City lights at night, from space
Astronauts photograph cities and discuss the lights, patterns, cultures.
I am cleaning out my reader and found this gem. Blake Stacey at Science After Sunclipse posted it 3 weeks ago. It takes 10 minutes, and the opening minute or so is a bit dull, so be prepared. After that, though, wow. Well worth it. Much more detail than the satellite images we are used to.
Blake has a nice discussion you can read.
Teacher Pay: Clifton, NJ
Clifton is a small city, 80,000, about 10 miles west of the G W Bridge, and just south of Paterson. It’s about three-quarters white, 20% Hispanic. Median family income is about $60k.
Clifton has 14 elementary schools, 2 middle schools, and one high school. Clifton High School has an enrollment of 3400, perhaps the largest public high school in the entire state of New Jersey.
A reader sent a link for teacher pay scales from 2005-06. Normally I wouldn’t want to put up older stuff, but it came in an easy format, and it was a gift… Salaries are below the fold –> Read more…
Carnival of Mathematics #31
is up at Recursivity. Nice discussion of 31. Some fun links. I like Denise’s figurate numbers. There’s a math/cat/calculus thing (they talk about it. To see the cats I suspect one needs to buy the book). And something else about Monty Hall, who has seen a resurgence since being included in 21
A day in Albany
Sick as I’ve been, I needed a lighter day to recover. So I did not cancel my trip to Albany.
Albany? Yup. 17 hours. Never left the hotel.
I was with State Ed, as part of a group working on an issue relating to the new Integrated Algebra Regents. There were people from all over New York State, and from State Ed (including a Deputy Commissioner, I think), and employees of the vendor, and for now the rest of what I write is highly limited as the vendor had me sign a confidentiality agreement. Most of this stops being confidential on June 27.
So what can I mention? Habillement! All the men were in jackets in ties, suits and ties, except… one guy with a tie, no jacket (turned out to be a teacher), one guy with jacket, no tie (turned out to be a supervisor who teaches a class), and one schmuck with neither (turned out to be me). I kept my stylish hoodie draped over the chair behind me.
Disagreeability Quotient. We were seated with our ‘groups.’ Early on one member turned to us and indicated that by the end we would all consider her the biggest pain in the ass in the group. I looked at her, shook my head no, and pointed at myself. In some people’s eyes, I won. :)
Meeting people. One woman worked in a district where someone I haven’t seen in 7 or 8 years teaches. I sent a message. A guy knew my friend from Siena and said some nice things. There were only a few teachers, but we found each other and had a few nice chats. One school board member from western NY was such a stereotypical good-guy – I just walked away thinking “I liked talking to him” – but I think it was just chit-chat. A woman from the Catskills told me a great deal about their problems with transient population. Big city problems are not the exclusive domain of big cities.
And that’s it, from me, until June.
I took the train each way. I was in Manhattan after the DA Wednesday, so not a problem. The return trip though… Late afternoon, early evening, the Hudson on my right, the sun across half my face. Slowly baking/basking. Grading? Nah. I just let the train rock me through the warm glow.
Bad shopping trip
Soup
cough drops
alka seltzer
Extra pressure – I’m taking Thursday off (meeting, some distance away). It doesn’t make life easier – it means I have to leave prepared materials (real stuff) for five classes, and I have to have it ready by 3 tomorrow. And then I need to grade or check whatever gets produced. So tonight I need to prepare for school and meeting, tomorrow day I need to teach and double-prepare, all while I have light fever, mild body aches, and a bit of congestion (actually, it feels like someone is hammering my head, and my nose stops solid when I put my head down).
Who knows this approach to cardinality?
Where did I take this analogy from?
My algebra classes have detoured into three days of set theory. Once upon a time (when I was in school) the detritus from New Math was scattered across the land, and 2nd and 3rd and 4th graders knew about sets and elements, subsets, union and intersection, the empty set. But no more.
So instead of walking my kids through 20 minutes of stuff that might show on the new state exam, I went for an ever-so-slightly deeper 3 days of sets. (seTS, with TS, not X, an easy slip you don’t want to make in front of 25 giggly, gangly teenagers).
On the last day, I get them to remind me, when sets are finite, 
Tell me if you recognize this:
Let’s walk into a huge auditorium, and notice that every audience member is seated, only one per seat. And let’s also notice that every seat is filled. No empty seats. No standing audience members. What conclusion do we draw?
