some thoughts on estimation

I'm intrigued by a recent article by Philip Ball, How natural is numeracy?

There are a lot of things in there that are really interesting, but it got me thinking about something I've pondered before: how we put numbers on a number line. I haven't really got any implications from this for classroom practice; it's more just something I've pondered because I do estimating a certain way with my classes: draw a blank number line, mark a number on the right that you're sure is too big (be brave!) and one on the left that you're sure is too small. Then mark on your estimate.

It's a good routine. A question arises: would we expect the estimate to be half way between the two other points, the bounds within which we think the actual number must be?

As Ball writes:

Some researchers have argued that the default way that humans quantify things is not arithmetically – one more, then another one – but logarithmically. The logarithmic scale is stretched out for small numbers and compressed for larger ones, so that the difference between two things and three can appear as significant as the difference between 200 and 300 of them.

In 2008, the cognitive neuroscientist Stanislas Dehaene of the Collège de France in Paris and his coworkers reported evidence that the Munduruku system of accounting for quantities corresponds to a logarithmic division of the number line. In computerised tests, they presented a tribal group of 33 Munduruku adults and children with a diagram analogous to the number line commonly used to teach primary-school children, albeit without any actual number markings along it. The line had just one circle at one end and 10 circles at the other. The subjects were asked to indicate where on the line groupings of up to 10 circles should be placed. 

Whereas Western adults and children will generally indicate evenly spaced (arithmetically distributed) numbers, the Munduruku people tended to choose a gradually decreasing spacing as the numbers of circles got larger, roughly consistent with that found for abstract numbers on a logarithmic scale. Dehaene and colleagues think that for children to learn to space numbers arithmetically, they have to overcome their innately logarithmic intuitions about quantity.

Attributing more weight to the difference between small than between large numbers makes good sense in the real world, and fits with what Fias says about judging by difference ratios. A difference between families of two and three people is of comparable significance in a household as a difference between 200 and 300 people is in a tribe, while the distinction between tribes of 152 and 153 is negligible.

Yes, it does make good sense, to me at least. So does that affect how we estimate? Is our uncertainty around our estimate logarithmic?

So I tweeted:

I'm thinking about estimation.
If you could answer these @Estimation180-style questions I'd be very grateful. What is:
🔸your estimate of the number of pieces of spaghetti?
🔸the largest number you're sure there's more than?
🔸the smallest number you're sure there's less than? pic.twitter.com/tD9lGrMpjD

— Simon Gregg (@Simon_Gregg) April 11, 2018

and got some replies. 

Professor Smudge pointed out that we would never do this though we might estimate how large a portion this would give. It's a good point, and it's worth thinking about whether an estimate we ask students to do is 'natural'. It also explains why we're quite likely not to get this right.

Here are the estimates and bounds. The estimate is where the blue and red meet, the lower figure is the left end of the blue and the upper figure is the right end of the red.

The top six are what I was expecting: the red is bigger than the blue. On the other hand, two (Lana and Kit) have red and blue the same length, and two (ThinkMath! and Erick) even have the red smaller than the blue. What other factors are at play here, I wonder?

As some estimates were still coming in, I sent out a neutral kind of tweet, with the third sentence adapted according to the answer I'd got:

The number counted is 344. What I was looking for is how your estimate is positioned between your lower and upper number. You've put yours closer to the lower one. Can you say anything about that? (No problem if nothing comes to mind.)

ThinkMath! said:

Using the phrase, you’re sure of, I chose 100 and 500, because I would be pretty confident (betting $ on it) that the number would fall into that range. By just looking at the jar, I divided into fourths, and figured 80-90 strands per section. 320-360.

Mary said:

My tendency is to be surer of lower limit than higher - not sure why. Interesting!

Miss Primary said:

I think I was more confident of counting a minimum number from looking so kept lower number close to my estimate but not so confident with the upper estimate so went bigger to be on the safe side!

Lana said:

Guess it looks like a lot of noodles, so I reeeeally wasn't sure abt the upper bound and pushed it up significantly.

Vincent said:

That's a good question, and don't think I have a scientific answer. I feel that I can grasp what roughly a 100 is, and I see there's between 100 and 400. But maybe up to 500. I think I had the 500g in mind, and a spaghetto weighs more than a gram.

The Chalkface said:

Tried not to think too hard about it at the time. In think maybe it has to do with thinking multiplicatively. As in, it's probably no bigger than twice my guess. So I guess on a log scale, I'm closer to my upper bound. Should have gone with 450 and been symmetric and less wrong!

I asked him more about the logarithm thing.

Probably depends on what's being estimated, but, say, guessing a person's salary we already talk in log 10 (5-figure salary, etc). I'd also argue that 600k is a better estimate for Nottingham population (300k) than 30, which it is multiplicatively if not additively.

That's interesting, isn't it. Our number system makes us think logarithmically: 1-digit numbers, 2-digit numbers, 3-digit numbers. And I like the example of population.

It would be interesting to try this on a larger scale, again with a sample of people that know how this works. (I tried looking at some of the Estimation 180 data, but it was clouded by some unthinking  and some completely out of the ballpark responses.)

It's not hard to estimate how many sleepers there are from the photographer to the 39 sign.
But it it gets harder to estimate, the bigger the number.