Ms. Lemily's Lesson: Looking Closer at Math Excellence (With Elves!)

Last week, PrichBlog shared some of the “Finding Solutions” account of Christa Lemily’s eighth grade students working on figuring out whether classic Corvettes make good investments. The question can be answered with several different mathematical strategies, so it's a great example of the fluent, flexible problem-solving called for by Kentucky's academic standards.

To explain that, let’s start by spending a minute with the Brothers Grimm and their tale of “The Shoemaker’s Elves.” The first night, the elves turned one piece of leather into a pair of shoes so fine that, after selling them, the shoemaker could afford to leather for two pairs. The second night’s two beautiful pairs brought in enough money to buy leather for four. The third night? You know the answer: enough for eight pairs.

So, how did you figure out that it was eight? Here are five respectable options:

1. Maybe you used addition.
2. Maybe you multiplied.
3. Maybe you’ve worked or played with numbers enough that you just know what happens as you double a small number. 
4. You probably didn’t use percentages, but you could. If the story was trickier, with two pairs yielding revenue to buy leather for three, multiplying by 150% each morning might be a good choice.
5. You probably didn’t use an exponential function either, but if the growth was 20% each night, and you wanted to know the result after 10 nights, you might end up using a formula like 
   

Fortunately, we don't need to unpack that formula to see the point about mathematical options.

Instead, let's swing back to Ms. Lemily's class at South Warren Middle and their question about Corvettes. If the shoemaker story is about growing (appreciating) value, the car story turns out to be about the opposite: depreciation or value going down. On average, Corvette convertibles lose 15% of their value the first year, and between 8% and 10% each year after that. With a $60,000 car, that’s going to produce values like this over time:

Under Kentucky's standards, Ms. Lemily’s students should be ready to work out those numbers by multiplying percentages, figure out that Corvettes are better as transportation than as investments, and notice that the graphed results do not look like a line. Kentucky's math standards for grade 8 call for students to be able to:

•  "Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.” [Emphasis added]

From the “Finding Solutions” description, Ms. Lemily's students may have reached that standard and also be closing in on some high school expectations, like being able to:

• "Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another," and
• "Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table)."

Do notice that the standards are not asking students to remember every formula they see in a math class. The goal is for them to recognize the kind of change involved and construct a function that works for that situation. That's one reason the Prichard Committee report came with a subtitle about how “Standards Push Students Toward Real-Life Problems.”

Overall, South Warren Middle School's strong focus on the math standards seems to be paying off. With Ms. Lemily in the lead, South Warren was one of the first middle schools in the country to join the work of the Mathematics Design Collaborative, and as of last year, their K-PREP scores impressively outpaced the state average for most groups:

One more thought. An even deeper goal in our standards is for students to master these key mathematical practices:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

The work happening in Ms. Lemily's classroom embodies those practices, and equips her students to use all eight in high school, in college, and across their careers. It's an impressive step up in how Kentuckians teach and learn math!

Will AMOs Ever Be Bigger Than One Point?

Question
Will Kentucky schools ever have AMOs (annual measurable objectives) that ask them to improve by more than a single point per year?

Short Answer
Under Kentucky's current accountability system, AMOs greater than 1.0 are effectively impossible.

More Detailed Answer
It is mathematically possible for Kentucky AMOs to be bigger than 1.0, but scores would have to change so dramatically that it is simply not a meaningful possibility.

For example, if the top half of elementary schools all raised their scores by 10 points and the bottom half all dropped their scores by 5 points, that would allow AMOs of 1.1 points.

A change like that, with elementary scores changing radically in both directions so that the strong and weak are pulled much further apart, is hugely improbable. Effectively, it's impossible.

Middle and high schools would need even more improbable-to-impossible changes, with even bigger growth in the difference between the upper and lower results.

Technical Answer 
(To Avoid Statistical Details, Skip the Fine Print Below)
In Kentucky, our AMOs are based on standard deviations. A standard deviation is a way of saying how results spread out around the mean.  If results all clump together close to the mean, the standard deviation is small. If they spread out a lot, the standard deviation is large.

Under changes voted on by the Kentucky Board of Education earlier this month, future AMOs will ask most schools to improve their next-generation learner scores by one-third of a standard deviation in five years, meaning one-fifteenth of a standard deviation a year.

That formula means a standard deviation of 15.8 or higher will be needed to allow a 1.1 AMO.

A web-based standard deviation calculator allowed some experiments about what scores would yield that 15.8.

Entering the 2014 next-generation learner results for our 720 elementary schools yielded a 9.2 standard deviation.  One-fifteenth of that would be about 0.6. With a result like that, the new minimum AMO of 1.0 will apply.

A next step was adding 5 points to the top 365 scores, subtracting 5 points from the bottom 355, and entering those results in the calculator.  That change was not big enough to yield a 15.8 standard deviation. 

Moving each group 6 points did not work either.

10 up and 5 down worked.

10 up and 4 down did not, showing 10 and 5 to be one of the smallest elementary changes yielding 15.8 as the standard deviation and 1.1 as an AMO.

