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Entries from CalculatorsAndComputers
GirlVsCalculator 11 Jun 2005 - 03:28 CarolynJohnston
Calculators aren't actually so reliable. Actually, if you were to compare calculations done with calculators to those done by humans one to one, I'd bet that calculators produce a much higher error rate; it's so easy to mess up and hit the wrong button, or to get a complex sequence of operations in the wrong order. My husband has a story that illustrates one of the consequences of overreliance on calculators in math classes. He was teaching a probability unit in a freshman college course in finite math, and most of the kids were struggling with it. These kids came into class every day clutching their calculators fearfully, as though they were talismans to ward off math demons (embodied by their professors, I guess). The unit test in probability happened, and one of the problems was as follows:
MathHorrorStories 09 Jun 2005 - 22:43 CatherineJohnson
I've been keeping a collection of math horror stories for awhile now. (So please! Send yours!) I got started on this little edu-sideline thanks to a friend of mine who's married to an architectural engineer. In grad school, she said, he would do pages and pages of calculations by hand, in teeny-tiny little print. These sheets would come back to him from his professor with equally tiny little red-pencil corrections scattered across the page. So today her husband is hiring students fresh out of grad school to work for him. These are grad students; they have MAs in architectural engineering. None of them does calculations by hand, ever. They use architectural engineering software. They'll bring in printouts of their work for him, and he'll look at it, spot a bunch of errors, and say, 'This is wrong.' They just stare at him. They have no idea what he's talking about, or where or what the errors might be. These are architectural engineers, folks. They build stuff.
TheCraftOfMath 04 Nov 2005 - 00:11 CarolynJohnston
Teaching mathematics as a craft seems to be waning in favor in elementary education. As evidence one might consider a quote from the infamous Steven Leinwand paper from Education Week, "It's time to abandon computational algorithms":
Shouldn't we be as eager to end our obsessive love affair with pencil-and-paper computation as we were to move on from outhouses and sundials? In short, we know and should agree that the long-division "gazinta'' (goes into, as in four "goes into'' 31 seven times ... ) algorithm and its computational cousins are obsolete in light of everyday societal realities.He claims that the requirement to be able to do math on pencil and paper has been rendered meaningless by the calculator:
Today, real people in real situations regularly put finger to button and make critical decisions about which buttons to press, not where and how to carry threes into hundreds columns. We understand that this change is on the order of magnitude of the outhouse to indoor plumbing in terms of comfort and convenience, and of the sundial to digital timepieces in terms of accuracy and accessibility.And so, in spite of Leinwand's accusation (in the same paper) that school districts make changes only in geological time, we are currently engaged in a huge cultural experiment testing his theory that kids can gain a knowledge of math without having to put pencil to paper (although, as I mentioned in this post, in some of the constructivist curricula kids are spending much more time learning to multiply than they would have in a classical curriculum). But putting pencil to paper is part of what I would call the craft of mathematics. I think you just don't get intimate enough with numbers and symbols by just watching them flash by on the computer or calculator. I've seen it over and over in students at the college level; the more they've relied on their calculator, the less of a feel they have for numbers and mathematics, and the less able they are at problem solving. They may feel that they understand you while you lecture, but when it comes to actually doing math, to getting the answers themselves, they can't do it; they're impotent. I may be misreading the situation. It may be that the same kids who have trouble with problem-solving at the college level had trouble learning the standard algorithms for computation, and therefore rely more heavily on their calculators. But we do have parallels in the employment world -- experienced engineers who find that junior employees rely too heavily on the answers given by their computer models, and designers who find that their juniors who have used CAD software have a weakened sense of design. We don't know whether the new tools are enabling people to enter these fields who wouldn't otherwise have cut the mustard, or whether the tools are actually weakening the skills of able people. Until we understand why kids who have relied too heavily on calculators for basic computation can't do math, and what is truly essential about the process of teaching kids to do math, we would be wise to continue making huge curricular changes in geological time. Afternote: in the spirit of knowing your opposition, here is a link to the Leinwand paper. You have to register at the Edweek site to access the article.
