Ever count ceiling tiles during a boring test? I have. And I run into a problem (besides that people notice me staring goofy at the ceiling). The tiles never fit. How do I count the broken tiles? Does each broken tile count as a full tile, since a full tile had to be consumed to make it? Or should I just be counting the area? My geek game, my geek rules – I count both ways.
One of the released PISA math questions sets up a similar problem – and then does not give even partial credit for a partial geek answer.
Before I share the question, I had every intention of fully ignoring PISA. The results are used to do things like justify attacks on teachers local unions in the US, or to set up charter schools whose kids would never learn to agonize over going with the Geek answer or the normal answer. The test is used to harm us, to create panic over a single data point (is there still a country between Sweden and Russia?)
But I was curious about what the questions looked like. Most of them seemed fine. But here’s the ceiling tile question:
Nick wants to pave the rectangular patio of his new house. The patio has length 5.25 metres and width 3.00 metres. He needs 81 bricks per square metre.
Calculate how many bricks Nick needs for the whole patio.
Hmm. The 5 x3.25 patio, there are 5 x 3 = 15 nice squares that could be filled. 15 x 81 = 1215. But now we have a strip a quarter of a meter wide (hate when they have fractional remainders, 1/4, for a metric problem. feels like mixed units), and 3 meters long. We should think about the shape of the bricks.
(by the way, the geek analysis is about to cost me credit).
Since there are 81 bricks to the square meter, the bricks could be little squares. One ninth (yeeks!) of a meter on a side. So two rows of them would be about .22 m, and we would have to get another row of partial bricks to fill our .25 strip (and lead to some wastage. or is it waste?) So it would take 3 x 9 = 27 bricks to complete each quarter square meter, or 27 x 3 = 81 of them to eat up the full remaining strip, so 1215 + 81 = 1296. My first answer. And I would not have gotten full credit. Because I forgot something.
What if the bricks were a different shape? We are tiling square meters, and the problem implies we can do that well. With 81 things – doesn’t that mean we are stuck with rectangles? 1 x 4 rectangles could work. That’s 1/18 of a meter by 4/18 of a meter. With one row long-ways and one row side-ways we could create a strip 5/18 m. wide. 5/18 ≈ .278, and now we are getting somewhere. Now, 54 long-ways bricks (stacked long side against long side) would create a strip 3 m x .22 m, and 14 side ways bricks would put a layer on top that was 3.11 m x 0.56 m, so there still is waste, but much less. 54 + 14 + 1215 = 1283.
And I still don’t have full credit. But now I have an approach to refine my answer.
Imagine bricks 3 m long and 1/81 of a meter wide. 81 of them cover a square meter. 20 of them would be just under a quarter of a square meter, so let’s go with 21 once, twice, three times. That’s a strip 3 m long and 21/81 ≈ .259 m wide, with minimal possible waste. Can’t get a better answer than this. That’s 3 x 21 = 63 + 1215 = 1278 bricks. Perfect answer.
So if the bricks are squares, we need 1296, but as the rectangles get further and further from square shape, we need fewer and fewer, with the minimum possible 1278.
And PISA would:
SCORING PATIO: QUESTION 13
Code 2: 1275, 1276 or 1275.75 (unit not required).
Code 1: 15.75 (units not required)
- 1215 bricks for 5m X 3m
(This score is used for students who are able to calculate the number of bricks for an integer number of square metres, but not for fractions of square metres.)
- Error in calculating the area, but multiplied by 81 correctly
- Rounded off the area and then multiplied by 81 correctly
Code 0: Other responses
Code 9: Missing
PISA would give me partial credit, while a Finnish child who was 2 or 3 bricks short of a full load would get full credit.
Math teachers, don’t let this one pass. Today is . These don’t happen so often, but they happen. We should take advantage of them.
Today paired up with this year’s other Pythagoras Day. When does that happen? Nice discussion, gives kids a chance to engage their smarts.
When’s the next one?
How many have they already lived through?
What would make a good PythDay greeting?
How many might they live through? (Hint: is there a highest year in two-digit format that works?) (Query: can we use the four-digit year?)
How can we change the rules?
I’ve played this before. I’ll play it again (skipping a year…)
That little inequality would be true in any district. But what would the numbers be?
