Calculator in math class
July 3, 2011 4:29 AM Subscribe
When should calculators be used in math class? What are good criteria for determining if a calculator should be used on a given set of problems?
Phunniemee: that isn't true anymore. Many $10 junk calculators you can get at the pharmacy now parse order of operations (and have many other features previously found only on high end TIs).
Personally, I think calculators should be avoided completely until absolutely necessary (things like logarithms and trig functions of non-standard angles), because even if they technically can do it, they just lose so many basic math and estimation skills by relying on them. I have honors high school students who whip out the calculator to multiply a small integer by 2, because they're so dependent on the thing. If only that were the exception, rather than the rule.
posted by Dr.Enormous at 4:51 AM on July 3, 2011 [4 favorites]
Personally, I think calculators should be avoided completely until absolutely necessary (things like logarithms and trig functions of non-standard angles), because even if they technically can do it, they just lose so many basic math and estimation skills by relying on them. I have honors high school students who whip out the calculator to multiply a small integer by 2, because they're so dependent on the thing. If only that were the exception, rather than the rule.
posted by Dr.Enormous at 4:51 AM on July 3, 2011 [4 favorites]
I'm another in the 'never, if at all possible' camp. It's basically impossible to lose the skill of how to work a calculator, but it's easy to see students losing or never developing the ability to work problems without a calculator. It's easy enough to develop problems that don't require the use of a calculator.
The big frustration I have with not allowing calculators is that the textbooks write problems assuming the use of a calculator, so I can spend as much time as I want preaching against calculators in class and then they look in the book and it gives an example or exercise with a number like .1257 or something. Okay, the real world does have decimals in it, but I want to break students of the idea that the 'right' answer has to be a decimal. I want to see ln 2, not 0.69 and so on. I want them to understand that the square root of two is a perfectly good thing to write down, not some mysterious thing that doesn't exist until it's calculated to two decimal places. (I'm not giving as detailed examples as I could, as I feel it's improper for a public forum.)
The one exception I can come up with is teaching Newton's method. If you really want to make them do the calculation, you've got to let them use a calculator. However, it's perfectly easy to ask conceptual questions about Newton's method. (Ask students to write down the formula for x_{n+1}, rather than having them compute it. Or, again, give easy numbers. Linear approximation for the cube root of seven works well.)
posted by hoyland at 5:28 AM on July 3, 2011 [2 favorites]
The big frustration I have with not allowing calculators is that the textbooks write problems assuming the use of a calculator, so I can spend as much time as I want preaching against calculators in class and then they look in the book and it gives an example or exercise with a number like .1257 or something. Okay, the real world does have decimals in it, but I want to break students of the idea that the 'right' answer has to be a decimal. I want to see ln 2, not 0.69 and so on. I want them to understand that the square root of two is a perfectly good thing to write down, not some mysterious thing that doesn't exist until it's calculated to two decimal places. (I'm not giving as detailed examples as I could, as I feel it's improper for a public forum.)
The one exception I can come up with is teaching Newton's method. If you really want to make them do the calculation, you've got to let them use a calculator. However, it's perfectly easy to ask conceptual questions about Newton's method. (Ask students to write down the formula for x_{n+1}, rather than having them compute it. Or, again, give easy numbers. Linear approximation for the cube root of seven works well.)
posted by hoyland at 5:28 AM on July 3, 2011 [2 favorites]
At risk of slight derail, the "West" would be a better place in the future if more children were taught the Abacus method for mental arithmetic [see Youtube links: 1 2
3]. Calculators would be less attractive unless they werereally much more necessary.
posted by dirm at 5:39 AM on July 3, 2011 [1 favorite]
3]. Calculators would be less attractive unless they were
posted by dirm at 5:39 AM on July 3, 2011 [1 favorite]
What age are we talking about? Kids need mental math skills to develop better number sense in the lower grades, and I'd say through Algebra. Precalculus classes benefit from calculator use, as does Calculus. No matter what the course, though, if calculators are allowed then material should be tailored to teaching concepts. Some of my favorite problems to assign are those that are strictly graphical. "A rubber ball is tossed down the stairs. Graph (a) its height versus time, (b) its vertical velocity versus time, (c) its acceleration, (d) an anti-derivative of (a)." Sure, calculators will be useful in "the real world", but people need conceptual knowledge before they start to punch anything in.
posted by monkeymadness at 5:39 AM on July 3, 2011
posted by monkeymadness at 5:39 AM on July 3, 2011
Response by poster: How about for a question like this?
