Tuesday, January 3, 2017

A class size math problem, part 2

A few weeks ago, I posted a “class size math problem” to illustrate how small elementary schools tend to generate outlier class sizes. The post received a lot of good comments—but nobody did the math problem! Specifically, the question was how many students you’d want to assign to a school that had ten (or, alternatively, fourteen) general education classrooms, and what the class sizes would be in each grade. The one constraint was that you had to assume that there would be a roughly equal number of kids in each grade level.

One purpose of the experiment was to show that it’s hard, in small schools, to have class sizes that stay in the neighborhood of the district’s class size goals. The goals, for most schools, are not to exceed 26 kids in grades K-2 and 30 kids in grades 3-6. We don’t want class sizes to exceed those goals, but if class sizes are too far below them, they undermine the district’s attempt to reallocate resources to higher-need schools. (More complete explanation here.)

Without mixing grade levels, this was the closest I could come to the class size goals, without exceeding them, in the ten-classroom school:

As you can see, many of the classes are very small—way below the district goal for grades K-2. That puts upward pressure on class sizes elsewhere and makes it harder to redirect resources to higher-need schools.

In the fourteen-classroom school, this was the closest I could come to the goals without exceeding them:

This school is significantly fairer to the district’s other schools, since its average class size is closer to the district-wide medians. It does less to undermine the district’s efforts to reallocate resources to higher-need schools. It accommodates a lot more kids. You could make those early-grade class sizes smaller—I’m sure many people would balk at the idea of a 26-kid kindergarten class—but only if you’re willing to live with smaller class sizes in the higher grades, too. As the upper-grade class sizes fall farther below the goals, class sizes elsewhere in the district would have to rise.

These scenarios assume an equal number of kids in each grade level. In reality, there is often a lot of variation in enrollment in the different grades. That’s a big part of what makes class sizes hard to manage in small schools. It’s actually hard to get much closer to the district goals than the “ideal” fourteen-classroom school, but the great advantage of larger schools is in their ability to handle that anomalous year-to-year variation, since they can spread the effects out over a greater number of classrooms.

In my next few posts, I’ll consider alternatives (such as mixed grades, paired schools, and the use of half-time teachers) that can help alleviate the class size issue in small schools.


Anonymous said...

Maybe I am not understanding your question, Chris. But the logical way to educate 210 kids in a 10 classroom building would be to fully use only 9 of the 10 classrooms in the small school and only hire 9 teachers instead of 10. This would allow you to serve either 224 or 230 kids and still stay within classroom size goals.

It's cheaper to let a classroom sit vacant than it is to underutilize it. Of course a better alternative is to redistrict so that all classrooms are fully utilized, but redistricting isn't a precise science either.

The myth that small schools are inherently more efficient is just wrong.

There is no way, for example, that the cost of tearing down structurally sound Hoover 1 and building larger Hoover 2 a short distance away will ever make sense financially.

Anonymous said...

Typo correction - 224 or 228 could be served by 8 teachers in 8 classrooms.

Chris said...

Anonymous: I agree that it sometimes makes sense to use fewer classrooms than are available. But I'm not following how you would divide seven grade levels (K-6) over eight classrooms. Are you mixing different grade levels together? (That's an option I'll discuss in a later post.) Even then I'm not seeing how you end up with eight, or how to total comes out to 224 or 228. Could you sketch out how that number breaks down into individual grade-level classrooms?

Chris said...

Maybe you mean seven classrooms of 26 kids each, for a total of 182? It's true that that would be cheaper, in terms of classroom-teacher-per-student, than the 10 classroom model, and would do less to drive class sizes up elsewhere. (You'd have to keep in mind, though, that there are other costs, such as that of hiring a principal and a front-desk staff person, that would cost more per student in a small-enrollment school.)

In that model, though, the main trouble would be the year-to-year variations in the grade-level enrollments. A school with an enrollment that small would have a hard time dealing with those variations. One year you might have only 16 fifth-graders but 32 first-graders, etc. In a bigger-enrollment school, where there are several classrooms for each grade, can handle year-to-year variations much more easily without having class sizes that are very small or very large.

Chris said...

By the way, I certainly agree that it makes no sense to tear down Hoover 1 and build Hoover 2 elsewhere--especially since the projected enrollment for Hoover 2 is pretty much exactly what Hoover 1's enrollment was before the closure vote.

Chris said...

