The problem with Maths is ‘English’ - 'twenty' examples

Maths is difficult to learn and hard to teach. But one BIG problem, which few parents fully understand, is language – the use of English.

Jack, do you know what volume means?

Yes.

Explain it to me.

Ok, it’s the button on the remote control that makes your TV go louder.

English is irregular

English is a magpie language, highly irregular and puts more load on working memory and that leads to more errors. Beyond working memory it is important to know that irregular terms have to be stored separately in memory, regular forms don’t have to be. This extra work, and extra steps, places an extra load on working memory, which has a limited capacity in terms of ‘registers’ and manipulation within these registers.

1. Number words

Chinese (Japanese & Korean are similar) has just nine number names from which larger numbers are generated, compared to English, which has more than two dozen unique number words.

2. Words shorter

Chinese speakers can easily memorise this sequence of numbers 7,3,5,6,9,8,4, compared to only 50% of English speakers. Why? Our working memory has to cope with words that are longer. Their number words are short, sharp sounds, like si and qi, not long-winded words.

3. 11 & 12

They say ‘ten-one, ten-two’, rather than the unique eleven and twelve. ‘Eleven’ and ‘twelve’ come from ‘ain-lif’ and ‘twa-lif’, meaning one-left and two-left (after counting up to ten) in Old German.

4. 13 & 15

English number names are more irregular than you’d think. For example, we say fourteen, sixteen, seventeen, eighteen, and nineteen. Wouldn’t it be easier if we also had oneteen, twoteen, threeteen, and fiveteen?

5. 20, 30, 40, 50

In counting by tens, we have a similar discontinuity. There are sixty, seventy, eighty and ninety but there are the irregular twenty, thirty, forty and fifty.

6. Teen reversal

With two figure numbers, if we keep counting up, the numbers above twenty will have the tens first (e.g., fifty-six), whereas for the numbers below twenty we put the ones first (e.g. thirteen).

7. Place value

So, rather than ‘twenty eight’, they say ‘two-ten-eight’. This hurdle for English speakers is ‘place-reversal’ as the English language reverses mathematical place: six-teen rather than ten-six, which causes problems when dealing with double-digit calculations. Partition, or breaking numbers down into parts then adding, subtracting, multiplying etc. is much easier if ‘making a ten’ is easy linguistically.

8. Hundreds & thousands

Take the number one hundred and four. The child may know one hundred is 100 and that four is 4, then say that one hundred and four is 1004. Similarly with one thousand and eleven and so on. This is not a problem in some other languages.

9. Addition & subtraction

In adding eighteen plus seventeen in your head you have to reverse both numbers first then add them. If you ask children to add seven hundred & forty eight and forty two, in English, they will need to convert those words to numbers (748 + 42) and then do the addition. In Japanese, this would sound like, “seven-hundreds; four-tens; eight plus four-tens; two.” There are far fewer things to interpret, hold in working memory then manipulate as ‘place’ is reflected in the structure of the language.

10. Numbers are not just numbers

I have a two baths of water at 25 degrees centigrade. What is the temperature if I pour a bath of water into the other? Some children will say 50 degrees. Why do kids double when they're meant to square? Because that little number hovering up tere is a '2'. Confusing or what?

11. Division & multiplication

Multiplication means things get ‘bigger’ and division means things get ‘smaller’ – right? No. if I multiply ten by a fraction the numer gets smaller and if I divide ten by a half the number gets bigger. It’s easy to teach surface maths, that teach real maths. This is just one of many examples where you have to ‘see’ the problem.

12. Fractions

Take the fraction four ninths in English, the same number in Chinese is ‘one part out of nine, take four’. The language literally unpacks the fraction, this makes the fraction not only easier to understand but also makes the addition, subtraction and other manipulations of fractions easier.

13. Shapes

The Finnish language has a lot of words which are easy to understand, if you're a native, even if you don't know the word originally. An example ss the shape ‘Hexagon’ in Finnish is ’kuusikulmio’, which means ‘a shape with six corners’. This allows the child to imagine and recall the shape with greater ease. In English, we’re lumbered with obscure Greek and Latinate prefixes.

14. Alphabet

Maths may seem like an exact language but its ‘conventional’ use of alphabetical letters can be confusing:

a,b,c tend to be constants (fixed values)

A,B,C points on geometrical figures

i,j,k,l,m,n tend to be integers for counting

x,y,z unknown variables

This can cause confusion in the interpretation of problems and geometric images images.

