kitchen table math, the sequel: 11/14/10 - 11/21/10

## Saturday, November 20, 2010

### from the SMO junior exam

(The competition is meant for 14 to 16 year olds.)

This was from the 2004 exam:
Find the number of digits in N where N is the product of all positive divisors of 100,000,000. (Note: For any positive integer A, we regard A itself as one of the divisors.)

Another one:
The number A=200420052006...2040 is formed when one puts the consecutive integers from 2004 to 2040 together. What is the remainder when A is divided by 9?

---

solution discussion for the first one:

Divisors come in pairs -- every divisor x, there must be a divisor n/x, and their product is n. (But if the divisor is a square root of n, then it is the same as the other divisor and it is only counted once.) How many pairs are there?

For n=10 there are two pairs (10-1 5-2), and we note the product is 10^2 = 100.
For n=100 there are five pairs (100-1, 25-4, 20-5, 10-10, 50-2) but one pair is degenerate and the product is 100^4 * 10 (=10^9).
For n=1000, there are eight pairs. Divisor product = 1000^8.
For n=10^8, the script I wrote in Mathematica tells me there are 81 divisors and the divisor product is 325.

The number of divisors for 10^n has an interesting trend, too. It seems to be (n+1)^2.

## Friday, November 19, 2010

I don't know what the heck has happened to the Comments sections or who ShirleyFenette2010 may or may not be.

What is a comment from whiteboydancefloor doing at kitchen table math?

Two parents discover Everyday Math:
As it turns out, our school district is using a controversial math curriculum called Everyday Mathematics, also known as "Reform Math." EM, as Everyday Mathematics is referred to by teachers, was developed by the University of Chicago, and according to their website, it is in use by about three million students nationwide. Here is one example of how simple addition "can" be performed using EM:

An example of what EM calls the "lattice method" for performing multiplication:

What becomes immediately clear is that several extra steps are now necessary to accomplish simple beeline computations. More steps will result in more errors -- only an idiot would claim otherwise. Eventually, EM students are taught four ways to add, five ways to subtract, four ways to multiply, and two ways to divide (traditional long division has been eschewed completely). Rote memorization is de-emphasized, and calculators (as well as estimating) are introduced in grade two.

Here is the basic rationale behind EM, directly from the University of Chicago website:
Research has shown that teaching the standard U.S. algorithms fails with large numbers of children, and that alternative algorithms are often easier for children to understand and learn. For this reason, Everyday Mathematics introduces children to a variety of alternative procedures in addition to the customary algorithms.

Links to or excerpts of said research are not provided -- we are to simply take these statements as fact. EM further claims to "make mathematics accessible to all students" by:

Incorporating individual, partner, and small group activities that make it possible for teachers to provide individualized feedback and assistance.

Encouraging risk-taking by establishing a learning environment that respects multiple problem solving strategies.

This couple is politically conservative, and as a result, the one thing they've got wrong is the idea that  liberal parents like this stuff when they don't. Not for the most part.

This passage took me aback:
What's worse, the methods purportedly being used to convince school boards to adopt EM reek suspiciously of Rules for Radicals: *
State that the traditional approach hasn't worked

Disparage testimony from those against the adoption as ideological and politically-motivated arguments

State that the success of any program depends on the teacher

Bring in teachers from affluent school districts as witnesses

Bring in a witness from a university
Never thought of these tactics in terms of Saul Alinsky.

Sheesh.

* Barry's article!

### On top

The top earner in Westchester County and the region, Scarsdale's Michael V. McGill, earns \$372,006 in total compensation.
Top-earner Scarsdale superintendent Michael V. McGill on our top students:
If you listen to people like Richard Elmore, who’s a teacher at Harvard, he says the very top top American kids are scoring about the 75th percentile on international studies. So we know our top performing kids are doing very well.
Source: A bully pulpit for the Superintendent of the Year | lohud blogs
Our very top students should be performing on par with Europe and Asia's very top students.

If they're not, then our very top superintendents are overpaid.