I guess you should know that this is my 17th year. I have a Masters. And I’ve collected a bunch of additional credits, but no additional degrees. Probably if we added up the advanced credits, it would be around 30, but I padded that with some credit by examination a few years ago, so in NYC I am closer to 50. Not that those credits matter elsewhere. You should also know that I am on sabbatical this year, so I am not drawing full pay. But for this exercise, let’s pretend I am.
The list includes all the towns and cities in Westchester and Nassau that border NYC, and I’ve thrown in the NYC rates from September 1997, when I started, to boot.
|District||Starting Teacher||jd2718||Top Teacher||(deg, years)|
|NYC 1997||$28,749||$28,749||$60,000||MA+30, 23 yrs|
|NYC 2013||$45,530||$85,426||$100,049||MA+30, 23 yrs|
|Yonkers 2010||$57,772||$118,709||$131,016||PhD, 30 yrs|
|Mt Vernon 2009||$51,540||$109,616||$122,275||PhD, 20 yrs|
|Pelham 2012||$52,931||$119,308||$137,433||PhD, 25 yrs|
|New Rochelle 2013||$54,969||$119,593||$131,839||MA+90 or PhD, 20 yrs|
|Great Neck 2013||$56,829||$119,270||$136,856||PhD, 25 yrs|
|New Hyde Pk – Garden City 2012||$53,620||$109,140||$119,702||PhD, 26 yrs|
|Floral Park- Bellerose 2011||$56,088||$105,768||$123,616||PhD, 25 yrs|
|Elmont 2012||$52,076||$106,275||$119,328||PhD, 22 yrs|
|Valley Stream 2010||$55,574||$112,362||$123,510||PhD, 26 yrs|
|Lawrence 2011||$51,432||$113,989||$130,072||MA+90 or PhD, 30 yrs|
I was playing math with a niece and nephew a few weeks ago. Really, just playing games. And challenges.
We played who can get to 12 (by adding 1s or 2s). I wasn’t going for a rule, but my niece was close, so we (sister-in-law helped) got her to discover that 9 was a good number. And so was 3. And my nephew (younger) wasn’t going to discover it, but once his sister announced it, he kind of sort of followed.
We played puppies and kittens (game I learned from Sue Van Hattum. Adopt as many puppies as you like. Or as many kittens. Or an equal number of each. And – here’s a twist – whoever adopts the last furry animal loses). The two kids played with each other (I watched), and while the girl discovered some strategy, it was not a complete solution, and the two seemed to enjoy it.
I broke out some wonderful dice that my games mentor gifted me. Blue dice have the numbers 5 – 10 on the sides. Red dice have 0 – 5. 2cm, wood. I gave my niece one red, and I rolled one blue, and we saw who got higher. I won two or three rounds before she called me on it. Then I gave her two red dice, and him one blue one, and they rolled against each other, sum of the red against the blue. And then I gave her five red dice, and him 2 blue dice, and they both had some quick adding to do. They played for almost fifteen minutes, and needed to be stopped. Completely engaged. (And no, not a fair game. I didn’t calculate the probability, but the expected value favored the younger child. Intentionally, to help maintain interest).
I pulled out some graph paper (1/2 inch) and some crayons. Here I didn’t involve my nephew (I asked him to draw me something), but I drew a rectangle for my niece, 4×3, horizontally oriented. I counted the boxes (12) and the lines on the outside (14). I asked her if she could make another 12 box rectangle. She copied mine. I asked her if she could make a different 12 box rectangle. She drew a 2 unit high rectangle, and counted, and closed it at 6 wide. I asked her to draw another 12 box rectangle. She drew a 6×2, but this one vertically oriented. I asked her to count lines for each rectangle – 14, 14, and 16.
My sister-in-law asked my niece if she could write a multiplication for each rectangle. Not where I was headed. But I understand that it is not obvious to non-teachers that not every encounter with mathematics needs to reach “fruition.” And it was fine, the girl knows a little bit about multiplication, so I sat back, and watched.
Next to the 4×3 she wrote 4 x 3 = 12. Next to the next 4×3 she wrote 4 x 3 = 12. Next to the 6×2 she wrote 6 x 2 = 12. Next to the vertical 6×2 she wrote 6 x 2 = 12, and started to cross it out. My sister-in-law started to speak, to interrupt the process, but I motioned to let my niece continue, and she did. And after crossing out 6 x 2 = 12, my niece wrote 2 x 6 = 12.
