sequestering carbon, several books at a time LXXXIX

The latest batch, including the eight we bought on holiday in Hay-on-Wye:

full moon at sunrise

The sun has just risen; the full moon is just setting.

7:07 BST, looking west

bits of ancient history

We drove from Westonbirt to Avebury, to see the the largest megalithic stone circle in the world.

some of the stones are big, taller than a person, ...

... some are smaller, and some are missing, marked by little concrete obelisks (like the one nearest the camera)

it’s possible to walk all the way round the circle (with some small detours), and to touch the stones; some people were clearly deeply communing with them (maybe because it is the autumn equinox today?)

here we can see the ditch and the bank (in the left foreground; I’m standing on the bank to take the photo)

Having walked round the circle, we followed the "Avenue" of stones (by car) down to Silbury Hill, the tallest prehistoric man-made mound in Europe.

this is as close as we were allowed to get

Over the road is the West Kennet Long Barrow, a Neolithic tomb.

view from the road: the barrow is that little line on the horizon

wow, we’re allowed just to walk into it

it’s dark in there (although there are a couple of skylights recently drilled through the rock ceiling)

view of Silbury Hill from the barrow

That’s enough tramping round the past: we’re off to our next B&B for some modern comforts.

Westonbirt, The National Arboretum

We poodled down from Hay-on-Wye to the Westonbirt Arboretum, arriving in time for an early lunch in the visitors’ centre cafe.  Then we set out on the “Autumn trail” (backwards, as we started from the cafe end, not the main entrance), which is a mile long.  The map says “allow 1–1.5hrs”; that’s because you will keep stopping to commune with so many marvellous trees.  Add yes, we took an hour to walk a mile.

the weather is considerably better than yesterday’s storm

gorgeous autumn colours

unretouched photos taken from roughly the same spot with two different cameras:
left, a Canon EOS 20D, right, a Samsung Galaxy S7

I love the fine tracery that the branches make

a beautiful pairing of colours and shapes

a magnificent specimen; it is hard to get a scale, but the point where the trunk narrows and the first branch comes outwards the camera is over 6 feet high, and something weird is happening there ...

... inosculation 

plane tree weirdness: this isn’t two branches growing up out from the ground; it is a single branch where the left part is growing down into the ground

another weirdly shaped tree

an absolutely magnificent birch

Highly recommended.  There’s a second, longer trail, which we did not tackle, since we were next off to Avebury for a spot of ancient history.

post storms Ali and Bronagh

Yesterday was very wet and windy: storm Ali hit the UK.  We spent some time driving around the Brecon Beacons, and some in Hay-on-Wye's famous bookshops.  (We bought only eight books.  The days of filling the back of the car with second-hand purchases are over, due to already having a lot of books, having already got them via Amazon.)

This morning, we saw some of the effect of the storm: the river Wye was somewhat higher than when we arrived, having flooded "the lower field".

The B&B proprietor said the river sometimes flooded that field, in winter, or in spring, but they had never seen it flood in September.

view from a car park

We have just arrived in Hay-on-Wye, for a short holiday.  Here is the evening view across the Wye, from the B&B car park.

view from a hotel window

I'm at a two day EPSRC meeting in Birmingham, near the NEC.  Standard view of a car park, although not the one I’m parked in:

view from a hotel window

I’ve just arrived in Glasgow, for a small workshop tomorrow.  The hotel is on a bit of a slope, and I’m on the ground floor, explaining the nice view up the rockery:

book review: Better by Mistake

Alina Tugend.
Better by Mistake: the unexpected benefits of being wrong.
Riverhead Books. 2011

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We often hear that we should learn from our mistakes (or, preferably, from the mistakes of others). Here Tugend digs down into this aphorism, and shows how we are actually given mixed messages: if mistakes are such a wonderful learning opportunity, why are we actually ashamed of them, punished for them, and why do we try so hard to avoid them to the extend of over-constraining our ambitions?

Tugend recounts some work showing that mistakes while learning can be valuable, for some people at least, and should be allowed, even encouraged, to enable the deeper learning that results.

[p93] Gully and his colleagues found that not everyone learned better by making mistakes, but those with certain types of personality traits—who are good at processing information, open to learning, and not overly conscientious—were more effectively trained by being encouraged to make mistakes rather than avoid them. In other research, Gully and colleagues found that telling people to perform well during training resulted in higher immediate performance; but it also resulted in shallower processing of information, more superficial learning, and less confidence. In contrast, those people who were told to learn—and nor worry about mistakes—during training did more poorly initially but ended up with deeper processing of information, more complex learning, and more confidence about performance. Those in the latter group also showed higher performance when faced with a really challenging version of the task they were trained to perform.

Of course, mistakes while learning are somewhat different from mistakes that happen during operational use. But we are actually learning all the time, and so these other mistakes should also be exploited as learning opportunities, both to educate the one who errs, and to make the overall system more robust to inevitable mistakes.

[p112] “when an adverse event occurs, the important issue is not who blundered, but how and why the defenses failed,” […] In life-or-death situations, it is important to set up a system in which, to whatever degree possible, one person’s error cannot sink the ship. […] focussing on active errors lets the latent errors remain in the system, and “their accumulation actually makes the system more prone to future failure.”

