Arithmetic (again)

A week ago, the Wiskundemeisjes('Mathgirls') posted that the debate on education of arithmetic and mathematics has come back from holiday again. The new season of fierce discussions has been opened by Joost Hulshof. Read his article in Dutch here: http://www.math.vu.nl/~jhulshof/TAL.pdf (CASE-sensitive!). Hulshof argues against the so called 'realistic arithmetic'. After reading the piece, I have to conclude that the proponents of realistic arithmetic have a major misconception of what cognitive psychology has to teach us.
(I sense polarisation at work.)

A mind without confusion

At the root of Yggdrasill, the Tree of Life, lies the magical Source of Urd. There dwell the three Norns: Urd (Wyrd), Verdandi and Skuld. Now, Urd is old Norse for 'fate'. It has become the root wert-'='to turn' to mean anything prone to change. The word 'weather' (='weer' in Dutch) originating from this, the same holds for the Dutch verb 'worden' (='to become') and also the word 'versus' ('turned against'). Because of the connotations with changing behaviour, it is easy to understand that the name Wyrd has developed into 'weird', strange and unpredicted.

This in itself is an interesting piece of etymology. But the story gets better. Last week I received a small gift from Oshita sensei. He passed this eighth dan exam iaido last year. He gave me a tenugui, a small towel, that was made for the occasion. It has a nicely calligraphed text on it. The calligaphy reads fudoshin, which normally is translated as 'an unmovable mind.'



The first and the last characters are very common: 'heart'/'mind' and 'not'/'no'. The middle one I had seen before, but I had forgotten its meaning. I used my Chinese/Japanese program Wakan and found out that the middle character means 'move, motion, change, confusion, shift, shake'. Thus, the calligraphy reads (as I translate it) something like 'a mind without confusion'.
In this way I found out that also in the Far East the bond between 'change' and 'confusion' is anchored in vocabulary.

Arithmetic

"Swordsmanship is just like this. If from the time the student is a beginner without having learned any techniques well, you say something like 'Technique will come naturally without intention; establish the hard by means of the soft,' or 'Techniques are only trivialities,' he will become empty-handed and lazy, and will never know of where to stand. Thus he will be at loss both in this world and the next."
From: Issai Cholzanshi, The Demon's Sermon on the Martial Arts (tr. William Scott Wilson).

In the Netherlands a debate on education in arithmetic and mathematics has been going on for a long time. On one side there are those who say that it is no use of learning techniques, if there is no understanding. On the other hand there are those who say that understanding will come naturally after technique has been mastered.

Prof. Jan van de Craats, an important spokesman of the latter group, argues that the didactic method of 'realisitc arithmetic' give children a variety of tricks, which leaves the pupils confused. Children need structure.

Those who advocate 'realistic arithmetic' claim that old fashioned techniques like the 'staartdeling' (an algorithm in essence equal to the English 'long division') are obsolete, because 'nobody does calculations with pen and paper any more' (Adri Treffers in De Volkskrant). That is a great loss.

I strongly feel that automatisation of arithmetic techniques, together with the learning by heart of the multiplication tables are necessary for a good sense of numbers. This is, I believe, in turn needed for a good grasp of and ability to handle with more abstract objects such as functions, variables and parameters.

As long as realistic arithmetic and mathematics remain the common way of teaching, our pupils may "become empty-handed and lazy, and will never now of where to stand."

Glacier water pockets

Some days ago, I was watching the episode 'ice' from the BBC documentary series Earth. In this episode Dr Iain Stewart shows us a 'back door' to a glacier ("he sees a glacier in action from below"). Here he saw that the glacier has holes filled with water, these pockets can contain up to about twenty litres of water. The commentator puts forth the hypothesis that these pockets are generated by the pressure of the ice mass above and that these holes make the glacier less rigid and hence move faster.

The beautiful thing about glaciers is that they are rivers of ice, really behaving like a liquid. The bottom meter (or so) is even filled with sediments (sand, gravel, stones) like they are picked up by the flow and then whirled into the ice. (We even have a word for these processes: glaciofluvial.)

A typical glacier will move several meters (up to about 30 meters) a day, with the bottom parts moving slower due to the friction with the bed. However, the enormous pressure in the bottom parts of the glacier, make the ice more fluid than that on the top layers. During the flow the top layers of the glacier will often crack, because this part (the 'fracture zone') is so rigid.

