wes@engr.uky.edu (Wes Morgan) (02/17/88)
The most original test I've seen came from Berea College in Kentucky.
It was announced in Calculus class that the use of calculators would
be permitted on exams. The day of the first exam came, and everyone
was whipping their button-punching skills into their peak level. The
tests were passed out, and surprise!
FOR EACH PROBLEM, GIVE AN EXPLANATION OF YOUR METHOD OF SOLVING THE
PROBLEM. DO NOT PERFORM ANY MATHEMATICAL CALCULATIONS WHATSOEVER.
YOU MAY USE COMMON THEOREMS BY EITHER WRITING OUT THE FORMULA OR
CITING THEM BY NAME (E.G. THE INTERMEDIATE VALUE THEOREM).
GOOD LUCK.
I thought this was an excellent examination method. It would prove
the students' knowledge while allowing a larger exam, since the ab-
sence of time-consuming math made longer problems possible.
Comments?
Wes
--
wes@engr.uky.edu OR wes%ukecc.uucp@ukma OR ...cbosgd!ukma!ukecc!wes
Ho! Ha ha! Guard! Turn! Parry! Dodge! Spin! Thrust! <*SPROING!*>
Actually, it's a buck-and-a-quarter quarterstaff, but I'm not telling him that! gore@nucsrl.UUCP (Jacob Gore) (02/23/88)
There is one important factor to consider when deciding whether or not to
allow calculators in an exam, that has nothing to do with the question of
whether or not people should be able to do arithmetic in their heads:
There are calculators, and there are calculators.
There is a great variety of calculators. If you give a test where it doesn't
matter to you how students do the multiplication, you may not mind if they use
the type of a calculator that does basic arithmetic. However, by allowing
students to bring and use their calculators, you are also allowing them to use
programmable calculators, many of which are powerful enough to be
preprogrammed with solutions for a variety of problems that a student expects
to be on your test.
I have taken too many tests where the tester wanted to know if you could solve
some problem by using a specific method. The intention may be to test if the
student understands the method, but in reality, many (probably most) students
will just memorize the formula or the procedure for the test. My personal
views are that such tests are bad. But as long as they exist, allowing
programmable calculators to be used in such tests is unfair.
Jacob Gore Gore@EECS.NWU.Edu
Northwestern Univ., EECS Dept. {oddjob,gargoyle,ihnp4}!nucsrl!goretjhorton@ai.toronto.edu ("Timothy J. Horton") (02/25/88)
In article <3900008@nucsrl.UUCP> gore@nucsrl.UUCP (Jacob Gore) writes: >... by allowing students to bring and use their calculators (in tests), >you are also allowing them to use programmable calculators, many of which >are powerful enough to be preprogrammed with solutions for a variety of >problems that a student expects to be on your test. With the exception of exams on mathematical techniques... Put one symbolic variable into just about numerical question (such that the variable doesn't just drop out) and it becomes unsolvable with a programmable calculator, unless it has MACSYMA in ROM and about 16 Meg of workspace. The hardest part of most any "real" problem is employing a conceptual basis to selecting an approach and find the way from "here to there". Why slow everyone down on the odd multiplication they need to do, when in real life you would just reach for a calculator? It's like having a cooking exam with no knives. As Oog says, with pride, "that's the way we did it in the old days." Does "B.C." really mean "Before Calculators"?
g-rh@cca.CCA.COM (Richard Harter) (02/26/88)
In article <1988Feb24.224849.928@jarvis.csri.toronto.edu> tjhorton@ai.toronto.edu ("Timothy J. Horton") writes: >....Why slow >everyone down on the odd multiplication they need to do, when in real life >you would just reach for a calculator? It's like having a cooking exam >with no knives. As Oog says, with pride, "that's the way we did it in the >old days." If you have a reasonable aptitude for arithmetic, it is faster to do the odd multiplication in your head than reaching for a calculator -- if you have been trained for quick mental arithmetic. I amuse myself and startle people by giving the answer to routine calculations while they are still fumbling with the calculator. I am inclined to think that quick mental arithmetic ought to be taught in the schools. Granted that not everyone has the aptitude for it. But the average person has much more capacity than you might think -- the lower schools seem to operate on the principle of teaching people that they can't do arithmetic very well. It has seemed to me that it would be better to operate on the principle that people can do arithmetic efficiently and to train them to do so. Quick mental arithmetic is simply a matter of a bag of efficient techniques and some drill until you can apply them as a matter of course. >Does "B.C." really mean "Before Calculators"? Correct. The BC period is divided into two eras, the age of sliderules, and the "Before Sliderules" era, also known as the time of B.S. -- In the fields of Hell where the grass grows high Are the graves of dreams allowed to die. Richard Harter, SMDS Inc.
