Why do schools still get kids to do so much arithmetic?

As lockdown III rolls on, parents are once again getting a glimmer into their children's education. One thing I've seen some parents asking is why, in an age of computers, children (particularly primary school children) are still asked to do so much non calculator work. While I don't want to claim the balance is perfect or that there are no problems with the primary school syllabus (does it really make sense to try and get children to learn to identify subordinate clauses? What's the deal with all this formal linguistics they seem to be drilling into 9 year olds?) I do think that there is real value to having a really good command of arithmetic, even if calculators can do it for you. Befitting to the topic, here are some reasons in numerated list form:

1. It allows you to do quick mental approximation

Being able to do approximate calculations mentally is a useful exercise. It allows you to spot if statistics used in politics or advertising smell fishy. It tells you if a calculation in every day life (e.g at a till) is obviously wrong. You might well do the actual calculation to double check with a calculator, but knowing whether to bother or not depends on whether something sounds about right or sounds implausible. Even if you are using a calculator, it's worth knowing if the answer seems obviously wrong, as you might have pushed the wrong button, as often happens with longer calculations.  You can't do this without being reasonably comfortable with approximated mental calculations. 

2. It helps with algebra

There is a whole lot of algebraic manipulation which is fairly easy and intuitive if you understand it by analogy to relations learned in arithmetic (algebra in the school sense is after all a generalisation of arithmetic relations in symbolic form). Knowing that fractions can be simplified when a common factor can be taken out of the numerator and denominator helps you understand why you can simplify an expressions like the one below on the left but not on the right:

Maybe this can be learned without ever learning how to manipulate fractions, but I think it would be a lot more difficult to do so (some things are easier to learn by doing, starting with numerical expressions allows the 'doing' to start earlier; it also means you can see intuitively why certain relations are true: 2/4 = 1/2 can be shown by sharing up pizzas, x+1/2x+2 = 1/2 less easily). Mistaking changing the form in which a fractional expression is written and actually multiplying that fraction by something is pretty much the bane of my teaching existence teaching algebra, and the difference is most obvious to children who have a good command of fractions. Of course, there could be a correlation vs causation thing going on here, but in my experience otherwise high ability students can struggle with certain aspects of algebra if they weren't taught arithmetic properly in primary school. 

3. It can be fun

Some arithmetic is unfortunately extremely dull. I still have bad memories of learning times tables. But some numerical problems can be fun and challenging, and would be utterly trivial and dull with a calculator. Consider the problem below, which is no fun if you use a calculator: 

4. It helps you actually use calculators well. I can't tell you how much time I spend helping children identify order of operations problems they're having typing stuff into a calculator. This stuff is a lot easier to spot if you know how bits of the calculations work without putting them into a calculator. Children who don't really get that  -5² isn't the same as (-5)² almost always make mistakes using the quadratic formula, and the explanation as to why these aren't the same don't hit home unless you are really really happy with the fact that -5 × - 5 = 25. 

5. There are plenty of numerical relations that calculators aren't good at working out. Sometimes, being able to work with exact values is really helpful. In the example on the bewl, the shaded area is exactly half of the total. Children who can only approach this problem with calculators will almost always miss this, because they will round their answers as they go along. This is even more true once you get on to calculus. Calculators can even be remarkably bad at certain kinds of problems.

Try getting your phone calculator to work out the value of

ln(e^9999). It can't do it, because it tries to evaluate the inside of the brackets first, which is beyond its computational power. A human can tell you the answer is 9999 straight off.

None of which is to say that arithmetic isn't overdone, or that there isn't enough focus on learning how to use new (frankly, not that new) technology well. But the purpose of some of the old fashioned stuff might be a little less obvious than it seems at first sight. 

                                                                                                            

 

Why is the government always late to act on covid?

As I write this, the government is probably a few days, if not hours, away from announcing another national lockdown. What's puzzling is why this wasn't done weeks ago. This is nothing new: at every stage, from the initial lockdown of 23rd March, to the November lockdown, to the cancelling of Christmas relaxations, the government has delayed measures by weeks for no obvious gain and at catastrophic costs, which I'm sure readers don't need further regurgitation. 