"You have 14 coins in a bag. 4 of them are unfair in that they have a 55% chance of coming up heads when flipped (the rest are fair coins). You randomly choose one coin from the bag and flip it 4 times. What is the percent probability of getting 4 heads?"
I'm taking my sons through Khan Academy (and myself for review and filling in gaps) and questions like this take a very long time to do by hand; and if you make one tiny mistake, you get it wrong, even if you understand the method very well.
If a question like this were on an exam, would the exam allow a calculator? Does anyone have experience with exams that don't allow calculators for questions like this? We've not used calculators for exercises at all yet, and doing well so far, but with a question like this, choosing the right level of precision at each step seems to affect the final answer in a potentially big way.
posted by strangeguitars at 5:59 AM on July 3, 2011
"You have 14 coins in a bag. 4 of them are unfair in that they have a 55% chance of coming up heads when flipped (the rest are fair coins). You randomly choose one coin from the bag and flip it 4 times. What is the percent probability of getting 4 heads?"
I'm taking my sons through Khan Academy (and myself for review and filling in gaps) and questions like this take a very long time to do by hand; and if you make one tiny mistake, you get it wrong, even if you understand the method very well.
If a question like this were on an exam, would the exam allow a calculator? Does anyone have experience with exams that don't allow calculators for questions like this? We've not used calculators for exercises at all yet, and doing well so far, but with a question like this, choosing the right level of precision at each step seems to affect the final answer in a potentially big way.
posted by strangeguitars at 5:59 AM on July 3, 2011
Because it's asking for the 'percent probability', I think it's assuming the use of a calculator. But you don't actually need one until the very last step. Work out the probability as a fraction, then find the decimal to check your answer against what they have.
I'd still stay that's firmly in the non-calculator category, though. For a start, I couldn't do it without writing down what I'd be multiplying first and at that point, you might as well just multiply the fractions.
I did math team competitions in high school. The NSML did allow calculators and the ICTM allowed calculators for the most part (not in eight-person team competitions). I want to say the probability questions always wanted fractional answers, though. I honestly can't remember what the calculator policies were in high school when we did this sort of thing in class. My university didn't allow calculators in math classes, as a general rule.
posted by hoyland at 6:13 AM on July 3, 2011
I'd still stay that's firmly in the non-calculator category, though. For a start, I couldn't do it without writing down what I'd be multiplying first and at that point, you might as well just multiply the fractions.
I did math team competitions in high school. The NSML did allow calculators and the ICTM allowed calculators for the most part (not in eight-person team competitions). I want to say the probability questions always wanted fractional answers, though. I honestly can't remember what the calculator policies were in high school when we did this sort of thing in class. My university didn't allow calculators in math classes, as a general rule.
posted by hoyland at 6:13 AM on July 3, 2011
Well, obviously they should be used to teach basic programming skills. Programs are really useful for visualizing concepts-- for example, I can't really imagine studying fractals without them--but students shouldn't use programs until they understand them. I have very fond memories of a 7th grade assignment to program an extremely simple Spore-like game of life.
In response to your example, calculators would typically be allowed on a test because just one component of the answer, .55^4, runs to something like eight decimals. In any time-sensitive environment it's silly for teachers to waste their time when they could be asking more questions. However, if your sons are repeatedly making calculation mistakes with unlimited time then sure, it wouldn't hurt to get some extra practice. They should definitely be able to express the penultimate answer in its uncalculated form, though. Calculating as you go is lazy and risky.
posted by acidic at 6:20 AM on July 3, 2011
In response to your example, calculators would typically be allowed on a test because just one component of the answer, .55^4, runs to something like eight decimals. In any time-sensitive environment it's silly for teachers to waste their time when they could be asking more questions. However, if your sons are repeatedly making calculation mistakes with unlimited time then sure, it wouldn't hurt to get some extra practice. They should definitely be able to express the penultimate answer in its uncalculated form, though. Calculating as you go is lazy and risky.
posted by acidic at 6:20 AM on July 3, 2011
"A rubber ball is tossed down the stairs. Graph (a) its height versus time, (b) its vertical velocity versus time, (c) its acceleration, (d) an anti-derivative of (a)."
