First, a little background. I strongly encouraged my students to get a graphing calculator for Algebra, but it is not a requirement. Just over half of them have one right now. The math department has a partial set of graphing calculators that I can check out if they're not being used, and they also have a set of 15 laptops running Windows XP.
I'm beginning to plan the lessons where we'll start looking more closely at linear relationships, including tables of values and graphing (in addition to equations). One of the things I hope to do is intro it to them as a recursive routine and hopefully help them see the connection between the rule, the table of values, the graph, and ultimately the slope of the line. The activity I'm planning right now has them use a graphing calculator and a recursive routine to help them generate points, then they graph those points by hand and we look at the relationship between the problem, the points, and the graph.
I could definitely do this with graphing calculators. Since the students are working in groups, the students have enough graphing calculators that every group could have one, and the math department has some that I can bring in to supplement in case any group doesn't happen to have one that day. But as I was planning on what I was going to show them on the projector/Smart Board, I found myself creating it in Geogebra (which is just a fantastic program, even though I don't know it very well yet).
I find it easier to use Geogebra than the TI Smartview that emulates the calculator on the screen, and I certainly like the clarity of the graphics much better.In Geogebra, I use the spreadsheet function to create the equivalent of lists and/or recursive routines on the graphing calculator, then use that to plot the points on the graph. I can then add colors to the graph, easily change my scales and views on the axes, and just manipulate the heck out of it.
I think initially the push toward graphing calculators was because they were relatively inexpensive (compared to computers) and portable, and therefore it was more realistic to expect students to purchase them, or for math departments to get class sets of them. Therefore I was assuming I was going to choose graphing calculators over computers for activities such as this. But now as the cost of devices has come down, and as more and more students have close-to-full-time access to some kind of computing device, I wonder if it might make sense to transition back to using computers for something like this. Aren't we now at the point where we can assume that students will most likely (at least as they get older) have access to computing devices more often then they will graphing calculators?
Certainly the idea of a recursive routine works just as well on Geogebra/a spreadsheet as it does on a graphing calculator, and perhaps better as they have to think more about the process. (On the graphing calculator we would probably use the ANS key to generate that recursive routine, which might obscure what was going on. On Geogebra, we would write the spreadsheet formula, which is more . . .ahem . . .transparent.) Geogebra is more flexible than the graphing calculator in many respects, although certainly the graphing calculator does things Geogebra does not. So I wonder if it makes more sense to try to get the laptops and have them use those one per group for activities such as this. (Also keep in mind that Geogebra is free for Mac OS X, Windows and Linux, so no cost to schools or to students if they want to install at home. It runs fine on our netbooks running Linux.)
So, this is a long-winded wind up to the point of this - what do you think? Does it make more sense to use graphing calculators, or transition to using computing devices for math learning/instruction/exploration? I'd love to hear your perspective, whether it's as a math teacher, an ed tech-type person, or some other interested reader.
I wonder what percentage of TI calculators are sold to non-students? 5%? 1%? Even less? My point: for most computing tasks awaiting your students in the future, they're going to use a computer, tablet/pad device, or phone. With those devices getting cheaper and more portable, TI must see that their days of dominance are numbered.
ReplyDeleteOn the other hand, education doesn't move as quickly as technology. Graphing calculators are allowed on some standardized tests and the devices above are not. It's not a good reason to only allow graphing calculators, but it's a reason.
I realize there are some significant impracticalities with this, but have you considered being tool-agnostic? Allowing students to explore multiple tools and compare their pros and cons is a very authentic experience. Perhaps you could just try it on select days, and make other days calculator-only when you need to focus on the capabilities they might need on future exams.
Raymond - Thanks for your thoughts. I truly think the test one isn't as good an excuse as most people think it is. Those standardized tests have to assume that students don't have graphing calculators when they take the tests. Therefore the questions they ask aren't designed to take advantage of the graphing calculator's capabilities. So while a really savvy students might be able to take advantage of those capabilities to help them answer some of the questions better on a standardized test, my guess is that a really savvy student would do just fine anyway. (And, if I'm wrong about the questions not taking advantage of the graphing calculator's capabilities, then we have a more serious problem - inequity in the tests and the tests would not be valid.)
ReplyDeleteWhile I certainly wouldn't force a student to use one or the other if they had access to them, my problem with providing them both and exploring is the same old one I'm fighting on lots of fronts - time. I just don't have the luxury of time to let them explore all this stuff (or at least that's what I'm telling myself).
I have a class set of graphing calculators and I'd use GeoGebra any day of the week and twice on Sunday. I think it just lends itself to the kids playing with the math and allows them to test their inductive reasoning regarding why the math works the way it does. Not sure if I have anything more pedagogically sound than that, though.
