Math or Technology: Take Your Pick (Sarah Natividad)
Recently Utah schools have been given an F for technology use in the classroom (or lack thereof). This is one area I hope Utah continues to fail in. Technology has been touted as a fabulous tool for teaching math and other subjects, but it’s not. Technology teaches technology; you still have to learn math separately if you want to know math too.
The misconception that technology can be part of learning math stems from the fact that there are calculators that can produce the same numerical result as mathematical calculations. Sadly, calculator use does not produce the same cognitive result as actually learning math. Just because a student can produce the answer to 23+56 on a spreadsheet does not mean he has mastered double-digit addition, any more than the ability to microwave a TV dinner constitutes knowledge of cookery. Too many curricula nowadays conflate the ability to get answers with the knowledge of how the mathematics works. And too many teachers fall back on technology use as a crutch, to help them “teach” students who for whatever reason are having trouble grasping mathematical concepts. I’d be rich if I had a dollar for every student who’s been in his college professor’s office, trembling with fear of failure at remedial algebra, for no better reason than that he can’t add fractions to save his life and is now being asked to add rational expressions using the same method as fractions. Only, the rational expressions won’t go into his calculator, see.
If all we want is the ability to get answers fast, then by all means let’s train our kids to be calculator or spreadsheet jockeys. Let’s give prizes to the one who can push the buttons the fastest or has the lowest error rate. But if we really do want our kids to go on to learn algebra and calculus and other nifty stuff, we had damn well better put some actual math into their heads, because the skills developed by the actual learning of arithmetic (the old-fashioned way) are the ones algebra builds on. Trying to build knowledge of algebra on the false foundation of technology-assisted arithmetic is like trying to build a five-story building on mobile home jack stands. Oh, jack stands work well enough– if all you want to hold up is a mobile home. You can live quite comfortably in a mobile home, but it will never be the impressive edifice we should be aspiring to construct.
We need to decide exactly what it is we are trying to build here, so that we can put the appropriate foundation underneath it. Are we trying to get our kids through as many hoops called “algebra” and “geometry” as we can, or are we actually trying to teach algebra and geometry? Are we aiming to produce calculator jockeys, or are we aiming to produce people with the ability to think logically? If the former is all that matters, then give up trying to make high school students pass a test of actual algebra, because they’re never going to be able to do it. And if it’s the latter we want, we ought to be able to weather the opinions of those who give us F’s for lack of dependence on technology. We can laugh at them all the way to the bank when it’s time for actual performance to be measured.
Sarah Natividad lives with her husband and four children in Utah, where she teaches math at the post-secondary level and runs a small business. She can be found blogging at Organic Baby Farm.
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Technology: “It Can Do More Harm Than Good”
Ryan Boots:I’ve been something of a cheerleader on the use of new media in the classroom, principally in the form of digital textbooks. But similar to what we’ve already seen with the calculator, such technology has the potential to inflict…
I’m the first one to admit that the vast majority of research-level mathematics is (currently) useless to everyday life. I beg to differ with Mr. Flener, however, that basic fractions and algebra are useless and should be replaced with calculator-based “explorations” of the type he describes.
The calculator is a Big Black Box With Buttons to anyone who doesn’t understand the mathematics behind it. Lessons like the one Mr. Flener describes only serve to reinforce that misperception. If you already know about compound interest, then a lesson like that one will definitely deepen your understanding. But if you don’t already know about compound interest, all it is is an exercise in button-punching that only serves to make numbers go up and down. What I am decrying is the outright _substitution_ of lessons like that one for lessons that would lay the foundation for that one.
Because we no longer teach the underpinnings of concepts like compound interest, mathematical knowledge is actually being LOST. If you don’t believe me, look up “conic sections” in any current Algebra 2 textbook. In most of them, you won’t find it, even though the properties of conic sections underly our communication technology, as well as lay the groundwork for future calculus coursework. There are no longer any teachers who understand it well enough to implement it in a curriculum, let alone enrich the experience of our future mathematicians and scientists, because the current generation of teachers has been raised solely on the thin gruel of “applications” that Mr. Flener seems to think constitutes a full enough meal. Meanwhile, a feast of real mathematics rots on the banquet tables.
Condorcet said, “[We want] to insure that in the future all citizens can be self-sufficient in all calculations related to their interests; without which they can be neither really equal in rights… nor really free.” Emphasis on the word ALL. All calculations, not just the few Mr. Flener thinks are interesting enough to hold his attention. If Mr. Flener thinks fractions and algebra are not useful, I can only imagine it must be because he does not know these topics well enough to get any use out of them himself. I know I use them every time I sew, cook, or shop for groceries.
Actually, Sarah Natividad doesn’t really get it does she? No one has ever said that the use of technology will eliminate a kid’s need to understand mathematics. It is the lack of “understanding” that has entombed mathematics education. That student “trembling with fear” has been around for as long as I have been in education. Not understanding the concepts of adding rational numbers, much less rational algebraic expression is not a new phenomenon. (Incidentally, you can add rational expressions with your calculator, if you have the right one.) Technology need not be a crutch allowing kids to avoid understaning. I should be a tool that enhances understanding. Let me share an example from one sixth grade math lesson that used spread sheet technology. Suppose your bank give 2.75% interest (per year) on money invested. Kids can either use 2.75% or they may choose to use the decimal equivalent, .0275. How much interest will they earn in a year if they invest, say, $100. Now with the technology, they can vary the amount they invest to any amount, say $437. Now comes the fun part, compounding it. Put that interest back into the bank and apply the interest rate to the new amount. First, they let the machine just do the calculations, then you suggest a new approach of using a formula approach (darn, that’s algebra isn’t it). Just by putting an “=” sign into the input box, it becomes a formula. Here’s the algorithm the kids seem to understand. First put the initial investment into the A1 cell. (“Let’s all use the same amount so it is easier to talk about later. Michael, how much do you want to invest? $137? Ok, everybody put 137 ito A1.”) Now in B1 put the interest rate. (“Shall we type in the ‘%’ symbol or use the decimal? I like the decimal because you really use that to multiply. Be careful where you put that decimal point. Put .0375 into B1.” Now we are going to let the machine do most of the work. Type =A1*B1 and see what happens when you hit enter. Son of a gun. It is a pretty ugly number, but we want it to be an amount of money, so let’s …”
Within a 45 minute lesson, the kids have learned about compound interest, how the amount of intitial investment affects the amount at the end of 10 years, they can even add an amount, say $100 at the end of each year, etc. They can have a much deeper understand than most adults have about compound interest.
Furthermore, they are engaged in a rich exploration of algebraic concepts. What does it take for a teacher to engage his/her kids in such a lesson? It takes a pretty good understanding of the mathematics and the technology. If Sarah thinks that these kids are losing out because they can’t add 3/7 + 5/11, she is quite wrong. If she ever saw the movie “Peggy Sue got married,” she might remember the line in which Peggy Sue ends returning to her high school days and is sitting in her Algebra class. Her comment is classic, and in a sense reflects the world that Sarah wants to maintain. “I never realized when I was here (in her algebra class) how useless this subject is.”
Much of Sarah world is useless, not because we don’t need to know what rational numbers are, but because we can do so much, much more with technology. Sarah’s view is highly limiting.
Math or Technology: Take Your Pick
Sarah Natividad:Recently Utah schools have been given an F for technology use in the classroom (or lack thereof). This is one area I hope Utah continues to fail in. Technology has been touted as a fabulous tool for teaching math…