## CC Algebra – Fewer Strong Scores – Part II – Untethering

A: When you are in New York State.

Q: When is your score not your score?

There’s something that anyone working with New York State Regents Examinations knows, that no one outside of education would assume. There are not 100 points on the tests. Common Core Algebra has 86 possible points. A complicated “conversion chart” changes this “raw score” (actual score) into the “scaled score” (reported score).

It used to be different. When we had Course I, Course II and Course III Regents, and at all times before, you earned points, and those points made your score. There were 100 points available. Earn 65 points? Your score was 65. Earn 85 points? Your score was 85. People understood this.

The first of a series of disruptive innovations in mathematics in New York State freed the test from the content (they called this “standards based testing” but we no longer knew what questions they would be asking), and freed the score from the points. Really.

Here’s old Course I exams. See, no conversion scale. But each question has a point value. As a kid worked this exam, they had an idea of how they were doing.

And starting in 2002, there were Math A exams. 85 points. And the last page in the answer key included a conversion scale. Teachers were not happy. Some of the exam was impossibly wordy, and hard on weaker readers, but the scale made it up by making 51 points (60% of the points) equivalent to a score of 65. The State was using scaling to fuzz over the fact that they could no longer write an appropriate test. And a kid taking the exam? They had no idea how they were doing, even if they attempted to keep track of points.

Oh, that 60% is passing? That went out the window quickly. In June 2003 the State gave a Math A exam that tons of suburban kids failed (it really was a poorly constructed test), and “fixed” the problem by jiggering the exam. They did not remove the inappropriate wordiness, the false contexts, or the over-penalization for rounding. They “fixed” the problem by dropping the passing score. By August 2004, 36 out of 84 (they changed the length) was now passing. 43%. They got kids to pass, but in the process convinced more teachers and administrators that they were incompetent.

Each new test had a slightly different chart. But the big changes were:

- Math A introduced (June 1999) (notice the 1999 – 2002 Math A’s are hidden in a “pre-1998” link at the bottom of this page)
- Math A rejiggered after the June 2003 fiasco (that the state has never accepted responsibility for)
- Integrated Algebra, (June 2008) and now
- Common Core Algebra (from June 2014)

We now have a full generation that works with exams out of 82 or 87 or 84 points. Do they accept that using an odd-ball conversion chart makes sense? Most of us, no. Does the public get it? No. Is it fair to kids that they don’t know how they are being graded? No.

I’ll provide more details on how the conversion charts are created in the next post in this series, the third.

But that system of conversions is key to answering the question: “Why did the top scores decline during the shift from Integrated Algebra to Common Core Algebra?”

That’ll be in the fourth and final post.

## CC Algebra – Fewer Strong Scores – Part I

When New York State changes exams, schools hold their collective breaths, trying to figure them out. We are not sure about content, about context, about difficulty, and most of all, we are not sure about scores.

Why New York State has been changing exams so frequently over the last 15 years (disruptive innovation) should be the subject of another post.

But my bread and butter exam has been Course I. Then Math A. Then Math A (adjusted). Then Integrated Algebra. Now Common Core Algebra.

What you do with these exams depends on who you are, and where you are. At Columbus I taught a course that took kids who had already failed Course I multiple times. Me and Bill Gerold taught it. And we figured out ways to get a kid who tried hard to break that 65. It was kind of amazing. And given our success, and it was success keeping the school off the SINI list or whatever it was called then for a year, the administration refused to offer the course again. At my school today I teach a fairly old-fashioned algebra course, with fairly heavy emphasis on mathematical understanding, challenging problems, lots of fractions, rich discussion of process, but not much emphasis on real-world connections. And then I build in the supplements for the exam. Obviously this means different supplements every time they change the exam.

It works fine. Kids get scores in the 80s or 90s. Once I had a kid with 100, but that was not my fault. The kids, the school, the parents, they all care about the scores. More than they should. Since my kids already know some math (they can all add fractions) when they arrive, and they all can take a standardized test (they get into the school by passing a test), the pressure is not on passing, but on getting scores that look good. And on getting the average for all of their regents exams to be at least 90, which qualifies them for an “honors designation” on their diplomas. It’s a sticker, and I give out nicer stickers, some more colorful, some scratch and sniff, some glittery, but the kids want this sticker in particular.

So the new Algebra regents (common core) hits last June, 2014, and I was on sabbatical. But I heard from around the state, from the AMTNYS listserve, and from talking to people, and from my school, that scores for strong kids were down 5 – 10 points.

So we set about scrambling to see why the scores were down, and what we could do to bring them back up. I went to the AMTNYS conference in Syracuse in October. I talked to people. Teachers, professors. Consensus was that those who used the “modules” ran out of time, those who taught the old curriculum watched the grades fall a full 10 points, or 10+ even, and those who used a reduced sampling from the “modules” did best. The modules are NY State supplied materials that probably would require a 300 day school year to teach completely (we have 180. There are 260 weekdays in a year).

