By The Numbers

What is Wrong With Everyday Math?

2012-02-24T16:38:00.001-05:00

I recently had the opportunity to sit down with the Math coordinator for Greenwich Public Schools, at his invitation, to discuss my thoughts on Everyday Math. We both realized from the start that we were not going to change each other’s mind (he is a supporter), but we had a excellent discussion on the issues (as I see them) and on the pluses (which I recognize exist and understand). I appreciate the openness shown by all members of the administration.

The primary issue with EDM is structural, i.e., how the material is presented. If the structure is flawed, anything you build on it will be weak. Educational theory holds that factual, procedural, and conceptual learning should occur in parallel. Problem is that not enough of the factual knowledge is being practiced, and the wrong types of procedural knowledge (i.e., non-standard algorithms) are being emphasized.

Structural

1. EDM spirals, does not have a clear progression of topics, and does not teach to mastery in a compressed timeframe. “A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.” (NMAP Finding 1) EDM may claim that research supports the distributed nature of its instruction, but research actually supports distributive practice, not distributive instruction.

2. Further, the actual distributed practice is insufficient to provide for a solid foundation in either the core math facts or the core algorithms. EDM spreads the practice over years, while it should be weeks or months. While the fact practice can be supplemented, the actual supplementing in GPS is inconsistent across schools.

3. As shown by the recommendations of the CCSS, EDM covers too wide of a set of topics, contributing to its lack of coherence and focus.

4. The language intensive nature of EDM presents an extra challenge for Special Educations and English Language Learner students. “Special Education students need structure and coherence, direct explicit instruction, more individual work, less group work, clarity of purpose, incremental goals, practice and repetition, regular assessment of skills and knowledge, a textbook, and meaningful graded homework. Homework assignments aligned with class work, textbook and tests will provide guidance and feedback to parents to support their children's home work and study.” (taken from a discussion with a NYC Special Education administrator, commenting about the adoption of EDM).

5. Homework problems stray far from mathematics (geography, art) in an attempt to be “everyday.”

6. Calculators – EDM requires and encourages the use of calculators.

7. Parents struggle to understand and help because the terminology/ language/ procedures are different than traditional elementary math and more advanced math. By learning non-standard terms and algorithms, the student will struggle when (1) they reach a traditional program (college mathematics) which uses only standard terms and (2) the alternative algorithms break down because of inherent flaws or excessive time requirements.

8. Lack of text book to aid students and parents. EDM reference book is not helpful, and in some cases in not even given out.

9. Time spent on a number of inappropriate topics (e.g., military time, geoboard designs, probability), when basic skills are not secure.

Factual Knowledge

1. Insufficient drill for addition, subtraction, multiplication, division facts. The schools are relying on parents to conduct the drills, in most cases only if they (the parents) realize there is an issue (i.e., only one of my daughter’s three teachers has indicated that she should drill at home, and that was her ALP teacher). The point is that if the teacher does not indicate that the student should drill, there is a large probability that the student and parent will not recognize the need, and will not drill at home.

Procedural Knowledge

1. Insufficient attention to standard algorithms

2. Too much attention to non-standard algorithms which are not general, not efficient and/ or are not applicable to advanced math

Conceptual Knowledge

1. Non-standard terms (e.g., “turn around rule” instead of Commutative Property) or made-up terms (e.g., cubes, longs, flats, big cubes), which can only lead to confusion in the real world. NOTE: these “made-up” terms are part of the world of manipulatives, which students use to visualize concepts, such as place value. Apparently they were made up after I went to school, but are accepted terms in elementary education. Probably won’t find them on your Chemical Engineering Thermodynamics final though!

Board of Education Meeting Recap - Making Progress

2012-02-24T16:03:00.001-05:00

Last evening I made my monthly visit to the Board of Education meeting. The principal of New Lebanon School, where the meeting was held, gave an interesting recap of their efforts to improve the literacy level at the school. It is obvious a large amount of work has gone into the thinking and execution of this plan.

The President of the PTA Council, Lisa Beth Savitz, indicated in her remarks that the PTAC was “discussing Everyday Math.” She also expressed concern about the focus on comparisons of Greenwich to DRG A and B districts, which I have used extensively to bolster my arguments. I understand her concern, but this is one means of getting the attention this issue requires (especially if you mention real estate values in the same line). Once the focus moves to the curriculum, I will be happy to stop using that tool, but also feel we should be sharing thoughts with other districts about what they are doing regarding math curricula and Common Core Standards. In addition, she questioned the lack of actions in several Monitoring Reports related to male/female gaps.

The Board of Education accepted the Math Monitoring Report. The recommendation contained in the report relating to the Common Core Standards implies that the next math curriculum review will be held, as scheduled, in 2014. Peter Sherr picked up on this, and noted that the actions in the report fell short of changing the curriculum. He shared my concerns about Everyday Math and the slow rate of improvement, and wanted to go on record to ensure that the “actions we are taking are the maximum actions we can take.” He also indicated he was not willing to defer to 2014 for a review. Dr. Lulow reiterated that the administration has made a commitment “to look at ways to move the timeline forward and not wait until 2014.” Board Chair Leslie Moriarty also indicated that the Board wants to get a deeper understanding of the transition to Common Core.

Thanks Peter!

Following is a copy of my comments (I left out the Shakespeare quote).

My objective in making these presentations over the last few months has been to get the Board of Education and the administration to seriously consider accelerating the Mathematics curriculum review, with the end goal of eliminating the Everyday Mathematics program.

By itself, the stagnation in elementary school performance on the CMT merits attention. Some schools are doing okay, but some are struggling to make progress, with or without coaches. However, the decline in performance relative to comparable districts indicates serious issues. Based on the last five years of data, if DRG A and B, and Greenwich all continue on their trend-line, it will take about 5 years until all DRG A students are at the Proficient level, 7 years until all DRG B students are Proficient, and 13 years until Greenwich reaches that level. At the more challenging but more appropriate Goal level, those figures are 10 years for DRG A, 15 years for DRG B and 196 years for Greenwich. The US is falling behind the world, CT is falling behind the US, and Greenwich is falling behind these districts. Something must be done now.

