and that leaves the generation in today's schools--who never marveled at a million--unimpressed.
Maybe teachers should drop the catchy handles "one million", "one billion", and "one trillion". We could talk in terms of what students understand. When I was 16, I thought $1000 was a large amount of money. Now put a million in that context. A million is one-thousand one-thousands. That is a dollar for every dot on this diagram:
Have students circle the amount of dots that represent a new computer or car; the tiny circle will provide perspective.
If that is a million, then a trillion would mean that we would have to replace every atomic dot in this image with a duplicate of the whole. We essentially would have a million, millions. Such a large quantity begins to warp our minds. Once quantities get so large, people begin to lump them all together as "big". In fact, the US government is probably lucky the debt is over 1 Trillion, because people would probably view 999 billion as larger. Nine hundred ninety-nine carries more weight that one.
I was watching a game show where the contestant was asked to estimate how many people watched the 2010 Superbowl on T.V. She guessed 2 million. When asked to detail her thought process, she said that there were 300 billion people in the US, and only about 2 would watch it. She said that she meant to put 2 billion, but mixed up the terms. This tells me two things: 1) This lady has no idea how much a billion is because there is no way 300 billion people would fit on earth comfortably. And 2) She has lumped million and billion into the category of "Big" numbers. Besides her horrible estimation skills, she showed horrendous large number numeracy.
So how can we, as teachers, support the understanding of large numbers? I would start with small, lab-like activities. They fit great when there is a little extra space before the school bell. I once brought my class to the library and asked them to estimate, by sight, the number of books in the library. Then they had to come up with a way to justify. I used the librarian's records to check answers; all solutions held some merit. Once I ended class by asking the class how long it would take to type out one-million semi-colons into a single document file. We timed 10, 20, and 50 trial runs, and then extrapolated the data. Such an activity would be excellent for large number numeracy, linear data, scatter plots, line of best fit, interpolation and extrapolation, linear functions, etc. I will leave this up to you, and your students, to solve.
Allow your students to spend significant time with large numbers. This will only ease the transition into the infinite when that eventually occurs. Use an up-to-the-minute calculator of the US National Debt, and attempt to create a visualization of the monstrosity of the number. How high would a trillion page stack of paper be? How long would a trillion centimeters be? Allow students to pose their own problems and seek solutions and representations. They will be creative in their representations, and impressed with their own ability to comprehend large numbers.