There was my answer – adopt portions of modules. Of course I did nothing of the sort. I took an already rich function unit, and expanded it. I added a couple of stats topics, taught differently than I had in the past, emphasizing equally what the stuff means, and how to calculate it. I ended the year with a week and a half of intensive test prep. And my students did well. But the scores were 5 – 10 points lower than I would have expected. Mid 70s through mid 80s. Something was wrong.

I’ll give you important background in the next post, some technical information in the 3rd post, and tell you what went wrong in the fourth and final post in this series.

Teaser: NY State lowered top scores intentionally.

## UFT “Endorsements” – a frustrating year – Part II

In the first part of this post, I wrote a bit about the governor’s race. The UFT helped Cuomo secure the Working Families Party line, then refused to support his opponent in the primary (and the AFT President made calls on behalf of Cuomo’s running mate), and watched Cuomo take the general election. And as a reward for not interfering with his election, we got nothing but problems from the guy.

But were there other problems with UFT endorsements this year?

In the fall:

Charlie Rangel. They endorsed Adriano Espaillat against Rangel in the primary, and tried to sneak it through the exec board without mentioning who Espaillat’s opponent was. Espaillat lost.

Robert Jackson. This guy has been a champion for public education, instrumental in winning the CFE case, our friend, our ally, John Dewey award winner. We endorsed Espaillat against him, and Espaillat won, 50% to 43%. But which one of these guys is still out there working for public education?

John Liu. Our friend. Damaged in a financial scandal that looks like it was intentionally dragged out to hurt him. Running for state senate against Tony Avella, one-time progressive who joined the semi-Republican IDC to keep the state senate in Republican hands. We have a clear side in this race, right? Wrong, the UFT sat it out. Avella won, 6,813 votes to Liu’s 6,245.

Jeff Klein, turncoat democrat who leads the IDC and keeps the Senate republican. He got primaried, about time. The UFT supported him anyway.

Tea Party backed, about-to-be-indicted congressman Michael Grimm from Staten Island was challenged in the general election. The UFT sat out the race.

In the spring:

When Karim Camara took a job with Cuomo, his assembly seat opened up. The UFT pushed an endorsement for Shirley Patterson, running democratic and independence parties (which should have been a sign). The endorsement pitch did not mention her Independence Party connection, her connections to landlord groups, or that her chief opponent, Diana Richardson was running with tenant organization support on the Working Families Party line. Richardson won, the first assemblyperson who won on the WFP but not Democratic Party line, and without UFT support at that.

Grimm’s seat opened up, due to Grimm being indicted, and once again the UFT sat out the race (they would not have even mentioned it had we not asked).

## UFT “Endorsements” – a frustrating year – Part I

At the June 2014 United Federation of Teachers Delegate Assembly, thirteen months ago, I asked a question.

Paul Egan had moved the contingency endorsement resolution (any endorsement questions that come up in the summer get referred to the Executive Board), MORE delegate Megan Moskop rose in opposition, saying that a special DA could be called, or that electronic voting could be used, and objecting to back room deals (the phrase “back room” drew hoots from the audience and a comment from Mulgrew).

We’ve had the contingency resolutions in the past, they are generally non-controversial. But we had the governor’s race sitting in front of us.

I rose to ask if a Cuomo endorsement could happen under the contingency resolution. And Mulgrew said no. For something that big, he said, we would not do it without the Delegate Assembly. Satisfied, I voted yes.

And then, over the summer, there was no official UFT or NYSUT Cuomo endorsement. Case closed?

Hardly. My question was two weeks too late. The UFT had already “watched” the Working Families Party deliver their line to Cuomo – ensuring that the best route for a strong challenge had been blocked off. One report said that the UFT (must mean an officer or top official) threatened to destroy the WFP if it didn’t endorse Cuomo.

Our role in the governor’s race did not improve. In September Regina Gori’s motion to endorse Zephyr Teachout over Cuomo in the primary was defeated by the UFT leadership’s caucus. And on the eve of the primary Randi Weingarten, AFT President, made phone calls, not as AFT President, to support Cuomo’s running mate. The UFT officially remained neutral in the general election. But not working against a powerful governor is not very different from working for him.

The rest is history. The WFP/Cuomo deal? Every promise Cuomo made, he broke. Cuomo and the teachers? We don’t need to ask.

If we had fought Cuomo, could we have stopped him? Wrong question. We didn’t get beaten. We lay down, and got kicked repeatedly in the teeth.

## A bunny problem, or a bunny challenge

Depends on you.

This spring a student (her real name is not Nancy) posed a problem for herself: Starting with one newborn pair of bunnies, after one month the pair matures, after the next month the pair produces a new pair, and continues doing so every subsequent month, until after six months the pair dies. Describe the number of living bunnies after n months.

Problem

If you are not sure how this is going to work, the problem is for you. See if you can figure out how many bunnies will be around for the first few months, and then see if you can describe the relationship mathematically.