By the way, one of the ironies of the Hoover closure is that Hoover is not the kind of “small” school that is particularly problematic in terms of enrollment size. Hoover has enough classrooms (21) to function comfortably on the fourteen-classroom model, with rooms to spare. It currently houses fourteen gen-ed classrooms, two preschool rooms, one or two special ed rooms, a music room, and an art room, with the remaining rooms serving other uses.

Anonymous said...

Hi, Chris, I'm 959 and 1004. Figure three classrooms for K to 12 @ 26 each and four classrooms for 3-6 @ 30 each plus one more classroom for either 26 or 30.

You will have the possibility of classes being below or above average size regardless of whether it is a small school or a big school so no size of school will come out exactly fully utilized.

Adjust the boundaries to use the classrooms as fully as possible and staff accordingly.

Overbuilding (which clearly is what is being sought by the proposed $200 million dollar bond issue) makes the problem worse instead of better because you will have more empty classroom space which will lead to more and more pressure to close smaller schools, especially the ones in older parts of Iowa City and the one in Hills.

Chris said...

Anonymous -- Minor quibble: There's no way to draw an attendance zone that would reliably produce slightly smaller K-2 cohorts than 3-6 cohorts. So if you're shooting for 26-kid K-2 cohorts, you have to also shoot for 26-kid 3-6 cohorts.

More major quibble: I don't think you're sufficiently acknowledging the class size problems caused by year to year variation. In a school that's big enough to have two classrooms (of, say, 26) per grade, if 70 fifth-graders show up, you can handle it by having three classrooms of 23 or 24. But in a school that just has one classroom per grade, if 35 fifth-graders show up, you've really got a dilemma that's not so easily handled. That's a big part of what causes problems with class sizes in very small schools.

I agree with your last paragraph.

Anonymous said...

Me again, Chris. Thanks for your reply. There is no way to know for sure how many students you will have on the first day of school until the first day of school. It may be more or it may be less than predicted. But unless we have completely wasted the $100,000s we have spent on enrollment projection consultants (which maybe we have - but that's another topic) we should be able to make staffing decisions based on these projections.

Therefore (and to address your very valid point about cohort size) in order to educate 210 kids in a 10 classroom school, staff 9 rooms of 23 students each.

Chris said...

You're completely losing me there. How are seven grades going to break down over nine rooms? If you have one classroom of 23 kids in each grade--that's seven rooms so far--what grade are the eighth and ninth classrooms in?

Anonymous said...

Me again Chris. I think your possibly flawed assumption is that a different size school building will make things better.

As I see it the perfect elementary school size would be either 198 or 396 or 594.

Why? 3 full grades (k-3) @ 26 plus 4 full grades (3-6) @ 30 = 198. And 396 and 594 are just 2 and 3 times 198. Under these ideal/perfect scenarios your classrooms are fully used and your teachers are fully utilized and your class size guideline "caps" are met.

The problem is that no schools are going to have enrollments which fall exactly at 198, 396, or 594. But having larger schools doesn't make this situation better - you will still have the exact same staffing/classroom problem.

In your hypothetical 210 student elementary, if your theory of cohort size being equal is correct, use 7 classrooms of 27 each and two classrooms as "floaters" (each with an aid or teacher) to teach underperforming or oveperforming kids.

amy said...

All I see is people continuing to kid themselves about how many children a teacher can teach well at one time, and refusing to look at the reality that if you want better education, it will cost money, primarily in the form of teacher salaries.

My kid's in the middle of an application to the U of Chicago Lab school, which is ranked somewhere near the ceiling and is where Arne Duncan went and sent his kids, rather than sending them into the Bosch painting that is the Chicago Public Schools system. Over half the kids at that school have parents who work at U of C, meaning that they're already not only bright but have been stuffed with education from the second they hit air. They go home and have supper with the people who do the talking that NPR dumbs down for the rest of us.

Even there, the teachers' contracts cap the classes at 24. Even there, with all the privilege and help those kids get. It's not merely a matter of gold-plated snowflakes; it's the reality that the school will not be able to maintain its standing if the teachers don't actually have time to teach all the kids. With, yes, a lot of one-on-one. Because that's what it takes if you actually want the kids to learn well.

I don't know how many ways there are to say that the classroom cannot be fantasized into a factory.

Chris said...

Anon—I agree that it’s hard to beat multiples of seven for the “ideal.” However, for purposes of deriving a “target enrollment” for a building, I do think you have to assume equal enrollments in each grade level, and if you don’t want to exceed the goals, you’d be stuck with 26 * 7 = 182 in the seven-classroom school.