Now, I was wondering. May it not be possible that the pockets of water in the lower parts of the glacier are a result of a solid phase water being forced into the liquid phase by the energy the glacier has to absorb on its way?

I tried to find an answer on this wonderfully informative site: Martin Chaplin's Water structure and science (here: phase diagrams). however, I could not find the answer there easily. Probably, because most of the information on the site is too specialised for me (or maybe even just too difficult and complicated).

By the way, it seems that the water in these pockets can't be colder than -20°C; looking at the phase diagram from professor Chaplin's site.


What a wonderful place we live in!

Empty Seas

We really ought to worry about our influence on our precious Mother Earth. I take this very serious, because I believe our world is sacred. The main problem is that we do not understand fully the impact of our actions.

However, since this is the case, I tend to scrutinize all warnings and all solutions. Of course, I do not nearly have enough knowledge of the subject to come up with solutions myself, let alone penetrate the complexity of all consequences. The reason I write this is not to cause doubt, on the contrary! I want to spread awareness of the problems we are facing and show that we should not make rash decisions.

Lately people started warning us for the danger of empty seas: our pollution and our fishing may very well tip the population size of fish, causing them to spiral to extinction. This also may cause famine between animals such as sea birds and dolphins. I strongly hope that fishing becoming unprofitable because of the small catches, will happen before fish population levels dropping below a critical sustainable level. Even then there is hope: from that point on, fish population may reach a new 'stable region' (I mean in phase space, not in the waters of the Earth).

At first I had hoped that fish in the oceans may migrate vertically: they may adapt and start to live at depths where they are save from fishing nets. (I have to find out: Will there be enough nutrients and inflow of energy to sustain a new ecosystem on that depth?) Unfortunately, deep seas are already targeted by trawlers, that can easily catch fish at depth at over 1km.

Who knows, maybe fish will become extinct locally or even globally. That would be terrible. If 'only' certain regions of oceans will become completely depleted, new pioneer species may step in, followed by others. Or maybe, the extinction of fish will cause algae and plants to flourish enormously in the oceans, resulting in a gigantic increase of CO2 absorption from the atmosphere thus relieving part of the climate problem. On the other hand, maybe there will be collapse of the global ecosystems causing the world to wander on a chaotic path. Who knows where that will end up.

Flawed renewable energy

Some days ago, the Dutch media reported that small windmills are no solution for the energy demand; today I read these articles on Slashdot and on Low-tech. It came as no surprise to me that the only relevant factor to windmill power yields is the diameter. This I already heard when I was a child as an answer to why some windmills only have two rotors, whereas others have many- more rotors catch more wind, I thought. (I believe my parents took me to a Greenpeace meeting in the mid-eighties, or 'then-about'.)



After a few clicks, I stumbled upon this Low-tech article: The ugly side of solar panels, where you can read that PV solar panels...

"... with an expected lifetime of only 15 years, the worst case scenario [in which they produce] 207 grams of CO2 per kilowatt-hour – [are] just 2 times better than gas. Agreed, this is the worst case scenario, and even in that case solar panels are still a better choice than fossil fuels. But it becomes quite hard to describe them as a “clean” source of fuel."

There are much cleaner types of solar panels, but we still have the issue of their influence on the albedo factor of their environment.

Apparently, we really need some revolutionary new ideas on energy production.

Kendo match schedules are fair

Some time ago I posed the question wat the probability distribution was of X, the number of groups with more than one fellow dojo member, when m members of our dojo are randomly assigned to any one of g groups, each consisting of n group competitors.

The reason was that in the last competition eight members of our dojo competed. The first round was a group competition. In total there were 29 groups, all but one with three competitors. To our surprise two of our dojo members were in the same group. All the rest were divided over different groups, so the rest of us didn't have to fight friends from the dojo.

The reason why it surprised us was that until today, we were much less fortunate. Some began even to suspect that there was some evil scheming at work.
Finding the answer to the probability distribution of the number of groups with more than one fellow dojo member seemed quite a simple task. So, even though I have been quite busy, I wanted a generalisation of the question (see Kendo match schedules fair? (rev.)). However, this made the question much harder to answer and not at all clearer.

