explicar l'esperança de vida / Explicar la esperanza de vida.

Els desenvolupadors de gapminder han preparat un PPT molt xulo per explicar l'esperança de vida, que és un indicador fonamental del nivell de salut d'una col·lectivitat. Utilitzen una comparació entre 5 nounats burundesos i 5 suecs, que representen la mitjana dels cinc quintils d'edat corresponents ordenats dels que falten primer als que arriben a més majors, i em pareix que resulta molt clara per als estudiants, des del rigor i la bellesa de les dades. Com que està sota llicència Creative Commons (thanks, pals) m'he pres la molèstia de traduir-ho al castellà i al català, i de pas l'he passat a OpenDocument, de tal manera que qualsevol pot no només baixar-se la presentació, sinó també el programa per exposar-la, per exemple a Libreoffice.

Presentació en català.

Presentació en castellà.

Qualsevol millora o comentari serà ben rebuda. Ja sabeu, difon la idea. Per suposat es pot emprar, modificar i/o transmetre sense més límits que mantindre l'atribució als autors.

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Teaching Pills 4 - Applied Financial Maths: the implicit grant of the ICO student loan

Many overgraduate students in Spain have applied during the last 4 years to a new kind of student loan, called ICO Renta Universidad, in order to complete their studies with a Master's Degree. This is pretty new in the Spanish system, despite being very common in other Western countries, specially in the UK and the USA. In this particular case the loan had quite favourable conditions: a 3 years grace period (up to 6 years, under certain circumstances), a 0% interest rate and 10 years to be repaid.

Some of my fellow friends, who meet the conditions to defer either one or their first three years of payments, have asked me whether it would be better to do so or not. Of course there is not a general answer for that, because each one of them would have different current situations and, more determinant, future expectations of income and, besides, there are some other uncertainties such as the internal devaluation in Spain, the labour market dynamics and the prospects of inflation. Thus I give NO advice. Nevertheless it tourned out to be a nice sample to explain how financial maths work. I'll take a made-up case study and I'll do some assumptions about unknown variables: I'll suppose Spain keeps up within the euro -nobody knows what could happen with these loans otherwise- and inflation will be low, but not negative*. I'll take inflation rate in a 2%, which is commonly accepted as the usual target for central banks and quite consistent with a depressed economy with an extremely high unemployment**. Finally, I'll take a 15.600€ loan, amount very frequently applied for, since it was the maximum amount for a one-year master's degree.

The three options they face are:

A) Regular payment. Starting in the year t to t+9 and paying 130€ monthly:

130 (€/month) *120 (months in ten years) = 15600€.

B) Defer one year. Starting in the year t+1, paying 130€ monthly every year up to t+8, and 260 €/month the last year, t+9:

130*96 + 260*12 = 15600€.

C) Defer three years. Starting in the year t+3, paying 130€ monthly the first four years and 260€ monthly the last three years:

130*48 + 260*36 = 15600€.

By all means it's the same NOMINAL amount because it is a 0% interest-rated loan. Does it mean that they are just indifferent? Absolutely not. Here is where inflation shows up: the same nominal money today is not worth the same 10 years afterwards -in terms of the goods and services you could buy with it. As most of us can intuitively deduce that means that, provided we face any positive inflation and with a 0% rate, it would be better to delay the payment as long as possible but, how much benefit would we take of that strategy? As I said, it depends on the actual inflation rate which we can't know beforehand, but supposing this 2% average inflation rate for the whole period and that it keeps constant during the period -this is a strong condition but it allows us to calculate and it's impossible to foresee any trend in advance. You should take it as an average- the figures would be:

B) 130*12/(1+0.02)^9 = 1560/1.195 = 1305.34.

It is to say, the money we don't pay in the year t and we pay in the year t+9 is worth 1305.34 in terms of real (year t based) money. 1560-1305.34 = 254.66€, or a 19.5% saving of one-year's part of the total credit for that deferred year (therefore for a 1/10 of the total amount, an extra overall saving of 1.95%).