For middle and high schools, the 2014 standard deviations were smaller than at the elementary level, so it would take even bigger changes to get to a 15.8 standard deviation and a 1.1 AMO for most schools.

Why did the experiments try scenarios with some schools moving up and others down?  Because, again, standard deviations get larger when results get further apart. Bigger AMOs can only happen when the distance between the top and bottom results gets larger.  

Bottom Line
Under Kentucky's current accountability system, AMOs greater than 1.0 are effectively impossible.  The scale of change that would be required just isn't going to happen.

--Posted by Susan Perkins Weston

Against revolutionary folly

Here's a bad argument: "No serious change can happen without full-blown revolution.  Until we change our whole economic structure, no important reform will be possible."

If I heard that once in my college years, I heard it a hundred times.  Maybe half the people who tried it out meant to align themselves with Marxist analysis, and the rest were offering some other form of radical thinking.

If I heard it a hundred times, I never believed it once.  I knew too much about the forward march of individual liberty, abolition, voting rights and civil rights.  I knew too much about how Jim Crow fell, and too much about the strategies of decency and solidarity that were even then eroding both white rule in South Africa and communist rule across eastern Europe.  I knew much too well that smart and steady work by committed individuals can indeed generate worthwhile change.

I haven't heard that revolution-or-bust rhetoric much in the last two decades, but I have heard something close.

The argument that comes close says this: "Schools can't change student performance unless we eliminate poverty and racism first."

The Washington Post published a piece today that tries that tired approach.  This time, Robert Samuelson points out that American white students do about as well on the PISA test as largely white countries like Canada, Australia, and New Zealand.  Then he adds that our national weakness can be traced to the weaker performance of other American children, especially those who are black and Hispanic. His argument is that:

These persistent achievement gaps demonstrate the limits of schools to compensate for problems outside the classroom - broken homes, street violence, indifference to education - that discourage learning and inhibit teaching....What we face is not an engineering problem; it's overcoming the legacy of history and culture. 

Yet again, the claim is that educational progress is essentially impossible without radical change to other institutions.

The problem with that position is that it ignores the data.  It cannot account for all the examples of schools that deliver student progress right in the teeth of the poverty odds.   It cannot escape the systematic evidence that key changes in instructional practices can raise achievement and narrow those haunting gaps.  And it cannot explain how other countries, many with ugly history of their own, are successfully changing their futures by changing their education systems.

This kind of argument looks erudite, but it's ignorant.  It sounds sophisticated, but it's crude and false.  It mimics past version of revolutionary talk remarkable well, and its case for radical despair is just as mistaken this time as it was in decades past.

The way to defeat this sort of willful despair is by doing the work they say is impossible, educating the children they say are doomed, and building the society they say is out of reach.

The boy (almost) next door goes for glory!

Ralph Morrison, of Danville High School's  Class of 2006, was also Williams College's Phi Beta Kappa Speaker at their 2010 graduation. 

Charters without champions

The Department's new charter school proposal confirms an insight from my favorite fictional President. Jed Barlet told an imaginary student audience that "Decisions are made by those who show up." 

The KDE revisions reflect careful consideration of feedback from those who have already shown up to lead.  Unsurprisingly, that feedback and the resulting legislative draft offer charter schools less flexibility than the earlier House Bill 109 amendments.

That's because, in this state, strong political voices in favor of charters are remarkably hard to find.

Kentucky does have a few seedling groups (mentioned here and here,  for example) that may someday become forceful charter advocates--but they haven't done it yet.   

If they've registered legislative agents, the Ethics Commission hasn't posted that information on-line.  

If they're creating membership rolls and accepting donations, they aren't doing it at websites I can find. 

If they're mobilizing to support the Senate amendments, the KDE draft, or any other charter legislation that could be considered this year, they're doing it very, very quietly.

Meanwhile, the Department is negotiating--as it should--with those who are available to negotiate.

The Speaker and Jar Jar Binks

The Courier-Journal reports that the House is considering an approach to school funding that would compel districts to draw down their fund balances:

Cutting funding that local school districts receive from the state and forcing them to spend money in their reserve funds is still an option to help balance the state budget for the next two years, House Speaker Greg Stumbo said Tuesday.

School districts have a total of about $800million in their reserve funds, although Stumbo said not every district's fund would be tapped.

“There's a significant amount of money there that could be (used) to balance the budget,” the Prestonsburg Democrat said. “I'm not saying we are going to do it, but it could be used to supplement the (school funding formula) in this one-time occurrence.”

House Democrats are scrambling to find about $780million to balance the next two-year budget.

The $800 million figure is not, in fact, accurate for district reserve funds. It's roughly right for the amount districts had on hand on July 1, but that amount simply is not all reserves. Remember that districts rely heavily on property taxes that come in starting in November. As a result, they save all spring to be able to pay their bills through the summer and fall. It's part of their regular money management to have the money on hand on July 1, expecting it to be gone again by January 1.

That said, some of the money is indeed reserve money districts have saved for dangerous financial times. Surely those times are near at hand.