I did an unscientific poll on my blog. I asked parents: If you recognize the issues with Everyday Math, how do you fix them? 13 out of 16 respondents were either tutoring at home, or hiring a tutor, or both. This was tutoring beyond just helping with homework (which most parents did also). Greenwich is a competitive place, so I recognize that some of these folks may be tutoring to advance their children. However, many of those thirteen are just trying to ensure their children get the education they need by filling in the gaps left by Everyday Math. How do I know? The parents have told me. My child goes to Riverside, as do many of the children of these parents. Think about the schools where the parents may not have the time to tutor, or the money to pay a tutor. What is this doing to their subgroup performance gap? For the sake of all of these students, something must be done now.

I realize that data can be interpreted in many different ways, and my interpretation may suit my goals and the administration’s interpretation may suit their goals. You are being asked to approve the Math Monitoring Report tonight. Before you do, ask yourself if you have examined the data, not just read the words. Think about your interpretation of the data. Are you comfortable with the analysis and the actions, or are you concerned about the direction our elementary schools are headed?

The Math Monitoring Report is going to reiterate the Administration’s recommendation that the next review be held as scheduled in 2014. I understand the conflicts with other reviews, and the burden change puts on the teachers. But I also understand the burden a failing program puts on a child. Sometimes a parent is helpless to fix that. The administration and the Board need to consider what can be and should be done. Perhaps a pilot program with another curriculum can be run with Math ALP classes or at an underperforming school. Let us not put ourselves in a position where we say, to paraphrase Cassius, “The fault, dear Brutus, was not in our stars, but in ourselves.” Something must be done now. Thank you.

Recognizing the Issue, and Doing Something

2012-02-20T13:14:00.002-05:00

In the latest blog poll here, I asked the question “If you recognize the issues with Everyday Math, how do you fix them?” Four options were provided, with multiple answers accepted:

1. I help with homework

2. I tutor at home using other math programs

3. I hire a tutor for my child (Kumon, Mathnasium, private tutor)

4. Right now, I am not doing anything

With the usual caveats and disclaimers (preaching to the choir, etc.), the results confirm my guesses. Ten out of sixteen respondents help with homework, and eleven out of sixteen tutor at home. Obviously a bit of overlap, as you would expect. Most parents try to provide guidance, as I do, to help their child understand some of the nonsensical homework provided by Everyday Math, and to provide the background in basic math which Everyday Math is missing. I am sure there are also other times when guidance is needed, like asking a question to help at the start of some of the word problems in Sunshine Math, or reminding your child to check their work (NOTE: if I were typical, my mom would say I started to do this sometime in high school).

Assuming no overlap between tutoring in house and hired out, thirteen of the sixteen respondents are providing tutoring. I see two main reasons for tutoring: wanting to move your child ahead of the class (or the child’s desire to do that), and/or wanting to fill in the gaps so that your child will not be behind when Everyday Math is over. I find nothing wrong with the first motivation, and actually wish it were the only reason.

I am concerned that the second motivation probably accounts for the lion’s share of the tally. While it indicates a wide recognition that an issue exists, the fact that parents have to respond in this way is troubling. Whatever the form of this tutoring, from flashcards to worksheets to entire home schooling efforts, that it exists at all indicates a failing by our schools. I obviously come down on the side which argues that the poor curriculum (Everyday Math) is the major factor contributing to the issue, but I don’t discount other contributing causes.

Conclusions? A high percentage of concerned parents, and (in my opinion) rightfully so.

The Important Questions

2012-02-09T21:54:00.001-05:00

After thinking about this for several months, two questions regarding our school math program and performance stand out:

1. Are we really preparing our students for success after high school, whether in college or in the work place?

2. Regardless of the answer to question one, can we do a better job?

Both of these questions deal with results: Greenwich student results and Greenwich students compared to other students.

I dealt with the first question already in a prior post:

http://greenwichmath.blogspot.com/2012/01/ready-or-not-here-they-come.html

That post figured we are failing about one-third of our students, by not providing an education that would allow them to take college level math without remediation. And I am being generous regarding “college” level, since many colleges and universities grant credit for “college algebra,” which has much the same content as high school Algebra 2. For example, see the course description below for the University of Connecticut’s first level math course, which satisfies part of the Quantitative course requirement for UConn’s College of Arts and Sciences general studies requirement, and earns credits toward graduation.

“Math 104Q is a course designed to serve as preparation for all the other Q courses offered at UConn. It emphasizes two components, the mastery of each is equally important for success in any course employing mathematics. The first component is made up of the collection of fundamental algebraic concepts and their manipulations. Most of this material is taught in High Schools and Community Colleges under the name Intermediate Algebra or Algebra II. Math 104Q covers this material using a college algebra approach. The second component consists of using these algebraic concepts for solving multi-step problems from other disciplines. This practice is called Mathematical Modeling, and is the part of the course that gives Math 104Q its unique interesting flavor, liveliness and usefulness beyond a usual Intermediate Algebra course.”

Interestingly, this course replaced another course which did not earn credit toward graduation. Can almost hear those standards slipping, can’t you?

So my answer to question one is: NO.

To answer question two, we need to determine if anyone in Connecticut, with access to the same resources (curriculum, teachers, technology, administrators) is doing any better. As part of my review of the Math Monitoring Report, I looked at the comparisons between Greenwich and the group of districts with comparable demographics (called DRG B), and between Greenwich and the group of districts outsiders think Greenwich is in, not realizing our diversity (called DRG A, which includes New Canaan, Darien, Westport, and a few other well-off, but homogeneous, districts).

In our (real estate seller’s) dreams, we should be performing as well as the average DRG A district. Ideally, we should be performing at the top of DRG B, as measured on an absolute basis (i.e., our average performance should be better than most if not all of the other DRG B districts average performance). In addition, we should be improving faster than the average improvement of the DRG B districts. In reality, however, we should be performing in the middle of the DRG B group on both an absolute and relative basis.

This reality, if it lasts a few years, will begin to impact real estate prices (negatively, in case you were wondering about direction).

Performance is measured by two Connecticut standardized tests: CMT (for Elementary and Middle Schools) and CAPT (for High School). These tests group student results into one of five categories: Advanced, Goal, Proficient, Basic, Below Basic.

The results for the Proficient level (i.e., percentage of students testing at or above the Proficient level) are used to judge Adequate Yearly Progress for No Child Left Behind purposes. All (100%) children are supposed to be Proficient by the 2014-15 school year. As previously discussed, Proficient doesn’t mean that a student is on track to be able to do college level math. Connecticut’s version of Proficient is barely better than the bottom of the national test’s (NAEP) Basic level.