swoop and swoop
notes on integer, subtraction, & absolute value study sheet
Wayne Wickelgren on why math is confusing, & Carolyn on procedural memory
KUMON & hands-on math
More Singapore math
Pencil and paper
The craft of math
SlideRules 30 Jun 2005 - 21:26 CarolynJohnston
Apropos of our discussion of Steven Leinwand's recommendation that we quit using pencil-and-paper computations because they are passe, it occurred to me today that we do have a sort of a precedent. When my Dad was in school, everybody used slide rules to do logarithms (also multiplications). You needed them to do the computations, but they were also a kind of a math manipulative (as one could argue, I suppose, that pencil-and-paper computations are!). Over time, learning to use slide rules, you learned about how the logarithmic scale worked. By the time I got to school, slide rules were gone. I don't recall having big troubles learning about logarithms, but judging from my dealings with my students from both remedial and college-level courses, I was exceptional that way: nearly everyone had trouble with them, and even those who could manipulate logs correctly didn't have any feeling for how they behaved or what they were good for. So here's my question: in olden, pre-calculator days, when people used slide rules to do logarithms, did they understand logs better -- the point of them, and how to do computations with them? In other words, did we give up an important learning mechanism when we gave up using slide rules, and is Leinwand proposing that we make the same mistake with all the other types of computations?
TwoMathEdBlogs 27 Jul 2005 - 19:26 CatherineJohnson
Stephanie just sent me a link to a fascinating list of prerequisites for college math, which includes a terrific Comments thread, at Tall Dark and Mysterious, a blog written by "Twentysomething curmudgeon seeking employment teaching college math in BC." And btw, these are not prerequisites for a serious college math course:
A year ago, I would have posted that list under a heading more along the lines of “Things Students Should Know By Grade Nine”, but alas, experience as extinguished such optimism on my part.
This is long, but it's so valuable I'm quoting the entire list, which I'll probably 'archive' over on....the 'math lessons' page? Another Content Question for the folks at Information Architecture, Inc. (Definitely read the Comments section as well):
Based on my experiences, students graduating from high school should, in order to succeed in even the most basic college math classes: 1.Be able to add, subtract, multiply, and divide fractions. Moreover, they should understand that the horizontal bar in a fraction denotes division. (Seem obvious? I thought so, too, until I had a student tell me that she couldn’t give me a decimal approximation of (3/5)^8, because “my calculator doesn’t have a fraction button”.) 2.Have the times tables (single digit numbers) memorized. At minimum, they should understand what the basic operations mean. For instance, know that “times” means “groups of”, which will enable them to multiply, for instance, any number by 1 or 0 without a calculator, and without putting much thought into the matter. This would also enable those students who have not memorized their times tables to figure out what 3 times 8 was if they didn’t know it by heart. 3.Understand how to solve a linear (or reduces-to-linear) equation in a single variable. Recognize that the goal is to isolate the unknown quantity, and that doing so requires “undoing” the equation by reversing the order of operations. Know that that the equals sign means that both sides of the equation are the same, and that one can’t change the value of one side without changing the value of the other. (Aside: shortcuts such as “cross-multiplication” should be stricken from the high school algebra curriculum entirely - or at least until students understand where they come from. If I had a dollar for every student I ever tutored who was familiar with that phantom operation, and if I had to pay ten bucks for every student who actually got that cross-multiplication was just shorthand for multiplying both sides of an equation by the two denominators - I’d still be in the black.) 4.Be able to set up an equation, or set of equations, from a few sentences of text. (For instance, students should be able to translate simple geometric statements about perimeter and area into equations. ) Students should understand that (all together now!) an equation is a relationship among quantities, and that the goal in solving a word problem is to find the numerical value for one or more unknown quantities; and that the method for doing so involves analyzing how the given quantities are related. In order to measure whether students understand this, students must be presented, in a test setting, with word problems that differ more than superficially from the ones presented in class or in the textbook; requiring them only to parrot solutions to questions they have encountered exactly before, measures only their memorization skills. 5.Be able to interpret graphs, and to make transitions between algebraic and geometric presentations of data. For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x (y) when y [x] is set to zero in the function”). 6.Understand basic logic, such as the meaning of the “if…then” syllogism. They should know that if given a definition or rule of the form “if A, then B”, they need to check that the conditions of A are satisfied before they apply B. (Sound like a no-brainer? It should be. This is one of those things I completely took for granted when I started teaching at the college level. My illusions were shattered when I found that a simple statement such as “if A and B are disjoint sets, then the number of elements in (A union B) equals the number of elements in A plus the number of elements in B” caused confusion of epic proportions among a majority of my students. Many wouldn’t even check if A and B were disjoint before finding the cardinality of their union; others seemed to understand that they needed to see if A and B were disjoint, and they needed to find their cardinality - but they didn’t know how those things fit together. (They’d see that A and B were not disjoint, claim as much, and then apply the formula anyway.) It is a testament to the ridiculous extent to which mathematics is divorced from reality in students’ minds that three year olds can understand the implications of “If it’s raining, then you need an umbrella”, but that students graduating from high school are bewildered when the most elementary of mathematical concepts are juxtaposed in such a manner.) 7.More generally: students should know the basics of what it means to justify something mathematically. They should know that it is not enough to plug in a few values for x; you need to show that an identity, for instance, is true for all x. Conversely, they should understand that a single counterexample suffices to show that a claim is false. (Despite the affinity on the part of the high school text I am working for true/false questions, the students I am working with do not understand this.) Among the educational devices to be expunged from the classroom: textbooks that suggest that eyeballing the output of a graphing calculator is a legitimate method of showing, for instance, that a function has three zeroes or two asymptotes or what have you.
