Category: Education

How Many Pairs of Rabbits Are Created by One Pair in One Year?

The Fibonacci numbers are often illustrated geometrically, with spirals or square tilings, but the nautilus is not their origin. I recently learned that the sequence was first reported as the solution to a dynamic modeling thought experiment, posed by Leonardo Pisano (Fibonacci) in his 1202 masterpiece, Liber Abaci.

How Many Pairs of Rabbits Are Created by One Pair in One Year?

A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Because the abovewritten pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in one month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; in this month 3 pairs are pregnant, and in the fourth month there are 8 pairs, of which 5 pairs bear another 5 pairs; these are added to the 8 pairs making 13 pairs in the fifth month; these 5 pairs that are born in this month do not mate in this month, but another 8 pairs are pregnant, and thus there are in the sixth month 21 pairs; [p284] to these are added the 13 pairs that are born in the seventh month; there will be 34 pairs in this month; to this are added the 21 pairs that are born in the eighth month; there will be 55 pairs in this month; to these are added the 34 pairs that are born in the ninth month; there will be 89 pairs in this month; to these are added again the 55 pairs that are born in the tenth month; there will be 144 pairs in this month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.

Source: http://www.math.utah.edu/~beebe/software/java/fibonacci/liber-abaci.html

The solution is the famous Fibonacci sequence, which can be written as a recurrent series,

F(n) = F(n-1)+F(n-2), F(0)=F(1)=1

This can be directly implemented as a discrete time Vensim model:

Fibonacci SeriesHowever, that representation is a little too abstract to immediately reveal the connection to rabbits. Instead, I prefer to revert to Fibonacci’s problem description to construct an operational representation:

Fibonacci Rabbits

Mature rabbit pairs are held in a stock (Fibonacci’s “certain enclosed space”), and they breed a new pair each month (i.e. the Reproduction Rate = 1/month). Modeling male-female pairs rather than individual rabbits neatly sidesteps concern over the gender mix. Importantly, there’s a one-month delay between birth and breeding (“in the second month those born to bear also”). That delay is captured by the Immature Pairs stock. Rabbits live forever in this thought experiment, so there’s no outflow from mature pairs.

You can see the relationship between the series and the stock-flow structure if you write down the discrete time representation of the model, ignoring units and assuming that the TIME STEP = Reproduction Rate = Maturation Time = 1:

Mature Pairs(t) = Mature Pairs(t-1) + Maturing
Immature Pairs(t) = Immature Pairs(t-1) + Reproducing - Maturing

Substituting Maturing = Immature Pairs and Reproducing = Mature Pairs,

Mature Pairs(t) = Mature Pairs(t-1) + Immature Pairs(t-1)
Immature Pairs(t) = Immature Pairs(t-1) + Mature Pairs(t-1) - Immature Pairs(t-1) = Mature Pairs(t-1)

So:

Mature Pairs(t) = Mature Pairs(t-1) + Mature Pairs(t-2)

The resulting model has two feedback loops: a minor negative loop governing the Maturing of Immature Pairs, and a positive loop of rabbits Reproducing. The rabbit population tends to explode, due to the positive loop:

Fibonacci Growth

In four years, there are about as many rabbits as there are humans on earth, so that “certain enclosed space” better be big. After an initial transient, the growth rate quickly settles down:

Fibonacci Growth RateIts steady-state value is .61803… (61.8%/month), which is the Golden Ratio conjugate. If you change the variable names, you can see the relationship to the tiling interpretation and the Golden Ratio:

Fibonacci Part Whole

Like anything that grows exponentially, the Fibonacci numbers get big fast. The hundredth is  354,224,848,179,261,915,075.

As before, we can play the eigenvector trick to suppress the growth mode. The system is described by the matrix:

-1 1
 1 0

which has eigenvalues {-1.618033988749895, 0.6180339887498949} – notice the appearance of the Golden Ratio. If we initialize the model with the eigenvector of the negative eigenvalue, {-0.8506508083520399, 0.5257311121191336}, we can get the bunny population under control, at least until numerical noise excites the growth mode, near time 25:

Fibonacci Stable

The problem is that we need negarabbits to do it, -.850653 immature rabbits initially, so this is not a physically realizable solution (which probably guarantees that it will soon be introduced in legislation).