However, for comparison to other districts, we will use percentage of students at Proficient, Goal and Advanced levels, and we will group grades 3-5 together, grades 6-8 together (all of these grades take the CMT), and keep grade 10 by itself (CAPT).

Comparison of current results (2010-11 school year) shows that, across all three grade groups (3-5, 6-8, 10) and for all three performance levels (Proficient, Goal and Advanced), Greenwich is substantially below the average of the DRG A districts (see charts below). This is to be expected, given Greenwich’s diversity. When comparing Greenwich current results to the average of the DRG B districts, only at the Advanced level for the Middle School (6-8) students and the High School students does Greenwich surpass DRG B. In most of the other comparisons, Greenwich is more than two percentage points behind.

So on an absolute basis, Greenwich is significantly lagging the well-off, DRG A districts, and in most comparisons is lagging the DRG B districts, including across the board at the Elementary School level.

The administration used five years of data to look at trends in Greenwich performance on an absolute basis and on a relative basis to DRG A and B districts. Note that I calculated the 2006-07 High School, DRG A and DRG B figures from raw data, as they were not included in the draft Math Monitoring Report.

There are several ways of comparing trend data. The simple way is to calculate the change from the first year to the last year. This is what the administration did. This gives a crude indication of the trend, since it uses only two data points.

Based on this measure, Greenwich is closing the gap to both DRG A and DRG B districts only at the Proficient level for High School and the Goal level for Middle School. In all other cases, both DRG A and B districts are pulling away from Greenwich. Again, the Elementary Schools lag behind the comparable districts’ improvement at all three performance levels.

What does this mean in conjunction with the absolute measure discussed above? We are behind, and we are falling farther behind.

And remember those two instances where we were still ahead of the DRG B districts for the current reporting year (Advanced level for Middle Schools and High School)? In both cases, we started significantly ahead in 2006-07, but are only barely ahead now. For the Middle School Advanced results, Greenwich had 47.8% of its students at Advanced in 2006-07, versus 46.3% of DRG B students. In the current year (2010-11), Greenwich is barely ahead (54.0% versus 53.9%). For the High School Advanced results, Greenwich started with 43.2% at Advanced versus 34.5% for DRG B. Now, Greenwich is barely ahead (38.5% versus 38.3%). Not much of an achievement, given that the students percentage at the Advanced level actually dropped 4.7% for Greenwich while jumping 3.8% for DRG B. Makes you wonder why the 2006-07 High School data was omitted from the draft report.

Realizing that the crude method used may hide some improvement actually made by our students, I used two other methods, which I feel give a more accurate indication of progress. The first method I call “Progress Toward Perfect” (PTP), which takes into account the starting point, since it is usually harder to make progress as you get closer to a target. That is, it is easier to improve from 80% Proficient to 85% Proficient, than it is to improve from 90% to 95%. PTP measures the percentage gain from the starting point to the target, based on the current performance level. So if the target for everyone is 100% Proficient, and you started at 80% and made it to 85%, PTP = 25% = (85-80) / (100-80). The district that started at 90% and had the same 5% gain has a PTP = 50%, meaning they have closed half their gap to the target.

The comparative results from this one are easy to present. Greenwich had slower progress, at all school levels and at all performance levels, than the average DRG A or DRG B district. Last place, across the board. So even where we had a higher absolute percentage, or a larger gain from start to finish, we trail in closing the gap to the target level, at Proficient, Goal and Advanced. And guess what. The Elementary Schools were not even close.

The second method is more sophisticated, and presents a more accurate depiction of progress than any of the other methods because it takes all five years of data into account. The method utilizes the slope of the line fitted to the five data points. This Slope gives an overall trend for the data. A positive number is an upward sloping line (meaning progress toward the target) and a negative number means you are moving away from the target. To give a better feel for the meaning of the Slope, I calculated what I call “Years to Target (100%)” which uses the current performance level and the Slope to determine how many years it will take, at the same rate of progress (same slope), to achieve the 100% target (i.e., all students at 100% Proficient, Goal or Advanced).

For this result, a glimmer of good news. The Middle Schools at the Goal level and the High School at Proficient actually beat DRG A and DRG B district improvement rates, and the High School at Goal level beats DRG A district improvement rates. However, it looks like it will take about 12 years until all our students are at the Proficient level (NCLB target is 100% by 2014-15, four years away). And again, the Elementary Schools fare very poorly.

So what is the point of this elaborate analysis? First, if anyone tries to tell you that our schools are doing well relative to other comparable districts, check their math. There are a few dim points (i.e., low key bright points), but overall we are behind and falling farther behind. Second, try is I might, I could not find any good news when it comes to the Elementary Schools overall. You have to dig deep to find that only Riverside (at Advanced, Goal and Proficient) and North Street (at Goal and Proficient) beat the average of the DRG A and B districts. Old Greenwich and North Mianus beat the average of the DRG B districts at all levels, as did North Street at Advanced and Dundee at Proficient.

A lot of districts are doing better than we are. And some of them use Everyday Math. A crude analysis I did of fifth grade scores for 2005-05 to 2010-11 school years indicates that districts using Everyday Math had the lowest average performance compared to districts using all other curricula.

Question two: Can we do a better job? YES

Could it be the fault of the curriculum? Is it time for a change? Yes to both questions.

______________________________________________________

Elementary School – DRG Comparison

Advanced	2006-2007	2007-2008	2008-2009	2009-2010		Five Year Change	Progress Toward Perfect	Slope	Years to Target (100%)
Advanced	2006-2007	2007-2008	2008-2009	2009-2010	2010-2011	Five Year Change	Progress Toward Perfect	Slope	Years to Target (100%)
DRG A	46.5	50.2	53.1	53.3	55.8	9.3	17.4%	2.17	20.4
DRG B	42.2	44.8	46.4	48.4	49.1	6.9	11.9%	1.74	29.3
Greenwich	42.5	42.2	45.9	45	44.9	2.4	4.2%	0.76	72.5
Goal
DRG A	85.6	86.8	88.3	89.7	89.6	4	27.8%	1.09	9.5
DRG B	81	81.6	82.9	84.4	84.8	3.8	20.0%	1.04	14.6
Greenwich	81.1	78.7	79.7	81.1	80.4	-0.7	-3.7%	0.10	196.0
Proficient
DRG A	95.2	96.6	97.6	97.4	97.5	2.3	47.9%	0.54	4.6
DRG B	92.9	93.7	94.7	95.4	95.5	2.6	36.6%	0.69	6.5
Greenwich	91.6	92	92.5	93.4	93.5	1.9	22.6%	0.52	12.5

