Math

My outlook on teaching math

One of the best pieces on this blog, reprinted in advance of writing more about teaching. Read, critique, praise. Your call.

• Best mathematics teaching inspires. At all levels and all ages it is possible to communicate some of the elegance, power and beauty of this most abstract subject.

• Mathematics encapsulates abstraction from the real world. A child learns to count spoonfuls, learns to count people, learns to count fingers, learns to just plain count, and in the process acquires the abstract concept of, for example, “two.” The child takes ownership of this concept, and can reapply it freely. As adults we may take “two” for granted, but we have never met it, never touched it, never tasted it. It is one of the first completely abstract concepts that we ever owned.

• Learning mathematics involves skill acquisition, drilling, repetition, and instruction by an authority. It also involves independent construction of knowledge, connection to physical or real world situations, reflection by the learner, and independent reapplication to new situations. Traditional instruction has been overwhelmingly weighted to the former list, standards based instruction to the latter. Neither by itself gives learners adequate opportunity to take ownership of the abstract concepts that make mathematics beautiful and powerful. Best mathematics instruction carefully blends traditional and standards-based techniques.

• I strongly believe that instruction should be adjusted or modified to meet the needs of the current students. This entails a constant process of carefully planned experimentation, reflection, adjustment, and evaluation. Further, I have found it valuable to share with students information about modifications (pacing, depth, styles of instruction, balance of traditional/non-traditional work), and to solicit additional feedback from them.

• Concept ownership takes place more readily when the learner considers him or herself a stakeholder in the process. To this end it is desirable to foster a sense of control or ownership of other aspects of the classroom, including, as appropriate, involving the students in some decision making (see above). It is also possible to make students part of the subject itself, whether through data studies of the class or students’ families, or the creation of geometric figures based on the students’ own birthdays.

• An effective instructor is also a learner. I continue to take courses in mathematics and to study on my own. I am an avid problem-solver. I have never stopped trying new techniques in the classroom, and modifying, or rejecting them based on actual experience. As a role model it is necessary to share this love of learning with students. I freely admit when I do not know, and gladly share with students how I intend to search for “the answer.”

• A teacher of mathematics must be able to distinguish between right and wrong answers. A teacher evaluates alternate approaches, and distinguishes between minor and conceptual errors. The teacher can place a topic into a broader mathematical context, and answer the questions “Where is this topic next applied within mathematics?” and “Where is this topic applied outside of mathematics?” (if there is an answer) Grade level curricula are a subset of mathematics as a whole. It is the teacher’s responsibility to ensure that what is being taught not only leads to a correct answer, but is mathematically valid and will not need to be corrected at subsequent levels. It is not acceptable to know just a bit more than the students. An effective teacher’s knowledge of mathematics must be extensive.

Integrability of matrix functionals

It’s almost Nanowrimo month! That crept up on me …

Anyhow, two interesting questions here, relating to integration, which I have no idea how to be about. I believe $e^{-\text{tr}(M^2)}$ and $e^{-\|M\|^2}$ are integrable over $\R^{n^2}$ (here is an $n\times n$ matrix)– both are exponentially decaying on the rays $\alpha M$ , and continuous on the unit ball. The first quantity pops up in relation to random matrices, and the second popped up in a tail bound I was attempting to make. More precisely, let $h : \R^{n \times n} \rightarrow \R$ be given by $h : A \mapsto A \cdot M$ where $\cdot$ denotes Schur (Hadamard) multiplication, then the Lipschitz constant of is $\|M\|$ . If we consider as a functional on Gaussian space, we have the deviation bound

$\displaystyle P(|\mathbb{E} h - h| > \epsilon) \leq 2 e^{\frac{-\epsilon^2}{2\|M\|^2}}$

Now if we consider to be a function of a random also, then the above bound is conditional on , so to get the desired inequality, we have to integrate out the Hence the question about integrability.

