Does more means less in the US?
I arrived in the United States in 1999 to take up a teaching post at an independent selective high school in Washington DC, having been head of a successful mathematics department in a selective UK girls' independent school for several years. I came with preconceived ideas about maths education in the US - everyone has heard how low Americans score in world ratings. I was in for a shock.
Not only were the students there far more motivated and success-driven than in England, but their academic level was, in some areas, far higher. The average US student will study maths for far longer than UK students do, usually well into college, and within a very abstract and rigorous syllabus, too. This is excellent training for those going into pure maths, but for the rest it is too easy to slip through the net, and not become even solidly numerate, let alone have a useful mathematical foundation.
The US operates on two very different systems: the divide between state and private education is even greater than it is in the UK, where different types of school are kept broadly together by the national curriculum and the constraints of the public examination system. In the US, every independent school is free to develop and teach curricula and courses of study entirely their own.
Nationally, the most standard US high-school curriculum offers algebra 1 in Year 10, geometry in Year 11, algebra 2 with trigonometry in Year 12, and pre-calculus in Year 13. Some schools will do algebra 1 in Year 9, so that students can get to calculus before they leave high school.
Naturally, as a mathematician, I agree that maths is probably the best training in thinking, in logical deduction; but its value is not diminished if set in context. Creative thinking needed to extract mathematical information from a complex situation is a stimulating challenge - the perfect training for budding scientists, engineers or even economists. In the US this is eschewed for a rigorously abstract pedagogy. In contrast, able students in the UK will have studied more mathematical applications and the weaker ones will have more practical mathematical tools to apply to problems later in their lives.
In my rather elite school, racially and culturally diverse, full of confident, self-assured young people, I was amazed to find juniors (UK Year 12) studying very advanced and rigorous calculus, such as Rolle's theorem and the differentiability of functions; seniors (UK Year 13) learn linear algebra and multi-variable calculus. Lower down the school, 14-year-olds were solving equations in complex numbers, or studying Euclidean geometry from first principes, including a thorough look at proof, and 15-year-olds were learning about groups and fields.
It is a more complex picture if we look back at a younger stage. At middle school (10 to 14 years), for example, the students could take a pre-algebra course, or a first year of algebra, as well perhaps as a geometry course. They will have impressive skills on the computer, but ask them to manipulate simple fractions and they will reach for a calculator. Mental arithmetic, so essential to get a sense of number, is hardly practised at all. Take trigonometry, for example. This is approached, in England, at perhaps 12 or 13 years old, from a right-triangle point of view. Simplistic, maybe, but look at all the science that opens up. Forces can now be resolved, and all sorts of interesting problems set and solved.
In the US, trigonometry is approached from the point of view of the unit circle, and of all six circular functions. Far more complex and demanding: more satisfying, perhaps, from a pure mathematician's standpoint, but far less practical.
Such a very theoretical maths syllabus may be why so many US students "fail" in maths. Abstract thought is essential to pure maths. Yet my experience has been that most people have a ceiling in maths, beyond which concepts do not fall simply and neatly into shape, making sense in a logical way for the individual. This is the point at which one cannot follow any more - and the problems become overwhelming.
And then there is the fact that the great majority of US pupils study no statistics - or applied maths - at all. Recently an Advanced Placement (AP) has been developed in statistics. AP is the examination set by college boards and taken at high school so that students can gain credit in a "college-level" course. For many years AP Calculus has been the gold standard for college entrants. It is an excellent course, and a pretty good exam, but an unrealistic goal for everyone to reach. More students could attain statistics - you really only need an Algebra 2 course (usually completed in the sophomore year, UK Year 11) as a pre-requisite. Furthermore, making sense of statistical data and applying tests may be more useful in later life. Realistically, how many students will ever need to integrate functions? But unfortunately statistics is the "AP for dummies"!
In the US, then, there is a sharp divide between the high achievers and the average or struggling students. Such students are not given enough practical mathematical tools and are left unnecessarily feeling failures on their abstract path towards calculus.
Plaxy Arthur was head of the math department at Georgetown Day high school, Washington DC