I was delighted. My sister-in-law was concerned. She wanted her daughter to see that 4 x 3 and 3 x 4 were the same thing. I did not. I thought the girl was in a good place, was developing a strong sense of multiplication, and would transition nicely, later. So I intervened to assuage her mother’s concerns while only denting, not exploding, her non-commutative model. I turned the paper, and let her conclude that a 4 x 3 could be a 3 x 4 if you looked at it differently. And I asked if 3×4 had the same number of boxes as 4×3. She answered without pausing. And 2×6 and 6×2? Ditto. Right, 4×3 and 3×4 in her mind were different things, with the same answer, and that’s ok.
I dragged out the rectangles challenges by asking if there was a different rectangle with 12 boxes with even more lines. She was stuck, so I drew a 1×12. She carefully counted. 26 boxes. I asked if that was the most, she was not sure, I began to draw 1/2 by something, counting half boxes with her along the way (she was good at counting by halves!), and she was certain that there were more lines. She counted anyhow. My brother, who had only watched part of this, asked if we would ever be done (with the most lines) and she articulated nicely a “keep cutting in half” approach.
Then I taught them Set (or rather, what makes a set. I turn teaching someone how to play Set into an enjoyable game itself. I’ve done this with high school and middle school students for years. The first day we never play). And then I sent them some turn-taking rules that I thought would be better for adults playing with little kids, and kids of different ages playing together. (I played these rules on Thanksgiving, 2 math teachers and a 2nd grader, fun for all).
I never wrote about going to the Math Circle 2013 summer conference a few months ago, at Notre Dame. But I believe my experiences there had some influence on this story.
There’s been much discussion of who the next chancellor should be. I’m not going to name names. I don’t know that anyone would listen to me. And I don’t know that I really know enough.
Instead, I think I can describe things that would be good, and things that would be problems. And where someone who wanted to look for a candidate might find one.
- Experienced educator. Some real time as a teacher. I’d like ten years, might settle for five without complaint, but five is pushing it. Would be nice if the person had some time with extra responsibility before becoming a principal, but that’s not necessary. Like being an AP, or playing some extra role(s) while still a teacher. Needs to have been a successful principal. I’d like ten years, might settle for five, but that’d be kind of weak. Might have gone on from there in any number of directions…
- Experienced public school educator. And “privately-managed public school” is not a public school.
- New Yorker. An out of towner is certainly not a huge problem, and there are plenty of icky New Yorkers, but all else being equal, a New Yorker is better.
- TfAer? Absolutely not. Absolutely unqualified on the basis of career path alone. Let them run their anti-education think tanks, their testing companies. (quick nod to the TfAers who have turned on TfA. Like this good one. Or that brave one. Good people, not for Chancellor.)
- No active “reformers.” The landscape is littered with anti-public education reformers, jumping from job to job, seeking new cities and communities to victimize. We don’t need one here.
- No one who has personally done grave damage to NYC’s schools.
An experienced educator, taught ten years, principal ten, might have gone on to bigger and better. Worked/works in NYC. Not TfA. Not an anti-public education reformer.
Where should we look?
- Current, long-serving, sitting principals? Would a principal who had never served higher in the bureaucracy be able to handle such a huge system? Probably not, but, with a strong team…
- Someone at the top of the current bureaucracy? The higher we look, the more likely they’ve played the role of an active anti-public education reformer. Plus, at the top today, few have sufficient experience as educators. Suransky and those who have worked closely with him should not be considered. House needs to be cleaned.
- Someone who rose in the bureaucracy, but not all the way up? That gets interesting. How many real educators, with real experience, are there, mid-level. And how many are good people, and good educators? And how many are high enough up to have a handle on running a big system? Some. I think there are some there, and I think it is a good place to look. Just recently 7% of NYC principals signed a poorly considered letter in support of keeping the network structure in place (the networks need to go, the principals were wrong). Leave the letter aside, these principals were writing in support of network leaders who actually support their schools in ways that the schools appreciate. These network leaders, I’m sure there are others, may be a very good place to look.
- Someone who rose in NYC, but left for other educational pursuits? Depends on who, but this might also be a fruitful place to look. One of the names being bandied about is Kathleen Cashin, fits this profile (some people are fans – I’m not wild about her curricular choices – but I’m not naming names, so, enough). Betty Rosa does as well. And I’m sure there are more.
