02 Oct 2005 - 20:03

it's an outrage

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.

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This, too, has to be in the Greatest Hits section, wherever that section ends up being.

99% of all parents wouldn't be able to catch this.

Horrifying.

-- CatherineJohnson - 02 Oct 2005

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I'm totally opposed to calculators in the classroom until a child has mastered a concept or procedure, and then I'm more-or-less totally for them.....though even there, I'm not sure. I've worked my way through 9000 hugely complicated computation problems in RUSSIAN MATH without a calculator.

I use the calculator to check my answers.

Lately, towards the end of the book, I've used it a couple of times when I wanted to focus on the concept.....and get through problems....and at this point I feel I've earned it, thanks to my 9000 hardcore pencil-and-paper calculations.

-- CatherineJohnson - 02 Oct 2005

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As a former engineer turned physics teacher who loves technology I have to agree with you. The problem is teachers going technology happy and not understanding the impact it is having. The students must first know how to do the problems without the technology, second that the technology, no matter how good, has limitations and third how to handle those limitations so the technology can ENHANCE the learning process not take the place of it.

A student must realize that a calculator may take the equation x = (1/3)*3 and give 0.999999999999 not 1 and not realize the answer is 1 not 0.999999999999.

The other side of this is that the teachers must also know the limitations of the technology they are using and I am afraid in a large number of cases this may not be true. There are two reasons why. One if the teacher is older they may be afraid of trying to learn the technology and know as little as the students, maybe less, and two if they are younger they were taught using the technology themselves and know no better.

I count myself fortunate. I learned math and science in high school before calculators. The first time I saw a calculator was the middle of my junior year of high school. My parents got me a SR-10 for Christmas. It could add, subtract, multiply, divide and do 1/x. That was it.I remember it was about 85 dollars in 1974. The SR stood for Slide Rule by the way. How many out there remember those. I still have one.

When I went to college a few years later I had a TI-58 and then a TI-59 but knew how to do the math without them. They were just nice to have to get through the math quicker on tests. That was the idea at the time at least with my phyics and engineering professors. They figured the students could get through the math quicker so they could present more and varied problems on a test so that they could test you on more of what they covered.

Now that I am a teacher I am appalled at the lack of math sense of a lot of my students. I have noticed though since I teach at an Catholic all girls college prep school, the students who went to a Catholic grade school used calculators much less in grade school than those who went to public schools. I have been doing my own informal research on this for the past few years.

I teach physics so in my class I do let them use calculators on tests because physics is concepts not math. My tests are about 75% concepts and 25% math type problems so they are using the calculators as a tool not a crutch. I am pleased to report that at my school the math department does not over use the calculators and still expects the students to understand the basics before they introduce the calculator in the class.

I do surprise them though. I will put a problem on the board and then will race them to the answer. I do not use a calculator and I nearly always win. Several reasons. One I know the math cold, two I do things like use g = 10 m/s^2 not 9.81 and I have the sine, cosine, and tangent of 30, 45 and 60 degrees memorized so I always use those in my example problems. It is an attempt to show them that they can rely on thier own brainpower.

This lack of math sense will come back to haunt us. If you remember back when you first started this blog I was the first person to leave a comment. Mine was about the young engineer just out of college who had to write the take off and landing program for a short take off aircraft I was designing. He had a take off roll time of 300 seconds and saw nothing wrong with it. His comment to me was "That's what came out of the computer." I had to remind him that he wrote the code the computer used and the output is only as good as his input. SCARY

Jeff Hetzel

-- KtmGuest - 02 Oct 2005

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Jeff, welcome back! I remember that comment well. Have you been reading the discussions we've had about engineering school? Here are some of the relevant links:

EngineeringSchool
EngineeringSchoolPart2

and, last but not least, TourDeForce.

Hope you'll stick around!

-- Carolyn

-- CarolynJohnston - 02 Oct 2005

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Carolyn,

I have been following it here and over at Joannejacobs blog. I leave comments there under the name Jeff_H. I was the one the who recommended the book "Gone for Good"

I will try to get here more often to see what is going on but with school starting I am busy with 4 science preps, the robotics team, JETS, Astronomy club, and my own life.

Jeff Hetzel

-- KtmGuest - 02 Oct 2005

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I'll second Jeff's observations. The calculator is a good tool when used by trained hands. It shouldnot be used as a teaching tool, especially when it impedes mastery of the material.

I'm all for a moratorium on technology until our educators learn how to use the scientific method. This would do the least amount of damage in the short term. But in the long term they need to learn how to integrate technology into the curriculum to increase learning.

