Saturday, December 10, 2016

A class size math problem

In my last post, I discussed how smaller elementary schools tend to generate more outlier class sizes. Before I explore possible ways to mitigate that problem, here are a couple of hypothetical questions that help illustrate the issue:

1. Suppose you have a school that has fourteen rooms available as general education classrooms. What would be ideal total enrollment at that school? What would the individual class sizes be?

2. Suppose you have a school that has ten rooms available for general education classrooms. What would be ideal total enrollment at that school? What would the individual class sizes be?

I chose fourteen classrooms and ten classrooms because those numbers work especially well for class size. In a fourteen-room school, you can have two classrooms for each grade level from K through 6. In a ten-room school, you can have two classrooms in each grade from K through 2 (where we usually try to have smaller class sizes), and then one classroom in each grade from 3-6 (where we usually tolerate larger class sizes).

The only constraint: You have to assume there will be an equal number of students at each grade level. So if there are 30 total sixth graders, there are also 30 total kindergartners, though you’re free to split some grade levels into multiple classrooms while not splitting others. (The reason for this constraint is that there is no good way to draw attendance zones that would result in consistently different enrollment totals in the lower grades than in the higher grades.)

Assume that the school is not a high-need school that would qualify for lower-than-average class sizes under the district’s weighted resource allocation model. So the district’s goal would be to keep the class sizes below 26 in grades K-2 and below 30 in grades 3-6. You can exceed those caps if you think that’s the best solution. But you should recognize that anything falling significantly below those goals will put your school below the district averages and make it harder to allocate teachers to high-need schools.

What are the best answers?


Anonymous said...

Does the curriculum allow for mixing grades (for example, K and 1st or 1st and 2nd) or not?

Chris said...

Anonymous -- Sure, I'd be interested to know whether you think that's the best solution. But I'd also be curious to know what you would do if that option were not available.

smallschools said...

Our district used to have a solution to this. Until 2012ish our K-12 schools paired 1st/2nd, 3rd/4th, and 5th/6th grades. The smallest schools like Lincoln & Shimek put three grades together; 1st-3rd and 4th-6th. The kindergarteners were in their own class(es). I was told at the time this was done not based on any educational philosophy but specifically to manage class sizes and to reduce the need to add teachers if one particular grade was too big/small. This idea may make parents who are not familiar with it uncomfortable, but at the time it seemed to work very nicely. It allowed for developing "buddies" between older and younger kids, the whole school population knew each other better, and I would expect could allow for flexible learning by grouping by ability rather than age for math & reading sections. Science and Soc. Studies subjects were taught on a rotating year-by-year basis. It worked and produced smart, successful graduates. My understanding is we abandoned it due to Common Core's requirements, but not sure about that.

Anonymous said...

Teachers are supposed to differentiate so pairing grades will work, with or without the Iowa core. Unless Superintendent Murley and Board President Lynch have already decided to shut our older schools down? I believe the stopping of paired grades occurred under Murley so he created this problem.

Anonymous said...

Teachers are supposed to differentiate so paired grades will work no matter what the curriculum and despite the Iowa core. I believe paired grades stopped under Superintendent Murley. He should be required to make them work again unless he plans to shut down more schools?

Anonymous said...

This problem is like the test James Kirk faced before he was Captain Kirk. Ask how to fill the classrooms? *Set up magnet schools*Pair grades*Set up classrooms so students are educated with students at their own academic level instead of by age*Set up a Central Academy*Alternative high school needs a gym, think out of the box & sell Tate perhaps to Kirkwood. Move students to an existing school with a gym when new ones are built.

Chris said...

My oldest (now a high school junior) had paired grades in first/second, third/fourth, and fifth/sixth at Hoover. As I understand it, the district is trying to avoid that model whenever possible. I do think it has some curricular disadvantages, especially now that the Common Core has separate content standards for each grade, which makes it hard to combine, for example, 3rd and 4th graders for science or social studies, etc. Still, there is no solution to the problem that does not come with disadvantages.

It's a separate question, though, just how the numbers would work with grade-pairing. I suppose you could do this:

K: 15, 15
1-2 (combined): 20, 20, 20
3: 30
4: 30
5: 30
6: 30

Then you could teach 210 kids with nine classroom teachers instead of ten, which would at least be a savings. But the lower grade class sizes would still be significantly below the district average, for what that's worth.