Another blog by a college calculus professor: Learning Curves
GraphingCalculatorsBadBadBad 05 Oct 2005 - 17:11 CarolynJohnston
I just got a nasty wake-up call. For those of you who just came in -- Colin is my stepson. He's 17, a junior in high school in an International Baccalaureate program, no intellectual slouch. He's got a math book called "Precalculus: Graphical, Numerical, Algebraic" by Demana and Waits, who are known for writing very 'technology-centric' books that make heavy use of tools like graphing calculators (with the crack investigative skills this gang has become known for, it took us no more than ten minutes to determine that Demana and Waits have financial ties with the Texas Instruments company, the dominant manufacturer of graphing calculators. In fact, Vlorbik has liveblogged one such technology-in-the-classroom lovefest featuring Demana and Waits; his article, with updated links, is here). Yesterday, Colin blogged for us about his semi-constructivist high school math class, in which the teacher announced on the first day that the kids would all "be teaching themselves this year." According to Colin, his math classes have disintegrated into desperate question-and-answer sessions in which the kids try to get a clue about the homework that they failed to do the night before. Now, this afternoon I was looking over a worksheet Colin had done about the domain and range (i.e., the set of all valid 'inputs' and 'outputs') of some common functions. He had correctly written down that the domain of the function f(x) = ln(x-3)+2 is the interval (3, infinity), but he had written that the range of the function was [4, infinity), which was distinctly odd, since there is nothing special about the number '4' in connection with f(x)=ln(x-3)+2 ("ln" stands for the natural logarithm function). "How'd you get this 4 here?" I asked him. Then I noticed something even funnier on the line above that one; Colin had written that the range of the natural logarithm function was [2, infinity). "Wait, the range of the natural logarithm function is all of the real numbers," I said. "How'd this 2 come into it?" "Look at this graphing calculator," he said. Sure enough, the d*mned graphing calculator was cutting the range of ln(x) off at 2. It's not the calculator's fault; it's unavoidable that it will cut the range off somewhere, because calculator displays are limited by the resolution of the screens. It's not a problem if you're an engineer with a good understanding of the basic properties of the natural log; but in this case, it's being used as a TEACHING tool, and it's teaching Colin something completely incorrect. This is a kid who is having no trouble with the usual intellectual challenges that domain, range and natural logarithms offer. That's a big deal, because those are notions that ALL of the students who ended up in my college algebra classes struggled with. He could easily learn this stuff right, but he is learning it wrong. Why? Because he's got the GRAPHING CALCULATOR in his face all the time -- the authors of this textbook push it hard (and yes, it's a Texas Instruments calculator). The problem is made worse by the fact that his teacher is not paying any attention at all to what the kids are actually learning. OK, before this, I didn't have a set opinion of graphing calculators and other forms of technology-in-the-classroom. There's been argument for years that weak students are overly dependent on their calculators, but I wasn't sure that was a reason to ban them. And then I get this wake-up call today; calculators are actively misleading even the strong students. Even if you have a good teacher working with the kids on these problems, catching their misconceptions early, the graphing calculator is going to be throwing a completely unnecessary extra layer of confusion into the mix. I am now officially against Technology-in-the-Classroom, and by extension the curricula that push them. A graphing calculator might be a good tool once you know what you're doing, but it's a terrible tool for teaching.