The key point here is that a systems approach to error does not look just at the “active error”, the immediate surface error that seems the cause of the problem, but also digs down to find the “latent errors”, the systemic issues that allowed the active error both to occur and to have such poor consequences. Just fixing active errors results in a patchwork of arbitrary rules and regulations that can make the system more complex and fragile.

How to use errors during learning requires a supportive culture. Tugend gives an example of the culture in Japanese schooling.

[p191] Making a mistake, therefore, isn’t a reflection of your lack of ability or intelligence, but simply that you haven’t learned something yet. “You have to show you’re trying hard—they have this expression for ‘facing the desk,’ ” […]

So there are two aspects. First, the learner has to be trying: their mistake is a good-faith error, not the result of carelessness, or even deliberate sabotage; the learner genuinely thought that their mistake was the right thing to do. Second, the mistake can then be recognised as a signal that the learner needs to learn something, maybe a new fact (surface error), or a revised mental model of the problem space (latent error). Occasionally, it is the teacher who has to change: the learner’s mental model might be more advanced than the teacher had realised, and their “mistake” is due to tackling an over-simplistic problem in their more sophisticated manner. A systems approach to diagnosing the latent error should be able to distinguish these cases.

Of course, not all mistakes occur when we are trying our hardest: we might be tired, or distracted, or careless, or selfish. A systems approach can help diagnose these problems as well: we don’t need to learn new procedures, rather a new attitude. Although Tugend doesn’t explore this aspect (other than in the chapter on apologising), this links to the rationale behind punishment of errors: the assumption that the error is in some sense “deliberate”, and the way to learn the required new attitude is through some sort of pain. But a culture change is needed, to distinguish good-faith learning opportunity errors, from sloppy poor attitude unnecessary mistakes – and to treat them differently.

There is more in the book, including how the air transport industry learns from mistakes and its use of checklists, how men and women react differently to their mistakes, and apologising for mistakes. There is a lot of food for thought here. I would have liked more guidance on the productive exploitation of mistakes, but maybe that is for another book.

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For all my book reviews, see my main website.

book review: The Irrationals

Julian Havil.
The Irrationals: a story of the numbers you can’t count on.
Princeton University Press. 2012

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The Irrationals – real numbers that aren’t rational – have many fascinating properties. Havil takes us on an historical journey, from the Ancient Greek mathematicians’ geometrical investigations, to present day deep algebraic concepts. The earlier material is, unsurprisingly, less technically challenging, but no less interesting, than later results.

I thought I knew a reasonable amount about irrational numbers before I started (possibly more than the target audience), but I still learned many interesting new facts. Some of these are quite elementary (presumably I missed these back at school due to having studied a “modern maths” curriculum), and some are more advanced.

So I learned the rational roots theorem (see p.131), which (in its simplest form) states that, given a polynomial equation xⁿ + a_n−1xⁿ⁻¹ +…+ a₁x + a₀ = 0, where all the coefficients a_i are integers, then any rational root must be an integer that divides a₀. That’s extremely useful, as it gives a very quick constructive way to test for the existence of rational roots. The theorem is presumably known to all children studying a more traditional maths curriculum. This result, along with the majority of the others, is accompanied by a proof; I confess I skimmed most of the proofs, but they are a useful resource.

We get discussions of constructing geometric figures, algebraic numbers, continued fractions, approximating irrationals with rationals, transcendental numbers, a “positive” definition of the irrationals (that is, not one like “all reals except for the rationals”), a way to quantify how irrational a number is, randomness, and much more.

One result I particularly liked is the following (see p.262): Let α be a positive irrational. Define β = α / (α−1). Define the sets A = {floor(nα) : n=1,2,3,…} and B = {floor(nβ) : n=1,2,3,…}. A and B partition the natural numbers; that is they have no elements in common, and together contain all the natural numbers. Different choices of α yield different partitions. Havil provides examples, for α = √2, π, and e. For e, the sets are: A = {2,5,8,10,13,16,19,21,…} and B = {1,3,4,6,7,9,11,12,14,15,17,18,20,…}. That would be fun enough but itself, but it leads on to even more fascinating results. It is obvious we can write a formula to generate the multiples of three: f(n) = 3n. But we can also write a formula that generates the non-multiples of three: g(n) = floor(n/2 − ¼)+n. The approach is general, and can be used to write formulae to generate the non-squares, the non-triangular numbers, etc. (I just love this kind of stuff.)

From the playful subtitle, to the Appendix on how to find the tomb of Roger Apéry in Paris, the whole book is full of gems, and is written in a very accessible manner, provided you know a little bit of maths to start with. Highly recommended if you like reading about maths.

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For all my book reviews, see my main website.

not suspicious at all

I've just received an email.

It contains the following 4 images:

Yes, an email that looks like a load of text is actually four images.  The first bullet in this pseudo-text has the excellent advice "Please do not click on any links you do not recognise."

The middle two images (the last three bullets, with the bold telephone number and the bold "here") not only are links, but you are explicitly invited to click on one.  These links have extremely unrecognisable URLs like "http://links.mkt529.com/servlet/M= ailView?ms=3DMjE4MTcxMTAS1&r=3DMjExMTk3OTQyNDQwS0&j=3DMTMwMzMzNDgwNwS2&mt= =3D1&rt=3D0"

So, an email warning me about a potential data breach that is itself either (i) a phishing attack; or (ii) from someone who does not understand their own security advice!

Sigh.