I’ll save the fun stuff for a follow-up.
McRib and Taxes
Me and McRib share an accountant (his, really). This weekend we took the long drive out to let the guy do our taxes.
Once upon a time I prided myself on being able to do my own 1040 and NYS100 or 200 or whatever that form was. But year by year it got harder and harder, took longer and longer. Finally, one year, overwhelmed, but with a refund coming, I filed for an extension, and then… forgot. Completely. For months.
McRib scolded me, and dragged me off to the accountant. Now I believe. I would have spent quadruple the time and do not as good a job… if I could do them at all. Time back, piece of mind.
And a refund. And a gorgeous drive through Harriman. Perfect, really. Except I got a spring cold.
Note from former NY State Ed Math Guy on Math A
Comments from an “old” SED Associate. (found on the NYState Math Teachers’ Listserve)
Since retirement, I have been reading this list serve with much interest. If you will bear with one of the “old guys,” I have just a few things I would like to say.
Prior to the Regents Action Plan and the Standards movement, the SED had a written philosophy that the regents exams were for the average and above average student. Hence, we had both local and regents level courses, exams, and, diplomas.
With the introduction of the first MST standards and all the interations since,that philosophy has changed. All students must take what is still called (unfortunately in my opinion) the Regents exam. In my opinion, “exit exam from high school” would be a better name. From this point on I shall just call it the exam. Since the exam is now for all students, there is no way it could be of the same high rigor and content of the previous Regents.
I do believe, however, that for the total student population it has raised the level of learning. The former “non regents” student is getting a better math education than in the “old” days. Unfortunately, if we use the new exam as the bar, we will not be providing our average and better than average kids with the mathematics education they deserve. We must all provide the best math educaiton that we can for all of our students.
Our bigger challenge is to educate new teachers, administrators, parents, students, and the communities at large that the new exam and its scoring method is a minimum for graduation and does not reflect what we need to do for all of our students. I know this is a big challenge, but we must succeed or we will fail our students.
I don’t think that there will be much of a change with the new Integrated Algebra exam. All students have to take this to get a diploma.
Although I’ve not been at SED for over 12 years now, I do know what it’s like in the schools as I continue to work with many schools in NYS as a mathmatics consultant. I appreciate the hard work that I see from so many dedicated teachers. I also see many students doing all they can to perform at the highest level. Let us stand together and set our own high standard. Then and only then will we be happy with our profession.
If you read this whole piece, thanks for letting me babble. Hope to see many of you at the AMTNYS meeting in the fall.
Ben Lindeman, Retired Associate
Bureau of Mathematics Education
New York State Education Department
Past President, AMTNYS
Article on looming NY geometry teacher problem
From “Math in the News” at gogeometry.com, this article by A. Posamentier originally appeared in the Buffalo News, February 23, 2008:
As if mathematics teachers did not have enough to worry about with the constant focus on student performance, beginning September 2008, New York State high schools will be introducing a new geometry course that is part of the new New York State mathematics standards initiative.
Instituting a new geometry course would not be a problem in any of the other 49 states, where geometry has been taught consistently for the past century. However, more than two decades ago New York dropped the tenthyear mathematics course (as the geometry course was then called) in favor of a sequential mathematics course, which was a rough attempt to integrate the previous high school courses of algebra I and II, geometry and trigonometry.
Couple this with the fact that the large number of math teachers in New York have less than four years of teaching experience and you find that there will be many relatively inexperienced teachers faced with teaching a course — geometry — that they have not even studied as a high school student. Further exacerbating the lack of preparation to teach geometry is the fact that most math majors do not take a course in Euclidean geometry.
It was bad enough in the “good-old days” when most math teachers — even the better ones — did not study geometry beyond the course that they were teaching. (Imagine teaching Shakespeare, having read none of Shakespeare’s works beyond Julius Caesar.)
The problem New York schools will face is not only providing teachers of the new geometry course with the content that they will be teaching and supporting material, but also making them aware of some of the subtle differences between the new geometry standards and the geometry topics they taught as part of the sequential-math sequence.
Even teachers who recall the tenthyear mathematics course will notice differences in emphasis on such things as the forms of writing geometric proofs and the enhancement of topics such as transformations in geometry and three-dimensional geometry.