That's bothering me now. Is the ball bouncing down the stairs, or was it thrown hard enough that it never actually touches them?
posted by Net Prophet at 6:20 AM on July 3, 2011
That's bothering me now. Is the ball bouncing down the stairs, or was it thrown hard enough that it never actually touches them?
posted by Net Prophet at 6:20 AM on July 3, 2011
The answer to that in my combinatorics class would have been:
((4/14)(0.55^4) + (10/14)(0.5^4))*100%
We were allowed to use calculators, but 7.07875% wouldn't have been "the answer", the mess above would have been the answer... And the problem would probably have been worded differently, reflecting that (no percentage, for one thing). I do think a calculator is "fair" in combinatorics in that case, since you can only use it to check your answer if you already have an idea of what the answer should be. Here I would compare to (0.5^4)*100% = 6.25% and see that it's slightly better than flipping a fair coin, which is what I expect, so it's probably a good answer.
We generally don't have calculators in calculus classes at my university, because the numbers given are either very small or recognizable (perfect squares and such).
posted by anaelith at 6:21 AM on July 3, 2011 [3 favorites]
((4/14)(0.55^4) + (10/14)(0.5^4))*100%
We were allowed to use calculators, but 7.07875% wouldn't have been "the answer", the mess above would have been the answer... And the problem would probably have been worded differently, reflecting that (no percentage, for one thing). I do think a calculator is "fair" in combinatorics in that case, since you can only use it to check your answer if you already have an idea of what the answer should be. Here I would compare to (0.5^4)*100% = 6.25% and see that it's slightly better than flipping a fair coin, which is what I expect, so it's probably a good answer.
We generally don't have calculators in calculus classes at my university, because the numbers given are either very small or recognizable (perfect squares and such).
posted by anaelith at 6:21 AM on July 3, 2011 [3 favorites]
That's the sort of problem where you should work out all the math first, and maybe estimate the answer, then a calculator is no big deal.
That said, if I were creating that problem, I would make the unfair coins 75% to make the fractional form and the powers thereof easier, and ban the use of a calculator.
posted by Dr.Enormous at 6:25 AM on July 3, 2011
That said, if I were creating that problem, I would make the unfair coins 75% to make the fractional form and the powers thereof easier, and ban the use of a calculator.
posted by Dr.Enormous at 6:25 AM on July 3, 2011
I'm taking my sons through Khan Academy (and myself for review and filling in gaps) and questions like this take a very long time to do by hand; and if you make one tiny mistake, you get it wrong, even if you understand the method very well.
In the problem you describe, you use a calculator in "real life" but for the purposes of school work, the entire point is to give the students enough practice in doing it by hand so that they know how to get it right without making a tiny mistake.
This isn't anywhere near the point where the math is so ugly that a calculator is necessary. Calculators became expected when we were expected to evaluate complicated polynomial functions.
posted by deanc at 6:36 AM on July 3, 2011
In the problem you describe, you use a calculator in "real life" but for the purposes of school work, the entire point is to give the students enough practice in doing it by hand so that they know how to get it right without making a tiny mistake.
This isn't anywhere near the point where the math is so ugly that a calculator is necessary. Calculators became expected when we were expected to evaluate complicated polynomial functions.
posted by deanc at 6:36 AM on July 3, 2011
Maybe a calculator isn't the right tool for that job. (Ignoring what may or may not be allowed on any particular test.)
I would first have written down what anaelith did, above. Then you want to convert that to a simple number. You can use a calculator to do so, and students definitely should learn how to do so, but it's not the best. Like you said, one slip up and the whole thing is botched. What about using one of the computer algebra systems out there (Maple, Mathematica, etc.) and just type the damn thing in? You can see if you entered it properly because all of the elements are displayed for you, plus it's easy to cut-and-paste any sub-expression you may be curious about.
I'm curious what the math educators out there think of this. Talking with my math teacher friends makes it clear that they don't yet have all the answers when it comes to integrating the new technology into teaching math.
posted by benito.strauss at 6:52 AM on July 3, 2011
I would first have written down what anaelith did, above. Then you want to convert that to a simple number. You can use a calculator to do so, and students definitely should learn how to do so, but it's not the best. Like you said, one slip up and the whole thing is botched. What about using one of the computer algebra systems out there (Maple, Mathematica, etc.) and just type the damn thing in? You can see if you entered it properly because all of the elements are displayed for you, plus it's easy to cut-and-paste any sub-expression you may be curious about.