Back when I was in high school calculator were powerful enough to calculate logs and trig functions, but we still wasted lots of time learning how to interpolate them from tables in the appendix. If anything is a buggy whip skill, it is this, and our time would have been better spent using the calculator to do these problems and learning more trig and pre-calc.

Once everyone has mastered the important basic skills, additional practice becomes tedious and the time spent past this point is better spent elsewhere. This is how you increase student performance past the point where it was 40 years ago before we had calculators, excel, and computers.

This is how almost every other industry has increased efficiency and improved their output. But our current crop of educators don't get it, they are stuck on stupid.

-- KDeRosa - 03 Oct 2005

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Jeff

A student must realize that a calculator may take the equation x = (1/3)*3 and give 0.999999999999 not 1 and not realize the answer is 1 not 0.999999999999.

Great example.

I'm putting that in Wit and Wisdom.

-- CatherineJohnson - 03 Oct 2005

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I had a student I tutored in calculus who was hugely dependent on his Texas Instruments 58 graphing calculator.

I gave him the following problem. There is an L shaped corridor. One part of the L has a width of "a", and the other part has a width of "b". What is the largest ladder that one can carry down the corridor and be able to make the turn? State the answer in terms of a and b.

His calculator was of no use in this problem. You could almost hear him whimper. We worked through the problem together. At the end of it he started asking me questions about the difference between "pure" and "applied" math. This problem seemed like a "pure" math problem to him. Why is that? I asked.

"In pure math you use a's and b's, and in applied math you use numbers," he said.

I assured him that letters and numbers are used in both pure and applied math. I talked to his mom the other day. He is a junior in college now. He has chosen to major in history.

-- BarryGarelick - 03 Oct 2005

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how high was the ceiling?

-- VlorbikDotCom - 03 Oct 2005

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The great physicist Richard Feynman once said that to know the name of something is not the same thing as knowing something. Students need a deep understanding of something, such as math, before they can turn to things like calculators.

As an engineer I relied heavily on my calculator and our computer aided design program (which by the way we wrote in house, we did not use a commercial program, a) because when we started to put it together one did not exist and b) because we knew what was in there and the limitations and we could change it if we needed to) to help me do my job, but in the end I needed to know what to put in the calculator to come up with the right answer.

Calculators and computers are tools like a car. If I want to go to the store I need to know how to get there. Once I know that, then how I get there is irrelevant. I could walk if I want to enjoy the scenery and get some exercise at the same time or I can take a car if I need to get there in a hurry.

This is, for example, the difference between the use of calculators in math and physics/engineering. In my physics and engineering classes I expect the students to know how to get to the store, in other words use the math. They are then free to use a calculator because I am not trying to teach them a mew math concept I am teaching them the physics/engineering concepts. The calculator is now a tool to help them get to the next problem faster. As I said in my earlier post I can now cover more and varied types of problems on a test because of the calculator. That's why I am a fan of them in courses like physics/engineering but not in math.

I also have them do problems using EXCEL. They have to set the problem up and understand the physics behind the problem to first get it set up. Once it is set up and they understand what is going on then we can quickly play "what if." What if I throw the ball up faster/slower, what if I do it on the moon, what if I account for friction/air resistance. The technology allows for doing things like this, things that would have taken too long before, but they still have to know what the physics is to write the EXCEL model in the first place. I never make a "black box" out of any EXCEL program they use, they must put it together and make it work. This requires them to do hand calcualations to make sure the numbers coming out of thier model are correct before they can use it.

The other thing that a computer program can never capture is experience. On my first assignment as a new aero engineer I was designing a structural memember to modify a C-141 aircraft and one of the old timers came by. This was 1983 or so and he had actually been in on some of the work on the DC-3 which flew first around 1933 or 34. He looked at my layout and my calculations and said that it needed beefed up in a particular area. I said that I had done everything by the book, I was not trying to be sarcastic or anything like that I was just trying to find out why. He looked at me and said that 50 years experience just made it look wrong to him. So I beefed it up by the amount he suggested and in the stress test before we did the modification the part passed with about 5% to spare. He had nailed it that close. I spent the next year or so at his desk every chance I got learning what he knew before he retired.

One last thing in this post, back to Richard Feynman. He used to teach an open class late on Friday afternoons, about 4 or 5 o'clock, back in the early 60's. The class was technically open to anyone who wanted to show up but because of the time and day only those truly interested in physics would show up. He would walk in to the lecture hall pick up a piece of chalk and ask if there were any questions about physics. He would then proceed to answer the questions with no notes at all. When asked if he had memorized every physics formula and that is how he could do it, he said no. He understood the underlying concepts and therefore could derive any equation he needed to use on the spot.