C) 130*36/(1+0.02)^7 = 4680/1.1487 = 4074.22.

In this case the money we don't pay in the years t, t+1 and t+2 and we pay in the years t+7, t+8 and t+9 respectively, is worth 4074.22 in real money (years t, t+1 and t+2 based) instead of the nominal 4680. So far, so good. Now if we want to put them all in the same base-year (we should!) we need to financially move the gaining of the years t+1 and t+2 to the year t (that's called discounting when it's backwards and capitalizing when it's forwards) -> for t+1: (4074.22/3)/1.02 = 1331.44; for t+2: (4074.22/3)/(1.02^2)= 1305.33. And for the year t the deferred payment was worth 1358.07 (No extra calculations needed, it's already expressed on the base year). If we consider the nominal payment was 1560 every year we have: 1560*3 - (1358.07+1331.44+1305.33) = 4680- 3994.84 = 685.16€ That's the overall gain of the option C expressed in real terms, taking t as the base year. It means an extra 4.4% overall saving.

I keep saying an extra gaining. Why? Because either way or even not delaying any payment we are already benefiting from the 0% interest rate, provided any positive inflation, and we are, therefore receiving a covered transfer from the government or the bank, whoever is covering the rates -I guess it must be the government, banks are not known by their philanthropic approach to lending. We could, for instance, calculate what's been the actual gain of the grace period for the first payment year, because we know the actual inflation 2.989% (2010), 2.377% (2011) and 2.868% (2012). Supposing someone who had the loan approved on the 1st January 2009, the grace period has supposed, so far: 1560 / (1.02989)(1.02377)(1.02868) = 1560/1.0846 = 1438.31. -> 1560-1438.31=121.69€ (year t based) or 8.46% saving for the first year of payments. In conclusion, the money he or she asked to finish the studies has received a sort of 8.5% grant, so far (for the next years we'll need to multiply this discount factor by (1+i)^t+n, being i= real inflation and n= number of years). Obviously this "implicit grant" will be higher as times goes by, provided positive inflation (for the 13th year, taking a 2% average inflation rate it would be 1.02^13 = 1.2936 -> a 29.36% "implicit grant": the 1560€ you have to pay then would be worth roughly a 30% less in real money than when you got them, given this inflation rate). The higher real inflation during the period the higher implicit grant you get, and vice versa. Whether it is fair or not has nothing to do with financial maths (although I can't help saying that I'd rather go for a public and affordable -ideally free- university system and a proper grant scheme for the poor, which allows everyone to get the education they want, regardless of their current or future income and their lower or higher opportunity costs. But this is another story).

I have only exposed the differential figures between option A and option B and C, which was the relevant information to make a decision in this case. Thereby year t is the year you start to pay the loan and not the year you received it, which would be the proper "year zero" to know the total amount of the implicit grant. It is indeed interesting to calculate the global gaining of the loan -to know which percent of it is a sort of implicit transfer- and if you feel curious, given a 2% inflation rate it would be  roughly a 19-23% depending on the payment option you choose, and in our study case it would represent an implicit grant of 3.000€ to 3.800€, but we can't infere the actual inflation rate and such long term predictions are rather unreliable. We know, however, that given any positive inflation we'll have this implicit grant and that the higher inflation the higher grant. Now we just need to patiently wait until 2023 to check which the actual grant will have been***.

So, in conclusion, the decision about paying now or delaying the payments depends, as I said at the beginning, on unknown factors such as the prospects of inflation, the risk aversion of the lenders and their future employability and/or other income resources. The figures are based on crucial and strong assumptions, specially referring to inflation, but it may be worth and it's always interesting to take a look to what numbers have to say.

*deflation is a real danger, though, if the crisis carries on for very long, Germany is still sickly obsessed with inflation and/or more bubbles burst, but this is another point (macroeconomics and economy policy issues) and one of the reasons I DON'T give any advice.