I'm curious what the math educators out there think of this. Talking with my math teacher friends makes it clear that they don't yet have all the answers when it comes to integrating the new technology into teaching math.
posted by benito.strauss at 6:52 AM on July 3, 2011
After you've/they've done it once by hand.
For example, when my students do multiple repeated calculations in their lab books, I make them write out the work for the first calculation, then the rest can be omitted and I assume they are doing them on their calculator or in a spreadsheet.
This rule comes from my own experience as a student.
posted by msittig at 7:21 AM on July 3, 2011 [2 favorites]
For example, when my students do multiple repeated calculations in their lab books, I make them write out the work for the first calculation, then the rest can be omitted and I assume they are doing them on their calculator or in a spreadsheet.
This rule comes from my own experience as a student.
posted by msittig at 7:21 AM on July 3, 2011 [2 favorites]
The beauty of Anaelith's answer to the coins problem is that I can translate it into words.
((4/14)(0.55^4) + (10/14)(0.5^4))*100%
4/14 is the probability of drawing an unfair coin;
0.55^4 is the probability of the unfair coin coming up heads 4 times consecutively...
If it's instead written out as .286*.092... it doesn't look like concepts anymore, it's just a bunch of numbers. Much less beautiful.
posted by Net Prophet at 7:28 AM on July 3, 2011 [1 favorite]
((4/14)(0.55^4) + (10/14)(0.5^4))*100%
4/14 is the probability of drawing an unfair coin;
0.55^4 is the probability of the unfair coin coming up heads 4 times consecutively...
If it's instead written out as .286*.092... it doesn't look like concepts anymore, it's just a bunch of numbers. Much less beautiful.
posted by Net Prophet at 7:28 AM on July 3, 2011 [1 favorite]
In almost all of my university math classes, no calculators were allowed. We just left our answers in forms such as anaelith has above. Even where calculators were allowed, graphing calculators were typically banned.
If in doubt, I'd practice getting the answers into such forms, then using the calculator to convert them into decimal form, but leaving both answers.
posted by ZeroDivides at 7:54 AM on July 3, 2011
If in doubt, I'd practice getting the answers into such forms, then using the calculator to convert them into decimal form, but leaving both answers.
posted by ZeroDivides at 7:54 AM on July 3, 2011
I teach my classes to distinguish between approximately equal and exactly equal. When I want an exact result they need an answer in terms of radicals and the symbol for pi because the calculator will only give them an approximation.
posted by Obscure Reference at 7:58 AM on July 3, 2011
posted by Obscure Reference at 7:58 AM on July 3, 2011
Best answer: One more little side note, one of the horrible things I see students do is something like
posted by anaelith at 8:28 AM on July 3, 2011 [2 favorites]
2 11 22 11 - * -- = -- = -- 3 2 6 3without realizing that they can cancel the 2s as the first step. So doing this kind of long arithmetic by hand is good for practicing that kind of simplification. For this problem:
4 0.55^4 10 0.5^4 4 (55 )^4 10 (1)^4
-- * + -- * = -- * (---) + -- * (-)
14 14 15 (100) 14 (2)
4 11^4 10 1
= -- * ---- + -- * ---
14 20^4 14 2^4
2^2 * 11^4 2 * 5
= ----------------- + -----------
2 * 7 * 2^8 * 5^4 2 * 7 * 2^4
11^4 5 2^3 * 5^4
= ------------- + ------- * ---------
7 * 2^7 * 5^4 7 * 2^4 2^3 * 5^4
11^4 + 2^3 * 5^5
= ----------------
7 * 2^7 * 5^4
11^4 + 10^3 * 5^2
= -----------------
7 * 10^4 * 2^3
11^4 (calculate on scratch paper): 14641
14641 + 25000 3.9641
= ------------- = -----
56*10000 56
long division gives 0.0707 R 41 or about 7.08%.
Which is a precise answer with only two "hard" calculations: 11^4 -- which is actually not hard, and then the long division at the end.posted by anaelith at 8:28 AM on July 3, 2011 [2 favorites]
And if you buy a notebook sized whiteboard, you won't even worry about how much paper you're using up. Or you can type it, but in that case I recommend using something like LaTeX and for the love of god not ASCII art, because that was easily the most painful part of this problem. ;-)
posted by anaelith at 8:31 AM on July 3, 2011
posted by anaelith at 8:31 AM on July 3, 2011
Here I would compare to (0.5^4)*100% = 6.25% and see that it's slightly better than flipping a fair coin, which is what I expect, so it's probably a good answer.
