## Sunday, August 28, 2011

### What Makes a Student Public? An Alternative Outcome for the Douglas County Voucher Program

Although the ruling came in about two weeks ago, lately the Douglas County voucher program hasn't been far from my mind. I credit George Will's column in Friday's Washington Post for making me rethink the case and its outcome, and finally motivating me to organize some loose thoughts that have been floating around in my head.

If you aren't familiar with the case, it basically boils down to this: The Douglas County School Board believed so strongly in school choice that it voted to give 500 "choice scholarships" (vouchers) to its own students to attend area private schools. The scholarships are worth \$4,575, or 75% of the district's per pupil revenue. (The district is keeping the remaining 25%.) Several groups representing the interests of taxpayers and those worried about public funding of private religious schools filed lawsuits, and on August 12th Judge Michael Martinez ruled in their favor, saying the plan "violates both financial and religious provisions" of the Colorado Constitution. Some of the consequences of this ruling are unclear, as many students have already accepted part of their scholarships and are enrolled at private schools.

I was a bit surprised by Judge Martinez's ruling. Not because it came down on the side of the plaintiffs, but because it used the public funding of religious schools as a primary reason. In reaching this decision, Martinez seems at odds with Zelman v. Simmons-Harris, a 2002 case where the Supreme Court upheld an Ohio voucher program primarily because the vouchers went to parents and not directly to religious schools. Although I don't know many of the details surrounding either Zelman v. Simmons-Harris or the DougCo case, George Will's assertion that the two are "legally indistinguishable" seems to have some merit. But that got me thinking -- what if Martinez had found different reasoning for his decision, one not relying on religion at all?

I think we tend to focus the debate on public vs. private schools. Instead, let's focus on students. Previous Supreme Court decisions have upheld both students' rights to receive a free appropriate public education and attend private schools, as well as receive vouchers. Students who attend public schools are public school students. Students who attend private schools (without vouchers) are private school students. But what are students who use vouchers to attend private schools? Public or private? Can we have something in-between?  If so, what rights do those students have?

In Douglas County, students who receive the voucher are still required to take the CSAP, Colorado's annual standardized test. Participating private schools are required to provide information to the district about their attendance and the qualifications of their teachers, as well as be willing to waive requirements that participating students attend religious services. It's clear that these provisions are included to meet various requirements of federal education law, namely No Child Left Behind's testing and "highly qualified educator" requirements. These are requirements of public schools and public school students. And let's not forget that Douglas County is keeping 25% of each student's share of state funding. Could it do this if it didn't claim the students were not, at least in some way, students of the Douglas County School District?

Instead of essentially upholding the Establishment Clause, what if Judge Martinez had instead declared the DougCo "scholarship" students as public students and decided that voucher students had the right to simultaneously receive both a free and private education? In essence, what if he had told Douglas County that the vouchers would be legal so long as they covered the entire cost of each student's education at their chosen private school? If the court decided that (a) by benefiting from public monies, the students were public school students (and there is no murky in-between), and (b) public school students have the right to a free education, then (ironically!) Douglas County would be facing a difficult choice about whether or not a voucher program was in their best interest. As proponents of school choice it would be awkward for the district to back away because they couldn't afford the vouchers, although the high cost is surely what keeps many families away from private schools, voucher or no voucher.

At this point I realize that my knowledge of the law is rather limited and this issue was probably dealt with a long time ago. Still, I find it an interesting perspective and it makes me want to hunt down Kevin Welner (who knows a thing or two about vouchers) in the hallways next week to ask him about it. If any of you have any knowledge or thoughts you'd like to share, I'd love to hear it in the comments below.

## Monday, August 15, 2011

### Modeling Dimensional Analysis

I generally ask myself two questions when I examine the design of a mathematical task:
1. What is the context?
2. How can we model the mathematics?
Mathematical concepts with tasks for which these two questions can be answered easily tend to be easier to learn, while teaching and learning generally becomes more difficult when one or both of those questions can't be answered. For dimensional analysis (sometimes called the unit factor method or the factor-label method), the first question is easy to answer. It doesn't take much of an imagination to design a measurement conversion task that is set in a real-world context. A model, however -- whether visual, mental, or a concrete manipulative -- is generally absent. Typical dimensional analysis problems look like this:

Q: What is 60 miles per hour in meters per second?

A: $$\frac{60 \mbox{mi}}{1 \mbox{hr}} \times \frac{5280 \mbox{ft}}{1 \mbox{mi}} \times \frac{12 \mbox{in}}{1 \mbox{ft}} \times \frac{2.54 \mbox{cm}}{1 \mbox{in}} \times \frac{1 \mbox{m}}{100 \mbox{cm}} \times \frac{1 \mbox{hr}}{60 \mbox{min}} \times \frac{1 \mbox{min}}{60 \mbox{sec}} = \frac{9656064 \mbox{m}}{360000 \mbox{sec}} = \frac{26.8224 \mbox{m}}{\mbox{sec}}$$

For those who successfully learn dimensional analysis this way, there's a certain beauty to how the units drive the problem and how the conversion factors are nothing more than cleverly written values of one, the multiplicative identity. Unfortunately, many students struggle with this method. Some are intimidated by the fractions, some can't get the labels in the right place, and some just can't get the problem started.

