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.