When I try to explain to students how to check answers like this, they look at me like I have two heads. I think they're not learning this sort of thing in high school.
posted by madcaptenor at 8:45 AM on July 3, 2011
When I try to explain to students how to check answers like this, they look at me like I have two heads. I think they're not learning this sort of thing in high school.
posted by madcaptenor at 8:45 AM on July 3, 2011
What kind of math test? Because I've taken a lot, and my opinion is "it depends."
Is the exam testing skills that the calculator can replace? Then don't allow one, because students need to learn to do the problems on their own.
Is the exam for a lower-level class, and can the problems be designed so that the "incidental" calculations don't take up too much of the students' time? Then don't allow one, because students at this level need practice to become comfortable doing calculations by hand.
Would students using a calculator (and therefore saving themselves time) allow the instructor to include more problems on the skills that are actually being tested and get better coverage of the course material? Then allow a calculator.
There are some things for which a calculator is necessary, unless you allow students to leave certain expressions unevaluated; in that case, I think that requiring a calculator is a good idea, so you can be sure that students do have the tools to fully solve those kinds of problems. (If not, requiring them to use a calculator on the homework or something.)
How about for a question like this?
I had a probability exam on Friday which had questions similar to--but a little more complicated than--your example. The professor allowed a calculator. While all of his students should be able to do such calculations by hand, it does take longer, and that's not what he wanted to test. Allowing a calculator meant that he could make his problems more complicated/add more problems to the test.
Also, in most of my higher-level math courses, the grading philosophy seems to be to take only a small amount--or nothing!--off if the student follows the right procedure but makes an arithmetic mistake. The grading is weighted heavily towards "does the student know how to do this kind of problem" and away from "did the student successfully multiply 8*13"? Often, you are not even required to find the "final" answer, but an expression that is equivalent to the answer.
posted by Kutsuwamushi at 10:38 AM on July 3, 2011
Is the exam testing skills that the calculator can replace? Then don't allow one, because students need to learn to do the problems on their own.
Is the exam for a lower-level class, and can the problems be designed so that the "incidental" calculations don't take up too much of the students' time? Then don't allow one, because students at this level need practice to become comfortable doing calculations by hand.
Would students using a calculator (and therefore saving themselves time) allow the instructor to include more problems on the skills that are actually being tested and get better coverage of the course material? Then allow a calculator.
There are some things for which a calculator is necessary, unless you allow students to leave certain expressions unevaluated; in that case, I think that requiring a calculator is a good idea, so you can be sure that students do have the tools to fully solve those kinds of problems. (If not, requiring them to use a calculator on the homework or something.)
How about for a question like this?
I had a probability exam on Friday which had questions similar to--but a little more complicated than--your example. The professor allowed a calculator. While all of his students should be able to do such calculations by hand, it does take longer, and that's not what he wanted to test. Allowing a calculator meant that he could make his problems more complicated/add more problems to the test.
Also, in most of my higher-level math courses, the grading philosophy seems to be to take only a small amount--or nothing!--off if the student follows the right procedure but makes an arithmetic mistake. The grading is weighted heavily towards "does the student know how to do this kind of problem" and away from "did the student successfully multiply 8*13"? Often, you are not even required to find the "final" answer, but an expression that is equivalent to the answer.
posted by Kutsuwamushi at 10:38 AM on July 3, 2011
I think they're not learning this sort of thing in high school
They just don't care. In their minds, it's black and white: you either have the right answer or the wrong answer. Personally, I think it's a result of the culture of standardized testing, but that's a whole discussion in itself. I try to grade with partial credit weighted upon whether your answer is reasonable, and even give full credit for some reasonable answers with slight mistakes, and they just don't care.
I have students who solve for temperature and get answers thousands below absolute zero, and never think twice. Doing math problems out the long way is at least part of learning these skills.
posted by Dr.Enormous at 10:39 AM on July 3, 2011
They just don't care. In their minds, it's black and white: you either have the right answer or the wrong answer. Personally, I think it's a result of the culture of standardized testing, but that's a whole discussion in itself. I try to grade with partial credit weighted upon whether your answer is reasonable, and even give full credit for some reasonable answers with slight mistakes, and they just don't care.