It seems unreasonable to expect there to be some first principles approach to calculating those integrals, but maybe there’s some clever argument that’ll get you there. I wonder if it helps that $\det e^A = e^{\text{tr}(A)} ?$

Updating links

A flurry of link changes. Most involved moving between categories: in particular, there is now a NY Teacher category (since I was following links from Gotham Schools, that’s no surprise), and I swapped between Teachers, Math Teachers, and NY Teachers.

I added a few nice links:

They call me teacher, NYC teacher
The Washington Teacher, union activist
Lorri Giovinco-Harte, staff developer, good read, really good teaching-oriented blog
Have a Gneiss Day, been around for a while, don’t know how I missed him
Is our children learning? new, teaches elementary, in the Bronx

I also restored some links I had previously dropped: Miss Malarkey (don’t know why I dropped her) Queen of the Quadratic (she had stopped posting for a long time), Chaz, Polski

I dropped Rocking the System. She announced last winter that she had arthritis, and would stop blogging, but I was hoping she might come back. I’ll keep an eye out.

And I should drop Text Savvy, but I’m still thinking it over. And Michael Paul Goldenberg, possible, but less likely. I’ll probably just add warning text, the way I do for Mathematically Correct and NYC Hold. There’s a story here, and it involves Devlin, and it involves me walking into a Math Wars battlefield, but for another time.

But please, go visit the new links. Go visit the recently restored links. Read. Be happy.

Orlicz spaces

If is a nonnegative convex continuous function on $[0,\infty)$ such that S(0)=0, then the Orlicz space L_S consists of all measurable functions x = x(t) such that

$\displaystyle \|x\|_S = \inf \left\{ u &> \,:\, \int_0^1 S\left( \frac{|x(t)|}{u}\right) \, dt \leq 1 \right\} < \infty.$

I don’t know much about these spaces, except that certain classes of random variables are characterized by being in an Orlicz space (defined slightly differently– for one, the integration is over $[0,\infty)$ , and w.r.t. the appropriate probability measure, and I believe there’s another condition on that constrains its behavior at $\infty$ – but the idea is the same) for some Subexponential random variables (ones whose tails fall off faster than those of a mean-zero gaussian with a fixed arbitrary variance), for instance, are in the Orlicz space corresponding to $S = e^{t^2} - 1.$ I suppose that might be one cause of interest in these things, another is that they generalize the L_p spaces.

An interesting question is, why are these Banach spaces?

What I learned this summer

Five weeks late, but with two days off last week, one this week, one next… I am finally catching my breath, deep breath, and reflecting.

Summer vacation is important. I didn’t take one. And I am suffering.

I am no longer 20. I am no longer 25…

It is possible to reach new teachers - but it takes an organizer’s mindset and an organizer’s tactics. Over 100 this summer, with a little help, it could be ten times that next summer. And I already have been approached by 5 people (non-partisan) who would like to help.

New teachers, at least Fellows, can be opened to a pro-union message. I don’t know how true this is for TFA. And I don’t think the message carried in the NY Teacher is effective.

I am no longer 20. When I speak to 20 and 25 year olds, it’s not the same as when I was 20. And not the same as speaking with people older than me. I can lecture. Which means, I need to be careful. IRL I am a rabble-rouser, a crank, a careful (and rather annoying) critic. But a 20 year old sees an older guy speaking with authority. It sounds different. Careful, careful.

I also learned that I REALLY like going far away once or twice a year, and New Orleans and Edinburgh (April) didn’t come close t making up for missing Turkey, Bulgaria, Italy, Greece, Germany, or wherever my strange summer plans to foreign lands should have taken me.

Summer is for reading. Not enough reading = a gap in summer.

I’ll try to make some of this up through this year, taking it a little easy here and there, maybe a real trip (or two) and some serious planning so I can get both volunteer work and vacation in next summer, both well-organized, both in serious quantities.

Advice to New Teachers: Your mailbox (and paper)

One of a teacher’s worst enemies, especially in the first year of teaching, is that little pigeon hole with your name pasted under it (or over it) in one of the administrative offices.

The first day you saw it, I bet there was a little smile, a sign you were really a teacher, or a little grimace, they spelled your name wrong. But whichever, there was no theme music from Jaws to warn you about what was lurking in plain sight.