Superintendent salaries

## Thursday, November 18, 2010

### sitting in a class on the elementary classroom

unintentionally. I'm in the Curry School of Education library at UVA, working on a physics paper. (Wikipedia tells me the Curry School was apparently ranked #19 in the nation, #7 for secondary education in 2007 by the UNSWR.)

Now there are friends of mine who are Curry School students, and I've seen one tutor kids on Reformation history in Clemons library -- but they're going into secondary education. This elementary classroom situation is a little different. The entire thing is a simulation of a 3rd grade (math?!) classroom, with the instructor simulating the teacher and the ed students simulating...3rd grade students. The teacher to his credit, is engaging, and his rhetorical strategies seem useful.... just the activities they're doing don't seem all that intensive.

I really can't see how participating in a national healthy foods recipe competition is all that useful. (And I love food science.) I suppose it must be about the ingredients-measuring. And calendar-time planning.

Now there's another instructor (less charismatic) coming in to discuss how to teach young kids fractions; a new piece of software -- "designed for preschoolers but has been used in elementary schools sometimes up to 4th grade". I must say I like some aspects of what I see -- they give me a, "oh hey! that might be useful!" reaction. I'd keep the sound effects and the trusting kid narrative voice, but use really powerful, engaging diagrams.

I just see many ways in which their approach could be scaled so much up and made so much more intense. One thing I do see is an underestimation and a consistent underexpectation of the imaginations of children. A reminder of the kind of elementary school classroom I grew up in. (And my first grade teacher was a graduate of Harvard.) Certainly if they do mean to inspire children to discover concepts for themselves there isn't a lot of inspiration going on.

I currently do not see a lot of inspiration in the ed school students going on either, in a "imagine the possibilities!" sort of way.

Now the lecturer is showing how the software can be used, and how it can be used in Spanish -- but not showing them for example, how it might be customised or programmed to suit a lesson -- even when the lecturer was the leader of the team responsible for its development! (Though apparently other students did write the program.)

I don't know. I imagine that an elementary school teacher could be taught how to use Mathematica and write simple scripts (fun slider bars!) to create kid-friendly yet very engaging, powerful plots.

### OK allows disabled students vouchers for private schools

Getting the vouchers, though is a "whole nuther ball of wax".

From Jay. P. Greene's blog post: Violators of OK’s Special Ed Voucher Law Get Good Mocking

## Wednesday, November 17, 2010

### TED, global health: does economic prosperity lead to good healthcare -- or is it the other way round?

The results may surprise you.

The other takeaway message of this is to take a critical look at things, and question assumptions. Averages can be deceiving when they mask distributions and what is really happening. You may be inclined for example, to think most of Sub-Saharan Africa have approximately the same issues with healthcare or economic development -- at least in the public mind they do.

I thought this would be relevant when discussing cost-effectiveness and development. Correlation does not imply causation, cause and effect and all that, etc.

Also the way he presents the data is really worth looking at. It's simply mind-expanding. Data presentation is far from making it look pretty -- it can tell you something new, that you might not have noticed.

The students here at UVA love TED talks -- from Commerce School students to biology majors to engineering students to Foreign Affairs students.

### "teachers' unions, exposed"

These are interesting ads...

I am not against the principle of teachers' unions. (Are any of us?) Recently there does seem to be a movement among teachers who dissent against the current practices of their union and wish to nucleate around alternate paradigms...

Btw, I did NOT know that Michelle Rhee was forced to resign last month, after the incumbent mayor lost the elections. It never appeared in any news article I saw...

---

How is a 20-year-old student to assist in teacher union reform? It is not a particularly hot topic here on Grounds, in the face of healthcare reform, the DREAM Act (which students recently marched in support of) and the like. I am friends with several education majors and pre-TFA hopeful (I myself being one), and the Curry School is a major ed school here at UVA. Massive chalking? Printing random ads in black and white? I deplore stagnation so so much.

