Intelligent Estimation: An Essential Skill
Note: In this post, Sukanya introduces what the skill of estimation is about and how she goes about helping children to develop this ability in her 5-hour course.
During the first test of an atomic bomb on 16 July, 1945, merely three weeks before the real bomb was dropped on Hiroshima, one important question was the yield of the weapon. During the test, the Nobel-prize winning physicist Enrico Fermi, one of the leaders of the team, estimated that it was about 10 kilotons. This was more than a mere guess. As the shock-wave from the explosion hit the Base Camp where Fermi was observing the test, he threw a handful of paper scraps into the air and watched how far the shock moved them. Then with a few straightforward assumptions, he made an estimate that turned out to be reasonably accurate . The actual yield turned out to be 19 kilotons.
Fermi's report about his estimation procedure for the intensity of explosion
A hallmark of such questions was that they were open-ended, with no definite “correct” answer and were presented with incomplete data, with no definite formula to plug in numbers. Solving them required a different way of thinking, integrating multiple skills and intuitions. One had to separate relevant from irrelevant data, round off numbers cleverly, often perform simple analogue experiments, have a keen awareness of rough magnitudes of physical quantities in the real world around us. Finally, one had to come up with a number which in most situations could not be verified in practice, yet one had to have the confidence that it was accurate to a factor of ten !
Working on such problems often requires innovatively solving them by two different methods and see that the answers agree broadly. In addition, such estimations rarely require mathematics beyond multiplication and division . The calculations are so simple that they can be performed on the back of an envelope.
In the GenWise course on estimation that I offered for children from grade 8, 9 and 10, I tried to follow the spirit of Fermi.
First, was to develop a rough quantitative awareness of the world around us. When we walk past a tree or a building, we tend to describe it as “ very tall” or “ very short”. Rarely do we use statements like, “ I think the tree was about 30 ft high”. Making a habit of roughly estimating sizes, weights, areas etc. without actually measuring them adds to the collection of estimation tools.
Second, to recognize that it is okay and often good to approximate. Children are unfortunately fed the idea that doing science means always being exact and accurate and they have to give answers spewed out by their calculators to five places of decimals. It took a couple of sessions to break the habit and get comfortable with being accurate to an order of magnitude.
Next came how to deal with being presented with a problem with insufficient data and yet demanding an estimate. This is often the case in real life, and is a worthwhile skill to have not only in STEM fields, but in all walks of life. I urged students to use Google minimally, try to draw from their own experiences, not use the calculator at all and observe connections carefully. They realized how a party game of estimating the number of candies in a jar could be connected to estimating the number of stars in a galaxy or finding the total mass of Corona viruses of the infected population of the world. The counter-intuitive numbers often came as big surprises.
We estimated the most outrageous things that I believed was not possible to estimate
Siddharth, Grade 9 Homeschooler
It was a fun roller coaster ride of estimating hairs on a human head, realizing how much water is wasted when you leave a tap dripping, estimating animal populations while they are moving around and speculating on the number of planets hosting alien civilizations.
Since many problems were open ended, children found diverse methods of solving them.
Extract from a student's assignment, estimating the number of hair strands on a human head
One of the things I hoped to achieve in the course is a change in outlook and a critical view of the numbers that are thrown at them. For example, if the newspaper said that the dry waste generated in Bangalore had dropped by 35 % during lockdown, I would like them to look at this news and wonder , “ Can that be right ? Let me try and estimate the dry waste generated every day ….’’
Notes:
The best curricula identify estimation and having a sense of scale as important capabilities to be developed. 2 such examples are:
The art and science of approximation in engineering, taught by Prof. Sanjoy Mahajan at MIT; see our related blog post.
Benchmarks for Scientific Literacy, published by the American Association for the Advancement of Science (AAAS) - a brief extract below from here on developing a sense of scale and a range of values for middle schoolers...