Alternatively, you could imagine having 45 kids per grade, as follows:

K: 22, 23
1: 22, 23
2: 22, 23
3-4 (combined): 30, 30, 30
5-6 (combined): 30, 30, 30

Then you could serve 315 students, but you'd need 12 gen ed classrooms, which would be tough to do in some of our smaller schools. And there might be even more curricular objections with combining upper grades than with combining first and second grades.

In a school with 14 gen ed classrooms available, I'm not seeing much potential for grade pairing. If you put two classrooms of 26 kids in each grade, you could accommodate 364 students, with the lower grades being at the upper limit of what we'd want, and the upper grades being somewhat below the upper limit. Anything that involved combining grades would either require at least 16 classrooms, or would look like the example above, reducing the number of students you could accommodate.

And of course year-to-year variations in any one grade would create their own problems in these scenarios, especially if there isn't an extra classroom that can be used to accommodate them when necessary.

I'm still interested in hearing people's specific ideas about how the numbers might break down in a smaller school. And yes, let's hear more of those outside-the-box ideas!

Anonymous said...

I think it's an error to frame it in terms of "classrooms" - meaning the physical space.
Doing so causes confusion among many who think it's about the actual room itself and also generalizes many of our schools that don't have cookie cutter rooms to fit the math you and others are proposing. Given class size is more a function of teachers via the general budget the focus, imo, should be on maximizing the number of the teachers and how class sizes will continue to go up regardless of the physical space when raises outpace new state money year after year after year.

Michael Tilley said...

Hi, Chris. Excellent description of some of the challenges we are facing. I appreciate the way you've expressed it. The one problem I think you'll find when you look at actual numbers of students rather than hypothetical, however, is that wide variations from one grade to the next are much more problematic for smaller schools. I know you know this and you briefly mentioned it, but I think it can be easily missed.

As for constructive solutions, I have two: the first is one you've heard me propose previously. That is, having two smaller schools become sister schools. Having one campus serve as, say, a K-2nd campus and the other campus serve as a 3rd-6th campus. I've talked about that pretty extensively, and it would work out numerically for a number of our smaller schools.

The second is to come up with magnet schools based on curricular choices that function best with multigrade classrooms. I believe Montessori classrooms would be very attractive to a lot of people in the ICCSD and they would most likely see significant classroom savings if implemented in a smaller school. In fact, Montessori typically puts 3 years-Kindergarten together (which could provide additional income to the ICCSD), 1st-3rd, and 4th-6th together. Another similar sort of approach would be to have a Progressive Educational magnet (similar to Montessori in multigrade classrooms, but it works with natural landscapes and the natural environment more having a more heavily curated Montessori-environment---see here: http://prairiecreek.org/).

That's what I've got. I understand the common core challenges, but I doubt students in either program would have difficulty doing well on summative standardized tests.

Anonymous said...

There are several elephants in the room which need to be recognized. First, many parents (for good reason) will assume that the administration/Mr Murley have an objective other than the best educational interests of their kids with any radical major proposal. Second, the problem is lack of money to hire enough teachers. You can blame lack of funding from the legislature. Or you can blame teachers being paid too much. Both are to blame. In any event, the results of the recent election don't bode well for any increase in state funding. And third, the foolish FMP has resulted in unnecessary schools being built in the wrong places, which will make staffing worse.

What I see happening, out of necessity, for better or worse, is moving away from one teacher per classroom and instead to something like two teachers per three or four classrooms with noncertified assistants making up the difference.

Unless there are cuts in the high level that IC compensates its teachers or an unforseen increase in state funding, that's where we are at.

Mary M said...


On a related topic, the administration was clear at a recent DPO meeting where the "weighted resource allocation model" (mentioned above) was a main topic that the model was NOT financially sustainable long term without additional funding from the state. School districts haven't gotten as much funding as hoped for in recent years, and are unlikely to get desired funding in the next few years given election results. Therefore, the board should review feasibility of the WRAM now so short sighted planning doesn't result in more budget surprises after a bond vote.

