@Corin again:

Ah, I’ve found the LingQ forum post which quotes the article which I can’t find. It was by David Martin (thanks David) and I’ll reproduced it here (because I can’t think how to link to just one forum post in a long thread):

This is an excerpt from an article called ‘Learning Languages Like Children’ from Blog - ALG World Automatic Language Growth

Learning usually depends on the varying levels of intelligence, motivation, and hard work of the students, and the usual way to measure this learning is to test each student. But natural language acquisition depends only on exposure; so it’s a lot easier (and a lot more accurate) to just measure the amount of exposure (actually the amount of understanding). With babies we measure their progress by their age. If someone says her little boy is 21 months old, that tells us more about how much language he knows than any test could. For children and adults, though, the rate of input is far less constant than it is with babies, and we have to find a way to count or estimate the number of hours of talk they have understood. (This is the subject of the next section.)

As a result of years of study and more than 40 years of observing the progress and abilities of literally thousands of students of second languages from over 50 different countries and cultures, we have found the following equation to be remarkably accurate. (Please note that as you read through this section and follow the development of it, you may be tempted to discount our conclusions based on your own experience or that of someone you know. If you save your exceptions until later and follow our reasoning however, you will probably see that we account for such factors as our thoughts develop.)

The BASIC LANGUAGE ACQUISITION EQUATION: y = 1-e-kx

where y is how much language they know (1 = native).

x is how many hours they have understood.

k is the acquisition constant: .0018

e is the natural logarithm base: 2.718

If a student accumulates 1000 hours of understanding Thai, for example, his acquisition of Thai will be 83%.

Or if we want to know how long it would take a student to get to 90% (this is a degree of fluency that structural students hardly ever attain), the equation tells us 1300 hours of understanding.

We usually think of complete immersion as the ultimate in exposure, but let’s look at a typical example. Suppose you’re exposed to speaking situations for 8 hours a day (meals, chatting, games, etc.). This isn’t non-stop talking, though, and it may come to only 4 hours of actual talk. And if half of this talk is your own, that’s only 2 hours of listening. An if you’re understanding 50%, that’s only 1 hour of understanding a day. It would take you almost 4 years to accumulate the 1400 hours needed to become ‘fluent’. (We use ‘fluent’ to mean ‘speaking correctly and without hesitation about everyday matters’ : roughly, y = 88%).

More often, foreigners live with their own families, and their exposure consists only of managing their daily affairs in the new language. This may seem like a lot of exposure, but when you add up the few seconds here and few seconds there and multiply this by your percentage of understanding, it rarely exceeds 10 minutes a day. At this rate, fluency would take 23 years.

Hours of understanding isn’t always clear in terms of months and years: normal life is so irregularly packed with talk, and talk is so irregularly understood. But ALG classes consist of non-stop talking and offer a much higher percentage of understanding than real life does.

Our first equation assumed that the student was doing everything right. This always works for children, but only occasionally for adults. For even though adults can do it right, they usually don’t. So the measure of how correctly an adult does it (we’ll call it C, for ceiling) becomes a crucial addition to our equation. It will be convenient to express C as a percentage, so y will also be a percentage; and 100 instead of 1, will be the measure of a native speaker.

Now to figure out how much a student knows (y), we’ve got to know how many hours (h) the student has experienced, how much he has understood (u), and how he has been processing those experiences. (C).

h is simply the student’s attendance.

u can be estimated from the student’s ‘responses’ during each hour. (We all tend to monitor a person’s understanding in normal communication though we are normally unconscious of doing this.)

C can be estimated from how much or little the student tries to repeat what he hears, the sort of questions asked, etc.

Periodically, the guides enter grades for the students based on their own perceptions. Once entered into the computer, we are able to monitor student progress. The average understanding grade for students is around 80%. Ceilings vary much more, but for a typical, adult student who begins with ALG the ceiling average is around 95%.

The first 13 students to show signs of natural speaking in our ALG classes were Chinese and Southeast Asians – even though the majority of our students were Westerners. It was only after we expanded our course to more than 1000 hours that other students started to reach this level. We soon saw that any level that required 1000 for Westerners and Japanese could be reached in about 800 hours by Chinese and about 600 hours by Southeast Asians. This suggested a ‘language ease’ factor (L) for our equation. For the Chinese learning Thai, L = .8; and for most Southeast Asians, L = .6.

y = C(1-e-kx/L)

The language ease has come to be called the Native Language Factor but there is more and more evidence that culture rather than language is the bigger influence.

So far, we have had little experience with the native language factor from English to French, German, and Spanish; but if Malaysian-Thai is .6 without the help of cognate vocabulary (the languages aren’t related and the only similarities are in culture and type of grammar), we would expect something more like .4 for these European languages. These and other guesses are shown below. Readers with better information can sharpen up these guesses. The hours and weeks refer to the amount of time required to reach a fluency of 88%. For the calculations below, the understanding factor is set to .8 and the ceiling factor is set to .95.

L Factor Examples Hours

1.0 English-Thai 1800

1.0 Japanese-Thai 1800

.8 Chinese-Thai 1450

.6 Malay-Thai 1100

.4 English-French 720

.4 English-German 720

.4 English-Spanish 720

.2 Portuguese-Spanish 370

.1 Thai-Laotian 180

.1 Norwegian-Swedish 180

.06 Norwegian-Danish 110

The original thread and post may be read here: http://www.lingq.com/forum/1/4224/