ReplyDeleteI think once students leave school they will probably NEVER uses a graphing calculator again. BUT everyone will have access to a computer. I think that using Geogebra instead of graphing calculator is a great move. Then, in the future, they can just download Geogebra onto whatever computer they have access to and they have the tools they need to solve problems.
ReplyDeleteWe need to be teaching and using the tools that our students will have access to in the future. The graphing calculator is OLD technology that is still trying to hang on.
With the Programming Language Python (a relatively easy language to learn and use) they could create a function (idea prompted by Wikipedia):
ReplyDeletedef fibonacci(n):
if n ==0: return 0
else if n==1: return 1
else: return (fibonacci(n-1) + fibonacci(n-2))
I have found with my students creating their own calculators (e.g. writing an algorithm that adds two fractions together). Even if Python isn't used. Pseudocode is helpful in determining the steps and understanding the underlying processes.
Why not do something more concrete? For example: Students mass candies like M&Ms, Tootsie rolls, Kisses, etc. Keep adding 1 candy to the electronic balance, record the mass and number of candies in a data table. Repeat for a different candy.
ReplyDeleteGraph the data. What does the slope mean?
Put a cup on the balance to put the candies in and repeat. What is different about data and the graph? What is the same?
Use the equation of the line, the graph, and the data table to predict how many candies are in a cup, given the mass. Use the equation of the line, the graph, and the data table to predict the mass of a cup w/ candy, given the number of candies.
You could do this with paper clips, poker chips, metal nuts, nails, etc., instead of candy.
You could hook them at the beginning with a perplexing question like "How many candies in the cup?"
-Frank Noschese
@Frank LOVE that activity idea exploring the slope concept!
ReplyDelete@Karl I agree that using geogebra is far more powerful and practical in terms of it's "after high school" usability and as a learning aid.
For me the question is really about assessment. I think students should have access to the same tools they use on assessments as they have in their learning experiences. So, if you're use geogebra exclusively will they have access to geogebra on your assessments? The state assessments? Are they more likely to be able to use a graphing calculator?
One other thought: what's happening in their mathematical lives in the next few years? Will their math classes (or teachers) after yours require the use of a graphing calculator? Might you be doing them a favour by teaching them how to use it now?
Frank - Thanks for the ideas. While perhaps not as "concrete" as your examples, we will be approaching linear equation through a variety of semi-real examples (distance/rate/time with several different scenarios, calories burned, alcohol blood levels, etc.), but perhaps I'll see if I can borrow some balances from the science department.
ReplyDeleteDavid - Thanks for your thoughts. What kind of access do you have to computers in your classroom?
ReplyDeleteOh, I'm pretty sure my students already think I give them enough to do, so I think I'll skip the twice on Sunday part.
Darren - That's a tough question. Certainly on state assessments they can have a graphing calculator (as long as it's not too good of one) but, as I said in an earlier comment, I don't think a graphing calculator will help with many (if any) of the questions on the exam.
ReplyDeleteAs far as future courses at my school, that really depends on what teacher they get. Very little computer use, and some graphing calculator use depending on teacher. But part of my thinking here is modeling for other math teachers in my building, assuming I can make a computer with geogebra and other stuff really work well with/for students. I really see that as the direction we're going.
Coming from IB schools, a graphing calculator is a necessity an expected for those exams. It is a tool with a learning curve so I do like to use it in class, but I have also used Autograph and Excel and other math programs depending on what we are exploring. I do like the TI for basic stats stuff in terms of math modeling.
ReplyDeleteFor what it's worth, I like the portability of the the graphing calculator. An iPod Touch with the right set of apps would probably be good too.
ReplyDeleteThis Graphing Calculator is AWESOME!!! =)
ReplyDeleteYay, four years late!
ReplyDeleteGraphing calculators are good. Obviously something like a laptop/netbook is going to be a more robust device, but I feel like it's a little bit silly to question the efficacy a little microcomputer with solid algorithms (I wouldn't trust your iphone for anything more complicated than basic addition/subtraction/mulitplication even if you get a bunch of neato graphing apps) just because there are neater looking toys with neater looking applications. I agree that the Texas Instruments monopoly needs to go (and that those new-fangled pretty-screen calculators people are selling nowadays are crap) but I can't stick my laptop in my pocket and whip it out everytime that I start wondering about what that function would look like if I were to plot it. I love gnuplot and matplotlib, but sometimes it's a damned headache to do something which I could have gotten done on my grapher by setting a variable. I do hate TI-Basic though. I want me an HP :/
I'll also say that students should know how to use abaci and slide rules. And I don't mean spend a week using it and never touch the things again. I mean that they should actually be integrated into the way in which we teach math to little kiddies.
ReplyDelete