A start toward managing larger class sizes might be to make sure sufficient paraprofessional support is in place in all classrooms where needed. Toward this goal, the recent state special education audit required the district to "Cease implementation of the paraprofessional request process. IEP teams should make the decision regarding paraprofessional needs." This requirement should be independently monitored by the board.

I also see no reason why paired schools, multi-age classrooms, and/or magnet schools shouldn't be seriously evaluated.

Thanks. Mary

Mary M said...

As an added comment to my comment above, the board should ensure that IEP teams have the authority to assign paraprofessional support where necessary to provide a free and appropriate public education without regard to classroom numbers. The school district doesn't have a choice about this.

Anonymous said...

Des Moines has a multi-age elementary school, the Downtown School. http://downtownschool.dmschools.org/about/.

Des Moines also has a couple of International Baccalaureate elementary schools.

Anonymous said...

Lincoln still has multi-grade classrooms and it seems to work alright since they are consistently a high performing school academically. I think there may be other schools in our district that still do as well. I think in some ways this may be a benefit since levels vary greatly among grades and everything seems to be 'dumbed down' to the lowest common denominator anyways. This would allow for more stratification of levels. For example if 3rd and 4th are combined the more advanced 3rd graders can work with 4th and conversely the lower level 4th with 3rd grade.

Amy said...

Let me offer some perspective from the university level.

Last spring I taught a grammar course for undergraduates. Nothing fancy at all: this was essentially 7th-grade grammar. And it was news to almost every kid in there - mostly Iowan kids from our famous Iowan schools. Subjects, verbs, objects, nouns, adjectives, phrases, clauses, all the usual: they didn't know what we were talking about.

About three weeks in I started the class by sitting down at the teacher's desk and asking them what was going on. What it was that they found so difficult about grammar, why this stuff was hard to understand. I thought maybe I was just being an exceptionally boring or difficult teacher, but I wanted to hear from them what they needed and what was standing in their way.

What they told me was that this stuff was indeed all new to them. Nobody had corrected their writing on a sentence level in years and years. Nobody had told them what was wrong. Which I understand, because when you're teaching 200 kids a trimester, then no, you don't have time to help them like this at all. And the result is that they show up in college without being able to marshal a cogent paragraph.

It's as simple as that. Teaching requires a lot of small-group and one-on-one contact time. It's a fantasy to believe it doesn't. And yes, this is expensive.

I wouldn't want to see an elementary classroom with more than 20 kids. Ideally, not more than 15. But more than 20, you've got kids falling off the edge, and there's no way to really help the kids who're capable of much more, either.

Karen W said...

The failure of kids to learn grammar is a curriculum and instruction problem, not a class size problem--at least not at the elementary level. I think my elementary classes were typically about 30 kids but we had a structured grammar program and elementary teachers marked up our writing assignments. There was no small group teaching in class and kids who needed more help went to the resource center.

Do secondary teachers need smaller student loads and/or more prep time? Probably. But they also need students to have mastered elementary-level grammar in elementary school.

One of the institutional constraints of public schools is that they cannot operate as one-on-one tutoring centers. We can't make curriculum and instructional decisions based on the idea that we're going to get funding for that, or even for class sizes of 15 or 20. It isn't going to happen.

Anonymous said...

Yes to what Karen W said!

Administrators will need to support teachers who have classroom behavior and management issues so the class as a whole can pay better attention.

Amy said...

Karen W, that might have worked when Iowa schools served a fairly homogeneous population of kids who were reasonably well-supported at home, particularly by a large number of stay-home moms. These things are always easier when the world outside the school is shouldering some of the educational burden. Unfortunately, that's not what we've got now -- kids are much poorer than they were, we've got many more non-English-speaking households, many transient families. The erosion of the middle class turns out to have an effect on how many kids a teacher can teach at one time in a public school.

I cannot pay serious attention to 30 students at once, and those are relatively self-maintaining college students who've volunteered to be there. In fact I wouldn't teach a writing course with that many students in it. 12-14 is a reasonable size, and in fact that's where the courses were capped, at 12, before the university started playing ranking games seriously.