Or, as a famous Gungan once asked Qui-Gon Jinn and the young Obi-Wan Kenobi as they hurtled through the waters of Naboo: "What yet? Monster out dair! Leakin' in here! All sinkin' and no power! Whena yousa dinkin' wesa in trouble?"

I hate the idea of tapping districts' reserves, but I'm not sure I hate it more than than I hate the rest of this exhausting, massive fiscal crisis.

ACT, CPE, and bit of Frank Baum

This story takes a little telling.

In November 2004, the Council for Postsecondary Education required all public universities and KCTCS schools to place students with an ACT English score of 18 in a credit-bearing writing course, with an ACT mathematics score of 19 in guaranteeing a credit-bearing math assignment.

In November 2007, CPE amended its admissions regulation to specify that:

• Students would need an ACT reading score of 21 as well as the 18 in English and the 19 in mathematics.
• Students scoring below those levels would have to take either a non-credit developmental course or a credit-bearing course with "supplementary academic support, such as extra class sessions, additional labs, tutoring, and increased monitoring of students, beyond that usually associated with an entry-level course."
• Schools would have to provide those courses in a student's first semester.
• The new rules would go into effect in the fall of 2009.

Only, in January 2009, CPE voted to amend that regulation again to say that:

• Students would only need a reading score of 20.
• Students scoring below the standards could spread their special course work over two semesters.
• The requirement for that coursework would go into effect in the fall of 2010.

And in May 2009 (tomorrow, in fact), President Bob King will present his regular report to the Council, including an update on Senate Bill 1 work that says the next step will be:

to review the current systemwide public postsecondary placement policy in English and mathematics and, working with institutional representatives and KDE, to determine whether revisions are needed in those content standards.

That raises the possibility that by the time the new placement policy goes into effect, every number, date, and timeline will have been replaced at least once, like the memorable tin woodsman Dorothy met on her way to visit the Wizard of Oz.

Is CPE heartless, to be changing the rules so often? Probably not.

Instead, the November 2007 decision may have been made without enough active discussion, meaning that too few brains contributed to figuring out what would be needed to make the changes work.

If so, I'm glad that CPE has the courage to give each element a new round of careful attention.

Rumplestiltskin to the rescue?

Jefferson County has been wrestling mighty budget challenges, with the school board approving cuts of $26 million for 2009-10 last night. The CJ article is here, with this paragraph deserving special note:

Cordelia Hardin, the district's chief financial officer, told the school board that the district still faces a $44 million shortfall in its general fund and will use about $15 million in reserve funds to make up for the rest of the shortfall.

Spending $15 million to cover a $44 million shortfall is mighty close to spinning straw into gold. Here's hoping the governor and the legislature can do as well with the $1 billion state level challenge.

Reading and culture

Although I haven't seen anything formally published on the links between email messages and tribbles, I consider it only a matter of time.

A perfect example of how cultural experience at 9 can enhance reading comprehension at 49, spotted in DeanDad's post on "the inbox police" here.

Death of an imaginary playmate

In case anyone doesn't know Martin Cothran, he teaches at Highlands Latin School, writes logic and rhetoric texts, manages an online academy, and analyzes policy for the Family Foundation of Kentucky. For most of two decades, he’s spun wondrous tales of an imaginary KERA-dragon.

Initially, Martin’s dragon tempted small children with dangerous elixirs called “valued outcomes.” Next, it ate several hundred educators when it caught them teaching spelling, and later still, it set fire to an entire village because a couple of local citizens dared suggest using multiple-choice questions in statewide testing.

The real reform story has been dull by comparison. On valued outcomes, legislators agreed in 1994 not to test for self-sufficiency, teamwork, and other marks of good character. On spelling, my oldest started waking me up for pre-dawn drills in 1995, and I know exactly which word my youngest missed at his schoolwide bee in 2007. On multiple-choice, we started accountability testing in 1992 and were working on adding multiple-choice items by 1996.

In Martin’s looking-glass commonwealth, the KERA-dragon cast educational spells far beyond the skills of the left-wing sorcerers of Berkeley, California, and Cambridge, Massachusetts.

In our real state, our elected leaders (from places like Richmond, Prestonsburg, Liberty, Burkesville, and Danville) voted in the Kentucky Education Reform Act. KERA delivered stronger and fairer school funding, reduced political corruption, and vastly improved facilities and technology. It nurtured more focused teachers, better instructional leaders, and a big step up in justified pride in public education. We've still got work ahead to strengthen classroom work, not because the primary program, extended school services, or sustained professional development were mistakes, but because we didn’t put in the hard work to help them succeed.

Martin’s fascinating narrative ended recently, with the final installment ("The Death of KERA") published here at his Vere Loqui blog. I admit to preferring the old song where:

A dragon lives forever but not so little boys
Painted wings and giant rings make way for other toys

Still, if slaying his personal Puff frees Martin to focus on strengthening the schooling he supports, I wish him all the best.

Myself, I'll keep doing my best to build up the public schools of our beloved commonwealth, along with thousands of educators, parents, and engaged citizens. While I’m working on that, I do hope Martin will stop by my own choice of "headquarters." The lights are on, the coffee’s hot, and the (magic-free) muffins are truly delicious.