From: lane@math.utexas.edu (David Lane) Newsgroups: sci.math,misc.education Subject: Re: Teaching H.S. Math #3 Date: Sat, 12 Aug 1995 21:54:44 Organization: UT Math Dept In article donp@cv.hp.com (Don Pettengill) writes: >I have seen little of worth, and a good deal of harm, come from >calculator availability in the classroom. Here's the result: a typical TA session... TA (trying to explain how to graph a rational function): "Explain to me ... What does it mean for a function to have a vertical asymptote?" Student: "Uh... I dunno" TA: "Well OK, let's do something that ought to be familiar. Let f(x)=1/x. Do you know what the graph looks like?" Student: "It..uh...well.... I dunno" TA: "Not to worry. We'll just figure it out. Lets do the plug and chug thing...Make a table of values...Figure out what's going on. What do you get if you plug in x=1? Student: "One..." TA: "Alright. How about 2? Student: "One half" TA: "Good. Now what happens as x starts getting really big?" Student: "Uh... I dunno" TA: "Well lets do some more points: How about x=100?" Student (silence) TA: "What is 1 divided by 100?........ .................................................. Oh come on. Don't reach for your calculator! In decimals... What is one one hundredth?" Student: "OH...Uh Yeah, sure.... it's .01" TA: "So the numbers are getting small?" Student: "Yeah I guess" TA: "Alright. Now lets go the other way. What happens as x starts getting close to zero?" Student: "It goes to zero?" TA: "Well...No, actually...Lets do more values [sigh]. What do you get if you plug in 1/2?" Student: "Can I use my calculator?" TA: "No! ... OK. Listen: Picture a piece of the number line between 0 and 1, so it's some physical line segment sitting there [Drawing on the board now] Here's 1/2 sitting here. It's this distance here...So how many of these things fit in this thing?" Student: "Two..." TA: "OK..So 1 divided by 1/2 is...?" Student: "Uh....." TA: "How many times does 1/2 [pointing] "go into" 1 [pointing]?" Student: "Oh...Two.. Right. I've got it.... Say... is this related to that law that tells you to flip the -- like -- the denominator -- like, upside down, and bring it upstairs -- or have I got it backwards? It never made sense anyway. TA: "Yes that's right. If you've got a fraction like [blah blah blah] ..... Great! Now that we know how to do that, what would we get for x=.01? What is 1/.01? Student: "Well I don't know how to do that.." TA: "Use the rule you just told me" Student: "Uh ... Yeah... But it's in decimal" TA: "Right...But we know this! .01 is one over what?" Student: [silence] TA: "We just did this right? It's 1/100? ....... Look: Do the number line thing again... Here's .01. This little bitty piece here...How many of these would fit into here? Lots of them, right? So what is it?" Student: "It must be one hundred" TA: "That's right. Now lets plot all these points... [5 minutes of pain and agony deleted] TA: ".....So it looks like the graph is doing this. As x gets closer and closer to 0 the values of f get bigger and bigger so it goes up here like this [etc, etc].....In fact, does it even make sense to plug in x=0?" Student "No" TA: " So here's the graph...Just looking at it, does it have a vertical asymptote?" Student: "Uh yeah, at y=0" TA: "You mean x=0, I think.." Student: "Do I?" TA: "Well, totally separate from all this...If I were to ask you to draw the line y=2, for example... [10 more minutes of pain and agony deleted] TA: "So the point is that vertical asymptotes have to do with the behavior of f as x gets close to a point where f is not defined..... You know, I'm kind of surprised that you haven't seen this function before." Student: "Well now that we've done it, it does seem kind of familiar" TA: "What kind of math did you have in high school?" Student: "Oh I did really well in math in high school. I had pre-calc" TA: "You didn't do asymptotes there?" Student: "Oh yes, we did lots and lots" TA: "So how did your teacher explain what a vertical asymptote was?" Student: "It was easy back then. I really understood it. To find the vertical asymptotes you graph it on the calculator and the asymptotes are the little dotted vertical lines you get..." I won't even try to talk about doing the graph of y=x/x! Dave ============================================================================== From: Kurt Foster Newsgroups: sci.math Subject: Tales from the Front Date: 14 Jun 1999 16:40:40 GMT The following story is true. Information has been left out, to protect the innocent, and the guilty. I'll call it: ------------------------------------------------------------------------- THE JUNIOR AND THE JANITOR Once upon a time, long, long ago, in a major American University far, far away... A college junior was complaining to her Teaching Assistant about points taken off on an exam question - "But I got the right answer!" The T.A. tried in vain to explain that the "right" answer was only because of luck - the argument leading to the answer was not valid. An unfamiliarity with English was not helping matters, so the T.A.'s officemate, another T.A. and a native of the U.S.A., tried to help out. He went to the blackboard and chalked the equatiion X^2 = 25 which he then "solved" by cancelling the twos, leaving X = 5, and then asked, "Now, is that method valid?" The junior muttered, "Let's see, 5 squared..." and replied, "Five squared IS twenty-five - yeah, that's right!" He gave up trying to explain the trouble with "right" answers by wrong methods. Later that same day, (a LOT later - it was 10 at night, actually) the janitor walked into the office. The defeated office-mate was working, but took a break and told the janitor, "Hey, can you answer a question for me?" The janitor asked incredulously, "A *MATH* question? I couldn't pass a math course to save my life! Why do you think I'm a janitor?" Undeterred, the T.A. chalked X^2 = 25 on the blackboard, and then cancelled the twos, leaving X = 5, and asked, "How do you like THAT method of solving that equation?" The janitor looked thoughtfully at the board for a moment. Then his eyes widened, and he shouted, "WAIT A MINUTE! YOU CAN'T DO THAT!!" ------------------------------------------------------------------------- Any other Tales from the Front? ==============================================================================