Here is an article about changes in the way some schools are administering tests. The title of the article is "Legalized 'Cheating'". There are lots of different kinds of "cheating" described here, some of which I think are perfectly fine and an obvious improvement.

For instance:

Twas a situation every middle-schooler dreads. Bonnie Pitzer was cruising through a vocabulary test until she hit the word "desolated" -- and drew a blank. But instead of panicking, she quietly searched the Internet for the definition.

...

In Bonnie Pitzer's case, teacher Becky Keene says using the Internet helped the seventh-grader, but in the end, she aced the test because she demonstrated she could also use the word in a sentence. "I want the kids to be able to apply the meaning, not to be able to memorize it," says Ms. Keene.

I actually have no problem with this. I think I have a fairly broad vocabulary, but I look up words from time to time. Who doesn't? If the student can demonstrate that she can look up the meaning of a word on the internet, and then apply the word appropriately, I think that's about the best we can hope for. You can't expect to memorize every word there is.

It's the same with formulas and tables and other things that can easily be looked up. Why make a kid memorize these things, when they're probably going to forget them after the test? Isn't it better to have the kid show that he can get that information any time he needs it?

But I do have a problem with this:

At Ensign Intermediate School in Newport Beach, Calif., seventh-graders are looking at each other's hand-held computers to get answers on their science drills.

It's OK in my book to look up a formula. How often in your life are you going to need the ideal gas law? (PV=nRT, for those of you who were going to look it up.) But the student still needs to be able to figure out how to plug values into that formula and get results. Some things you have to struggle with to learn. You have to pull it out of yourself. Somebody else can show you what they did but it's not the same at all.

And then this reminds me one of the things that I observed during my daughter's odyssey through middle and high school that I thought was a really Bad Thing.

The changes -- and the debate they're prompting -- are not unlike the upheaval caused when calculators became available in the early 1970s. Back then, teachers grappled with letting kids use the new machines or requiring long lines of division by hand. Though initially banned, calculators were eventually embraced in classrooms and, since 1994, have even been allowed in the SAT.

Calculators are one thing. Graphing calculators are something else again, and wisely, they are not allowed on the SAT. Early in the year that F took Algebra II, I picked her up from school one day and she started telling me how frustrated she was. She could not understand how to do her homework. She'd asked her teacher for help, and her friends, and she could not get it. She was just about to the point of tears telling me this.

I took her to Starbucks and got us drinks, and then I said, "Show me." F pulled out her book and her notebook and her TI graphing calculator and started trying to do the homework. I realized right away that her mental focus was splintered between trying to understand the math concepts, and trying to make her calculator work the problem. In the example problems, the book even approached the whole thing in terms of making the calculator do the work. (Do they get kickbacks from TI?) Some people can learn step-by-step instructions when they don't make any sense, but not my F; she has to understand what she's doing. (I think that's a good way to be.) I shoved her calculator aside and took a piece of paper and drew an X and a Y axis, and started working the examples that way. Here's the y-intercept. Here's the slope. Shazam. Now, here is what this graph is telling you about the relationship of these things in the problem. Well, the light bulb went on. She worked all of her homework problems, with increasing speed and confidence, drawing the graphs on a piece of paper like I did. We checked the answers to the odd-numbered problems in the back, and they were all correct. Once she was done, I pulled the calculator back over and said, "Now make the calculator do it." Now that she understood what it was the calculator was supposed to do, that was a piece of cake.

I think teaching kids math by way of those graphing calculators is a big mistake. Math teachers that I've spoken to don't agree. They say that they can introduce advanced ideas more quickly because the kids aren't spending time plotting points and drawing graphs. But for some kids (like mine) I think that's where the learning occurs. I think it's harmful to rush past that process. It splits off the kids who can get all that stuff intuitively, which probably describes the math teachers and is why they don't see the problem, and it makes all the other kids think that they just aren't good at math. Unless they have parents who can teach them at home.

It's just like people who started doing chromatography after the chromatographic software became available. I started doing gas chromatography back in the day when you had a strip chart that moved at a constant speed, and a little pen that drew the chromatograms on it as it went; you made an injection and wrote on the chart what you were injecting, and later drew your baselines and measured retention times and peak heights with a little six-inch plastic ruler. (We used millimeters, of course.) I'd never want to go back to that, but I think people who never did it that way tend to think there's something magical about the way the software identifies and measures those peaks and manipulates the data. I wonder if they really understand what's happening.