If you want a real education, you're going to have to pay for it. My kid's in the middle of applying to the U of Chicago Lab school, where the sticker price (for non-staffers) is about $38K/yr. Class sizes are small, and it's essentially the Princeton of high schools. Local competition includes the northern-suburbs schools, including New Trier, which spends about $23K/yr on each kid, and the money shows in what comes out of those schools; they're among the very top public schools. Here we spend what, about $12K? And we do pretty well with that, but we'd be doing a hell of a lot better if we recognized that yeah, if you want an education, your teacher needs to have a close eye on what you're doing, and needs to have time to teach you now and then. Not the class: you. If you look at what the ed reformers want, the ones who're big on discipline-and-3Rs-solves-all, they look to dole out vouchers of around $5K max. They're living in a dreamworld, or they just don't care if the kids come out illiterate.

We pass bonds all the time for sports fields and new big schools and whatnot; one that was about hiring teachers, that I could support.

Karen W said...

Amy, I think we're going to have to agree to disagree about how much one-on-one attention is required for most children to learn that a noun is a person, place, or thing. When large numbers of students aren't learning something, I think it's reasonable to ask whether that something is (still) part of the curriculum and whether the teaching methods in use are effective. In fact, we should probably start with questions about curriculum and instruction before we assume that the problem is demographics or class size.

For example, I attended a conference during which Iowa 6th grade teachers were explaining that none of their students, not one, had the multiplication tables memorized. This wasn't a demographic problem or a class size problem and there was no reason to think the students were incapable of memorizing multiplication facts. This was a problem directly related to a curricular and instructional choice in earlier grades that memorization of math facts was no longer important in the 21st century.

More money can be helpful, but it can also be spent on curriculum that still doesn't cover what we'd like students to learn or on ineffective instructional methods. In any case, I think we have to be careful about attributing results at selective schools to the level of spending and not selectivity.

Anonymous said...

Just some observations based on what I have personally seen with my kids in the ic school district. They are in third and sixth grade. The teachers are too busy dealing with disciplinary issues that there is limited time to actually teach that want to learn. Often the teachers are using prepackaged videos to teach and just handing out worksheets that accompany the text. Those students who are ahead of the lesson plans are expected to help those that are not up to speed yet rather than advancing any further. I really don't understand what they are doing with the math curriculum at all. When I was in school we memorized our basic math functions and had one algorithm that always worked to come up with the correct solution quickly. Now it seems like they spend much of the time talking about many different elaborate abstract methods to solve a simple problem and make things much more difficult than they should be. We have had some exceptional teachers and some that were not so good. The teachers can only do so much when there are so many kids in each classroom and many of those kids do not have basic respect for education and have no desire to learn.

Anonymous said...

To Karen W and Anonymous at 11:05-

From anonymous at 11:05-"Those students who are ahead of the lesson plans are expected to help those that are not up to speed yet rather than advancing any further." Teachers using students as helpers in the classroom is a big problem in this district. It's not a teaching strategy for academically advanced children when they already know how to do the work.

I agree with Karen W about asking the right questions and concerns about the curriculum. Elementary students can memorize math facts and basic English and grammar rules if the teachers know it and teach them.

Discipline is also a huge problem. Chris, can you comment on what the administration is doing to support better behavior in classrooms?

amy said...

Karen, oddly enough, I had a memorable and quite depressing class session many years ago -- before the 21st century, in fact -- teaching community-college students that a noun is a person, place, or thing. These were people ranging in age from about 18 to about 30, meaning that they'd mostly gone to school long before whole-language approaches were in vogue, when good ol' hick'ry-switch 3Rs were still the rule, and they still didn't have it.

In fact we spent a good 20 minutes on the concept. "Is a chair a person, place, or thing?" No. Yes. "Is a chair a person." No. "Is a chair a place." Yes. Etc. You can see the problem: they're mentally awol. Remarkably often the problem comes down not to knowing facts or algorithms but to learning to think, to the development of a prehensile and questioning mind, even to the recognition that this is allowed. That's not a simple thing.