### the spartan school and the low-income student

Far from an advocate of minimalism or the Lacedaemonian style of discipline, I am often struck by how schools elsewhere in the world can still retain excellent educations on so little a budget.

I admit, I have an ulterior agenda, one that pertains to immigration policy. I often hear the justification for stricter immigration controls (that is to repress even legal migrants) because additional migrants are overloading the resources of school systems. My answer is that a lot of districts have no idea how to manage low-income students or to construct low-cost schools.

Low-cost schools with excellent education, if you ask me, is a lot better than the alternative -- buildings that look like their high-income counterparts but the teaching does not. Schools in Singapore, ranging from the government-run "neighbourhood schools" to the independently-run distinguished names share some common design characteristics. They are not shamed by bare concrete floors; they are easy to spray-wash. I estimate that my primary school's (FMPS) current building cost at maximum \$12 million SGD in todays' dollars, with a capacity of 1800 students. Contrast this with my high school district in Maine, which a few years ago wanted to take out a \$57 million USD bond to renovate a high school that sees a student population between 900-1150 students. Where does the money go? I mean, Singapore, materials have to be imported from overseas; I imagine that cost of materials in New England is considerably cheaper. On top of this, FMPS managed to build several koi ponds and other feng-shui pleasing things in the courtyards -- a delight at every recess.

But the physical space is really a minute fraction of overall cost.

Is it the structure of the classes? From ages six to sixteen, schools do not try to imitate universities, though there are differentiated teachers even at primary one. So you don't have a single first-grade teacher trying to cover mother tongue languages, English, math and social/civics. But at the same time you're not shuffling kids from room to room in an imitation of the college environment. Yet I imagine a lot of American schools have this too. One thing my school did was to have morning and afternoon sessions -- from 7:50 am to 12:50 pm, and from 1:15 pm to 6:15 pm (with a 30-minute assembly before each session), so each classroom was effectively used by two different cohorts each day. Economization of space, I suppose.

How much expense is the food budget? Schools traditionally lease their canteen to eight vendors, which are in competition with each other, but the rent was cheap, such that the food prices were usually half that of the food prices found in the market across the road. But American schools can just as easily lease their premises to ARAMARK or some similar outsourcer, although a poor district may find itself paying for a large amount of free lunches.

Discipline perhaps. A large part of discipline was managed by peer leaders known as prefects -- but then again, it was needed, because a Singaporean teacher will typically manage a class of 40. Prefects were usually popular, respected students who could influence their peers, and were nominated for this reason. (I remember my form teacher once told the class monitor -- also a student discipline role -- she did not nominate him for prefect because he was "too quiet".) In my class, 11/41 students ended being prefects. (I was not the model student, and so, I was not one.) The system has its complaints; sometimes the prefects were not perfect instruments of administration and would sometimes conspire in mischief. But at the same time, perhaps the system takes considerable work out of classroom management, reducing stress and turnover. Does the system work well in Britain, who we inherited it from?

Books. The MOE much prefers thin textbooks and workbooks. Explanations in the textbooks are rather concise and visually illustrated (not with like, full-colour photographs, but with well-annotated diagrams). In primary school for each semester, my math textbook was at max a 100+ pages and was a low-cost paperback at \$3.50 SGD. The workbook was a thin tome, costing like \$2. And after exhausting the workbook, the teacher (with the assistance of the department) easily made up her own problems, or used past exams, and assessment books if need be. In secondary school the textbooks and workbooks were significantly thicker, but the prices were not ridiculous -- on the range of \$10-\$15 SGD usually.

Perhaps it's because publishers do not have a monopoly. Schoolbooks are meant to be worn; publishing in hardcover is ridiculous. I believe MOE drafts the material (with the help of participating teachers) and then has publishers bid to publish it the most cost-effectively. On the other hand, I have never heard of the Department of Education ever publishing anything besides reports, "standards" and "rubrics". I have never seen a Department of Education approval stamp on say, a textbook.