** It is true that inflation in Spain had been traditionally high and we have quite a dysfunctional economy able to sustain high unemployment and high inflation at the same time (a phenomenon known as stagflation) but the amazingly huge unemployment suggests it won't be the case the following years: before the crisis it reached an average 3% and the last 3 years it's been constantly over 2% (2.99%, 2.37% and 2.87%), while at 2008 and 2009 it was only 1.4% and 0.8%. The slowdown of global markets, the decreasing GNP, the still overpriced housing sector and, specially, the unbearable 26% unemployment rate suggest it won't raise any more than 2%. Finally, I've taken into account that the loan is for a total 13 years and, hopefully, the depressed conditions won't last that long, so considering deflation for the next 13 years seems just unreasonable. A 2% will do.

*** And I found a way of writing a future perfect somewhere ;)

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Teaching Pills 3: teaching through ICT

"In God we trust; all others must bring data" Dr. W. Edwards Deming

In my last TP post I wrote about how you can teach with almost no material resources and in this one I'd like to show just the opposite, how you can use in a sensible way some of the huge amount of resources currently provided by the ICT. There are many websites, blogs and even apps which may help you to prepare materials, to decide teaching strategies and to support your work. The few times I have had the chance to teach myself I have relied heavily on powerpoints, dynamic graphs and maps. I have also made internet research about the subject (no matter I had it very clear in my mind), keeping an open-minded approach not only because I am a novice (which I clearly am) but because you may find extraordinary tools. Today I'm speaking just about two sites I've already used. However, before starting I'd better say two important things: firstly, these resources are a support to your lessons, but they can't substitute the teacher, it's important that you prepare yourself properly for every lesson and that you don't rely too much on your PPT or your internet connection. Questions will be asked and you ought to be ready to clearly explain them. Secondly, it's important that you adapt the resources you find to your objectives and to the stages you are working with, because a messy explanation may be worse than no explanation. Once these two warnings have been issued, let's go to the resources:

Gapminder.org: the beauty of stats

Gapminder is a website offering very nice resources. In my lessons of introductory economics to nurses (yeah, I know it sounds weird, it was nice, though) I used them to show some of the stats relating health and wealth, HIV and children by woman. They are very useful to start debates about the underlying causes behind some issues, and doing it from the scientific approach of data. However, they can be used at any level of teaching and they have this beautiful stuff about life expectancy:

We all know what life expectancy means, only that... do we know? Well, we know that it is an economic indicator very often used to measure the health of a society, and thereby, the development level of a country, but most people, specially young students, tend to think about it like if everyone in let's say Rwanda was going to live more or less 56 years, which is their current life expectancy. And everyone in the UK about 80. Of course this is the wrong way to interpret the indicator. Life expectancy is an average and, thus, it rather reflects how hard is to arrive to be old in a country, how easy is to die before your time. The guys in gapminder.org have prepared a powerpoint that explains it very clearly. Highly usable. It's worth to take a look:

http://www.gapminder.org/downloads/life-expectancy-ppt

Worldmapper.org: Put your stats on a map

This is a demographic-corrected UK map

We are used to see statistics in number or in graphics. And we are used to physical and political maps of the world. In the worldmapper website we can find maps reflecting statistics. I really like them, they set a very interesting point of view, specially regarding to international development statistics such as poverty, GDP, health, education... they are somehow impressive when you compare different regions and they give you a different sight of both the world and your surroundings.

An adaptation I made of two worldmapper.org maps to provoke debate among students: the red one is the world sized by GDP, the coloured one is the world sized by people who lives with less than 1$ a day

Of course, this two are samples I knew about because I had used them before to teach. You could also explain the carbon footprint and let the kids calculate their own one; there is plenty of sites with Maths or English games... it's up to you to use these resources and make learning a bit funnier.

