Finding the Focus for Education Reform

I’ve recently had a series of discussions with a friend of mine who is an advisor on education policy to a member of Congress and a TFA alum (from a particularly hard-scrabble school in Philadelphia), and in general has an extremely good sense of what is possible both in schools — classrooms and administrations alike — and in the federal government.

I will say right now that this post could’ve been shorter if it simply reported the conclusions to which it eventually comes. I am instead presenting it in a more historically accurate sequence because I believe that the process of reconciling the two views below gets at the real work to be done in education policy.

In our discussions, we have disagreed about the appropriate focus for education reform  — and for the purpose of this note, I’ll restrict myself to mathematics education, both for concreteness and because it is arguably the biggest challenge.

My position in these discussions is related to the fact that effective mathematics teaching requires an actual interest and yes, academic preparation, in mathematics — one cannot effectively teach what one does not thoroughly understand. Such preparation is so sorely lacking in public American mathematics classrooms that it could make you depressed if you didn’t know that wouldn’t do you any good.

There is a whole host of studies involving value-added data — eg. the citations in this extraordinarily clear report [pdf] from the Education Trust — that support teacher effectiveness as the single biggest determinant of student achievement, especially in previously academically struggling students. I concluded that nothing less than a renewal of the mathematics teaching force will provide the mathematics education American students deserve.

My friend, who understands the teacher union side of education policy very well, responded that any such proposal will be viewed as unmitigated antagonism by the unions; furthermore, and more importantly, that the current teaching force — even the part that is demonstrably ineffective — has something valuable to contribute: namely, the willingness to show up. She points out that teachers in low-income, low-performing (and evidently hard-to-staff) schools have in that way demonstrated a key requirement of their very difficult job.

This might sound like the kind of false praise with which a fellow at the American Enterprise Institute might begin a hilariously Ayn Rand-ish attack on teachers unions, but if you think about it for a second, showing up at some of our worst schools is not at all trivial.

Will finish this soon (have to run now)…

Number Theory Freestyle

You probably got the sense that I like teaching mathematics, but it turns out I like doing it too. Below is a little narrative description of a number theoretical question I was fooling around with. It has minimal prerequisites for understanding — things you’ll probably know if you know about convergent series (a_1 + a_2 + a_3 + …. is finite). It’s fun, so check it out if you’re in touch with your inner mathematician, or just want to reconnect with her:

While considering strictly increasing sequences of natural numbers a bit ago, I wondered if the property that the pairwise differences be strictly increasing is enough to ensure that if you sum the reciprocals of the terms, the series converges; it turns out this is sufficient, since the largest sum that could possibly result is from the sequence (2,3,5,8,12,17,23,…) whose pairwise differences are the smallest they can possibly be (1,2,3,…). If one drops the first term (2), then the sequence (3,5,8,12,17,…) is bounded below by the triangle numbers (1,3,6,10,15,…), and consequently the series of reciprocals of the former is bounded above by the reciprocals of the latter.

It turns out that the latter’s reciprocal series converges (to 2, which can be seen by a neat little arithmetic trick using the fact that the nth triangle number can be written n(n+1)/2, and the fact that 2/[n(n+1)] = 2[1/n - 1/(n+1)]), and therefore the series associated with (2,3,5,8,…) does too. Since that sequence is the smallest (therefore having the largest reciprocal sum) strictly increasing sequence with the property that the pairwise differences are also strictly increasing, all sequences with that property have convergent reciprocal sums as well.

This made me wonder if all one needed was weakly increasing pairwise differences, with only infinitely many ‘increases in difference’, to ensure that the sum of the reciprocals would converge. Spectacular as that would be, it turns out not to be true:

Since for any fixed natural number k, the sum of the 1/kn diverges (if not, then the comparison test would imply that all the other (k – 1) such subseries converge too), one can always find finitely many n_1(k) > … > n_j(k) such that

1/k(n_1(k)) + … + 1/k(n_j(k)) > 1,

starting from any arbitrarily large n_1. Thus, we can just construct a strictly increasing sequence of natural numbers with weakly increasing pairwise differences, so that the pairwise differences only actually increase when the sum of the reciprocals of the numbers with the given pairwise differences adds up to at least 1. The associated series clearly then diverges.

A natural question is then whether having the sequence of pairwise differences be weakly increasing, and such that there are infinitely many constant pairwise differences (ie. infinitely many pairwise differences remain constant from one term to the next), is enough to ensure divergence. The answer is no, since we can just essentially splice two triangle number sequences into one, with pairwise differences (1,2,2,3,3,4,4,…), resulting in the sequence (1,3,5,8,11,15,19,24,29,35,…). This sequence’s reciprocal sum can be split into two series by taking every other term – one with denominator terms (1,5,11,19,29,…), and the other with (3,8,15,24,…) – each of which is bounded above by the reciprocals of the squares (which converge).

So by this point, I’d answered my original question and a couple natural follow-ups, and it’s becoming clear that what I’m really after is some condition on strictly increasing sequences of natural numbers that distinguishes those whose reciprocal sums converge from those whose reciprocal sums diverge.

More on this in a subsequent post (enough to chew on for now, yes?)…

Positive Ism

I recently came across an extraordinary book: Rob Brezsny’s “Pronoia is the Antidote to Paranoia.” Passages like the following seemed to me to be the embodiment of wisdom:

“Understand that the universe is fundamentally friendly. This will cure paranoia. Evil is boring, cynicism idiotic, and despair lazy. Joy is fascinating. Pleasure is our birthright. Receptivity is superpower, and always remember: fascination is joy.”