TomLovelessOnMathAchievementTrends 14 Oct 2005 - 22:28 CatherineJohnson
Tom Loveless, director of the Brown Center on Education Policy at the Brookings Institution, told a U.S. Department of Education "summit meeting" on math education in February that only in a couple instances - the ability of 13- and 17-year-olds to compute percentages - did students nationwide register gains in the 1990s. Discussing the performance of 9-year-olds in addition, subtraction, multiplication and division of whole numbers, he said: "All four areas reversed direction in the 1990s, turning solid gains that were made in the 1980s into losses. Not only that, but the declines came from levels that weren't very high at the beginning of the 1990s - certainly not at a level that is acceptable for such fundamental material."
Division flares up over math by Alan J. Borsuk SENTINEL JOURNAL 10-4-02
CoolMathAnimations 31 Oct 2005 - 16:06 CarolynJohnston
Check out this page of cool math animations by Lingfa Yang. Most of the animations are of constructions of some fairly obscure algebraic curves, but they are cool anyway. This one, the "Pursuit Curve", I've never heard of, but from the animation you can see pretty clearly what a Pursuit curve is about. The red dot is 'pursuing' the black dot that is moving on the y-axis (the red dot will never actually touch the y-axis though). Does anyone know the significance of this one, the "Logistic Map'? And check out this ellipse one. This illustrates one of the characteristics of an ellipse -- that if you tie a string to the two foci, and trace a pencil around the inside of the string (keeping it taut), then the shape you'll draw will be an ellipse. (hat tip, Bernie)
BooklessClassroom 10 Nov 2005 - 02:58 CarolynJohnston
Via Joanne Jacobs, I came across this article in the Deseret News about a middle school math teacher who has replaced his entire curriculum with activities on elementary math websites.
Jerry Mangus' textbook-less teaching has dazzled the U.S. Department of Education. Mangus, who teaches fifth- and sixth-grade math at Plymouth Elementary, uses only computers to teach fractions and other numerical concepts to kids. He's built computer labs in his school, each of his students has his or her own machine, and their test scores have leaped. For his efforts, the federal education department on Wednesday bestowed Mangus with its No Child Left Behind Act American Star of Teaching Award. The award, for which the department received some 2,000 applicants, goes to one teacher in every state and Washington, D.C. "He's someone who has gone far and above," said Carolyn Snowbarger, director of the department's Teacher-to-Teacher Initiative, who presented the award.Gone over and above? Even if this turns out to be the best thing ever, he's basically ceded his job to these web pages he sends his kids to. He's gotta be putting out less effort than ever before. But this does make me wonder. When Ben was about six, his spoken language was in trouble. He could speak, but there was often up to a 15-second lag between your asking him a question, and him responding. That's way too long for normal discourse. We got him a copy of a computer game called Earobics, which is marketed as an aid for weak readers -- the idea being if a kid can link the written symbol with the sounds he is hearing, he can learn phonics and start reading better. In Ben's case it worked in a sort of inverted way. Very quickly after we began with Earobics, he began responding more rapidly in his spoken language, because the computer would prompt him for a response, and then something unhappy-making would happen if he didn't respond (for example, in one game, a balloon would pop -- he HATED the sound of popping balloons). We decided that the computer was a good tool to use in this case, because no matter how we tried, we humans couldn't be consistent enough in dealing with his unresponsiveness. We'd give him a little more time, or another prompt, before giving up. The computer, conversely, was machinelike and unforgiving; the consequence came down 3 seconds after he'd been cued, invariably. So, in that case, computers were just the right tool. Here's Mangus's explanation for why they've been successful for his students:
The key is the immediate feedback computers provide, Mangus said. Kids don't waste time doing homework wrong, and then feel the frustration of bombing out when they get the answers the next day, Mangus said.Of course, I check Ben's math homework every night, so he gets almost-instant feedback. But I can see where he might prefer instant feedback from a computer, rather than me. And gee, even I might prefer that... But I still believe you've got to get a kid to do math with his hand. Maybe not all the time, but some of the time. And there are also conceptual problems that you run into when you try to get a calculator or computer to do the work of a teacher (check out this post for an example). So, if this guy is doing something right -- and there's so many questions left open here, such as: how is he assured that the kids are covering all the topics they are supposed to get in the grade they're in? -- then how is he getting around the downside of using computers in the classroom?