I have students who solve for temperature and get answers thousands below absolute zero, and never think twice. Doing math problems out the long way is at least part of learning these skills.
posted by Dr.Enormous at 10:39 AM on July 3, 2011
In their minds, it's black and white: you either have the right answer or the wrong answer.
Unless they got the wrong answer, in which case they start begging for more partial credit.
posted by madcaptenor at 10:51 AM on July 3, 2011
Unless they got the wrong answer, in which case they start begging for more partial credit.
posted by madcaptenor at 10:51 AM on July 3, 2011
I'm curious what the math educators out there think of this. Talking with my math teacher friends makes it clear that they don't yet have all the answers when it comes to integrating the new technology into teaching math.
I suppose I can count as a 'math educator' for purposes of this discussion. (Are we drifting too far off the point of AskMeFi?) Like I said above, I'm firmly in the 'no calculator' camp. It's possibly to effectively use 'technology' (Mathematica, etc) in the classroom, but it requires a lot of planning and a lot of thought to do well. Otherwise it's just a mind-numbing waste of time for everyone involved.
In this sort of situation, Mathematica isn't going to tell you anything. It'll give you a fraction to check against your fraction, but that's it. It's drastically easier to use a computer to do math effectively when you understand the mathematics that you're doing (if nothing else, you have to be able to know if the computer is behaving how you expect it to).
posted by hoyland at 10:55 AM on July 3, 2011
I suppose I can count as a 'math educator' for purposes of this discussion. (Are we drifting too far off the point of AskMeFi?) Like I said above, I'm firmly in the 'no calculator' camp. It's possibly to effectively use 'technology' (Mathematica, etc) in the classroom, but it requires a lot of planning and a lot of thought to do well. Otherwise it's just a mind-numbing waste of time for everyone involved.
In this sort of situation, Mathematica isn't going to tell you anything. It'll give you a fraction to check against your fraction, but that's it. It's drastically easier to use a computer to do math effectively when you understand the mathematics that you're doing (if nothing else, you have to be able to know if the computer is behaving how you expect it to).
posted by hoyland at 10:55 AM on July 3, 2011
I too fall on the side of no calculators. That's how I was taught, for the most part, and while it was somewhat unbearable as a student, now I know math.
This even applies to nutty things like calculating logs and trig functions. I think it is easier and more instructive to have a table with lookup values than it is to use a calculator. It forces the mind to *know* how it works, rather than just plugging numbers into formulas and hoping for the right answer. (Also makes correcting the tests easier- if all the students use the lookup table, the right answer will always be the same.)
Concepts can be taught without the need for calculators. You design worksets where the arithmetic is simple, and standardized tests where the right answer can't be "divined" from simply stabbing at a calculator until a sane answer appears. Unfortunately, lazy textbooks don't do this any more.
(Not to mention, forcing students to context switch between the work and the calculator messes up their concentration and adds stress rather than reducing it.)
posted by gjc at 11:39 AM on July 3, 2011
This even applies to nutty things like calculating logs and trig functions. I think it is easier and more instructive to have a table with lookup values than it is to use a calculator. It forces the mind to *know* how it works, rather than just plugging numbers into formulas and hoping for the right answer. (Also makes correcting the tests easier- if all the students use the lookup table, the right answer will always be the same.)
Concepts can be taught without the need for calculators. You design worksets where the arithmetic is simple, and standardized tests where the right answer can't be "divined" from simply stabbing at a calculator until a sane answer appears. Unfortunately, lazy textbooks don't do this any more.
(Not to mention, forcing students to context switch between the work and the calculator messes up their concentration and adds stress rather than reducing it.)
posted by gjc at 11:39 AM on July 3, 2011
On the more advanced math end, the linear algebra class I'm in has an interesting setup - the homework problems are set up so we don't need to use calculators, and we do our tests by hand BUT we get extra credit for (also) doing our homework in Matlab and submitting the documentation. I think it's a nice balance of minimizing context-switching during lecture and focusing on concepts vs. recognition that I will probably never do these problems by hand after this class.
posted by momus_window at 4:26 PM on July 3, 2011
posted by momus_window at 4:26 PM on July 3, 2011
Best answer: When should calculators be used in math class?
Do you mean in a testing environment, or just where (at all) in the classroom? I think they are very different scenarios, particularly by today's standards.