Middle School – DRG Comparison

Advanced	2006-2007	2007-2008	2008-2009	2009-2010	2010-2011	Five Year Change	Progress Toward Perfect	Slope	Years to Target (100%)
DRG A	55.7	58.2	60.8	63.6	63.3	7.6	17.2%	2.06	17.8
DRG B	46.3	48.1	51.4	53.8	53.9	7.6	14.2%	2.09	22.1
Greenwich	47.8	46.5	50	52.1	54.0	6.2	11.9%	1.80	25.6
Goal
DRG A	90	91.3	92.3	93	92.6	2.6	26.0%	0.69	10.7
DRG B	82.7	84.1	86.7	87.8	87.6	4.9	28.3%	1.35	9.2
Greenwich	78.4	79	82.8	82.7	83.6	5.2	24.1%	1.41	11.6
Proficient
DRG A	96.8	97.4	98.1	98.6	98.7	1.9	59.4%	0.50	2.6
DRG B	94.3	94.6	96.5	97	96.9	2.6	45.6%	0.76	4.1
Greenwich	92.2	92.6	93.8	93.8	94	1.8	23.1%	0.48	12.5

High School – DRG Comparison

Advanced	2006 -2007#	2007 -2008	2008- 2009	2009 - 2010	2010-2011	Five Year Change	Progress Toward Perfect	Slope	Years to Target (100%)
DRG A	52.2	53.1	46	47.4	53.4	1.2	2.5%	-0.33
DRG B	34.5	36.9	34.6	35.1	38.3	3.8	5.8%	0.58	106.4
Greenwich	43.2	34.4	37.6	39.3	38.5	-4.7	-8.3%	-0.45
Goal
DRG A	82.1	86.8	83	83.1	87	4.9	27.4%	0.61	21.3
DRG B	66.6	72.8	70.7	72.6	73.2	6.6	19.8%	1.30	20.6
Greenwich	70.8	68	70.2	72.3	71.9	1.1	3.8%	0.65	43.2
Proficient
DRG A	97.0	97.2	97	97.6	98.0	1.0	33.3%	0.24	8.3
DRG B	92.6	93.7	93.4	94.1	94.5	1.9	25.7%	0.42	13.1
Greenwich	89.7	89.8	92.2	91.2	92.3	2.6	25.2%	0.66	11.7

# - data calculated by Brian Peldunas, needs to be validated

Upon Further Review

2012-02-03T18:30:00.000-05:00

I had a meeting today with several of the curriculum folks with the Greenwich Public Schools. As with my previous visit with Dr. Lulow, everyone was very interested in my thoughts and ideas, and we had a good conversation. That is not to say I got an agreement to do an immediate curriculum review (that would be too easy, wouldn’t it?).

We agreed to disagree on that, but I understand their constraints (as I did before) and they know I will continue to push utilizing a variety of means. Only by having a clear and informative Math Monitoring Report (and reports for the other subject areas) will the Board of Education be in a position to judge where reviews need to be done and where resources need to be committed.

We reviewed my thoughts and ideas on the Math Monitoring Report, some of which will appear in the next version. Hopefully all of the data issues will be corrected. The format of the report will remain the same, and I will forward my suggestions for changes to the format/organization to the Board of Education members, who are talking about how to improve all of the Monitoring Reports.

The author of the section containing comments from Middle School teachers about “the grade 6 students’ preparation for middle school math: difficulty reading problems, basic fact fluency, operations with fractions, long division competency” clarified that the report should have stated that this “informal poll” pertained not to all students, but to a small sample of students with learning disabilities. Noted.

I am glad that most of our sixth graders are not having difficulty reading and, by extension, translating English word problems into math problems to be solved. But I will go back to a comment made by an administrator from another district who said that Everyday Math is a “language rich” program, and that they don’t even use Everyday Math for their special needs learners. Might this be a cause of our lack of progress for the English Language Learners and Special Education students?

And what of long division? It seems that the school has realized it is important and it is now being taught, although the lessons are being supplemented outside of EDM. Several parents of middle schoolers, however, including parents of honor students, have mentioned that their children don’t know how to do long division and that they (the parents) have had to show them. Maybe a catch-up session is needed.

And do you really feel comfortable with your fourth, fifth, and sixth graders’ command of their basic facts?

Upon further review, even with the clarification, I will stick with my 2.5 strikes against Everyday Math.

Strike 2.5

2012-02-02T17:05:00.000-05:00

In researching the best way to tutor my daughter at home, I ran across an article that provided a clear path, as well as an interesting structure for understanding what is wrong with Everyday Mathematics.

http://www.aft.org/pdfs/americaneducator/winter2009/willingham.pdf

Dr. Daniel Willingham is a professor of cognitive psychology at the University of Virginia (My nephew, who graduated from UVa as a Mechanical Engineer, assures me it is an excellent school. His brother, who graduated from Virginia Tech as a Chemical Engineer, respectfully disagrees). The article asks and answers the question, “Is it true that some people just can’t do math?”

Referring to the National Mathematics Advisory Panel report that argues that factual, procedural, and conceptual knowledge are required to learn mathematics, Dr. Willingham develops the linkages between the three knowledge types. “Factual knowledge refers to having ready in memory the answers to a relatively small set of problems of addition, subtraction, multiplication and division.” Procedural knowledge is the knowledge of “a sequence of steps by which a frequently encountered problem may be solved.” “Conceptual knowledge refers to an understanding of meaning; knowing that multiplying two negative numbers yields a positive result [the procedure] is not the same thing as understanding why it is true” [the concept] (brackets added). Dr. Willingham goes on to discuss the math wars, where the conceptual versus procedural question is at the battle front. He reasons that both types of knowledge should be taught together.

Which brings us to Everyday Math. When I read the article, most of the shortcomings of EDM came sharply into focus. We just have to look at the draft Math Monitoring Report to learn that our Middle School Math teachers are concerned about their sixth graders “Basic fact fluency,” also know as factual knowledge. Ask any parent who works with their child on their math whether their students know their multiplication tables cold, or their division facts well. I would wager the answer would be a resounding “No!” And these are kids where the parent works with them. Everyday Math doesn’t even pretend to take students to the required level of automaticity for these facts, despite the recognition that “Automatically knowing basic number facts is as important to learning mathematics as knowing words by sight is to reading” (EDM Teacher Manual). Strike one.