So, after a while I decided to post the answer to the original question.
Let X be the number of groups with more than one fellow dojo member; m is the number of members of our dojo; they are randomly assigned to any one of g groups, each consisting of n group competitors. In total the number of competitors is about N = n·g.

Then X is binomially distributed with the number of (Bernoulli) trials is equal to g (the number groups in the competition) and p the probability of more than one member of our dojo in the group:

Here:

This holds for all dojo in the competition, so it is much handier to calculate the probability distribution for each dojo separately; especially since we are only interested in proving or disproving that there is an evil scheme against our own dojo.

In the example of the last tournaments, this results in the following probability distribution*:




Now, X = 1 seems a perfectly reasonable outcome for the tournament. This makes me curious what is the probability of a (fictional, but close-enough to typical) other outcome. Typically our dojo competes with six members (m = 6), there are twelve groups (g = 12) of four competitors (n = 4; N = 48). We get two groups with multiple members of our dojo (X = 2). How unlikely is this typical situation?

P(X = 3 | g = 12; p = 0,0708) ≈ 0,159.

Now, I cannot be convinced that there are other forces at work than mere chance.

Renewable Energy and Chaotic Climate Systems

We use quite a bit of energy. In an attempt to reduce our impact on the environment, we are searching for so called renewable power sources as alternatives to nuclear power and fossil fuels. Often we hear about solar power and wind power as serious candidates.


Here in the Netherlands we used to have a commercial in which a child asks her father: “Will the sun ever stop shining?” and “will the winds ever tire?” Of course the message is that sun and wind can be used as limitless sources of energy.


We want to find ways to limit our impact on the climate, but is it sensible then to do that by extracting energy from the dynamics of the atmosphere and even from sunshine, the great engine behind the climate? There are also plans of extracting energy from the sea: there is a lot of energy stored in currents, the tidal movements and even in ordinary waves.


I do not know much (if anything) about the subject, it’s just that I am so curious and always I am scrutinising. The problem I see here is that the climate is a highly chaotic system. Changing the trajectory of such a system through its state space may result in a trajectory towards just about any unforeseen equilibrium. Extracting energy from for example the tides may have little effect at first, but no one knows what will happen after a while.


Either I am not looking in the right places, or there is not much known on this subject, in any case I cannot find much on the Internet. One of the few articles on wind power addressing the question of the impact on the climate (David Keith, (2004) Wind Power and Climate Change) states that large parks of wind mills may have an effect on the local climate. Keith does not seem to be worried by the effect of wind farms on the global climate, however. Another article states that the turbulence generated by the rotors causes different layers of air to mixing (S.B. Roy, et al. (2004) Can large wind farms affect local meteorology?), also P. Rooijmans (2003, 4) did some studies that "show different effects of the wind farm [a large-scale (9000 km²) offshore wind farm in the North Sea] on the meteorological variables, especially for cloud formation."


Also, I am sceptical about parks of solar power plants. These panels have quite different properties (how they warm up and reflect light) than the ground that they are standing on (more here:albedo). For example, when one thinks about the ground of the Sahara desert, typically, there will be quite a lot of reflection of light. Solar panels do not reflect much light; that is the whole idea behind them, so maybe the presence of solar panels in desert areas may cause a local increase of temperature. The planting of crops in these areas (e.g. in Saudi Arabia and in Nevada) may have similar effects: vegetation absorbs sunlight much better than rock and sand. This will cause an increase in regional temperature (and it's quite hot there allready! Furthermore, these crops need water, so there will be displacement of water and an excess of evaporation in normally arid regions.


Now here is my next quest: what kind of power source is not itself part of the climate system AND has no known nonlinear properties (i.e. does not have chaotic behaviour)?

Kendo match schedules fair? (rev.)

I have not yet had time to think things over, but I realised that a generalisation of the problem might be more interesting:

What is the probability distribution of X, the number of groups with more than one fellow dojo member, when there are d dojo competing with each m_d members and all competitors are randomly assigned to any one of g groups, each consisting of n group competitors?

Kendo match schedules fair?

So once in a while I go to a kendo match in a delegation of our dojo. Today, I went to the Ijima-cup tournament; named after Ijima sensei, who has meant a great deal for the development of kendo in the Netherlands. Our dojo sent 8 members to the competition.