You can pretend -- and in fact we do, quite effectively -- that when 70% of the kids are scoring all right on the standardized tests, what we're doing is working, more or less. But the proof is what happens when they leave K12. And my experience, in classrooms, as the teacher who's responsible for seeing that something sticks in their minds, is that teaching is extremely labor- and time-intensive work, and that a thing once taught generally does not stick unless it's reinforced many, many times, and that even when you've gone through all that, it sticks much better after one-on-one. Which should come as no surprise to anyone who has children. How long did it take, how much one-on-one, before your children behaved like anything but savages at the table? You showed them, you helped them, you corrected them, over and over and over. You answered their questions. You modeled behavior for them. It took a lot of work and a lot of time, and that was for a handful of children.

It takes time and individual attention because the things we teach are not, in general, easy. You want the kids to know their times tables. I do too. I would like for them to be able to spit out the answer to 4x8= ? when woken out of a sound sleep. But why do you want them to know their times tables? Do you have reasons, or is it just an eat-your-veg sort of thing? I find that very often the people most devoted to the 3Rs/agin-the-gubmint-school line don't really have a clear idea of why these things might be important, what they might be important for. They're just regarded as signs of virtue. I don't think they're signs of virtue. Nor do they automatically lead to "good-paying jobs".

Why might it be interesting, 4x8=32? Can you say?

Until you have time to give to individual students, too, you might see problems in their work, but you don't know the source of those problems. What misunderstandings might obtain. What crucial part of thinking or doing the kid's regarded as unimportant. Nor do you have a genuine sense of their potential. I spend a lot of time meeting with students, in part because they haven't had this attention in K12, and for many of them it's the first time a teacher's spent half an hour listening to them talk about their interests, concerns, troubles in school, troubles outside school that give them problems in school. Depressingly often I show them something perfectly ordinary and it's a revelation to them, a door they'd never known existed. It's not because their teachers were inept, necessarily, or that the teachers didn't care. It's that they don't have time. They've got class time choreographed to the minute in there, and there are repercussions for falling behind. Is method important, yes, some. But so is the teacher's education, and so is the teacher's freedom to teach, which means having time to give each student, and flexibility to teach to where the students actually are.

It costs money, education. Money and time, which are often the same thing.

Frank said...

Unless one is going to use a calculator at all times, how could a student even begin to do more advanced math without memorizing math tables? Obviously before they have memorized these they would understand the concepts of what addition or multiplication is. Without memorizing these would they count on their fingers or draw blocks on paper to figure out what 4x8 equals?

Karen W said...

Amy, I think we're in agreement that what we want kids to learn at school isn't easy or natural for them to learn and getting things to stick in long term memory does require reinforcement (spaced practice) over time. That's why I think curricular and instructional decisions are absolutely worth talking about. I don't think we have optimized curricular and instructional decisions to the point that only class size reductions can yield further improvement.

And I do think that time and attention in a group setting can be effective, especially if teachers have sufficient prep time to review and mark up student work. Besides, what realistic option do we have for public education except to make group settings work for the largest number of students possible?

I guess I'm guilty of believing that the 3Rs still matter, even in the 21st century. I also want kids to study history, government, geography, science, and literature. I don't think that makes me ignorant or thoughtless about what purposes public education might serve or what might make for a well-educated person.

If your times tables questions wasn't rhetorical, without making an exhaustive list, I would say that knowing the times tables to automaticity makes it much easier to do long division, do arithmetic with fractions, recognize least common multiples and greatest common factors easily, and factor equations.

Anonymous said...

ICCSD most definitely needs to start talking about curricula decisions. One only has to look at some of the syllabi to figure out the the focus is on behavior, not on what is being or should be taught in the classroom.

My child's mathbook is confusing, and there isn't much explanation about how to do the math problems.

Chris, how are textbooks selected? Who decides who gets put on a curriculum review committee?

Anonymous said...

Lot's of good discussion here. What's interesting to me is that in the last progress report that ICCSD administration presented to the board it is obvious that for most students ICCSD is lagging behind other Iowa districts, especially if these students are non-white and non-affluent. Why is that? These other districts have the same (or worse) funding constraints. And they would seem to have less community resources. Yet they improve, and we decline.

Unless you start with an administration that cares about educational advancement and has the technical skill to create and implement changes nothing is going to happen.

Our administration is so bogged down in cheerleading a doomed-to-fail $200 million bond issue that they have lost track of the kids.

amy said...