The money must be going somewhere. The Singapore administration would probably like the opportunity many districts in America are facing, with a large influx of migrant students. From a purely economic perspective, more students means more productive citizens in the future, and new opportunities to strengthen the state machinery, especially in the face of a declining birth rate. We don't have to be that cynical, but I really dislike how many local governments view their low-income students, treating them as liabilities rather than as future solutions to the poor neighbourhoods they came from.

## Monday, November 15, 2010

### do principals know who the best teachers are?

If principals and other administrators are doing their jobs, they already know who the best and worse teachers are. One year, Brookline administered one form of the Iowa Test of Basic Skills in the fall to third graders and the state administered a different form of the same test in the spring. I calculated the mean gain score for each teacher and asked the Assistant Superintendent for Curriculum and Instruction to tell me which teachers would have the highest and lowest gain scores. She got the group of highest teachers exactly right and missed only one of the lowest. I would expect building-based administrators to do even better. More formal analysis confirms my experience (Murnane, 1975; Jacob and Lefgren, 2008).

by Kevin Lang

### 21st century skills

One of the goals of No Child Left Behind is to increase the availability of data. Part of the implicit model underlying No Child Left Behind is that with improved information, parents will recognize good and bad schools. Principals will identify good and bad teachers. District administrators will identify weak and strong principals, and state  administrators will recognize struggling school districts. Armed with this information, parents will choose with their feet, and the other actors will undertake the necessary reforms to improve education.

As an empirical economist I am, of course, sympathetic to the use of data, and as a school board member I pushed for more thorough evaluation of our programs. But the gap between the rhetoric and the ability to use education data effectively is large.

Principals, district administrators, and even state-level administrators generally begin their careers as teachers, and relatively few teachers have strong backgrounds in statistical reasoning. In my experience, the people who rise to senior administrative positions in public education are smart. They understand in a general sense that estimates come with standard errors attached, but faced with a report that last year 43 percent and this year 56 percent of black students in fourth grade were profifi cient in math, few could tell you whether with 75 students each year, the change was statistically signifificant.

When I stepped down from the school board, one of my colleagues joked that they could all go back to treating correlation as causality. In education policy settings, one repeatedly hears statements like: “Students who take Algebra II in eighth grade meet the profifi ciency standard in grade ten. We must require all students to take Algebra II in eighth grade.” “Students taking math curriculum A and curriculum B get similar math SAT scores. The curricula are equally good.” “Students who are retained in grade continue to fall further behind. Retention is a bad policy.”2

School administrators may understand at some level that they are only looking at  correlations, but almost none have the training to address the issue of causality, and faced with a correlation, they will often interpret it causally in the absence of evidence to the contrary. The capacity to address causality, weaknesses of various measures, and other strengths and weaknesses of statistics is very limited. The Public Schools of Brookline recently recruited for a Director of Data Management and Evaluation. Although school board members generally are not (and should not be) involved in personnel decisions other than those involving the Superintendent, in this specific case the Superintendent asked me to participate in the candidate interviews. Many of the candidates held or had held similar positions in other districts. I asked each candidate how we could decide whether a math curriculum used by some, but not all, of our students was effective. Many of the candidates did not think of this question in statistical terms at all. Only one addressed the issue of selection—and we hired him.

by Kevin Lang

Apparently they don't cover Excel in ed school.

### Journal of Economic Perspectives

back issues are free online
The Journal of Economic Perspectives (JEP) attempts to fill a gap between the general interest press and most other academic economics journals. The journal aims to publish articles that will serve several goals: to synthesize and integrate lessons learned from active lines of economic research; to provide economic analysis of public policy issues; to encourage cross-fertilization of ideas among the fields of thinking; to offer readers an accessible source for state-of-the-art economic thinking; to suggest directions for future research; to provide insights and readings for classroom use; and to address issues relating to the economics profession. Articles appearing in the journal are normally solicited by the editors and associate editors. Proposals for topics and authors should be directed to the journal office.

Online issues of the Journal of Economic Perspectives published since 1999 are now publicly accessible at no charge, compliments of the American Economic Association.

Goody.