There were no less than 29 groups in the first stage of the tournament. All but one group contained three kendoka. To our surprise two of our dojo members were in group number 20. All the rest were divided over different groups, so the rest of us didn't have to fight friends from the dojo.
The reason why it surprised us was that until today, we were much less fortunate. Some began even to suspect that there was some evil scheming at work. I told them that it was probably just a matter of probability. However, I have not yet come to make a model.
So this will be my next 'challenge':

What is the probability distribution of X, the number of groups with more than one fellow dojo member, when m members of our dojo are randomly assigned to any one of g groups, each consisting of n group competitors?

At the moment, I am tired. I still have some preparations to do for tomorrow, so I will not try to figure out the answer just now.

P.S. The tournament went well. I stranded in third round. After a period in which my kendo seemed to get less flexible, it slowed down, cramped up and only felt frustrating, today I did get "in the flow" again and I could do some fluid, strong and fast kendo. Most of our dojo members did well.

The human footprint solved

In this post I will give you what I think is the answer to the question in my previous post (The human footprint).


We wondered how to calculate the average frequency of human sexual behavior (in sexually active adults) given a set of answers to the question "how long has it been since the last time you had sex?" Here is an attempt to find the solution.


To simplify things a bit, we assume that the event (sexual intercourse) occurs perfectly periodic for each individual, with period I (interval). Another simplification is that the period is an integer (discrete stochastic variable) and that the minimal period equals one day. The only data we have are the number of days (D) since the event occurred last. Obviously we try to estimate the expected value of the interval period E (I). If N is the size of the data set (the number of respondents), then the estimated value of the average interval equals:



Bayes states that:


We are able to estimate P(D=d | I = i). Here is how. What we do know is that for a person who has sex every day (I = 1) there will be a 'fifty-fifty' distribution between the answers "just today" (D = 0) and "that was yesterday" (D = 1). Even though we cannot know the value of I for any individual, we do know that for all levels of I the chance of D = d equals P(D = d | I = i) = 1 : (i+1).


This yields the following table:

Distribution of P(D = d | I = i)
		D number of days since last event
		0	1	2	3	4	...
I (interval between events)	1	1/2	1/2	0	0	0	...
	2	1/3	1/3	1/3	0	0	...
	3	1/4	1/4	1/4	1/4	0	...
	...

To estimate P (I = i) we have to know P (D = d ∧ I = i). At first I had hoped to use the above table for this, but we cannot use it directly. We would like to estimate P (D = d ∧ I = i) with our given set of n_d and P (D = d | I = i), but we cannot use the table here because if we allow I to be unbound then:



P (D = d ∧ I = i) remains unknown. Note, however that for all levels of i (with d < i) all P(D = d ∧ I = i) are equal, so P(D=0∧I=i)=P(D=1∧I=i)=P(D=2∧I=i). This means that the actual number in cell (d, i) should be about equal to the numbers is the non empty cells in the same row (with the same level of I). I will call the estimated number in the non empty cells of the first row .


Now we are getting somewhere; we do know is n_d, the number of people in the sample who answered D = d. We would expect n₀ ≈ n₁. The important thing is that , et cetera.


We can now estimate .


In general a (crude) estimation of n(d,i) would be

.


Now we know enough:



thus




This estimation is very crude indeed. At this stage I have not tried to integrate the information of n_i+2, n_i+3, and so forth in the estimation of n(d,i).

The human footprint

Some time ago I saw the British documentary The Human Footprint. This documentary shows u how much milk an average British person drinks, how many chickens they consume, the total length of fingernails such a person grows during their lifespan and how often they have sex.
About this last topic an interesting remark was made: researchers don't just ask “how often do you have sex?” to a large group of individuals, that would give very unreliable results. To estimate the average frequency of sexual activity in British adults, the researchers asked: “how long ago did you last have sex?”
How does one calculate the average frequency of sexual activity on the basis of data on time since last activity? For example, if a man answers “the day before yesterday”, we know he doesn’t have sex every day. But we do not know whether he has sex about every two days, once a week, or once a month. It may even be a once-a-lifetime experience for the man.
Because of this asymmetry, one cannot simply assume that the given length of time is half way the average inter-intercourse period. If I want to learn how to calculate average frequency of sexual activity (the average occurrence of volcanic eruptions, or other semi-periodic events), I need to find out about waiting time theory. It is a topic I have not had to deal with since 1994; luckily I still have most of the university books I used.