Challenge

If this set up is not a problem for you (if you can write the recurrence relation directly from the problem set-up), then I have a challenge for you: what interesting new problem can you create out of Nancy’s problem that would take someone who can already write the recurrence relation and make them think?

## Extending Fibonacci with Death

I had a student this year play with Fibonacci, then modify the problem, and give a partial solution to the modified problem. The modified problem is well-known and completely solved. You can try your own hand at it, (see next post). but here’s the student’s story:

Late in the fall of this past year (November 2014) I assigned freshmen the task of taking a problem that we had solved and discussed in class, and proposing a new problem as a modification or extension of the original. Some found it fun, and at least one remembered it later. (I’m sure it was more than this one.)

This February I started a special one-day-a-week class for freshmen (number theory and arithmetic, special topics of their choice, I did this once before).

Nancy (not her real name) worked in a team on Euclid’s algorithm. They did a very nice, very clear presentation, most of the students in the room were able to follow and perform the steps and work out a simple example. And then the team broke up.

Nancy decided to play with Fibonacci on her own. I was a little worried about real-world examples, but she stuck to the traditional “a pair of bunnies is born. In its first month it matures. In each month after that it produces a new pair. And she played it out and let the recursion and the problem statement match up fully. (My Ghost the Bunny is just word play)

And then she got bored, and played what-if. Nancy modified the problem – her bunnies would now have 6 month life spans. She carefully worked out what this would mean: 1, 1, 2, 3, 5, 8 all stay the same, but 13 – 1 = 12, and it gets interesting from there. Nancy identified the quantities that needed to be added (the two previous) and subtracted (six back) but had not written up a recursive formula when the class ended (we only met one lunch period per week, ate before we worked, and homework was not allowed).

But see why I’m excited? She played with a problem, then posed her own problem? Because she was curious. Ninth grader. Cool, huh?

## “Looking Back ” by Extending Problems

I like posing and solving problems on-line. Is this reflected in my classroom? Not enough, but yes.

In the fall term of algebra classes I carve out a day here or there, or maybe a few half-periods, to work on extended problem solving. It is generally not on-topic. On-topic would allow the kids to know before they start HOW they should solve the problem, and that would spoil the joy. I usually choose problems with multiple paths to success. And I certainly do not choose problems that have an accessible formula – that would spoil the challenge.

This fall I did different problems with different classes – but all did the checkerboard, and some did Ghost the Bunny too.

I use an Understand / Plan / Carry out the Plan / Look Back approach with the kiddies, but too often “looking back” for them just means “check.” Over the years I have pushed “find another approach” or “find the relationship between two successful approaches” or “generalize a solution.” But this year I pushed in a new direction.

“Use your work and solution to think of a new, interesting problem.” The idea is not to simply make the problem bigger, or generalize it, but to come up with something related, but new, probably closely related and more complicated, but not necessarily so. And it was quite possible for the new problem posed to be easier than they realized or harder, to yield to a similar approach as the original problem did, or not to yield at all. After all, if they knew there would be a solution of appropriate difficulty, it would mean that there was not original problem posing going on. And after practicing generating ideas on earlier problems, we hit the checkerboard, and I assigned them to extend the problem, gave them time in class and at home, and required them to write up a problem solving “experience”:

- Understand the Checkerboard
- Devise a Plan
- Carry out the Plan
- Look Back (include posing a new problem)
- Devise a Plan
- Carry out the Plan
- Look Back (since many new problems were not solved, this included commentary on obstacles. Where problems were partially solved, we got suggestions for the next team to pick the problem up. Where problems were solved, we got ‘normal’ generalizations, but also suggestions for future work. From 9th graders. )

So, post-checkerboard, what problems got posed? Here’s a few that I recall:

- Solve for an abnormal 8 x 9 checkerboard. Generalize to squares on an m x n checkerboard.
- Solve for a checkerboard with the four corners missing. Try again with the four 2 x 2 corners missing. 3 x 3. Generalize to an n x n checkerboard with four m x m corners missing
- Variation (different group). Solve for a checkerboard with one corner missing. Then a 2 x 2 corner…. Generalize to an n x n checkerboard with a single m x m missing.
- Variation (there was a lot of removing squares going on). Solve for a checkerboard with a 2 x 2 hole in the center. 4 x 4. 6 x 6. Generalize to an n x n board with an m x m hole in the center.
- Solve for an 8 x 4 checkerboard. Account for the difference between two 8 x 4 boards and one 8 x 8 boards (the write up for this was beautiful)
- Solve for rectangles on a checkerboard.
- Leaving the board out of it, count trimonos, tetrominos, pentominos, hexominos. (I think this group got side-tracked into some fascinating but for the moment fruitless discussions of symmetry and handedness. Product? Nah. Discussion – excellent.

So, when you get an answer, are you at the end? For most of the kids the response is still “check, and that’s enough, unless the teacher makes you go on” – but for a substantial minority I think they got used to the idea that mucking around further is a good idea, and potentially fun or interesting.