I scribbled my reaction down in a notebook:

We must remember to maintain an appropriately cosmic perspective. To avoid getting too caught up in tasks or projects or obligations just for the sake of completing them, of productivity. Productivity ought to be our slave. We must remember not to be weighed down by expectations of doing this or that, of getting this or that done. Sure, it’s nice to get things done, to produce things, to affect the change we want to see in the world, to leave a legacy. But it would be cheating ourselves to allow this obsession to overshadow – to crowd out, for you economists – the simple joy of living. This basic pleasure, the taking of joy in what we do, being fascinated by something or someone, is our birthright, and should be engineered, hardwired into any proper way of organizing human life; if not, well then that’s grounds for civil disobedience, for a quiet kind of rebellion, the kind where we just do what should have been expected and for which accommodations should have been made but somehow wasn’t and weren’t. This rebellion cannot be confused with laziness by someone with the appropriate perspective – by the connoisseur – but it wouldn’t hurt to have some evangelists, some vocal believers to disseminate the manifesto, The Gospel of The Interesting.

New Policy Section

I just added a new section of policy writings. Currently it has two notes I’ve recently written, and I plan to update it with pieces I am currently incubating.

Industrial Agriculture vs. Ecosystem Robustness

Industrial farming behemoths like Monsanto and Archer Daniels Midland seem to be in a battle for hearts and minds — even New York Times-reading, subway-riding, environment- and health-aware borderline communists, judging from their ad placements. I’ve recently seen a series of advertisements from Monsanto extolling their good works, one even featuring a dignified Ivy League Professor. His red herring rhetorical question was “why would we refrain from using in food production the kind of technological solutions that have, in medicine, vastly diminished disease and extended human life?” — here is my response:

Industrial and synthetic techniques aren’t bad by definition, but the analogy between medical engineering and agricultural engineering obscures the reason to pay close attention to our agricultural practices: unlike the unintended effects of experimental medical treatments, industrial/chemical farming techniques can have consequences far beyond the subjects to which they’re applied. Farmlands must be viewed as interconnected ecosystems, as nurseries not just of food, but all kinds of plant, bacterial, and animal life whose contribution to, say, maximizing corn production may not be immediately apparent, but which can nonetheless be ignored only at the peril of the ecosystem’s robustness — including its ability to sustain corn production in the future, among the many other services it provides.

The problem isn’t Monsanto’s or Archer Daniels Midland’s pro-active goal of alleviating yield variability and sensitivity to climate shocks — that’s as civic a goal as one can imagine — and nor is it their desire to profit from this worthy endeavor. But there are at least two fundamental problems, one with industrial farming corporations’ approach to their worthy goal, and the other with their effect on farmers’ livelihoods.

The first problem is that industrial agriculture’s ’solutions’ have in an important sense been too hasty — what do I mean too hasty, when millions of people are starving every year? I mean that its actions — eg. genetically modified invasive seed varieties or industrial pesticides more like buckshot than sniper fire — have not been adequately understood for their eventual consequences. Many factors determine the functioning and robustness of an ecosystem, but even more important than the number of determinants of ecosystem health is the fact that those determinants interact in complex and interwoven chains of dependence.

For that reason, optimal use and treatment of farm production systems — ecosystems — must incorporate comprehensive risk assessment; in the presence of those complex interdependent webs, this requires prudence in the intervention into and potential disruption of the ecosystem. Simply put, we’re not as good at playing God as Mother Nature is, and we therefore need to be cautious and prudent in our management of her ecosystems. To the extent we do intervene, we certainly need to understand the ramifications as best we can.

The second problem is that industrial farming corporations have done their best to extract as much of the surplus financial benefit from feeding the world as they can, squeezing farmers’ bottom lines to fatten their own. For instance, regardless of the ostensible purpose of a terminator gene in seed varieties, it has the insidious effect of necessitating that farmers buy seeds every year, rather than using the seeds produced in the plants in year T for planting in year T+1. Industrial farming solutions are also developed and optimized in suites, so that if you use X’s seeds then you’ll be compelled to use its fertilizers and pesticides as well, and vice versa.

Of course these practices make business sense, in its most callous, monopoly-aspiring form — but in a way, that’s exactly the point. Farmers, and the communities they nurture, would be better off if their livelihoods — along with the industrial farming corporations’ — were a factor in the calculus of optimal products and techniques.

A Problem in Self-Similar Tilings

This [pdf] is the note on that remarkable series of lessons I mentioned – it has pictures, which is why it’s not posted directly here. The link takes you to the draft currently under review for publication in The Mathematics Teacher.

Welcome!

Hello, Dear Readers:

I plan this weblog to be a collection of short notes (ranging from a paragraph to 3 or 4 MS word pages) on subjects I’m interested in and about which it’d be fun to have my thoughts out there in the ether (damn you, Michelson and Morley). I like mathematics, politics, economics, and education (esp. mathematics education), and mostly it’ll be about those things.

The first post (other than this introduction) will I hope set the tone. It is a note on a remarkable series of lessons I taught to my 6th graders when I was a mathematics teacher — what was remarkable was not the lessons themselves, but the students’ reaction, as you will see.

Finally, I would like to invite interested readers to participate by commenting, and by writing their own posts. After a few weeks and a few posts have gotten out there in that ether, so that the idea for this collection of writings is more clear, feel free to email me if you’d like to contribute (maylward at university of michigan email domain).