Jeff Hetzel

-- KtmGuest - 03 Oct 2005

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I have mentioned elsewhere that I have seen this hand calculation versus calculator/computer use question develop over the last 30 years. I think calculators and computers can work very well in the learning environment. First, just a few comments.

1. Little to no class time or credits should be given to learning how to use a calculator or computer program. Presumably, the course is about a subject (math, physics, engineering), not the computer. I mentioned before that in my son's 4th grade EveryDay^? math reference guide, there are more pages devoted to using the calculator than there are describing fractions.

2. The calculator/computer should make the course more difficult/challenging, not more easy. The idea is that with a calculator/computer you can tackle more complicated tasks/theories that you couldn't attempt by hand.

3. There are times where learning to do things first by hand before using the computer is not necessary. I just can't think of any cases before high school.

4. No calculators should be allowed for K-6 math. Maybe I could come up with a few examples, but why is there a need to work hard to prove this?

I saw an interesting use of the computer on Saturday night. My son's piano teacher (college music professor) gave a formal recital and one of the pieces was a new composition by another professor. The Steinway grand was moved to the side of the stage and they had a laptop projecting the music (score) on a screen. They started a program when he began to play and the program had a vertical line that followed along in the score. The problem was that everyone was intent on following the vertical line through the score and I was trying to figure out how the program matched the sounds to the music. Sometimes it was behind and sometimes ahead! It may be a great teaching tool, but not a very good listening tool.

-- SteveH - 03 Oct 2005

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I don't want to take on the entire world of technology-in-the-classroom; just in math classes.

Today's story has convinced me that the use of a graphing calculator -- and similar graphing computer programs -- as primary teaching tools in mathematics classes is misguided. It lets in too many opportunities for misunderstanding.

Colin was also exercising his calculator extensively for his physics homework, however, and I don't see a problem with that. However, I didn't see THIS problem coming until today, either.

The use of even a simple, non-graphing calculator in a math class is bound to bring up some deep mathematical issues. An inquisitive kid who experiments with his calculator is going to notice that 1/3 times 3 comes out as .9999999 (out to 7 or 8 places), and not 1; and his teacher is going to have to explain to him how, in the limit, .9999.... really is equal to 1. Bernie and I have both had to do this in classes. We have also had to explain that the 'pi' key on their calculators brings up a number that is really only an approximation to pi and not the genuine article.

Any teacher who wants to use graphing calculators extensively in the classroom is going to spend a lot of time fighting misconceptions like these, that the technology introduces. If the teacher is really good and the kids are (all of them) really strong, I could see it actually being a good experience for the kids; but let's face it, usually someone is not up to it, and some kid is going to leave the room still confused. The misconceptions snowball, too. It's just not worth it.

-- CarolynJohnston - 03 Oct 2005

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Vlorbik,

This was 2-D calculus, but if you'd like, the ceiling height is "c".

-- BarryGarelick - 03 Oct 2005

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This whole math textbook being written by calculator manufacturer reminds me of the Simpsons episode when the "Buying Actionfigure Man" episode of Actionfigure Man was being nominated for some award at the same time as Lisa and Bart's cartoon.

-- KDeRosa - 03 Oct 2005

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Long ago logarithms were used heavily for computation at the high school level. This is of course obsolete. Today the log function needs to be taught as the inverse of the exponential function. If we sketch the graph of e^x and then reflect it across the line y=x we get log base e.

If y=e^x then x=ln(y). This is how natural log is defined.

The notion of the inverse of a function is very important, essentially it is a function that reverses the action of the "primary" function.

The word inverse is sometimes misunderstood. The reciprocal is sometimes called the inverse of a number and it is the multiplicative inverse. The negative of a number is its additive inverse. Using inverse for reciprocal should be avoided. Teachers should make sure that the notion of the inverse of a function is not confused with the reciprocal of a function.

-- CharlesWilliams - 04 Oct 2005

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Once we know that log is the inverse of exponent the domain of the exponent function becomes the range of log.

(Couldn't resist one last thought.)

-- CharlesWilliams - 04 Oct 2005

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You gotta try harder to resist that thinking stuff.

-- CarolynJohnston - 05 Oct 2005

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Someone recommended a couple of edu blogs on graphing calculators--I think it was Anne or Becky.

Do you remember who?

-- CatherineJohnson - 05 Oct 2005

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Tall Dark and Mysterious

-- KDeRosa - 05 Oct 2005