Having served on the New York State Math Standards Commission, which prepared the new standards, I am particularly sensitive to the need to prepare teachers appropriately.
These are not overwhelming challenges for any properly prepared math teacher, yet they deserve special attention well before the fall 2008 school-year begins. Take this as a wake-up call to begin intensive in-service training throughout the state, so that teachers can gear up gradually, appropriately and in a meaningful manner.
I hope other schools of education as well as the Department of Education will support other such efforts. School districts would be wise to secure in-service training for math teachers slated to teach geometry in the fall to make a smooth transition to this new course, thereby preserving the excellent teaching of this most important subject.
Easy AIME?
What do you think?
Two problems from the regular administration of this year’s American Invitational Mathematics Examination look, to me, unusually easy. However, I have not carefully followed this exam ever year. For those of you familiar, what do you think? I’m also vaguely curious about variations on solutions (for #2, not really #1)
These two problems are copyright 2008, Mathematics Association of America
1. Of the students attending a school party, 60% of the students are girls, and 40% of the students like to dance. After these students are joined by 20 more boy students, all of whom like to dance, the party is now 58% girls. How many students now at the party like to dance?
2. Square AIME has sides of length 10 units. Isosceles triangle GEM has base EM, and the area common to triangle GEM and square AIME is 80 square units. Find the length of the altitude to EM in triangle GEM.
Who do you think…
is going to pry that rifle from his cold, dead hands?
Blig Blug – Fresh Plug
Faxes for tenure reform?
I’m stuck. I have an e-mail from my union, asking me to tell the members in my school to fax legislators, and I don’t know how to handle it.
Usually I just post the info, or stick it into a letter (I write a Chapter Leader’s letter, something like an update bulletin, every few weeks or so).
It’s trickier when I think the info is wrong, but not that tricky. For example, our union endorsed Clinton. I’m not supporting anyone in the primaries. But I put up a notice about Clinton phonebanking. During the discussion of the 2005 contract, I clearly expressed my opinion (we should have rejected it) but distributed literature from my DR in support of it as well. Members have a right to information.
But here? There is a nasty trend. Districts are looking at “value added” stuff with students, really just standardized test scores, and are proposing using them in tenure decisions. The UFT opposes this. I oppose this. NYSUT opposes this. All of you should oppose this, too. That’s the easy part.
There is a bill in the NY State Legislature, attached to the budget (read: no open debate), that would restrict the use of test scores in making tenure decisions. But it would allow tenure decisions to be based, in part, on the way teachers use test scores. Huh?
Here’s Randi Weingarten: “teachers can be evaluated on their use of student test scores and other data and how they adjust their own teaching to help students improve.” Huh?
I don’t know if we should be saying this. I don’t know if we should be supporting this legislation.
Are we trying to block an attack on tenure by offering to chip away at it slowly? Is it reasonable to expect teachers to be data specialists?
I don’t know what to tell my members. For now, I am sending no fax.
Parent-Teacher Conferences, Day 2
Thursday evening (after double checking) I spoke with parents of 32 kids + 1 a bit earlier, in two and a half hours. But the night is always heavier, right? Friday afternoon, same pace. In two hours, 27 conferences.
Most common messages:
- child is doing great
- child needs to do more homework
- child should come to tutoring
But that does no justice to what happened. The conferences were personal. I really know how each kid is doing in my class. I was surprised how infrequently I went to the grade sheets. There was an overall very positive, very respectful tone, in both directions. Not universal, but close enough. Even where there were ‘issues’ the meetings did not feel adversarial. And in most cases there was a clear sense that teacher and parent were working together for the best interest of the child. I walked away feeling that a lot of good, useful communication had taken place. But after 60 conferences in two days, I was exhausted.
Logic Texts
I teach a logic course (elective) in high school.
(Link at bottom to paper on history of logic text books!)
In past years it has been open to juniors and seniors; this year, because of slightly greater demand, there are seniors only. It is one term. In our school major subjects meet four days each week (extended periods). We award one math credit (a NYC credit is equivalent to half a credit elsewhere in New York State).
The grading guidelines are generous. And one day of every four is devoted to games or puzzles (for example, we examined Monty Hall last week. We played 3d tic tac toe the week before. And we solved LSAT logic games the week before that.