Two other concerns the Middle School teachers have are “Operations with fractions” and “Long division competency,” which point to issues with procedural knowledge. The standard algorithms are given limited attention in Everyday Math, but a variety of other algorithms (e.g., lattice multiplication, partial-sums addition) which have limited application in more advanced math are given time. And the practice to ingrain any of these procedures, much less the standard procedures, is limited. Strike two.

So at least Everyday Math is teaching the concepts, the whys (ever wonder why that word is not spelled “whies?”), right? Dr. Willingham discusses the conceptual problem with understanding that an equal sign refers to equality (mathematical equivalence). “Students often think it signifies ‘put the answer here.’” Could it be because they are more commonly referred to as “number sentences” so as to include inequalities? Ask your child what an equal sign means. If they are above sixth grade, let’s hope they understand. If so, according to one study, they would be better off than 72% of their classmates, meaning only 28% understood. If they were in China, 98% of their fellow students would understand.

http://people.cehd.tamu.edu/~rcapraro/Publications/Capraro_CI_Equal_Sign.pdf

Everyday Math or teacher preparation? I’ll give half a strike to EDM.

All-in-all, a reasonable explanation why Everyday Math is failing our children.

Take the Survey

2012-01-28T12:56:00.000-05:00

If you recognize the shortcomings of Everyday Math, you are probably doing something with your child or children to supplement their math education. Take the survey at right to record how you help. You can choose more than one answer.

Report from Pelham

2012-01-27T00:37:00.001-05:00

A group of parents in Pelham are working to get the school board to replace the current elementary school math curriculum, TERC Investigations. A link to their website is located at right.

They held an open forum with a panel of two real math and one real computer science professors (not education professors). These professors see the result of the elementary school, middle school and high school mathematics education.

This is the recap of the event published in the Pelham local paper, and the Pelham Patch.

http://www.thedailypelham.com/schools/parents-professors-protest-pelham-math-program

http://pelham.patch.com/articles/pelham-k-5-math-curriculum-criticized-during-math-forum#photo-8972158

Dr. Akin equated math to music and sports, and brain memory to muscle memory. Not everyone is going to be a professional musician or basketball player, but with some practice everyone can play the song or play the game. Same with math.

Math is a language, used to solve problems. One must be able to translate the English (the word problem) to the other language (math symbols), then the training/practice takes over and the math problem can be solved easily IF YOU HAVE PRACTICED. Everyday Math and Investigations lack the practice needed to build the memory, so the ability to solve the problem is diminished. For a more in-depth discussion, see Dr. Akin’s paper entitled “In Defense of Mindless Rote” at:

http://math.sci.ccny.cuny.edu/docs?name=MindlessRote.pdf

This is an excellent view of the need for basic instruction and a solid foundation if a student is to be successful in math.

Dr. Ocken discussed the progression from standards to assessment to curriculum, and the components required to build an effective curriculum. He discussed his work on a panel to advise NYC on their curriculum revision (prior to Common Core State Standards).

http://math.sci.ccny.cuny.edu/document/show/2010

I quote at length from this document, given that Greenwich and NYC both use Everyday Math (my comments are in [brackets]):

“In terms of establishing procedural fluency as recommended by NMAP [National Mathematics Advisory Panel], the content standards [NYS standards] are particularly evasive and lay the groundwork for student failures in understanding how to manipulate decimals and fractions. The operative phrase in the standards from grades 3 through 5 concerning the teaching of basic arithmetic is "use a variety of strategies to…" [Similar to the CT standards], in contrast to "learn how to…" Students are encouraged to create their own methods for multiplication and division [same in Everyday Math], rather than learn the standard methods upon which their understanding of more advanced mathematics will ultimately depend [this is the fatal flaw in learning algorithms like lattice multiplication and partial quotients]. In contrast, the California standards, anticipating the NMAP report, consistently and wisely omit reference to any algorithms other than the standard ones.

The “use a variety of strategies” perspective of the New York standards is implemented in many curricula [Everyday Math again], notably those that present multiple alternate versions of the standard algorithms. These curricula encourage the use of inefficient methods to solve straightforward problems and should not be adopted. If they are being used, as is the case in some New York City schools [Everyday Math], they should be replaced.”

Dr. Ocken also warned (see the Pelham Patch article) of the difficulty of using supplemental material at home to repair the damage of Investigations or Everyday Math. Simple drilling will not make up for the structural deficiencies of Investigations or Everyday Math. You children are being taught two different programs, one at school and one at home, usually in different sequences. Confusion, not clarity, may result. He also warned of the failures of the Connected Math Program (used in our middle schools).

Dr. Siegel brought the prospective of a more advanced view of the result of our math education, being further down the line as a Computer Science Professor, where the course work assumes competency with calculus. The higher level thinking required in this area (as opposed to what Investigations or Everyday Math may claim to teach) relies on a firm foundation in the basics.

Food for thought for everyone.

Math Monitoring Report Update

2012-01-24T18:13:00.000-05:00

The administration pulled the Math Monitoring Report from the agenda prior to the Board of Education meeting last Thursday evening. Applause to Dr. Lulow and his folks for taking the time to get the facts and conclusions correct. I am hoping that once the data issues are cleared up, the conclusions will become evident.

The inconsistencies seen in the tables showing elementary school results were the result of two different sets of data being used. One included English Language Learners; the other excluded them. The problem was there were no explanatory footnotes. There were also several mistakes in presenting the numbers. Very confusing to anyone trying to use the report as a tool to determine what needs to be corrected to achieve the best result.

Expect a revised version for the next BoE meeting.

Off to Pelham

2012-01-24T18:05:00.001-05:00

I am off to Pelham this evening to listen to three college professors (CUNY and NYU) speak about the curriculum used there: TERC Investigations. If it is possible to believe, this program is usually regarded as worse than Everyday Math. A group of parents have banded together to petition the school board to change the program. Sound familiar?

Ready or Not, Here They Come

2012-01-22T17:53:00.000-05:00

You may have seen the recent articles about graduation rates from college for Greenwich High School students, and the follow up today with some of the class’ students. For the class of 2004, 53% graduated from college (two year or four year) after six years, although the administration believes the figures are incorrect and are closer to 67% (which they also admit is not great).

http://www.ctregents.org/files/pdfs/p20/057%20Greenwich.pdf

Today’s article asks the question, "Are GHS students are actually prepared for college?" It really didn’t answer it. So I will give it a try.