I brought this up with my kids, and they immediately went to the physics of the problem: “Rabbits don’t live forever. How big is your cage? Do you have rabbit food? TONS of rabbit food? What if you have all males, or varying mixtures of males and females?”

It’s easy to generalize the structure to generate other sequences. For example, assuming that mature rabbits live for only two months yields the Padovan sequence. Its equivalent of the Golden Ratio is 1.3247…, i.e. the rabbit population grows more slowly at ~32%/month, as you’d expect since rabbit lives are shorter.

The model’s in my library.

Statistics >> Calculus ?

Another TED talk argues for replacing calculus with statistics at the pinnacle of mathematics education.

There’s an interesting discussion at Wild About Math!.

I’m a bit wary of the idea. First, I don’t think there needs to be a pinnacle – math can be a Bactrian camel. Second, some of the concepts are commingled anyway (limits and convergence, for example), so it hardly makes sense to treat them as competitors. Third, both are hugely important to good decision making (which is ultimately what we want out of education). Fourth, the world is a dynamic, stochastic system, so you need to understand a little of each.

Where the real opportunity lies, I think, is in motivating the teaching of both experientially. Start calculus with stocks and flows and physical systems, and start statistics with games of chance and estimation. Use both to help people learn how to make better inferences about a complex world. Then do the math as it gets interesting and necessary. Whether you come at the problem from the angle of dynamics or uncertainty first hardly matters.

Math >> Calculating

This TED talk by Conrad Wolfram, of Wolfram Research, will really resonate with anyone who follows system dynamics and learner-directed learning.

He asks, “what is math?” and decomposes it into four steps:

  1. Posing the right questions
  2. Translating the real world problem into a mathematical formulation
  3. Computation
  4. Map the mathematical answer back to the real world, and verify it

He argues that 80% of conventional education is expended on step 3, which is boring if you do it by itself. Instead, he says, we should use increasingly-ubiquitous computers for step 3, and focus on the fun parts – 1, 2 & 4.

This is basically a generalization of the modeling process and the SD approach to education. I do millions of calculations per day, but not more than a few by hand or mind. The real wrangling is with steps 1,2 & 4 – real world problems that Conrad describes as knotty and horrible, with hair all over them.

Out with the bad, in with the good

A while back Obama jumped on the fire-bad-teachers bandwagon:

“You’ve got to have radical change, and radical change is something that’s in the interest of students,” he said. “We’ve got to be able to identify teachers who are doing well. … And, ultimately, if some teachers aren’t doing a good job, they’ve got to go.”

Politico

This is all well and good, but some of what I’ve read about this idea seems naively linear. Bad teachers gone >> students learn more? Just contemplating the stocks and flows gives me pause. If we accelerate the outflow of bad teachers, what happens to the stock of teachers? Does it go down, causing class sizes to go up, inadvertently making things tougher on the remaining good teachers, who might then also leave? If not, where do we get the inflow of replacements, and what makes them any better? Is there an infinite source of potential good teachers out there, waiting to be exploited, or do we have to do something to create it?

Certainly there are some good reasons to think that getting rid of bad teachers is part of the solution. Anecdotal evidence of exceedingly low turnover rates in some districts suggests an opportunity. More importantly, there are positive feedbacks around quality. Good teachers make good colleagues, so a dolt-free school should be able to attract more good teachers. Good teaching reduces inspires, reducing behavior issues, so schools can focus resources on teaching, not discipline.

But at the end of the day, retention of good teachers has to be part of the picture as well. That means caring for them appropriately: giving them the flexibility to develop their own teaching style, not making evaluation obtrusive, providing slack time for development and continuing education, and – god forbid – paying them well. Many other education initiatives run counter to this purpose. For example:

Obama also said “nothing’s more important” than education, and he said if students stayed in class for one more summer month every year, they would retain more information. “I think we should have longer school years,” he said.

