Language and mathematics Dick Hudson Godolphin and Latymer School September 2016

My main messages You started to learn maths when you learned your first language. So language provides the ‘folk foundations’ for maths. The folk foundations are better than none. But they’re also misleading. So the better you understand them, the better for your maths. And in any case, they’re interesting.

Plan Grammar helps: Numbers Grammar hinders: Bases Grammar helps: Folk algebra in language Grammar fun: Danish numbers

1. Grammar helps: Numbers All numbers have three different meanings: Serial: 1, then 2, then 3, ..... This is how small children learn numbers e.g. dates, house numbers: “My house number is 3.” Quantity: 2 = 1+1, 3 = 2+1, ... This is the stuff of maths e.g. “3 x 4 = 12” Sets: two = a set with 2 members, ... This is the stuff of counting e.g. “I have three pencils”

The linguistics of numbers Some languages have no numbers at all! So mathematics is impossible. English has: simple one-word numbers: one, two, three, .... complex many-word numbers: one hundred and twenty-three most languages are like this.

The grammar of complex numbers and = + (a hundred and three) number1 before number2: if number1 < number2, then = number1 x number2 (three hundred) otherwise, = number1 + number2 (twenty three) So twenty-three thousand, six hundred and fifty seven = (20 + 3) x 1,000 + 6 x 100 + 5 x 10 + 7 = 23,657 But the rules used to be different: five and twenty so + is always and no need for <, > And German still has five and twenty!

So what? When we learned to count, we learned about addition multiplication <, > This is part of the ‘folk mathematics’ built into our language. But the rules vary between languages, like the rest of grammar. eg in Latin, 18 = duo-de-viginti (‘two from twenty’) NB So Latin grammar includes subtraction.

2. Grammar hinders: bases The grammar restricts multiplication to a few ‘multipliables’: English: dozen, score, hundred, thousand, million, ... three dozen/score/hundred/thousand/million ... But none of these is the ‘base’ for complex numbers! For a learner, the base isn’t obvious from the grammar. As it would be if we said three ten and four for 34. In fact, the grammar makes the base really hard to find: thirty is an irregular replacement for three ten maybe the base is the multipliable that’s grammatically irregular?

Is the base the only irregular multipliable? No Our base could be 12 because eleven and twelve are super-irregular. Our base could be 20 because thirteen, fourteen, fifteen, ... nineteen are irregular. Many languages have base 20 (or remnants thereof) e.g. French quatre-vingts = 4 x 20 = 80 One language has base 15!!!

So what? Many different multipliables are available. They’re all grammatically ‘collective nouns’: dozens/scores of ..., like groups/crowds/herds of ... Just one of these is the base, used in building complex numbers. But for young learners, the base is hidden by the grammar because it doesn’t even appear to be a multipliable three hundred/thousand but three ten thirty

3. Folk algebra in language unknown same 3. Folk algebra in language Algebra uses letters to show ‘unknown’ quantities and ‘sameness’ eg1. x + 2 = 5 eg2. x2 + y2 = 2x + 3y Both x’s have the same unknown value, and likewise for both y’s. Grammar too shows ‘unknown’ and ‘sameness’: Who hurt himself? ‘who’ is unknown, but ‘himself’ must = ‘who’. And a ... the ... e.g. Once upon a time, there was a king .... the king .....

Factorising in grammar Algebra: 2x + 2y = 2 (x + y) When adding two expressions A + B, extract any common factor F from A, B, leaving A’, B’ bracket the remainder: (A’ + B’) apply the common factor to the remainder: F (A’ + B’). Grammar: She came home and she went to bed. = She (came home and went to bed)

Unknowns and factorising But in grammar, A(B+C) may have a different meaning from (AB + AC). Who (has red hair and owns a bicycle)? Who has red hair and who owns a bicycle? This distinction is possible because each who names a different unknown. 1. X (has red hair + owns a bicycle) 2. X has red hair + Y owns a bicycle. Grammar teaches factorising even to small children.

So what? Psychology Biology Chemistry Physics Mathematics Grammar Mathematics is the queen of the sciences. But grammar is the queen of mathematics. Chemistry Physics Mathematics Grammar

4. Danish numbers: a puzzle subtraction! ‘half a score less than’ seven and ‘half’ three score 93 = ? Welcome to the UK Linguistics Olympiad!

So what? Grammar lays the folk foundations for mathematics. Different languages lay different foundations. Maths teaching has to start from the folk foundations. But mathematics has to rise above them. Grammar is interesting.