NegativeStudyComputersInTheClassroom 14 Apr 2006 - 17:09 CatherineJohnson
via This Week in Education, another study showing computers in the classroom don't improve learning:
Taxpayer-supported school computer and Internet giveaways are political gold, but studies have questioned whether they actually help student achievement. This research, presented at the American Educational Research Association's annual meeting, confirms skeptics' doubts. In one study, researchers from Syracuse and Michigan State universities examined a program that gave laptop computers to middle-school students in Ohio in 2003. Preliminary findings are mixed. "Overall, we don't know if it is a worthwhile investment," says Syracuse researcher Jing Lei. About 37% of the children say they stare at the screens for more than three hours a day; a few report more than five hours a day. Parents help kids with homework more often and students' grades benefit slightly, but teachers report more classroom distractions as students check e-mail. And students actually feel distracted: In the first year, their grade-point averages rose modestly, but when Lei and a colleague asked them to estimate their GPAs, students actually believed they dropped. "They felt that time is not used as effectively as before," she says. Laptop giveaways are the latest educational fad; five states either have or will soon have them. More than one in eight school districts have some sort of program in which every child gets a PC. Evidence has shown that computers are finding their way even into the homes and schools of the nation's poorest students. A Tennessee study found that schools serving low-income children had more computers than your typical school — 125 for poor kids' schools vs. 114 elsewhere, and computers in low-income schools often were more connected to the Internet. But using computers, for instance, to teach reading in primary grades actually showed negative results.
I'm going to go out on a limb here and say I don't think any child should be looking at a computer screen more than 3 hours a day. I think Richard Louv is right; U.S. children are suffering from nature deficit disorder. (Temple is a big fan of Louv's book, fyi. I'm ordering a copy.) And it disturbs me that these kids seem to have experienced a drop in confidence. In reality their grades were up; the kids thought their grades were down.
Ed and I have always contributed to the IEF, which has purchased laptops for students. Up until a year ago, I would simply have assumed that passing out laptops to a class of 5th graders was a wonderful idea. Now I realize I don't know whether it's a good idea or not. Irvington kids may be using laptops and computers to become more knowledgeable about the subjects they study. Or they may not — I have no idea! We all need to start evaluating inputs in terms of the outputs that do or do not result. And we need to evaluate the opportunity costs involved in any program or purchase. Opportunity costs are real. If you're doing 'X,' you're not doing 'Y.' Is time alone with a laptop the best possible use of our kids' precious and limited instructional time? I don't know the answer to that, and I wish I did.
Speaking of the outdoors, I'm going to go do Saxon Math on the picnic table. I'm almost done. I've studied my last Lesson and taken my last test. I've finished the Appendix lesson (Roman numerals and base 2) Now I have two investigations left. One on scale factor in volume and one on a proof of the Pythagorean theorem. And that's it!
computers in the classroom
ed technology never fails
another negative study
Steven Jobs on computers in the classroom
-- CatherineJohnson - 13 Apr 2006
StevenJobsOnComputersInTheClassroom 14 Apr 2006 - 20:46 CatherineJohnson
...Steven Jobs, one of the founders of Apple Computer and a man who claims to have “spearheaded giving away more computer equipment to schools than anybody else on the planet,” has come to a grim conclusion: “What's wrong with education cannot be fixed with technology,” he told Wired magazine last year. “No amount of technology will make a dent.... You're not going to solve the problems by putting all knowledge onto CD-ROMs. We can put a Web site in every school—none of this is bad. It's bad only if it lulls us into thinking we're doing something to solve the problem with education.”
computers in the classroom
ed technology never fails
another negative study
Steven Jobs on computers in the classroom
-- CatherineJohnson - 13 Apr 2006
TheBrainExerciser 18 Sep 2006 - 22:48 CatherineJohnson
So instead of working my way through the entire Saxon oeuvre, I could have just bought myself a Brain Exerciser.
Now they tell me.
-- CatherineJohnson - 18 Sep 2006
LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson
I think everyone here knows about Linda Moran's Teens and Tweens blog. I've recently (re)discovered that she has a listserv attached to the blog. I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.
-- CatherineJohnson - 09 Dec 2006