But the content? More or less what I learned in logic 101 at Brooklyn Polytech, and with the same text (Hurley). But look, there’s more texts out there. Sitting on my shelf are introductory books by Suppes, Tarski, Smullyan (and Copi at work). Dusty though.
In fact, I am violating my own philosophy of teaching math… I don’t know enough about logic to be teaching it. Summer reading. And a real course. My evening employer has a highly-regarded logician on staff. I will get advice (course recommendation?)
Back to texts. Myrtle asked me about logic, including which text we use, and was not familiar with Hurley. I found a great paper from by Francis Jeffrey Pelletier of San Diego State, comparing methods of proof in 3 dozen major texts (including Copi, Suppes, Hurley). A History of Natural Deduction and Elementary Logic Textbooks. If you don’t have the patience to read it all (ok, me neither), the intro tells the story of two mathematicians getting natural deduction rolling, independently, in 1934. And then on page 31 is an interesting comparison of the texts. The differences in approach to proof are relatively recent.
New Carnival of Mathematics
#30, at the Number Warrior. A dozen posts, wide variety. Don’t miss the anti-math post at Too Much Cookies Network.
Parent-teacher conferences Day 1
Last night was Round 1 of this Spring’s parent-teacher conferences. This afternoon (after a shortened day) is Round 2.
I am teaching a full load this term, for the first time in a while (five classes instead of three and a half). So, for two full freshman classes, there are parents who had never officially met me (though many have actually chatted with me before.) Plus, I am teaching all the freshmen in the building (freshmen parents are the most likely to come). Plus I have a new senior elective, with mostly different kids from the Fall (was teaching Combinatorics, now teaching Logic.)
So, a pretty busy night, from my point of view. Conferences with parents of 32 kids, about half from the new freshmen classes.
I was surprised how few mentioned the change in teacher. In fact, I ended up bringing it up several times. But only three wanted to discuss it in relation to their child’s performance.
Now, the teacher who left was wonderful. Her development of material is slow and clear (in comparison to mine). Mine is more rapid, and I am more likely to jump off topic. I have more “other stuff” going on. Soothing vs engaging? Maybe. But kids did well with both of us. And the content was identical.
So one of the first couples to sit down, unsolicited, says how much better it is with me. And then that subject dies until much later in the evening. I get two towards the end. For both the grade dropped, and in each case the comment is directed to “jumpy” topics. The comments were fair. We discussed how we could make a better adjustment. They really were both good kids, and I expect positive outcomes.
But one was interesting in a whole nother way. The parent made a polite preamble, respects me, good teacher, all that, and then for the important stuff says, “I know you understand Spanish” and proceeds to speak for a minute, minute in a half about the cambia de maestro in untempered Dominican Spanish. Now, I did take Spanish for one year in Junior High School. And I’ve lived in the Bronx for a while. But it is not often when I am put in the position of needing to listen for close comprehension. I could have stopped him, but he jumped right in and , in fact, I did understand, much more easily than I would have guessed. I did not know the topic in advance, but there was good context, and the complaint made sense. Plus, he repeated himself a bit.
The change of teacher sort of came up two more times. One child, high math grades until this year, explained to her mom that the drop was because she did not work for the previous teacher because she did not like the previous teacher. I asked how much work she had done for my class, and her answer (very quietly “none”) exposed as fiction everything else she had said.
Another child’s grade fell with me, and I asked if it was a rough transition. No, she liked both me and my predecessor very much, but she had trouble with the old fashioned mixture problems, and was doing much better with graphing and systems of equations. I liked that answer.
March blog numbers way up
Not just me. I’ve noticed other ed and math ed bloggers saying the same thing (here, and here, and here, for instance)
February was the busiest month here ever, with over 16,500 visitors and 23, 000 page views. March continued the upward trend, with an extra jump for pi day: 20,000 visitors and 29,000 page views.
Incisive writing? Umm, maybe, some. But lots of post. And, mostly, they come for salary schedules. And the recent additions of White Plains (550 views in March), Paterson (490), Fort Lee (25), and Yonkers (750) played a role. The jd2718 sitemeter is open. You are welcome to poke around.