How it attracts garbage! And important stuff. And stuff that might be garbage, but wait, maybe you need it? And what if you just leave it there and decide on your way out of school but oh, how tired you end up being and you find it (or don’t) buried under fresh garbage, er paper, the next day.

1. Don’t let things sit in your box. Take the extra minute, or two, or even three, once a day (morning is best), to carefully empty it out.

2. Identify what’s what. Throw out anything that can be thrown out. Save anything that needs to be saved. Complete anything that needs to be completed, and turn it in (how quickly? see below)

Urgent, but not important

Your Assistant Principal and Principal generate lots of this kind of stuff. They need it now. Ten minutes ago. And you know, and they probably know, that a sheet signing off on the fact that you have seen the postings on the bulletin board, everyone knows that this is not important. Doesn’t matter. They want it back. Assume it is urgent. Fill it in. Be done with it. Let it be gone.

Important, but not urgent

Materials related to pension, certification, requests for materials, etc, etc. No one is in a rush. There is time. No. These things matter to you. Fill them in as soon as you can. No one will push you, or harass you, like they will with “Urgent but not important” but you need to treat it similarly. Get a file system (see below) so that you can tuck these things away safely, and deal with them, not on the spot, but expeditiously.

You might be interested in…

You are a new teacher. You are exhausted. You are tired. You are interested in getting through your first year, and not very much else at all. Read the first few lines.

If the item steals your time (eg, a workshop) Into the trash! If the item relaxes you (weekend hike) you can hold onto to it. If it offers money to steal your time (tutoring opportunities!) Into the trash! (unless you are absolutely desperate for money, in which case do as little as possible. (If your principal wants you to sign off that you have received the notice, see Urgent, but not important, above).

If the item gives you something that you need (eg info about certification) think about whether you can afford to wait and do it at a much later date. If the item offers you progress towards certification, decide if you need it now.

And if it is a union notice: 1. meeting in your building - attend. 2. informational meeting in your borough office - weigh the importance, but probably no. 3. social event in your borough office - your choice, if you think it will be relaxing.

Rule of thumb, most goes into the trash. If you can’t decide, into the trash.

Saving for later: 3 (or 4 or 5) files

You will find things in your box that must be saved. I suggest getting one of those legal accordion folders (I am partial to pale brick red) that is divided into three sections. Or you might get three separate ones.

Anything related to pension, toss in one section.
Anything related to certification or licensure, toss in another.
And anything related to your progress in your school (letter assigning you there, rating sheets, observation reports, etc, etc, in the third.

It’s better if you read these things first, but note, you don’t have to. Do make time, when you are less stressed, to go through them. And please, get everything in there, without exception.

I could imagine up to two more files: a separate file dealing with pay, and yet another dealing with university stuff (masters, credits, etc)

A characterization of Schauder bases

Today’ll be a good day. I’m going to use the Gaussian-Rademacher trick in the same way I mentioned Latala did, then try to use results on concentration of measure for functions of gaussian matrices to extend our bounds on the expectation of the spectral error in approximating a fixed matrix with some random matrix of independent entries to tail bounds on the probability distribution of that error. Also, I’ll try something along the same lines with our $(\infty, 1)$ and $(\infty, 2)$ error bounds.

But first, this equivalence got stuck in my head this morning. Let $\{x_n\}$ be a sequence of non-zero vectors in a Banach space; we say it is a Schauder basis for that space if every vector has a unique expansion of the form $x = \sum_{i=1}^\infty a_i x_i.$ It’s true that $\{x_n\}$ is a Schauder basis iff there is a constant $K \geq 1$ such that $\left\|\sum_{i=1}^p a_i x_i \right\| \leq K \left\| \sum_{i=1}^q a_i x_i \right\|$ for every $p\leq q$ and all scalars $(a_1, \ldots, a_q).$

Give it a shot.

A ‘data’ agreement that benefits no teachers

This evening the New York City Department of Education released a letter explaining new “Teacher Data Reports”

The reports will link the teachers who taught a child with that child’s standardized test scores. The initial group of reports will be for 4th through 8th grade teachers, but the DoE intends (and the UFT consents) to reports being generated for all of us.