More importantly, it’s simply not true that you’ll have the exact same classroom/staffing problem in the larger-enrollment school. Again, if 70 fifth-graders come along in the fourteen-classroom model, you can make three classes of 23 or 24. If 35 fifth-graders come along in the seven-room model, you can either suck it up and have a class of 35, or you can have 17- and 18-kid classrooms. (Again, subject to alternatives I’ll discuss, all of which have their own downsides.) The larger school can handle those problems and still have class sizes that are closer to the median. That’s really where the additional class-size expense in smaller schools is.

This is borne out if you look at the number of teachers per student in schools with large enrollments versus schools with small enrollments. On average, the small enrollment schools spend more on classroom teachers per student. I certainly don’t think that means we should close them, but if there are other ways to mitigate the problem, they’re worth considering.

Chris said...

Amy -- not disagreeing. If we had the funding to give everyone a class size of 20, we'd do it. All of these posts are based on the assumption that there's a limit to what we can spend on classroom teachers, and that we're just talking about how to allocate it. Given the election outcome, it seems likely that we won't see much more funding from the state in the near future. So all we can do at the local level is talk about our allocation of the money. Now, I think it's a fair question whether we might be able to squeeze some more general fund money out of other parts of the budget, but those sacrifices would be painful in other ways. It would be great, though, if there were much more transparency about what those tradeoffs are; the board and the community might weigh them differently than our central administrators do.

Anonymous said...

Chris, if the actual student count exceeds the very expensive enrollment projections we have obtained by 50% you have bigger management problems than just a teacher shortage. More importantly, under both the big school and little school hypotheticals you propose, a new teacher needs to be hired who will not be fully utilized. Wouldn't the cost increase for each scenario be the same - the cost of the new teacher?

And I agree with 336 that 24 would be a great student number to have. But its not going to happen funding-wise.

One option, of course, is to decrease the cost of hiring a teacher in the ICCSD. But as I see it under the agreed upon wage packages that cost is going to be increasing, not decreasing.

It would be nice if we could afford to think of schools as not being factories - but when you have limited resources I don't know what else to treat them as if you want an efficient operation.

Chris said...

Anon -- The difference is that in the bigger school the extra teacher is teaching more kids. Over the entire district and over time, that adds up. That's why the smaller schools have higher classroom-teacher-per-student costs.

Our enrollment projections don't estimate on a grade-level by grade-level basis for individual schools. That kind of fine-grained prediction is likely impossible, no matter how good your demographers are.

Anonymous said...

Chris, I don't mean to quibble but hiring a teacher at $60,000 per year (or whatever) costs the district the same regardless if that teacher is teaching 26 kids or 16 kids. And the DeJong projections which we already have are broken down by grade for each school building. For example, on page 85 of the DeJong report it projects that Hills will have 37 kindergardeners in 2024-25. And on page 101 Mann is projected to have 30 for that year.

Anonymous said...

Discussing half of the equation, specifically building size, really misses the mark. As you and others point out, having the "right size" of schools based on desired class sizes can help -- mostly by smoothing the edges and decreasing the extreme outliers - high and low. It's also obvious larger schools can greater absorb year to year fluctuations. However, all of this is irrelevant if you cannot afford the teachers to staff the classrooms at those levels. The discussion must, therefore, include a discussion of teacher salaries. It's unfortunate this seems to be the "third rail" topic in this district and by the board. Perhaps it's about time for the board to actually examine and discuss how teacher raises have, every year for the past 10, surpassed the new money/growth from the state. What would class sizes be like if we would have just given them the same raises the state gave the district?

Chris said...

Anon (4:33) -- You're right about the grade by grade projections; I misremembered it. I still think expecting accuracy at that level is expecting a lot from demographic projections years out. Moreover, even if you can see the year-to-year variations coming, that doesn't help you figure out how to deal with them, teacher-wise.

But I continue to disagree about the cost of the additional teacher. To demonstrate why: Imagine two small-enrollment schools that pretty much track a seven-classroom model. Imagine that in both schools, there is a fourth-grade class of 35. Assume, for purposes of argument, that that's too high a number to tolerate. So imagine that the district breaks those classes down, in each building, into two classes of 17 and 18. Now you're hiring four teachers to teach 70 fourth-graders.