Comparing teacher pay in the US to other countries
A newly released study says teacher pay in the US is lower than in a host of other industiralized countries (OECD). On average, according to the study, US starting teachers receive 81% of per capita GDP. Starting teacher pay in the other countries studies ranged from 92 – 141% of per capita GDP.
The Economic Policy Institue – (Research for Broadly Shared Prosperity) also has links to “The Teaching Penalty” and its press kit, covering broadly similar topics.
Redirecting my blogging
Great ideas need landing gear as well as wings.
A reader has pointed out some problems with this blog. Serious criticism should be taken seriously
- There is an overall negative tone.
- Too critical, especially of the UFT’s leaders
- There is a know-it-all-ish feel, especially related to the teaching of math
- The travel/politics/and oddly assorted posts are rather dull.
Instead of responding as if these comments were an attack, instead of denying the problems, I am going to accept that there is at least some merit to each of these. There are a few things I can do.
- Inspirational messages. These are easy to find on the web. I will include one at the beginning and one at the end of each post. That should help with tone. Saw the one at the top of this post? Kinder, gentler, right?
- The political writing is dull. Enough, I’m done with it. The other off topic stuff doesn’t catch anyone’s interest. Why not just stop?
- Less math. Math is right or wrong. Absolute. Talking about math tends to be in absolutes as well, leading to some of the know-it-all feel, the sharpness, the pomposity with which this blog is occasionally associated.
- Travel leads to lousy writing, but also to negative comparisons of New York to other places. Maybe it’s time to go back to teaching summer school, and leave the summer travel to others. Two birds, one stone.
- Reduce negatives about union leaders and the DoE. My principal’s not abusive. I like my school. My problems get handled. Why rock the boat? Another increase in May. I got it made. No cubicle on 52 Broadway, but I’m already happy – if I learn to keep quiet, maybe a teacher center gig?
- Math blogging is not the whole problem. Math teaching makes me see the world in black and white. While I am tossing out all this negative energy, how about rechannelling my efforts? Certification in Social Studies? My writing might get much softer, much kinder, much gentler if I taught a subject where right answers don’t really matter.
Why not improve the blog, if I can? And if not today, when?
Change for the sake of change is good if and only if the change was good before the sake.
Simple divisibility rules – first explanations
It’s easy to dismiss the divisibility rules for 2, 5, and 10 as “obvious,” but let’s pause for a moment to look at them.
A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
First, let’s agree that 
A detail. If 2 goes into two numbers, it goes into their sum. Think, 2 goes into 4 twice, and into 6 three times, so it goes into 4+6=10 exactly 2+3=5 times. I should make a picture.
And one more detail, if 10 is divisible by 2, then any multiple of 10 (write 10k or 10n) is divisible by 2.
So now let’s look at some big ugly number: 123456. This is 123450 + 6. 123450 is a multiple of 10, so 2 goes in, and we already agreed about 6. Since 6 goes into both, it goes into their sum, 123456.
This works for any number at all: #$%**1 can be rewritten as #$%**0 + 1. Two goes into the ugly multiple of 10, but not into the 1, so it does not go into #$%**1.
The rule for 5 will work almost the same. 0 and 5 are multiples of 5, and 10 is a multiple of 5, so xyz5 can be rewritten as the sum of xyz0, a multiple of 10, and 5. Since both addends are divisible by 5, so is their sum.
The rules for 4 and 25 are easy extensions. Neither 4 nor 25 go into 10 evenly, but they both go into 100. Rewrite a large number, eg 1234 as 1200 + 34. We then know that 1200, a multiple of 100, is divisible by 4, so we only need to check 34 (nope, 4 doesn’t go into 34, and it doesn’t go into 1234).
25 works the same way, but as there are only 4 two-digit endings (00, 25, 50, 75) it’s even easier to check.
Let’s push on, just a bit. Aren’t 4 and 25 the squares of 2 and 5? And they require checking the last two digits?
What about the cubes of 2 and 5? Well, 8 and 125, neither divides 100, but both divide 1000. So for those two, we could check the last 3 digits.
Why is this working so nicely? hmm. Let’s factor 1000: 
Note: the language here needs lots of work, and I may have flubbed a few mathy details. I’d like this to be accessible to kids and people who don’t normally read math. I am willing to compromise language, but not to the point of being wrong or unnecessarily hazy. If you see something that could use fixing, please, say something. I’d appreciate it.