Part of the same report (see page 12) contains information about the number of Greenwich High School graduates (class of 2010) who attended one of the Connecticut State Universities (Central, Eastern, Southern, or Western), or who attended one of the Connecticut Community College, and had to take a remedial or developmental math or English course.

The report says there were 672 graduates. Of these, 15 went to Connecticut State Universities (does not include UCONN, because UCONN technically does not have remedial courses), and 56 went to Connecticut Community Colleges.

So how did they do? Of the 15 university attendees, nine had to take developmental or remedial math. Of the 56 community college attendees, 29 had to take developmental or remedial math, and six were unaccounted for (did not take or complete the diagnostic test, or did take the test, but the results were not captured). So nine out of 15, and 29 out of 50. Looks like just short of 60% to me (I didn’t even need to use a calculator). For comparison, 60.9% of the all Community College goers (less unaccounted test takers) and 55.3% of all State University goers had to take remedial or developmental math. So Greenwich High School is about average, based on this ten percent of the class. Ouch!

So can we figure how the rest of the class did?

- Roughly 10% of the graduates didn’t enroll in college right after graduation, and I have to believe a large percentage of them never will, and probably would struggle if they did. So, figure 90% of the 10% who opted not to go to college probably could not do college level math without remediation (my guess).

- 60% of the group discussed above (another 10%) needed remediation, but I would venture that this group was closer to the bottom of the class.

- Based on my reading, about 30-40% of college enrollees need some sort of remedial math, across the range of quality for colleges and universities. If we figure that half the class went to the “average” college, let’s assume 30% of them were not prepared.

- For GHS class of 2011, about 30% went to a “Most Competitive College,” so this is probably true of the class of 2010. Let’s say that only 10% of this group needed remedial math.

https://www.greenwichschools.org/uploaded/high_school/guidance/2011_College_Date.pdf

Doing the math yields about 33% of the graduates who would need some remediation before doing college level math, a bit lower than my previous guess of 40% (but this calculation is conservative).

When the class of 2010 took the CAPT in 2008, they achieved the following results: 34.4% at Advanced level, 68.0% at Goal level, and 89.9% at Proficient level. Likewise, when they took the CMT Math test as eighth graders, they achieved: 46.7% at Advanced level, 81.4% at Goal level, and 92.3% at Proficient level. Of course the two tests are not directly comparable, but the trend is interesting, no?

To really extend the math, compare the CAPT Goal percentage (68.0%) to my estimate of how many would or did need remedial math (33%). HHHMMMMM.

DISCLAIMER: I made a lot of estimates and guesses to arrive at the figure above, so my 33% estimate should be viewed more as a range (25-40%) rather than an absolute.

But the range doesn’t include zero, and it certainly doesn’t include 10.1% (non-Proficient on the CAPT).

Conclusion: Proficient really isn’t proficient, if Proficient means you are ready for college math.

Could we really be failing one-third of our graduates? But wait, you say. You included the 10% of the class who didn’t want to go to college. That skews the numbers. Yes I did. Why? Because they need math for their job, at the same level as a starting college freshman.

Goal level may be more appropriate as a measure of potential college readiness. Who might agree with that? Arne Duncan, US secretary of education, who said, in reference to No Child Left Behind, “It created an artificial goal of proficiency that encouraged states to set low standards to make it easier for students to meet the goal.” But the federal Department of Education accepted the CT Proficient standard for adequate NCLB achievement. Go figure.

Note: CT State Universities and Community Colleges use Accuplacer as the diagnostic exam. You can take a practice exam on-line. The math questions are at the end (I got them all right). Many universities use ACT or SAT Math scores to determine if a new student needs to take remedial math. From a quick survey of colleges, many use cutoffs between 500 and 550 (one required a 600 SAT to get into engineering without taking an extra course, and another required a 610 SAT to take calculus instead of college algebra). The average (not median) SAT Math score for the class of 2010 was 577, so that would put a good chunk of the class at or below 550. Not conclusive, but lends more credence to the 33% figure above.

Confused in Greenwich

2012-01-15T22:25:00.000-05:00

Last week Chris Fountain, on his blog For What It’s Worth (see link on right), commented about plans to use some of the Greenwich Public School budget for summer school. I responded:

How about using the 90k to start the overdue curriculum review for Math. Then we can get rid of Everyday Math, and get something that allows our children to learn.

“Grade 6 teachers recently listed the following areas of concern regarding grade 6 students’ preparation for middle school math:
- Difficulty reading problems
- Basic facts fluency
- Operations with fractions
- Long division competency”

I’m not making this up. This is a direct quote from the GPS administration’s own report. I am hoping that “difficulty reading” actually means difficulty understanding and solving math problems. The last three are only some of the important areas where the Everyday Math program is failing our children. In case you don’t speak educationese, “basic fact fluency” means being able to add, subtract, multiply and divide without having to think or use your fingers. Never thought this would be a problem in sixth grade! Thank Everyday Math.

Better curriculum will benefit all Greenwich students.

Someone wrote back the next day: ________________________________________________________________

Everyday Math is a huge success story in my opinion. My kids consistently score very high in math thanks to that program which keeps pounding the basics over and over. That is how you learn math – doing it over and over, again and again, so it is second nature.
Spelling – that is another matter, entirely.

_________________________________________________________________

I responded:

_________________________________________________________________

You are right. The best way to learn math basic facts is to drill. BUT, Everyday Math does not keep pounding the basic facts. If your kids are lucky enough to have a teacher who recognizes the shortcomings of EDM, the teacher is supplementing the program so that the kids will learn. Take a look through their Home Link or Study Link books. Do you see any real drill sheets in there? Check the copy write on the drill sheets they do get. I can guarantee they are not EDM.

I am happy your kids scores are high. The same can’t be said for the Greenwich students 3-5 where scores are dropping on the CMT at the levels that matter (Goal and Advanced). The good teachers are supplementing. But it is not consistent across the district, and soon our real estate values are going to be impacted.

I am fearful there are parents who do not understand the danger presented by Everyday Math. If you know such a parent, please have them read this blog.

Oh, and I agree spelling (and grammar) is an issue too.

Making the MMR Better - A Review

2012-01-15T15:35:00.001-05:00

An old boss said to me once, “Bring me solutions, not just issues.” The administration needs to be advocates for the students, so that issues are understood and the BoE can determine resources (within the budget parameters) to resolve the issues. The poorly written Management Issues section of the Math Monitoring Report is short changing the students of our schools.