15. Share & straight

These two words seem straightforward but research shows that children often interpret these words differently when learning maths. If I said ‘Ten sweets are shared between Rob and Jack but Jack has four more than Rob’ responses such as ‘But they’re sharing so they must have 5 each’ are not uncommon. Similarly, when children hear the word ‘straight’ they may interpret this as just vertical and horizontal and not regard a sloping line s straight. These linguistic traps are difficult for adults to spot but easy for children to fall into.

16. Instruction

You can be asked to: find, calculate, work-out, how many

Addition: Add, make, total, plus, addition, make, sum, altogether, fewer

Subtraction: Subtract, take-away, deduct, minus, leave, less, difference between

Multiplication: Multiply, by, times

Division: Divide, into

Surveys, where children voice their difficulties have uncovered many problems around the use of these terms for mathematical problems.

17. Literacy hits maths

Low levels of literacy may lead to poor or no understanding of the often convoluted problems that mats teachers and textbooks set in maths. Most are unlikely to ever have been heard by the child before, many using language that is beyond their reading age.

18. Poor, wordy test items

Many maths problems, set in exams, are more tests of complex literacy than maths. This is why over reliance on word problems may hold children back as they fail to untangle the linguistic traps that are inherent in English and poor assessment items. Too many obscure, word-based, test items involve unpacking tense, comparison and change models that are beyond the actual testing of addition or subtraction.

19. Vicious circle

Asian language speakers, from an early age, get more success from their efforts, This creates a virtuous circle, where learners get quick results and feel as though numbers are easily manipulated. Compare that to the vicious circle of English learners, who have to cope with the irregularity of the language problems and cognitive overload.

20. Culture of ability not effort

One last, but seriously fatal, cultural difference may be the fact that some cultures see failure in maths as a lack of effort, not ability. We have a culture that all too often uses the language of ‘talent and ability’ not ‘effort’.

Conclusion

Appearances are deceptive in maths. For most children it seems like a subject full of traps, deliberately set to fool you. The problems set are often badly worded, convoluted and unrealistic and often not enough variety of problems are used. This is exactly why teachers need professional training, as the effective teaching of maths needs a deep understanding of what has to be learned.

Only 60% attend lectures & plummets across semester & week – and that’s at Harvard!

Whenever I hear the word ‘pedagogy’ in HE, I reach for my gun. Again, institutions that have the ‘one hour lecture’ as their core ‘pedagogy’ need to have their heads examined, or at least read some basic psychology on attention, storage, practice and recall. In short, lectures go against everything we know about the psychology of learning. Of course, if you do question it, you need to duck as the stones from glass houses come at you like hailstorms. I know – I’ve been there. That’s why this Harvard study is a killer, but first some reflections.

University rankings are a lie

University ranking tables are perhaps the most mendacious form of marketing known to man. They are, quite simply, a lie. Why? They say nothing about ‘teaching’ the reason for most of this marketing effort is to attract students and funding (monetising). The reason they say nothing is that they don’t measure anything. It’s all proxies. The Times rankings are a case in point. They claim that their ranking scores include teaching. In fact, only 30% is based on teaching but they use NO direct metrics. The proxies include student/staff ratios (which is skewed by how much research is done) and, even more absurdly, the ratio of PhDs to BAs. It is therefore a self-fulfilling table, where the elite Universities are bound to rise to the top. There is no direct measurement of face to face time or student satisfaction. They don’t even measure how many students turn up to lectures, their primary form of delivery. Imagine a restaurant that doesn’t record the number of customers that eat its food or the film that doesn’t measure its box office numbers. This puts paid to the idea that HE is anywhere near the Big data world. It is closer to the no data world.

When you then ask why they don’t measure this stuff – the reason is clear – they’re scared. Studies on lecture attendance are rarer than dragon’s teeth. Even when you do find one, it is usually flawed by being self-reported data on both attendance and reasons for non-attendance. In addition, they are usually single courses and the selection of those courses does not eliminate bias in the selection process. In short – they’re fatally flawed.

We lecture because we always did

Schank, a major academic and learning expert, has been looking at this issue for over 30 years thinks he has the answer, and I agree. Fundamentally, we lecture “Because we always did. No one wants to change this really. We are all just used to it.”

Lectures are lazy

Beyond this “Lecturing is a lot less work for everyone….they are the lazy person’s approach to education. Both lectures and listeners agree that neither of them wants to do much work. Real work, and real doing, and real conversation, is all that matters for learning, but education is really not about learning”. Read this piece for further elaboration.

Harvard study

For once we have a solid study that covers a wide range of courses and relies, not on self-reporting but actual attendance. It’s smart, as they use a camera to identify and count the attendees. Go Pro cameras were used in four lecture halls on time lapse, with software that was verified as accurate in counting attendees. If you are in any doubt here, the count was validated against manually counted images and found to be astonishingly accurate.