- Someone from outside NYC? Besides the obvious (I just indicated that this is a deficiency), where in the country are there people in education who are looking for jobs, not anti-public education reformers, not TfA? It is possible, but perhaps unlikely.
I’m not picking a name. But I’ll judge the choice against these criteria, as soon as De Blasio becomes mayor and makes it.
Get up early tomorrow. Vote before breakfast, and as many more times as they let you.
I hear people talking about all the things De Blasio might not do. Sure. And it’s worth saying so… No illusions.
But tomorrow, you know that most important thing De Blasio won’t do? He won’t be Bloomberg. Vote as many times as they let you.
Also, Letitia James. Vote for her, too. Public Advocate. Sometimes they become Mayor.
And also, that casino thing? My union says to vote for it. The money will go for education. Just like the Lotto money… Hmmm. That didn’t happen, did it? And building casinos? To prey on those who are addicted to gambling? Look, if you want to fund education in NY State, tax the rich. They pay much less today than they did under Andrew’s father. Just bring the rates back to where they were. No on One.
As many times as they let you.
I shook up the curriculum for an off-track Geometry last Spring. It’s worth looking at how it went.
In this post I review what the changes were, and summarize the results. I will follow up with more detail in the coming weeks.
In my school, the “advanced” math group, as freshmen, do one term of algebra (usually harder stuff) in the Fall, and take the first term of geometry in the Spring. 2012-2013 I had both of the off-track sections, and rewrote chunks of what I was doing. More significantly, I restructured the course in a way that seemed to me to be a little radical.
(I clearly miss the classroom. My mind instantly goes to little radicals: me, the square root of two…. puns are better with an audience)
1. Open with an extended logic unit, with proof. Much more than the old Regents Logic. Include extraneous statements. More rules of replacement and rules of inference, and prove the rules before using them. Venn Diagrams and Euler Diagrams and truth tables. Consistency. And indirect proofs. 4 weeks.
2. Have students create their own glossaries/reference sheets. Allow/insist on constant revisions and updates. Allow/insist that the students bring their reference sheets to each quiz and test.
3. Construction. Fully one quarter of the class periods (every Friday) devoted to construction. Some standard construction. A lot of more creative stuff. A set of Michael Serra’s geometry books – a good resource. Students required to have the tools with them at all time.
4. Oddball theorems.
a. Most high school geometry proof is 1. diagram + 2. some given information = 3. prove something that is already obviously true.
b. The other kind of proof is to prove a theorem. The book does this for the students. Or I do it in class. And then we use the theorem. Sometimes the proof of a second version of the same theorem is offered as an exercise. If this were the 1970s or earlier, the students would memorize theorem proofs, and recite them on a test.
But this is wrong! Proving theorems is at the core of what mathematicians do. The students need to be asked to prove theorems. And all the good ones are taken. So I ask students to prove less-known, less-useful theorems. We practice doing the real thing. We talk about the difference between proving a theorem, and doing a proof-exercise from the book. We approach them slightly differently. And we write them differently.
So how’d it go?
1 We covered all the material I intended to cover. Some of the time given over to construction embedded other topics. At other times the experience with construction allowed the students to move through material more quickly.
2 Most students experienced success writing proofs. Students recognized the difference between theorem proofs and ‘exercise’ proofs. Some students were taken with proof by contradiction. One asked if he could use it all the time. (Irony here, on homework in a graduate course earlier this month, the professor asked me to not to use indirect proof where direct proof was easily available)
3 The construction experience was overwhelmingly positive, and added to course, without causing us to skip material. Most students were pleased with what they were able to produce.
4 The logic unit did not detract from the course. However, not all students ‘felt’ the connection between the logic proofs and the geometry proofs.
5 I got resistance from a small number of students to learning things that would not be on the Regents Exam, exacerbated by the difficulty of the material.
6 I polled both sections at the end of the term about what they liked best: Logic, Construction, or “Proof Geometry” – and was surprised to find that one section overwhelmingly preferred geometry, with logic second and construction last, and the other section was divided between geometry and proof, with construction last. This did not seem consistent with how engaged they were during the construction periods.
I will follow up, with much more detail, in the coming weeks.