This phenomenon is the subject of well deserved exasperation, but it is also just a bit of a riddle. It's well known and understood why delaying restrictions is harmful on a policy level (more people get sick, measures are in place for longer etc etc). What's less clear is why the government doesn't seem to understand this. What's driving these mistakes? Is it just stupidity, or is there some tangible gain, political or otherwise, which is accrued from these delays? I'm not sure I have the answer, but here are some possibilities I think are worth considering, some mutually supportive and some mutually exclusive. 

1. Stupidity. I've always assumed that even ministers of this government by in large understand the nature and dynamics of this crisis (the basic ideas aren't particularly complicated!), and I'm pretty sure Matt Hancock does, but you never know.  They are certainly under a lot of pressure from backbenchers and activists who really are stupid enough to not understand what is going on.

2. Blind optimism/motivated reasoning. This may be what drove reluctance to push for more control measures in the Autumn, when it is rumoured that Sunak and Johnson fell for some wilder ideas pushed by Gopta and Heneghan that the herd immunity threshold was already fairly close, perhaps due to cross immunity from other coronaviruses. This appears to have been largely an exercise in wishful thinking, as does the strong assumption that transmission from schools is negligible (there is good reason to think it may be lower in children than adults, but the belief that it was near non existent was probably largely just what people wanted to be true). Perhaps there is a similar kind of blind optimism that delays introducing lockdowns.

3. Blind pessimism. I think this is less important now, but in the early days of 2020, this was probably the crucial thing that prevented European countries from mimicking the success stories of East Asia, Australia and New Zealand. Quite simply, there was a widespread belief that it was naive and unscientific to think that the spread of the disease could be controlled to any meaningful extent (remember people arguing that the Italian government was being a bunch of crass populists by introducing a lockdown?). This kind of pessimism seems deeply inappropriate now (controlling covid in the short term is clearly possible, and vaccines are here, now!) but bad ideas have long shelf lives. Measures that are continuously too late can feed into this pessimism; we act to late, we do far too little to isolate cases when they are low, and people start to tire of measures that don't seem to work well enough. Moreover, the new strain certainly does raise questions about whether the measures the British government thus far been willing to contemplate will suffice (at no point have we closed non essential workplaces to the same extent Italy, Spain or France has). 

4. Public opinion. I don't think this is a real constraint on the government, but it is worth mentioning if nothing else in its negative form as this is frequently cited as a possible motivation for delays. Polling has consistently shown substantial majorities in favour of tougher restrictions sooner than they were announced, across the political divide (see here on school closures, for instance). I suppose you could still have a noisy group that opposes restrictions, and perhaps the government finds it convenient to be forced to act (probably why they tend to leak measures to select journalists hours before they are announced, so people are not hearing the measures from them the first time). There could be a bit of a bet that compliance to measures will be higher once people are afraid (i.e once the situation is obviously bad). But I don't buy that this can be a significant factor. It would make far more sense politically to bet that an already broadly supportive public would reward successful policy making.  

5. Poor management style in government. This for me is probably the most convincing part of the explanation. This government seems to favour a management system whereby individual ministers understand their brief to be about representing specific interests which they lobby for. The Prime Minister understands his job as an arbiter between these competing interests. Sunak lobbies for the Treasury, Hancock for the Department of Health, Williamson for his education priorities as they understand these. This is a uniquely bad way of running things in a situation where there are significant feedback loops and effective policy making relies on understanding the dynamics of a system of moving parts. The Treasury has come to understand (or feels it has to understand) its position as in conflict with covid restrictions, even though they are not. Williamson feels he has to lobby for schools to stay open, come what may, even though nowhere in the job brief for the Minister of Education did it say that better educational outcomes should come at the expense of large amounts of death. Still, this explanation does require a little bit of explanation 1. or perhaps something similar, as you'd expect that intelligent ministers would realise the interdependence of their departments and the mutual benefit of just going for maximum viral suppression. At the very least, competent leadership from the Prime Minister would enforce this view. Which brings us on to point 6.

6. Incompetent and weak leadership. Lockdowns may be necessary, but in the short term, they do have significant costs associated with them. These are hard decisions to make, particularly if you aren't that comfortable with thinking about mathematical and scientific models. Even if there is no gain from delay, incompetent and weak governments may simply put off decisions until they are out of options, as they are so put off by the short term costs that they end up just paying much higher costs when they are forced to.