Results

10 courses were studied:

1. Only 60% (average) of students attended any given lecture

2. Attendance declined over the semester, from 79% to 43%.

3. Attendance declined over week

2. Incredible variability between courses (38%-94%)

Why the difference between these two sets of courses? Attendance on the top three were graded.

Puzzle

Given the fact that Harvard has some of the best academics in the world, consistently hits a top spot in university rankings and that it costs somewhere north of $70,000 a year to attend, you’d think the students would be pretty full on when it comes to lecture attendance. If these numbers are true, imagine how bad the rest are! 

Finland - more myths on teachers, class sizes and selection

Finland - more myths

My last post on 'Why Finland is finished as a role model in education' generated some great contributions from people who do teach in Finland. Their comments were welcome and I don’t want to repeat them here. But I do want to include some other voices, with views on Finnish teachers, class sizes and selection, that run counter to that great Finnish saga, which presented Finland as an educational paradise. They largely blame PISA, as it was PISA that sparked off the mythology, exaggeration and cherry-picking. Finland is not to blame for this, indeed many I speak to seem a little embarrassed by the attention – it is not in the Finnish character to puff your chest out in public. Indeed, many Finnish voices have been drowned out by the evangelist Pasi Sahlberg. Note that this is not his fault, as he's quite realistic in seeing the Finnish system as non-transferable to other countries.

General myths

I mean people like myth-buster Fred Durvin, Professor of Multicultural Education at the University of Helsinki, who points to‘Mythologies’ (1957) by Barthes, to expose the process where myths (such as the Finnish educational myth) easily slip into being ideologies. That's exactly what is happening here, for example, with the UK government's adherence to PISA as the standard against which our education system will be judged. He does not see PISA as a good tool for assessing ability and claims, counter to the mythology, that the Finnish system is highly competitive, with children generally starting school at age two, not seven. He also points towards the lack of reporting around violence and major gun massacres, as a typical example of the selectivity of the evidence and mythologising. I had never heard of the school massacres at Rauma (2 dead), Jokela (9 dead) and Kauhajoki (11 dead) before reading Durvin.

Finnish teachers – the myths

The problems come when positions are exaggerated. Finnish teachers are NOT paid the same as Doctors. Indeed many, many teachers are paid more in other European countries and countries around the world (see OECD stats). The statistics are also skewed by the fact that teachers in Finland take 20 years to climb the salary ladder, compared to 12 in England and 6 in Scotland. Oh and they do have a National Core Curriculum and the country does teach to set textbooks.

Class size

Class size is also misrepresented. Finland has small class sizes, as they have a very low population density. This point was made in 'Finland’s PISA success: Myth and transferability' by Johann C. Fuhrmann and Norbert Beckmann-Dierkes, who showed that Finland, which is marginally smaller than Germany, has a population density of only 17 per square kilometre, compared to 230 in Germany. This is why 70% of Finnish schools have fewer than 7 teachers, with a quarter of all schools having less than 50 students and only 3% with more than 500 students. This is why the Finnish government has just announced a 260 million Euro cut to the schools budget. The stagnating economy can no longer afford to keep so many small schools open.

Local schools

Do Finns send their kids to the local school? Are they free from the pressures on parents in the UK to choose a school for their children? In fact, the usual sharp elbows of the middle class do select into certain schools. You may be surprised to learn that parents in Finland have had that choice since 1999, and that many do exercise that choice. In fact, in Turku and Helsinki, about 40% go to schools that are NOT the nearest. The exercise of that choice seems to be correlated with wealth and social class. This has caused problems for politicians who espouse a local schools policy, with no option to choose, then choose to send their own children to non-local schools. It may also surprise many, it did me, that pre-school education is NOT free but means tested and Finland DOES have many dozens of private schools.

Agendas

A lot of the so-called evidence comes from fleeting visits by observers who see what they want to see. In practice, the Finnish school system is far from being bad and, as Mark Cuchner explains, there is much to admire. However, it is far from being the utopia we have been sold. The problem is that the word ‘Finland’ has launched more ‘agendas’ than Viking ships. That agenda is either right-wing politicians getting obsessed with maths and PISA, teachers getting obsessed with higher pay and status or teacher-trainer organisations demanding higher and higher qualifications for teachers.

The truth is more prosaic. Finland has become a political football, where admirers cherry-pick the things they want to hear and ignore the rest. In an interesting experiment, Finland has been reported as scrapping cursive writing lessons in favour of keyboard skills and touch typing – it will be fascinating how both the right and left react to that one! Guess what, neither have jumped on that bandwagon – as it’s on no one’s agenda.