zwicky@pterodactyl.cis.ohio-state.edu (Elizabeth D. Zwicky) (02/27/88)
In article <24954@cca.CCA.COM> g-rh@CCA.CCA.COM.UUCP (Richard Harter) writes: >In article <1988Feb24.224849.928@jarvis.csri.toronto.edu> tjhorton@ai.toronto.edu ("Timothy J. Horton") writes: > If you have a reasonable aptitude for arithmetic, it is faster to do >the odd multiplication in your head than reaching for a calculator -- if you >have been trained for quick mental arithmetic. I amuse myself and startle >people by giving the answer to routine calculations while they are still >fumbling with the calculator. > I am inclined to think that quick mental arithmetic ought to be >taught in the schools. Granted that not everyone has the aptitude for it. I also calculate things in my head while people are still fumbling with calculators. And I had quick arithmetic drills in school. The problem is that the two are totally unrelated. I failed every single speed drill I ever took; they regularly reduced me to tears. What gives me my speed is the non-traditional but simple calculating techniques that I learned from my mathematician father - you know, the ones they used to fail me on math tests for using? All those math drills never taught me a thing, and my husband, who had the drills but not the mathematician at home, is utterly lousy at "routine mathematical calculations", even though he likes numbers and I don't. (I stopped letting him calculate tips long ago, having discovered that he was more often wrong than right. Now I let him do it again, having dicovered that the reason that he screwed up was that he was doing cross-multiplication, just like the books have you do. That gets good scores on tests, because it looks right. I was moving the decimal place to get 10% and then multiplying that, which math teachers fail because the intermediate steps aren't "right" but it sure works.) > Richard Harter, SMDS Inc. Elizabeth D. Zwicky
g-rh@cca.CCA.COM (Richard Harter) (02/28/88)
In article <7301@tut.cis.ohio-state.edu> zwicky@pterodactyl.cis.ohio-state.edu (Elizabeth D. Zwicky) writes: > >I also calculate things in my head while people are still fumbling with >calculators. And I had quick arithmetic drills in school. The problem is >that the two are totally unrelated. I failed every single speed drill I >ever took; they regularly reduced me to tears. What gives me my speed is >the non-traditional but simple calculating techniques that I learned >from my mathematician father - you know, the ones they used to fail me >on math tests for using? All those math drills never taught me a thing, >and my husband, who had the drills but not the mathematician at home, >is utterly lousy at "routine mathematical calculations", even though >he likes numbers and I don't. (I stopped letting him calculate tips long >ago, having discovered that he was more often wrong than right. Now I >let him do it again, having dicovered that the reason that he screwed >up was that he was doing cross-multiplication, just like the books have >you do. That gets good scores on tests, because it looks right. I >was moving the decimal place to get 10% and then multiplying that, >which math teachers fail because the intermediate steps aren't "right" >but it sure works.) Interesting. I don't know what they teach in speed drills in school but, from what you say, it's a mistake. What I had in mind was more the sort of thing that your father taught you. I taught myself -- one of my sources was an 1890's handbook of practical calculating. Really. Some of the stuff (quick commercial calculations for example) was totally obsolete, but a lot of it was general, even if the wording was a little quaint. One question -- what did you mean by cross multiplication. To me, this means the following method. Example: 523 469 --- 245287 Steps: 3x9 = 27, write down 7, start 2 as a running sum 2 + 18 + 18 = 38, write down 8, start 3 as a running sum 3 + 45 + 12 + 12 = 72, write down 2, start 7 as a running sum, 7 + 30 + 8 = 45, write down 5, start 4 as a running sum, 4 + 20 = 24, write down 24, done expanded: 3x9 = 27 = 7 carry 2 2 + 2x9 + 6x3 = 38 = 8 carry 3 3 + 9x5 + 6x2 +4x3 = 72 = 2 carry 7 7 + 6x5 + 4x2 = 45 = 5 carry 4 4 + 4x5 = 24 Essentially, this is formal multiplication of polynomials carried out in numbers. If you can running totals in your head, this way of multiplying is actually quite easy -- if the numbers are written down and you can write the answer down. If everything has to be done in your head, it's not so simple because you have to keep track of the answer in your head while you are also running through the partial sums. -- In the fields of Hell where the grass grows high Are the graves of dreams allowed to die. Richard Harter, SMDS Inc.
mchin@homxc.UUCP (M.CHIN) (03/03/88)
> I also calculate things in my head while people are still fumbling with > calculators. And I had quick arithmetic drills in school. The problem is > that the two are totally unrelated. I failed every single speed drill I > ever took; they regularly reduced me to tears. What gives me my speed is > the non-traditional but simple calculating techniques that I learned > from my mathematician father - you know, the ones they used to fail me > on math tests for using? All those math drills never taught me a thing, > and my husband, who had the drills but not the mathematician at home, > is utterly lousy at "routine mathematical calculations", even though > he likes numbers and I don't. (I stopped letting him calculate tips long > ago, having discovered that he was more often wrong than right. Now I > let him do it again, having dicovered that the reason that he screwed > up was that he was doing cross-multiplication, just like the books have > you do. That gets good scores on tests, because it looks right. I > was moving the decimal place to get 10% and then multiplying that, > which math teachers fail because the intermediate steps aren't "right" > but it sure works.) > > > Richard Harter, SMDS Inc. > > Elizabeth D. Zwicky Let's hear it for calculating things in our heads. I do this too, in many instances, especially when it concerns shopping. Even now, I still do some of the easier multiplication and division problems on paper because its faster than finding a calculator or calling one up on the computer. In the instance of tipping, if you live in a state where tax is 5 %, 15 % tip comes really easily. What I often end up doing is dividing by 10 (move the decimal) divide that by 2 (very easy). and add the two numbers. What can I say, its easier than multiplying by 3 and dividing a memorized number by 2. Just to add my two cents worth, I think what is needed is a good sense of abstraction. This means that after having done the drills, you are able to do the arithmetic. This doesn't mean you know the shortcuts. Word problems teach you the shortcuts. Once you know several different ways to attack a problem, you are able to take the shortest step. If this means reaching for a calculator to multiply 3 four-digit numbers, so be it. But, until you are familiar with every process, you won't know the quickest method. Now, word problems, by not stating the method or the formula, make you think more abstractly. This draws up, in general, a larger picture of the problem in your head. Therefore, you are able to view it from several different angles as opposed to through a microscope. Given the extra viewpoints, you are able to choose the easiest/quickest path to the solution. At least that's me view. Unfortunately, this always posed a problem when they started throwing in extraneous variables that had nothing whatsoever to do with the problem :-) Michael Chin ihnp4!homxc!mchin