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Teaching pills 2 - Fractions: Learning by playing

We are increasingly getting used to see our kids having access to a whole set of sophisticated (and expensive) tools designed to helping them study: laptops, next generation tablets and mobiles, digital blackboards, very advanced and complicated software, internet and so on. Nevertheless, I believe that the most important thing still is, and will always be, a passionate teacher and the will to learn of the students. You can encourage learning without any of these gadgets, and you can be very effective in breeding knowledge and attitudes with a pencil, somewhere to write on and a bit of creativity.

Two years ago I went to Nicaragua with an education Spanish charity and I had, among other goals, the objective of working with the local teachers in maths pedagogy. Specifically we had to work about fractions equivalency and operations, and we had to bear in mind not just the national curriculum but also the difficult conditions they face every day, almost without material means or support of any kind. That's the main reason the charity worked there. Because of the students' age, which was rather diverse, from may be 6 to young teenagers of even 15 -this is only one of the difficulties these teachers have to deal with, if you believe your job is tricky you should go there and see how they manage to teach in these conditions. Well, I wanted to focus it as a game, and one with different levels. After some weeks of thinking and reading about the question that's the game I proposed for learning fractions equivalency:

You need something to write with (say a pencil) and something to write on (say a paper). You don't need anything else. Not even a blackboard and not for sure a maths book, which they did not have most of the time. Then you make cards, or better, the kids make their own cards, and in these cards you (they) write fractions in different forms: 1/2, 2/2; 1/4, 2/4, 3/4, 4/4.... for the whole game you write percents: 25%, 50%..... numbers: 0,33; 0,4; 0,5; 0,66; 0,75....... and there is no rule saying you need to stop in 1, you can make fractions above the unit: 6/4; 12/6; 20/5........ and percents and numbers as well: 150%; 300%; 1,25; 2,5; 4............ and even mixed numbers: 2 1/4; 1 1/5....... The amount of cards you'll have at the end is up to you. It depends on the number of kids you are working with, the deepness you need to work and so forth. I like to have many, hundreds of them. You can also repeat them, make some sets, which will help you to work in groups.

And what can you do with these cards? Everything.

I proposed them 4 or 5 different games, but they could (and ought to) work out their own games with them: the more the kids played, the more used they'd get to the fractions equivalency, and the equivalency between fractions, real numbers and percents, and operations with fractions like adding and substracting. My idea was the kids working in groups -because they improve their social skills- but you can also do it individually if you believe that fits better with your conditions. Some of my proposals were: Giving a whole set of cards to each group and a) classify the fractions that are actually the same (like 2/4; 1/2; 5/10) -fractions equivalency-; b) find a number by adding fractions: (find 2? one group find it by adding 1/2 + 4/4 + 2/4 another group can use another cards 4/3 + 2/3...); c) and d) You can repeat these games with fractions and percents and numbers all together and so one can find 2 by adding 0,5 + 100% + 50/100, for instance. And in the game c they'll have a little mountain where 6/3, 4/2, 200%, 50/25 and the number 2 will be together representing the same amount. Another little mountain with 1/3; 0,33; 33%; 3/9, 2/6..... Nice, isn't it? The more they play, the better they'll understand what fractions actually mean. A second session of games, once they are used to the cards (which may as well be in the third or fourth lesson or later, depending of their rhythm) would be giving them an incomplete set of cards, and every group a different one. So, now you can create two new kind of games: 1) negotiation games, where to get to the result they'll need to negotiate with other groups to get the cards they need, while offering the cards they don't need. Interesting introduction to trade, bargain and so on -there is a sort of introduction to David Ricardo and trade theory here, eh?- and working a lot of social skills. 2) You can introduce substracting (and even other operations) to arrive to the goal number.

As a matter of fact you could make up many more games with these cards, focusing on different needs (discover the hidden card -intro to equations-; games to identify prime numbers, and so on). They'll pick it up very quickly, though, and the main target of the cards, let them get the hang of the fractions, will be achieved in just one or two sessions. Besides, they have made the game themselves, and they have learned all this while having fun, challenging their minds. You don't need  more than your will and a pencil.