In my last post, I offered a few suggestions about how to make the Math Monitoring Report (“MMR”) better. Why is this a concern? The MMR sets the tone for the use of resources within the district. It also serves (or should serve) as the main source to gain an understanding of the issues and achievements of our students. If the report is not accurate, or if it downplays failures or overplays successes, we may be ignoring issues or focusing resources in the wrong area.

How Bad Is It?

How well do the recent and previous MMR’s accomplish the task of issue management? Let’s look at the 2009-10 report first (which now appears on the BoE website in the correct place), then follow through to the draft 2010-11 report.

The 2009-10 MMR lists six Management Issues:

MI1 - Mathematics Interventions

MI2 - Mathematics Interventions – New Lebanon School

MI3 - Adequate Yearly Progress (AYP) – Secondary

MI4 - Middle School Instructional Time

MI5 - Increasing the Percentage of Middle School Students Reaching Goal or Higher on the CMT

MI6 - Increasing the Percentage of Students Reaching Goal or Higher on the CAPT

The first issue with the above issues, is that some of them are not issues. The last two really are targets. The discussion talks about how the Middle School and High School Math Departments will reach the targets. Neither identifies an issue which needs to be addressed with additional resources, or through some other means.

What’s an issue?

An issue is a failure to meet a target, or the likelihood that a failure will occur without additional action. Are targets good to have? Absolutely, but let’s see all the targets in one place and all the issues in a separate place.

For Management Issues 1-4, I count about nine actual issues. I say about since one section is written so poorly it is difficult to determine the true meaning. The issues are:

1. MI1 - Budget constraint (reduced coaching) may make it so 2009-10 growth will not be duplicated

2. MI1 - A gap of some sort, probably to the DRG B schools (this is the confusing one)

3. MI1 - A gap to DRG A schools

4. MI2 - Lack of sustained growth in Math CMT scores at New Lebanon

5. MI3 - Not meeting AYP in two subgroups at the High School

6. MI3 - Not meeting AYP in two subgroups at Western Middle School

7. MI3 - Struggle to maintain heterogeneously grouped on level classes with the remedial needs of certain subgroups (related to the two preceding items)

8. MI4 - Insufficient instructional time for Connected Math at the middle schools

9. MI4 - Difference in instructional time available for Algebra I and Geometry between the middle schools and the High School.

Looking at the 2010-2011 report, you would expect to see some reference to these issues, and whether they had been resolved or were on-going.

1. Budget constraint – no specific mention of the impact of the constraint, although there is a discussion of Hamilton Avenue’s 2009-10 success and of North Street’s 2010-11 success due, in part, to coaching. So did the constraint have an impact? Can’t tell, but this is the perfect place to discuss the continued drop in Dundee’s percentages (Advanced and Goal hit five year lows), and the drop in Hamilton Avenue’s 2010-11 percentages.

2. A gap of some sort, probably to the DRG B schools – there is some discussion, with similar wording as in 2009-10, about the gap to DRG B schools, and how to close it, with an update on the gap size after the four year trend.

3. A gap to DRG A schools – a virtual repeat of the wording from 2009-10, with an update on the gap size after the four year trend.

4. Lack of sustained growth in Math CMT scores at New Lebanon – no mention in the Management Issues, even though the percentages were up, but it was noted in the Executive Summary.

5. Not meeting AYP in two subgroups at the High School – update provided indicates that the two subgroups still did not make AYP, but the update failed to highlight the fact that the while the Proficient target level went up, the actual results for the subgroups declined.

6. Not meeting AYP in two subgroups at Western Middle School – update provided that Western had earned Safe Harbor status, but no indication whether the issue is now resolved.

7. Struggle to maintain heterogeneously grouped on-level classes while trying to meet the remedial needs of certain subgroups (related to point 6) – repeated the issue, with the same verbiage.

8. Insufficient instructional time for Connected Math at the middle schools – repeated the issue, with the same verbiage.

9. Difference in instructional time available for Algebra I and Geometry between the middle schools and the High School – repeated the issue, with the same verbiage.

The last two issues were specifically addressed during the BoE Work Session, and the administration is reviewing alternatives to resolve the issues. From conversation around the table, it appears the issue has been raised for about six years.

Of the nine issues, two (1,4) were not followed up, five (2,3,7,8,9) had updates which repeated the issue with minor changes, and two (5,6) had updates which were less than informative. No “issue closed.” No “on-going issue.” No meaningful comparison of progress year-to-year, where appropriate.

For 2010-11, the following additional issues were identified:

1. Central Middle School not making AYP

2. Middle schools not likely to hit two targets for 2012-13 (are there any targets for 2011-12?)

3. Grade six teachers’ concerns regarding grade six students’ preparation for middle school math.

4. The existing K-8 math material will need to be supplemented in order to meet the change to the Common Core State Standards.

5. The full impact that the new computer-based assessment system will have on our computer infrastructure is unknown.

Again, the High School “issue” was a restatement of a target. Credit to those who put together the new middle school points; they seem to get the meaning of “issue.”

How to Make The MMR’s Better

1. Fix the data

2. Analyze the corrected data to arrive at meaningful observations

3. Determine if the observations indicate an issue

4. State the issue clearly

5. State the remedial action to be taken, the required resources, and the expected results and time frame

6. Follow up on the issue – a yearly update should be a minimum until the issue is resolved or the remedial actions are judge to be ineffective

Next up, my summary of the Math Monitoring Report.

Math Monitoring Report Comments - Part 4

2012-01-13T14:21:00.000-05:00

This is the final part of four parts of my comments on the Math Monitoring Report. This section covers the management and governance issues, which means the previously identified or newly identified issues faced by our schools relating to Math.

I also include some comments about how to make the Monitoring Report better, i.e., more readable and more actionable. My next post will go into that in more detail.

_______________________________________________________

GOVERNANCE ISSUE

I have made myself clear on the need to begin a review of the curriculum now in light of the adoption of CCSS. So as not to appear to be taking things out of context, however, I will quote the italicized recommendation: “The recommendation to the Board of Education will be to adopt Connecticut’s Common Core Math Standards at the next scheduled mathematics curriculum review. We understand that the Common Core Standards represent the state curriculum document and that Greenwich may add grade level objectives to better frame the district’s definition of mathematically proficient.”

Given the amount of change occurring, waiting is not an option. I certainly hope that Greenwich will actually add, as opposed to “may add,” grade level objectives.

