Purposeful Ramblings Through My Mind

Thanks to amyura for the link.

The federal election isn't the only one on the horizon. There's a provincial one too, and that one will hit very close to home for teachers. The Conservatives are laying low, attempting to slide in as the only choice in the face of dissatisfaction with the current Liberal government. The probability of labour unrest, the likelihood that we will lose gains made over the last number of years, the likelihood of increased testing and decreased teaching - all of it is strong with a Conservative government.

What a lousy, cowardly cretin! He split the union-busting portion of the bill from the part that spends money. They don't need a quorum for any bill that doesn't spend money, so they could pass the union-busting portion even without the fourteen senators who went to Illinois.

I hope they recall the bastard, but I'm terribly afraid that if they do, he'll run for President and get it.

http://www.nytimes.com/aponline/2011/03/09/us/AP-US-Wisconsin-Budget-Unions.html?_r=1&emc=na

An excellent article about the failings of the film "Waiting for Superman." I know I'm preaching to the choir, here, but have a link.

"It's not the school's job to cater to [insert student with a specific difficulty which they may or may not have brought upon themselves here.] It's the student's job to figure out what they have to do and do it."

The specific scenario, in this case, was a teen mom who gave birth two weeks before the end of the school year. She got up from her hospital bed, left her baby in her mother's care, and went to write a couple of tests so she could graduate.

I have a really, really big problem with this.

First, I can't imagine who the school thought they were serving by requiring this. Most women are not at their best intellectually or emotionally a few days after giving birth, so it's not hard to imagine that the young woman in question might have seen her marks suffer when she wrote those tests. That makes the assessment invalid, because it doesn't match her usual abilities. If the test is not a valid measure of her abilities, then it's not serving her needs for her to write it.

She wanted to graduate and go to college in the fall, so the argument could be made that the college needed her marks to know exactly what she could do and to decide on admissions. I'm not buying it, again for the reason of the test's lack of validity: the college was getting a skewed view of her abilities unless she managed to pull some excellent grades on that test. So an invalid test doesn't serve the purposes of the institute of higher education, either.

So whose needs were being served? The school's, of course. The flexibility required to let her graduate without the week or so's missed work required extra work on the part of the school, and a lack of (what the school would call) fairness to other students. They might have to recalculate a GPA to exclude those tests, so she wouldn't be penalized for missing them, or they might have to give her an alternate, less-stressful assessment, or they might have had to plan in advance for her to finish her schoolwork (or at least finish enough of it that she could be said to have been evaluated on the full content of the course) a bit early due to the likelihood that she'd deliver around the time of her final exams.

I don't believe that's what real fairness looks like. Real fairness evaluates students in a variety of ways, giving them lots of opportunities to show what they know and can do. Real fairness can and should look different for different students. A rigorous adherence to a marking system based on tests and GPAs is inherently unfair, not just to our new mom in the example but to every kid who has test-taking anxiety, or a learning style that makes test-taking a problem, to name a couple of possibilities.

If some bureaucracy is inevitable in a public school setting (a debate for another day) then the least schools can do is ensure that what bureaucracy they have is essential to be fair to the students.

A U.S. court upheld the expulsion of a counselling student who could not reconcile her religious belief that being gay was immoral with her ethical requirement to counsel anyone who needed her help. There's at least one similar case elsewhere in the country, and if it gets decided differently, the matter could very conceivably go to the Supreme Court.

Kudos to cereta for this one.

Gender gap persists at highest levels of math and science testing

The authors of this study point out that the achievement gap between boys and girls, when testing gifted seventh-graders, has narrowed dramatically in the last thirty years. When it was studied in the eighties, the number of boys scoring above 800 on the math SAT outnumbered the girls 30 to 1, and that gap has narrowed to about 3 to 1. That happened in the first fifteen years - that is, the 3:1 gap has been consistent since 1995.