Now imagine a larger school that tracks the fourteen-classroom model. Imagine 70 fourth-graders show up. Assume 35 is too large a class size. You deal with it by having three classes of 23 or 24 each. Now you're hiring three teachers to teach 70 fourth-graders. That's why, over many schools and over time, the larger schools have, on average, lower classroom-teacher-per-student costs.

Chris said...

Anon (4:34) -- I can just tell you, from my experience, that the topic does not so much strike me as a "third rail" -- it's just that the outcomes are driven by factors that are not as much within a school board's control as people might think. If a board can't agree with the teachers about the compensation increase, the teachers can take the issue to arbitration. On any given contested issue (such as compensation), the arbitrator is not free under state law to "split the baby"; he or she must choose one side's position or the other. So, if a board tries to push the issue with a number that's lower than the teachers are willing to agree to, it's a gamble: the arbitrator may well choose the teachers' number, and then the board is spending even more money than it would have under a negotiated agreement.

Arbitrators, in deciding which number to choose, are influenced by what they see in negotiated outcomes elsewhere in the state. So the whole system tends to circle in on a particular range of increase from year to year.

So, in my experience, individual boards don't feel that they have a whole lot of latitude on the issue. Is it possible that they're underestimating that latitude? It could be, but the "gamble" aspect of the interaction keeps either side from getting too ambitious with its stance.

I want to pay our teachers well, and there are good reasons for paying them well even if we could get away with less, but I understand the point you're making about compensation increasing faster than state funding is increasing. I see that as an argument for better state funding, but of course others can see it differently. In any event, the conversation at the local level doesn't really end up being about any of that; it's more about circling in on what an arbitrator (whom we don't control) would likely do.

So the problem with the question, "What would class sizes be like if we would have just given them the same raises the state gave the district?" is that it acts like the board unilaterally decides what increase to give, which is not true in a collective bargaining situation. If the board had tried to limit increases in that way, class sizes might well be worse than they are now, because an arbitrator might have chosen the teachers' position over the board's.

Anonymous said...

Chris - doesn't that gamble work both ways; meaning the teachers and there reps could end up in a worse situation if the arbitrator chooses our "last best" offer as you describe? It also seems like, based on you analysis, that the district is essentially SOL and had to accept raises in excess of new state money ...thereby causing higher class sizes. If that's the case I would say that it's particularly difficult for teachers, many of whom are also parents, to complain about larger class sizes when the single biggest factor in class sizes is teacher raises as balanced against new state money.

Either way, still seems like a "don't talk about" topic since the effect of raises on class sizes has never been discussed at the board table and the board unanimously votes in favor of the raises year after year after year.

Btw, quick math using State aid vs raises for last 10 years rounds out to about 14 mil plus on the current budget since each subsequent year builds on the past. That's what, 200-250ish more teachers we would have?

Anonymous said...

To the extent your hearing comparable raises from other districts factor it sounds like you might be misinformed. While the settlements are factored, they are measured in terms of comparability of pay levels, not straight settlement percentage. This weighs in the ICCSDs favor as it has the highest paid teachers in the state. The appilicable code also expressly states that ability to pay is a primary factor.

Anonymous said...

Who represents the district in negotiations with the union? Is a board member or the superintendent at the table? Does the board have an expert?

Anonymous said...

Hi Chris,

I saw your comment that the arbitrator must choose one side or the other. Will the board make getting this changed a legislative priority?

Chris said...

Anons -- Yes, the "gamble" does work both ways, so it constrains what the teachers can ask for, too. I'm not at all convinced that we'd be better off under a system where the arbitrator doesn't have to choose one of the two positions on any given disputed issue. If the arbitrator could "split the baby," the parties would have an incentive to stake out extreme positions rather than an incentive to reach agreement, and there's no necessary reason to think that the arbitrator's decisions wouldn't still be driven by statewide trends.

Again, in a system of collective bargaining, no matter how you arrange it, the district doesn't unilaterally call the shots. The teachers have bargaining power (as they should!), which affects the outcomes.

Anon (9:24): There is a board member involved in the negotiations with each bargaining group, but the district's personnel director plays the lead role on the district's behalf. There's no question that the board relies to some extent on the expertise of the personnel director when it comes to evaluating options. Is it possible that he or she could be wrong, or overly cautious, in evaluating options? Sure, but it's just as possible (probably more possible, given their relative lack of experience with the system) that individual board members could be wrong, too. Again, any time you take an issue to arbitration, it's a gamble.