What we need is a model. Let's start with the most basic of unit conversion models, a ruler with both inches and centimeters:

 (Yes, I'm still using the same ruler I got as a 7th grader in a regional MathCounts competition.)
With only simple visual inspection, students should be able to use a ruler to estimate conversions between inches and centimeters. This is an informal model, one students can literally get their hands on. We can assist the learning by making the models progressively more formal. Here we model a trivial conversion from one inch to centimeters with a double number line:
 (Yes, you still have to know your conversion factors!)
Such a simple example looks almost too easy to be useful, but we can add number lines for more complex conversions. We can even abstract the model further and go beyond conversions of distance. Suppose we wanted to convert 3 gallons to liters. I could model that conversion with number lines this way:
 (I could have used any number of transition units, but I knew 1 quart was roughly 946 milliliters.)
Filling in the question marks from top to bottom, I'll see that 3 gallons, 12 quarts, 11,352 milliliters, and 11.352 liters are all the same volume. It's easy to see they're the same because on each number line those values are the same distance from zero. Because we're only converting one kind of unit (volume), we only need one dimension.

In our initial example we were converting 60 miles per hour to meters per second. That's two kinds of units, distance and time, so our model needs two dimensions. Furthermore, it can help to think of 60 miles per hour as a line, not just a point. After all, we often travel at a speed of 60 miles per hour without actually traveling a distance of 60 miles in exactly one hour.
Can you guess where our double (or however many are necessary) number lines will go in this model? The following video will demonstrate what I would call the graphing model or two dimensional model for performing conversions.
With the work shown in the video, we haven't just done one conversion. In fact, we're prepared to write 60 miles per hour 15 different ways, not that we'll ever be asked to do that. If we needed 60 miles per hour in centimeters per minute or feet per second, all the work is done. Just choose the appropriate quantity from the vertical and divide by the appropriate quantity from the horizontal. Of course, if we're in a hurry, we won't find all those intermediate figures and instead just proceed from miles to meters and hours to seconds as quickly as possible. Will that be quicker than the traditional method shown above? Probably not, but the purpose of using a model is understanding, not speed. Once the understanding is established, students can move on to a formal method or use technology when appropriate.

## Saturday, August 6, 2011

### RYSK: Butler's Effects on Intrinsic Motivation and Performance (1986) and Task-Involving and Ego-Involving Properties of Evaluation (1987)

This is the third in a series of posts describing "Research You Should Know" (RYSK).

As teachers, we care not only about what students learn, but why students learn. In a perfect world, we would all agree on what's important to learn and do and be self-motivated to learn and do those things. But our world isn't perfect, and students are motivated to learn and do things for many reasons. Understanding those reasons is important if we want students to be properly motivated and to perform well with the right attitude.

Ruth Butler earned her Ph.D. in developmental psychology from the Hebrew University of Jerusalem in 1982 and was a relatively new professor there when she teamed with veteran educational psychologist Mordecai Nisan, whose career includes time spent at the University of Chicago, Harvard University, The Max Planck Institute for Human Development, and Oxford University. Together, they sought to build upon studies that compared extrinsic vs. intrinsic motivation and positive vs. negative feedback, looking specifically at how different feedback conditions -- ones that can be manipulated by teachers -- affect students' intrinsic motivation.

Butler and Nisan conducted three sessions with the groups:
• Session 1: Students performed the tasks.
• Session 2: Two days after Session 1 the tasks were returned.
• Students in the first group got comments in the form of simple phrases such as, "Your answers were correct, but you did not write many answers," or "You wrote many answers, but not all were correct."
• Students in the second group got numerical grades that were computed to reflect a normal distribution of scores from 30 to 100.
• Students in the third group got their work returned with no feedback.
After students reviewed their previous work, they were given new tasks and told to expect the same type of feedback when they returned for Session 3.
• Session 3: Two hours after Session 2 students again reviewed their work and feedback (except for the third group, who got no feedback) from Session 2 and then got a third set of tasks. Students were asked to complete the tasks and were told that they would not get them back. The session ended with a survey of students attitudes towards the tasks.

Butler modified this study for her 1987 paper Task-Involving and Ego-Involving Properties of Evaluation: Effects of Different Feedback Conditions on Motivational Perceptions, Interest, and Performance. In it, Butler adapted a theory of task motivation used by Nicholls (1979, 1983, as cited in Butler, 1987):
• Task involvement: Activities are inherently satisfying and individuals are concerned with developing mastery in relation to the task or prior performance.
• Ego involvement: Attention is focused on ability compared to the performance of others.
• Extrinsic motivation: Activities are undertaken as a means to some other end, and the focus is that goal, not mastery or ability.

The study was similar to the 1986 study, with 200 fifth and sixth graders split into four groups (comments, grades, praise, and no feedback) with subgroups in each for high- and low-achieving students. Tasks were administered in three sessions, with no feedback given after the third session. The tasks this time were divergent thinking tasks, used as Task B in the 1986 study. Praise would come in the form of a single phrase: "Very good." An attitude survey was given after Session 3.