So the authors are postulating that the persistence of this 3:1 gap indicates a difference in innate ability between boys and girls in math and scientific reasoning (where the same gap is evident.)

I'm not buying it. Here's why.

First, for every elementary school teacher who is well-versed in constructivist teaching methods as they relate to math, there are a bunch more who aren't. The NCTM (National Council for Teachers of Mathematics) put out the original version of their constructivist curriculum in 1989; I suspect if one were to break down the changes further within that thirty-year time span, it would be the years between 1990 and 1995 that would show the biggest change. But the uptake is, at best, piecemeal. Teachers still teach from textbooks, which short-circuit the problem-solving process by their very nature. Manipulatives still start to disappear from ready availability in classrooms as early as grade four. The higher one goes in math, the more likely it is that manipulatives will disappear almost entirely from the classroom, to be replaced with purely abstract problems and procedural methodology - not because those are the end goal of mathematics instruction but because that's how the teachers were taught, and when they get out of the comfort zone of their pedagogical instruction which is generally aimed at the middle of the expected outcomes for their grade level, they tend to fall back on what they know.

In short, how much of this is the fact that girls learn mathematics differently, and their learning styles for mathematics aren't being supported in their gifted classrooms? My gut instinct says that's a huge part of the reason for the gender gap, but of course I don't have the stats to back it up.

Second, their base data is of twelve-year-old gifted kids. Leaving aside the selection criteria for giftedness (which honestly I question, knowing as I do dozens of people who are very clearly gifted academically but were not identified as such in school) there's the question of socialization. Girls are still socialized away from mathematics, more subtly perhaps than they used to be and less often by teachers, but it still happens. Twelve-year-olds are at the point in their lives when they're really struggling to figure out their place in the world. How many of those gifted kids have already decided that math isn't their thing, due to a couple of years of the poor teaching I mentioned above? How many of them will be talked out of that thinking once it starts to establish itself? Or will it simply be seen as her choosing what she's best at, and hey, there are great careers in language-based subjects, too, so what does it matter if she gives up on the highest levels of math?

In short, socialization has been downplayed as a reason in this study, probably erroneously. The cultural myopia of the data selection is in my favour, here: there is no gender gap in several Asian countries when it comes to mathematics, which makes me question why there should be a 3:1 gender gap here. But the study is done entirely on American students using American tests.

Third is the issue of NCLB. It started in 2001. It short-circuits attempted improvements in instruction because so much of the testing is knowledge-based rather than based in a problem-solving model. Because the testing has such very high stakes attached to it, teachers teach to the test, meaning that improvement in instruction has been stymied in favour of getting the test scores up. You'd think that wouldn't affect gifted education, but school culture affects everything, including the kids who otherwise might not have to worry about it. If the teachers' PD is all about getting test scores up, the teacher of the gifted students effectively is getting no PD at all. His kids are going to do just fine. But he's not then getting trained in the enrichment methods which would really serve everyone much better and are absolutely essential for the highest-functioning kids.

In short, if you want to see problem-solving in students, you have to ask for problem-solving on the tests. The US as a whole is not doing that, so the level of problem-solving isn't improving.

Should I email the authors of the study and point out the problems in their methodology? :)

I think maybe I'll make this a weekly feature, just for the heck of it. It reminds me that I'm doing a good job.

So, this week in math, we started by reviewing what little we'd learned about equivalent fractions before the March break. Since exploratory learning is starting to become de rigeur for my students (as it should be) I took them to the computer lab and let them play around with the fractions manipulatives on the national library of virtual manipulatives website. The differentiation was easy. Don't get this at all? Try the basic fractions manipulative. You've already figured out the pattern rule? Great, do the grade six one.

On Monday, I gave them an assignment to do. They would get four or five pieces of regular paper in different colours, their choice. On each one, they'd write a different common fraction. The example, which I made up with my six-year-old on Sunday, was one half. Then they'd come up with three or four equivalents for it, and represent those equivalents in pictures. They had to explain the fractions and how those fractions were connected to the original fraction. There is a rubric - let me know if you want it.