Final four? No, the furthest four
So Alabama (10) lost to Butler, who lost to Tennessee, who lost to Louisville, who lost to NC
Southern Cal (6) lost to Kansas St, who lost to Wisconsin, who lost to Davidson, who lost to Kansas
Kentucky (11) lost to Marquette, who lost to Stanford, who lost to Texas, who lost to Memphis
Belmont (15) lost to Duke, who lost to West Virginia, who lost to Xavier, who lost to UCLA
I lost when Baylor, lost to Purdue, who lost to Xavier, but Xavier squeaked by West Virginia, 79-75
I like the idea of a #6 taking it, but my gut says Kentucky. What do you think?
Teacher Pay – Fort Lee, NJ
Fort Lee is a town of about 35,000 in Bergen County, New Jersey, at the western side of the GW Bridge. It’s about two-thirds white, one-third east Asian (half of those of Korean ancestry), and median family income is about $72K. Fort Lee has four elementary schools, a middle school, and a high school, with a total of about 3,500 students.
The Fort Lee Salary schedule has too many columns to be included here: BA, BA+10, BA+20, MA, MA+10, MA+20, MA+30, MA+40, MA+50, and MA+60. The bolded columns are included, below.
Starting salary is $45k, with increases of about $7-800 per year for the first few years. Years 5-12 include increases of around $2k, and the largest increases occur after year 12. Moving across columns is generally worth about $1k per/column, with the exception of a bigger kick for the masters.
Maximum ($102k) is not reached until year 18 (still quicker than NYC by 5 years, but slow compared with other suburbs in this region).
Fort Lee’s Salary schedule is below the fold ———–> Read more…
Simple Divisibility Rules
We all know them… or many of them. But they are great fun (and useful)
2 – if the number ends in 0,2,4,6, or 8, it is divisible by 2
3 – if the sum of the number’s digits is divisible by 3, then the number is divisible by 3
4 – if the number’s last two digits are divisible by 4, then the number is divisible by 4 (if it is not divisible by 2, no need to check – it won’t be divisible by 4)
5 – if the number ends in 0 or 5, the number is divisible by 5
6 – if the number passes the test for 2 and 3 (ends in 0, 2, 4, 6, 8 or 0 and the sum of the digits is a multiple of 3), then the number is divisible by 6
7 – tricky, two methods, future post!
8 – tricky, future post!
9 – if the sum of the number’s digits is divisible by 9, then the number is divisible by 9 (if it is not divisible by 3, no need to check – it won’t be divisible by 9)
10 – if the number ends in 0, it is divisible by 10 (alternately, if it passes the tests for 2 and for 5, then it is divisible by 10)
11 – tricky, with two methods, future post!
13 – tricky, future post!
16 – tricky, future post!
and more and more for future posts.
Some examples:
78 – ends in 2, so divisible by 2. The sum of the digits (7+8=15) is divisible by 3, so 78 is divisible by 3. The last two digits… hm, the number is the last two digits! so we just divide to find that 4 does not go in evenly. The last digit is not 0 or 5, so it is not a multiple of 5. It passes the tests for 2 and 3, so it is a multiple of 6. The sum of its digits is not a multiple of 9, so 78 is not a multiple of 9. And it doesn’t end in 0, so it is not a multiple of 10.
1234 – ends in 2, so is divisible by 2. The sum of the digits (1+2+3+4=10) is not divisible by 3, so 1234 is not divisible by 3. The last two digits, 34, are not divisible by 4, so 1234 is not divisible by 4. It doesn’t end with 0 or 5, so not divisible by 0 or 5. It passes the test for 2, but not for 3, so it is not divisible by 6. The sum of the digits is not divisible by 9, so not divisible by 9. Doesn’t end in 0, not divisible by 10.
271845 – ends in 5, so not divisible by 2. The sum of the digits (2+7+1+8+4+5=27) is divisible by 3, so 271845 is divisible by 3. It is not divisible by 2, so it won’t be divisible by 4. It ends in a 5, so it is divisible by 5. It passes the test for 3, but fails for 2, so it is not a multiple of 6. The sum of the digits (27) is divisible by 9, so it is divisible by 9. It does not end in 0, so it is not divisible by 10.
JD2718 