This is classic “get a bigger hammer” thinking. Is one more month of school that isn’t working going to help? Are underpaid teachers going to provide 10% more hours on a volunteer basis, or do we cut their effective pay to implement this? Could the resources instead be used to reduce class sizes 10%, or raise salaries 10% to attract better teachers? Again, there may be a kernel of wisdom here, but it’s hard to separate it from its systemic context.

My half-baked view is that it’s unreasonable to expect a revolution in education without providing more resources. That money isn’t going to come from poor school districts. The physics of the distribution of wealth suggests that it would have to come from the rich. At times, the rich have been willing to ante up for education, in recognition that wealth is unsustainable without civil society. But currently we seem to be in a social Darwinian phase, in which wealth is exclusively personal (in stark contrast to the view of achievement in science). So, perhaps the first step would be to make the problem salient: internalize the costs of uncivil society. Let’s pay for policing and the prison system with a luxury tax on McMansions, sports cars, yachts, first class air travel, space tourism, fine art, vintage wine and Viagra. Then we can tackle the really hard stuff, like anti-intellectual culture (since lotteries, our tax on ignorance, don’t seem to be depressing the supply).

Would you like fries with that?

Education is a mess, and well-motivated policy changes are making it worse.

I was just reading this and this, and the juices got flowing, so my wife and I brainstormed this picture:

Education CLD

Click to enlarge

Yep, it’s spaghetti, like a lot of causal brainstorming efforts. The underlying problem space is very messy and hard to articulate quickly, but I think the essence is simple. Educational outcomes are substandard, creating pressure to improve. In at least some areas, outcomes slipped a lot because the response to pressure was to erode learning goals rather than to improve (blue loop through the green goal). One benefit of No Child Left Behind testing is to offset that loop, by making actual performance salient and restoring the pressure to improve. Other intuitive responses (red loops) also have some benefit: increasing school hours provides more time for learning; standardization yields economies of scale in materials and may improve teaching of low-skill teachers; core curriculum focus aligns learning with measured goals.

The problem is that these measures have devastating side effects, especially in the long run. Measurement obsession eats up time for reflection and learning. Core curriculum focus cuts out art and exercise, so that lower student engagement and health diminishes learning productivity. Low engagement means more sit-down-and-shut-up, which eats up teacher time and makes teaching unattractive. Increased hours lead to burnout of both students and teachers. Long hours and standardization make teaching unattractive. Degrading the attractiveness of teaching makes it hard to attract quality teachers. Students aren’t mindless blank slates; they know when they’re being fed rubbish, and check out. When a bad situation persists, an anti-intellectual culture of resistance to education evolves.

The nest of reinforcing feedbacks within education meshes with one in broader society. Poor education diminishes future educational opportunity, and thus the money and knowledge available to provide future schooling. Economic distress drives crime, and prison budgets eat up resources that could otherwise go to schools. Dysfunction reinforces the perception that government is incompetent, leading to reduced willingness to fund schools, ensuring future dysfunction. This is augmented by flight of the rich and smart to private schools.

I’m far from having all the answers here, but it seems that standard SD advice on the counter-intuitive behavior of social systems applies. First, any single policy will fail, because it gets defeated by other feedbacks in the system. Perhaps that’s why technology-led efforts haven’t lived up to expectations; high tech by itself doesn’t help if teachers have no time to reflect on and refine its use. Therefore intervention has to be multifaceted and targeted to activate key loops. Second, things get worse before they get better. Making progress requires more resources, or a redirection of resources away from things that produce the short-term measured benefits that people are watching.

I think there are reasons to be optimistic. All of the reinforcing feedback loops that currently act as vicious cycles can run the other way, if we can just get over the hump of the various delays and irreversibilities to start the process. There’s enormous slack in the system, in a variety of forms: time wasted on discipline and memorization, burned out teachers who could be re-energized and students with unmet thirst for knowledge.