The kids who were having trouble even accessing one half came for some guided math with me. We did fraction strips, and played around a bit with blocks, until they got to the point where they understood that if you divided all the pieces the same way, you had an equivalent fraction.

Then, because an abstract understanding is important to work towards, we took the equivalent fractions they'd come up with and analyzed them. How do you get from 1/2 to 4/8? If they expressed it as adding, I told them they were right, but they'd see the pattern faster if they thought of it as multiplication. So what would they multiply by? Some figured out the pattern independently; others needed to be shown several times, and finally got it with the help of more manipulatives. Then I encouraged them to use the pattern rule they'd found to come up with what they thought was an equivalent fraction. They had to come up with the proof that they were right by drawing a representation they could connect back to the original fraction.

As of this writing, most of my kids have finished two or three sheets of equivalent fractions. The rubric gives an A to kids who can connect their equivalents to decimals or percents, so I've explained the idea of repeating fractions to the kids trying that for 1/3. I suspect most will finish on time - the assignment is due on Monday.

Next Week: comparing and ordering fractions is so much more interesting when the denominators are different.

I don't think I posted this here before, so I'll do it now. It was one of the initial assignments for my specialist course.



The assignment has become a standard one for grade one students in Ontario: gather a collection of 100 items to bring in and explain to the class on the hundredth day of school, usually in mid-February. It filled my six-year-old with glee. She knew exactly what she wanted to bring in. I was less thrilled, because the collection she chose did not belong to her, but to me: my dice.

Nevertheless, we spent an hour one Sunday afternoon gamely dividing dice into baggies, categorizing them by number of sides, then by colours. Along the way, we explored concepts relating to the base-ten system and the absolute basics of multiplication.

My daughter has plenty of experience with dice. They’ve been a part of her life since birth. Plush dice, foam dice, and the vast array of dice used for roleplaying games by both her parents and all their friends, were her first introduction to numbers that weren’t on her fingers. For her, dice represent fun times with friends, groups of people laughing and telling stories around the dining room table, the adults who take an interest in her life even though they aren’t related to her – and math. So it was natural that when she needed a real-life collection to bring in for Hundreds Day, her first thought was dice. Dice are math as it is in her life.

There is a great deal of emphasis in mathematics education on making math real, on finding the ways to make the numbers concrete, tactile, visual. This emphasis is a vast improvement over numbers that never left the page, because it does facilitate a deeper understanding of mathematics, and that is the ultimate goal of mathematics education. (1) But my daughter’s experience with dice is evidence that it doesn’t go far enough. For her, dice are not something a teacher brings out to show how numbers work; dice are real life that we describe using numbers. The educational establishment has been getting it backwards. The goal is not to make math real. Math is already real. The goal is to teach how reality can be described using math.

Paul Lockhart, in his article, “A Mathematician’s Lament,” discusses how mathematics is the art of pure idea. (2) When we teach it procedurally, we strip from it the inherent creativity and beauty of it; but when we use it to describe our ideas, and engage students in describing progressively more complex ideas with mathematics, we find that everything is math. There’s no need to make it real because it already is. As teachers, our vision for our students should be to bring their mathematical understandings into the classroom. Where is the math in their lives? What forms of art exist in their cultures, and in the culture to which we’re introducing them, that can further their understanding of number and pattern and relationships? It is when we follow students’ mathematical understandings and extend them that we get the deep understanding of mathematical ideas that creates lifelong learners and problem-solvers.

Resources

1. Carpenter, T.P, Hiebert, J., Fennema, E., Fuson, K.C, Wearne, D., & Murray, H. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth. Heinemann.

2. Lockhart, Paul. (2008) “A Mathematician’s Lament.” Mathematical Association of America Online. http://www.maa.org/devlin/LockhartsLament.pdf