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Teaching pills 1: Primary Maths. The easy way to multiply

What I love of maths teaching is that you are, may be more than in any other subject, teaching a way of thinking. To be honest I don't really know how multiplying is usually taught in the UK, but in my country in most cases it's still regretfully taught as a memoristic skill, with tables from 1 to 10 that children need to memorize and repeat until they automatically can throw up the result of any single-digit multiplication. In my opinion this is not only a rough and boring way of learning something, it also denies the opportunity of logical and deductive thinking to the children and spoils a chance to reinforce their self-confidence and their understanding of how maths work, the relationship between adding and multiplying, substracting and dividing, and so on. In short, wrong way of multiplying teaching. Moreover, it is hard, takes time and can create an ugly -and mistaken- impression that maths are boring. I bet I can teach any kid who pays attention to multiply in just one session and making it fun. I have tried it before with kids in Nicaragua, kids who had a fairly large lack of previous knowledge -which might have been good, at the end- and most of them got it quickly and, with a few hours of practising, were able to do it by themselves. Of course I haven't "invented" (though "discovered" would be more accurate) this teaching strategy, I am sure it's broadly used, with subtle differences, all over the world. Twenty-five years ago I was taught to multiply with part of this at my school Aire Lliure, which used to be -at least at its origin- an impressive education cooperative, a school based on a sort of libertarian pedagogy, teching how to think instead of what to think. I became a fan and I have just completed it.

Here it comes: My easy way of multiplying that promotes deductive thinking is based in two well-known mnemonic rule -the 9 and 5 rule- which most teachers use to teach to multiply by 5 and 9; the natural ability of mentally adding few times (up to 4); and memorizing just two multiplications: 6x6= 36 and 7x8 = 56. Once you know this, and supposing you are able to add and substract, you have all you need to mentally deduce any single-digit multiplication. Children don't need to memorize anything else and, more important, they'll feel good every time they deduce an unknown one, reinforcing their confidence and their maths liking, which may be really useful for their future.

I guess most of you have got it straight away, but I'm going to explain it with a bit more of detail just in case anyone studied laws -sorry, solicitor and lawyer friends, I know the truth can be painful. First, the mnemonic rules: You all know multiplying by five is easy, just the half ten (v.gr. 8x5 = [the half ten of 8] 40). Most of you might also know that multiplying by nine is very easy indeed, just the previous ten and what's missing to get to nine. It is to say: if you want to multiply, for instance, 6x9, you take the previous ten of six (5, fifty-something), and the unit is what's missing to make nine: 4. So, 6x9= 54. This works. Ok, we've got every single number multiplied by 5 or 9. Then, adding up to four times can be done mentally without effort. So, we have deduced every single digit multiplied by 1 to 5 and by 9. Now we have these two ops, 6x6=36 and 7x8=56, which we'd better memorize (we could eventually add them but I have realized it's a bit tricky for children) and now we have a supporting point to get to every single-digit multiplication directly or by just adding or substracting the number once. Think about any single-digit multiplication, let's say 6x7? the child mental process would go: Ok, I know 6x6=36, so if I add 6 I'll have 6x7! 36+6=42. Good. 8x3? Ok, eight and eight, sixteen, and eight twenty-four. Easy. 8x6? Humpf, well I know 8x7=56, so if I substract 8 I'll have 8x6 -> 56-8=48. Or either I know 6x6=36 I add six twice and I'll have 8x6 -> 36+6=42 +6= 48. And so on, and so forth.

 It's a perfect system, and despite it may look a bit complicated in writing it's very easy indeed to do mentally. What I love of this teaching strategy is that it focus not in memoristic learning but in deductive abilities. It relates multiplying to adding, which is nice, and to substracting also, which is even better because it gives the child a more comprehensive understanding of mathematic relationships, but the important point is that it reinforces a lot their self-confidence and transforms the maths in a deductive game, something that is worth and fun. As it should be.

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