Readers of this blog knew that this was coming. You read it three weeks ago, you read it over the summer, you even read it last fall.

The UFT’s response has been to attempt to block the use of the data for tenure decisions. They got an agreement to that effect last winter. And now we, all teachers in NYC, will be getting a letter in the next day or two, jointly signed by Randi Weingarten and Bloomberg’s chancellor saying, among other things:

We wish to be clear on one point: the Teacher Data Reports are not to be used for evaluation purposes. That is, they wont be used in tenure determinations or the annual rating process.

On EdWize, Leo Casey blogs this. When he says what the data should not be used for, he also indicates its value:

…this data has the potential to enhance education by providing teachers with new tools to understand the educational needs of our students and to fashion our instruction to meet those needs…

That is, essentially, bunk. These reports will provide new a ways to discipline teachers, and new tools to bend all of our teaching to ‘the test.’

There is a lot that needs to be said, but later. Short form:

the data is no good. the tests not reliable
the data cannot be reasonably disaggregated by teacher
the data was already available to each teacher, and to each principal through normal, in school reports. These new reports make it readily available downtown, which is likely the real point
some of our chapters are strong and will howl the first sign that the DoE is abusing the agreement (which it will); but many of our chapters are weak or non-functional. We should be concerned, very concerned, about the members we have left in harm’s way.
the DoE has shown clearly that it cannot be trusted. Look what the DoE has done with the ATRs. They twist, distort, lie, and cheat. There is no reason to assume that this behavior is aberrant for them.

The UFT negotiating for the use of data in this way - shameful. Yelling and screaming at our leaders right now might make us feel better (it would make me feel better), but how can the damage be mitigated? That’s a much better question.

The agreement does protect us on tenure. We need to make use of that protection.
We need to watch for abuse, and report it, and tell our chapter leaders and members to watch for abuse, and report it.
The agreement doesn’t mention the merit pay bonuses. We should ask for clarification that these reports cannot be used for awarding merit pay (formally known as “school-wide bonuses”)
The agreement does not mention counseling memos. We should seek to expand the understanding to counseling memos.
The agreement does not mention the items that are later used in rating a teacher. We should seek to expand the understanding to prevent items from these reports from finding their ways into formal observation reports.

There is more to say. In particular, the joint DoE/UFT fetishization of data may put us at deeper risk than most of us realize. But for now,

let’s make sure all teachers are aware of the “Teacher Data Reports”
let’s make sure that teachers are aware of the danger,
let’s make sure that teachers and chapter leaders know about the limited protection the current deal offers, and
let’s get the above points clarified in our favor.

Math doesn’t suck

When the Penguin Group publishing company contacted me to see if I was interested in reviewing Danica McKellar’s popular books, Math Doesn’t Suck, and Kiss My Math, I jumped at the chance. No, it wasn’t for the free books. The time I spend reading and writing doesn’t justify the cost savings. I review products I believe in. Period. For the sake of full disclosure, I received a free copy of each of the two books. That’s it.

Danica McKellar is a well-known actress, a mathematician, and an advocate for Math education. I’m delighted to see people with a tremendous amount of influence use that influence to make Math more accessible.

(more…)

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More on buttons

Follow up to DoE saying no campaign buttons. My post here. Gotham Schools post here.

This e-mail raises an important issue. If you are in a school where the principal instructs your to “de-button,” you should do so, inform your Chapter Leader, who should inform the District Rep. We don’t need teachers and other UFT members getting written up for insubordination while UFT Central is trying to solve this through negotiations (or court)

DOE is taking the position that our members cannot wear candidate buttons in school. Randi is committed to fighting this, including going to court to protect our first amendment rights. Members who are told by their principals not to wear buttons should let their district representative know. Remind them that failure to comply with a directive could lead to disciplinary charges. [Our counsel] and others are in contact with the DOE and we hope to have this matter settled as soon as possible.

Expect the UFT to win this, and expect to see lots of Obama buttons in schools…