Belatedly, to Karen W:

Unfortunately, "spaced practice" is really not all that's necessary if you want students to learn well. You actually have to spend the one-on-one time. If I knew of a substitute for it, I'd use it, but the reality is that in the end either the student is capable of teaching him- or herself well or you, as the teacher, must help the individual student find a way for the ideas to enter the mind and take root. Seemingly simple ideas do not in fact turn out to be easy for all or even most students to recognize and pick up on their own. If you're unfamiliar with this phenomenon, I'd encourage you to spend a few years teaching.

The 3Rs certainly do matter, but the idea that they can be mass-delivered through a series of well-memorized catechisms is, alas, wrongheaded. You can certainly make a lot of kids cry that way, or make them memorize things and then forget them until the next test, but if you really want the kids to learn them well, it's going to take time and a lot more teachers than you're used to paying for.

Your answer about the times-table memorization...you realize that all you've said that automaticity is good for is the ability to take more math classes? To which I'd ask: okay -- so what's the value in that, particularly since, unless you become an engineer or work in physics, it's unlikely you'll actually use most of the mathematics you learn in school beyond elementary algebra and maybe a little geometry? (Even the physicists, it turns out, tend to skip the math when they read physics papers.) Yes, the automaticity is very important, but your reason, by itself, isn't it. It's not even one of the most important reasons. Try again: why would it be good to know, without even thinking, that 8x4=32, or that 7x5=35, or etc.?

Karen W said...

Alas, it is true. I want my children to be prepared to take more math classes. I also want my children prepared for high school lab science courses that use math, too. I don't think I'm alone in this. We simply disagree about whether this is an "important" reason to care about curriculum and instruction.

Honestly, what's the downside to preparing kids to take more math classes? A commenter, on other blogs I read, frequently says that K-12 education ought to be about keeping doors open for kids as long as possible. And I agree with that. We don't know which students will go on to major in subjects that require math and, in point of fact, there are more majors than just physics and engineering that require math.

I'm not at all bothered that I am unlikely to use all of the math that I learned in school. I enjoyed learning math in high school and college and I can't see that learning it did me any harm.

amy said...

Okay -- so you took a lot of math, but you can't articulate what's good about it, and you view it as just sort of generally nutritious. (There aren't actually a lot of subjects or jobs that require real math chops. You might be amazed how much of STE really never touches much M. The only people I know personally who really use math at work are those engaged in statistical analysis, quantum computing, designing graphing software, and, to some degree, physical science. They make very, very nice money, and not very many of them are needed.)

Almost nobody does recreational mathematics. Which is not to say that recreational mathematics is a bad thing; some of my favorite books ever are recreational math books. But it's a pretty tiny audience. If you're talking about something that's just generally nice and educational, then, there's no reason not to replace math with film studies. So no, that's not a good argument, either.

Like I said, there are real reasons to learn math and to learn it well. Keep on thinking about it. Think about what math has shown you that nothing else has. Think about the people who're really good at math, way off in PhDland, and what they seem to be able to see that other people don't. That's a little tougher, because they're not famously good communicators, but some good people to start with are Douglas Hofstadter, and Brian Greene (not a mathematician), and George Gamow (or him, really), and Cathy O'Neil, and Marcus du Sautoy, and Mike Lawler.

Like anything else, though, teaching math well -- even teaching *about* math -- takes time and attention, and one-on-one. Consider how much time I've put in here to pulling you around to thinking more seriously about what's interesting and special about math, and why memorizing the times table might be both trivial (though useful) and nontrivial.

Why does it matter whether or not you think about these things? Because -- among other reasons -- if you can answer those questions you have some idea of when and why it's important (or unimportant) to harp on times-table memorization when talking about math curriculum. As opposed to viewing the times-table memorization as a general marker of virtue and future success, but being cool with ignorance about the more important thing: why it's worth doing in the first place.

Anonymous said...

Amy, you are missing the main point. Karen W wrote "I don't think we have optimized curricular and instructional decisions to the point that only class size reductions can yield further improvement."

ICCSD can improve. This improvement can happen with or without class size reduction and even if class sizes increase.

You brought up math. Many people, not just your examples, use math at work and in their everyday lives. Our kids also need math to compete against kids from other countries.