I was working recently with Mr Davaasuren. He had written some teaching material about averages and we were looking together at what he had done. We discussed some of the issues involved in helping students to understand about mode, median, mean and range.
The next morning we met again and he asked if I would like to see the model he had made the night before. You can see it in the photograph
It consists of six transparent vertical tubes with a scale next to each one. The tubes are all connected via a further horizontal tube at the bottom. The tubes can be filled with coloured water and there is a stopper to go in the top of each one. Mr Davaasuren carved the stoppers from erasers.
This is how it works. Suppose you have the six numbers, say 16, 18, 12, 7, 16 and 9.
First fill the tubes to the level of the lowest number, in this case 7. Since they are all connected they will all fill to the same level.
Put a stopper in the fourth tube. The level of this will stay at 7. Add liquid to bring the rest up to 9. Stopper the sixth tube. Continue in this way until the levels in each tube are the six numbers in order and there is a stopper in each tube.
First we can demonstrate the range. It is simply the difference between the highest and lowest levels.
Next the mode. There are two tubes at the same level, so that is the mode.
Next the median. Find the highest and the lowest (18 and 7) and “discard” those. That leaves 16, 12, 9 and 16. Now discard the highest and lowest of those (one of the 16s and 9). That leaves the 12 and a 16. The median is halfway between the two. We can find this by removing the stoppers from those two tubes. The levels will even up so that they are both on 14.
Finally the mean. Remove all the stoppers and the liquid in every tube will adjust to the same level – 13 – and this is the mean.
Isn’t that brilliant? Range, mode, median and mean all demonstrated at the same time. Along the way it shows why the mode might not be a good choice of average and the fact that the median for an even set of numbers if the mean of the two middle numbers.
You could also use it to discuss what happens if you change the scales. Suppose, for example, you add 10 to every number on each scale. How does that affect the range, mode, median and mean? What if you add a different number? What happens if you multiply every number by 2? Or some other number?
Mr Davaasuren intends to make a video of his Mean Machine in action and put it on a website so that teachers will be able to show it in their classrooms. Unfortunately the website is in Mongolian and unless you have a working knowledge of that language you will have difficulty using it. On the other hand, if you have some plastic tubing laying around in your garage you might be able to make your own Mean Machine.
Students often find it hard to understand why the mean is defined in the way it is. I think this visualisation of evening out the different levels is a superb way to visualise it. It made me wonder if there are other tricky mathematical topics which could be explained easily if we just had the right visual aid. Any suggestions anybody?
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Chris Pearce was head of mathematics in an 11-18 school and later a National Strategy adviser. Now he works as an independent writer and consultant.