MANAGEMENT ISSUES

Mathematics Intervention

I applaud the focus on early intervention (K-3) as a means of closing the gap to comparable districts. However, while the report covers topics like coaching, instruction models, testing, and diagnostics, there appears to be a lack of attention to curriculum as a potential gap closer.

Student Outcome Indicators – Middle School

“Grade 6 teachers recently listed the following areas of concern regarding grade 6 students’ preparation for middle school math:

-       Difficulty reading problems
-       Basic facts fluency
-       Operations with fraction
-       Long division competency”

This seems to sum up the results of five years of Everyday Math very well.

REPORT STRUCTURAL SUGGESTIONS

1. There needs to be a much clearer link between the analysis (Data Review section), new issues raised by the analysis (Management Issues section, where I find the statement of the issues to be poorly done), and the actions to be taken (which, for the most part, are missing or nebulous).

2. The Management Issues section also needs to highlight and discuss last year's issues, and indicate closure or continuation:

Example: ISSUE (date raised) - ACTIONS TAKEN/TO BE TAKEN - RESULTS - STATUS (close or continue).

Note that last year's Monitoring Report is not posted on the BoE website, so I could not compare this to last year. NOTE: this was my comment when I wrote the review. I subsequently found the 2009-10 report attached to the agenda for the meeting in which it was reviewed. See my next post to discuss old issues and new issues.

3. Organizing the report by Elementary, Middle and High School segments (i.e., separate 3-5 and 6-8 as there is little overlap) would make the report flow better, and make it more user friendly.

Math Monitoring Report Comments - Part 3

2012-01-11T22:05:00.000-05:00

This is the third of four parts of my comments on the Math Monitoring Report. More of the same in this part of the Report: poor analysis, bad conclusions. So I came up with my own analysis, which indicates that the gap is shrinking between the Special Education, Hispanic, Black, English Language Learners ("ELL"), and Free/Reduced Lunch ("FRL") subgroups and the overall district percentage for students reaching the Goal level. This is good news. The bad news is it will take forever (forever means after the current second graders graduate) to actually close the gaps. We don't need incremental change. We need radical change. Ditch Everyday Math now.

One other interesting (?) point: Table Seven in the Report is titled "Connecticut Academic Performance Test - Five Year Trends (%)." The Table is on page 24. The interesting point is that it only covers four school years!

______________________________________________________________________________

NARROWING GAPS (3-8): AN ANALYSIS OF FIVE YEARS TRENDS

Subgroups

1. Point six should be the first point, as it establishes the issue to be examined. The issue is not whether performance gaps exist among the subgroups. That is obvious. The issue is whether past actions are actually closing the gaps. From the data presented, it is impossible to answer this question because there is no definition of what a gap is, no discussion of the size of a gap, and no analysis showing the rate of closure or widening of a gap.

2. I offer what could be used as a measurement tool. Since the state ascribes some meaning to the Goal level (and I have already discarded Proficient as a meaningful target level), I will use Goal as the basis for analysis. Does it represent the level which will guarantee success in college or on the job? That is a larger debate than I care to undertake here. With that side-stepped, what should that goal (small “g”) be? Nothing less than 100% of all subgroups at Goal level. Achievable? Probably not, but it is a useful construct against which to measure progress.

3. Based on the data, we need to determine if the trend in Goal percentages is up, down, or flat (is the slope of the trend line positive or negative) and how fast is progress being made for one group versus another (what are the relative slopes of the trend lines)? We also need to determine if the fit of the line is at least reasonable (correlation coefficient).

4. Applying this to the data in Table 2 yields:

Goal Percentages
	2006-07	2007-08	2008-09	2009-10	2010-11	SLOPE	Intercept	Corr. Coeff.
YEAR	0	1	2	3	4
District	79.7	78.3	81.1	81.6	81.8	0.75	79.0	0.80
Asian	91.8	87.9	92.1	91.9	94.0	0.84	89.9	0.60
Black	42.4	32.3	43.0	45.1	44.8	1.76	38.0	0.53
Hispanic	56.8	54.2	60.0	59.0	62.3	1.58	55.3	0.81
White	83.4	82.8	84.7	85.8	85.9	0.80	82.9	0.91
Female	79.8	77.8	80.0	80.9	81.6	0.67	78.7	0.74
Male	79.6	78.7	82.1	82.2	82.0	0.83	79.3	0.80
SPED	39.1	38.9	44.4	44.5	42.8	1.30	39.3	0.74
ELL	54.7	53.5	65.9	59.5	60.9	1.84	55.2	0.58
FRD	47.5	40.1	48.4	50.3	49.3	1.38	44.4	0.54

5. First the good news. The correlation coefficients are reasonable, so we may get some useful information out of this. The slopes (which can be viewed as the annual gain in percentage points) are all positive, so all of the subgroups are trending up, which is great. The slopes of the Black, Hispanic, SPED, ELL and FRD subgroups are all much higher than the district, White or Asian slopes, indicating that (at least by this measure) the relative gaps are closing.

6. Now the bad news. If the goal is to get 100% of each subgroup to Goal, it is going to take a very long time based on the progress being made, or more pointedly, based on what we have been doing. ELL will get there in 21 years, and FRD in about 37 years.

7. While the methodology may be open to criticism, the real take away is that if we truly want to make a difference, we need to make a radical change, not the incremental change we have had.

8. For point 7, the one point which quotes data, the conclusions (“there is no gap in performance of males and females on CMT mathematics at all levels” and “what is important…is that the overall gains are evenly matched over the last five years”) are debatable. To conclude that a 4.5% spread is not a gap is to ignore the actual number of students this percentage indicates. At roughly 625 students per grade across six grades, and assuming a 50/50 female/male split, means that about 84 more females than males are below Advanced. That is almost four full classes. And contrary to the second conclusion, the gap is widening by five females a year (because the actual percentages are 3.4% growth for males and 2.2% growth for females).

Benchmark Districts

1. As was shown above, the elementary grades are not closing any gaps between Greenwich and DRG A and B districts. Even a cursory look at Chart 14 (page 20) shows that the only gap being closed is at the Goal level with DRG A, and a simple calculation shows that the gap will be closed three years after DRG A reaches 100%, that is, DRG A is so far ahead (almost 10%) that it is difficult for them to gain the same amount each year as Greenwich. And it should be noted that Greenwich gave up the 1.5% lead it had over DRG B at the advanced level.