The key is, how to get started. I suspect that the conservative approach of privatization half-works: it successfully exploits reinforcing feedback to provide high quality for those who opt out of the public system. However, I don’t want to live in a two class society, and there’s evidence that high inequality slows economic growth. Instead, my half-baked personal prescription (which we pursue as homeschooling parents) is to make schools more open, connecting students to real-world trades and research. Forget about standardized pathways through the curriculum, because children develop at different rates and have varied interests. Replace quantity of hours with quality, freeing teachers’ time for process improvement and guidance of self-directed learning. Suck it up, and spend the dough to hire better teachers. Recover some of that money, and avoid lengthy review, by using schools year ’round. I’m not sure how realistic all of this is as long as schools function as day care, so maybe we need some reform of work and parental attitudes to go along.

[Update: There are of course many good efforts that can be emulated, by people who’ve thought about this more deeply than I. Pegasus describes some here. Two of note are the Waters Foundation and Creative Learning Exchange. Reorganizing education around systems is a great way to improve productivity through learner-directed learning, make learning exciting and relevant to the real world, and convey skills that are crucial for society to confront its biggest problems.]

Dumb and Dumber

Not to be outdone by Utah, South Dakota has passed its own climate resolution.

They raise the ante – where Utah cherry-picked twelve years of data, South Dakotans are happy with only 8. Even better, their pattern matching heuristic violates bathtub dynamics:

WHEREAS, the earth has been cooling for the last eight years despite small increases in anthropogenic carbon dioxide

They have taken the skeptic claim, that there’s little warming in the tropical troposphere, and bumped it up a notch:

WHEREAS, there is no evidence of atmospheric warming in the troposphere where the majority of warming would be taking place

Nope, no trend here:

Satellite tropospheric temperature, RSS

Satellite tropospheric temperature (RSS, TLT)

Continue reading “Dumb and Dumber”

Hadley cells for lunch

quinoa convection

At lunch today we were amazed by these near-perfect convection cells that formed in a pot of quinoa. You can DIY at NOAA. I think this is an instance of Benard-Marangoni convection, because the surface is free, though the thinness assumptions are likely violated, and quinoa is not quite an ideal liquid. Anyway, it’s an interesting phenomenon because the dynamics involve a surface tension gradient, not just heat transfer. See this and this.

Dynamics of … er … flatulence

I sat down over lunch to develop a stock-flow diagram with my kids. This is what happens when you teach system dynamics to young boys:

dynamics of flatulence

Notice that there’s no outflow for the unpleasantries, because they couldn’t agree on whether the uptake mechanism was chemical reaction or physical transport.

Along the way, we made a process observation. We started off quiet, but gradually talked louder and louder until we were practically shouting at each other. The boys were quick to identify the dynamic:

loud & louder

Jay Forrester always advocates tackling the biggest problems, because they’re no harder to solve than trivial ones, but sometimes it’s refreshing to lighten up and take on systems of limited importance.

If your kids are boring, you're doing it wrong

The other day I ran across a blog post (undeserving of a link, though there is a certain voyeuristic fascination to be had in reading it) that described children as boring little wretches, unsuited to inhabit the cerebral stratosphere of their elders. The mental model seemed to be something like the following:

Bad parenting mental model

The policy response to the misfortune of having children implied by the above is to foist them off on TV and day care until they grow up enough that you can tolerate their presence. That leaves you plenty of time for more intellectual pursuits, like tweeting, or speculating about the romance of the person in the next cubicle.

This reminded me of an earlier perspective on children, now thankfully less prevalent:

Their Hearts naturally, are a meer nest, root, fountain of Sin, and wickedness; an evil Treasure from whence proceed evil things viz. Evil Thoughts. Murders, Adulteries &c. Indeed, as sharers in the guilt of Adam’s first Sin, they’re Children of Wrath by Nature, liable to Eternal Vengeance, the Unquencheable Flames of Hell. – Benjamin Wadsworth

Untitled, Ansel Fiddaman, Pastel

Continue reading “If your kids are boring, you're doing it wrong”