Non-exam class time: First, I think calculators/technology should be encouraged when exploring something new. This definitely comes out of a "discovery-based learning" mindset for me, and there are lots of examples of this out there...using technology as a discovery device. This excites me because a computationally-heavy idea can be tested quickly, and the focus is placed on finding relationships and modeling an idea, and less on lengthy hand-calculations. I will not make a case for discovery-based learning here (as that is not pertinent to the question), but using technology in the classroom is so fitting for this purpose.
On that note, do any folks here actually shy away from technology when they are trying to figure something out? Are pencil and paper really the tools of choice for the math people in the room?
Exam time: Surely, there are particular things that each teacher and curriculum want tested without a calculator. What those things are will differ, but writing an exam with those things in mind takes considerable thought. With those things in mind, I always write exams with two parts: Part A - no calculator allowed, Part B - calculator allowed. I guess I'm lucky in that my department has described what things we want to test and how they should be tested, and for the most part, I personally agree with the assessment strategies. We have technology built into the curriculum in fairly specific ways, so I know that I need to assess, for example, whether a student can solve a max/min 1-variable calculus problem. I am possibly testing several skills in a problem like this, but I could allow the ti-89 and just drive at knowledge of the 1st and 2nd derivative tests. I try to look at every question and see if I have properly assessed what we say we want to, and I know that using technology is a way do distill the proper outcome I am looking for.
How about for a question like this?
What's being tested there is most likely whether you can write your process down for a solution, as in anaelith's ((4/14)(0.55^4) + (10/14)(0.5^4))*100% which clearly shows the reasoning. Yes, I've given problems like this on a no-calculator portion of an exam, with special instructions like "you do not need to simplify your answer". Reason? I am not testing arithmetic in a class like this. I might further ask a student to tell me if this was less-than, equal-to, or greater-than 6.25%, and for them to provide a reason.
I am asking myself your original question every day in my job. I used to be a no-calculator-till-you've-mastered-it kind of person, because that's the way I was taught. Using calculators was seriously frowned on when I originally learned this stuff, but I am not convinced that learning it one way is preferable to another. It seems to make more sense to push problem-solving with the technology of the day than hold "the way I was taught" as sacred. Not to be That Guy, but how many of us really learned what a square root was, or what division meant, by correctly using some by-hand algorithm? Moreover, if you want to know the value of log 3, what do you use to get it?
I think we're seeing a big change coming in how math education is treated in light of evolving technologies. I think your question is becoming more important. While I wasn't totally inspired with the way this talk approached the topic, I am excited to see what happens with the ensuing discussion.
posted by klausman at 4:36 PM on July 3, 2011 [1 favorite]
Do you mean in a testing environment, or just where (at all) in the classroom? I think they are very different scenarios, particularly by today's standards.
Non-exam class time: First, I think calculators/technology should be encouraged when exploring something new. This definitely comes out of a "discovery-based learning" mindset for me, and there are lots of examples of this out there...using technology as a discovery device. This excites me because a computationally-heavy idea can be tested quickly, and the focus is placed on finding relationships and modeling an idea, and less on lengthy hand-calculations. I will not make a case for discovery-based learning here (as that is not pertinent to the question), but using technology in the classroom is so fitting for this purpose.
On that note, do any folks here actually shy away from technology when they are trying to figure something out? Are pencil and paper really the tools of choice for the math people in the room?
Exam time: Surely, there are particular things that each teacher and curriculum want tested without a calculator. What those things are will differ, but writing an exam with those things in mind takes considerable thought. With those things in mind, I always write exams with two parts: Part A - no calculator allowed, Part B - calculator allowed. I guess I'm lucky in that my department has described what things we want to test and how they should be tested, and for the most part, I personally agree with the assessment strategies. We have technology built into the curriculum in fairly specific ways, so I know that I need to assess, for example, whether a student can solve a max/min 1-variable calculus problem. I am possibly testing several skills in a problem like this, but I could allow the ti-89 and just drive at knowledge of the 1st and 2nd derivative tests. I try to look at every question and see if I have properly assessed what we say we want to, and I know that using technology is a way do distill the proper outcome I am looking for.
How about for a question like this?
What's being tested there is most likely whether you can write your process down for a solution, as in anaelith's ((4/14)(0.55^4) + (10/14)(0.5^4))*100% which clearly shows the reasoning. Yes, I've given problems like this on a no-calculator portion of an exam, with special instructions like "you do not need to simplify your answer". Reason? I am not testing arithmetic in a class like this. I might further ask a student to tell me if this was less-than, equal-to, or greater-than 6.25%, and for them to provide a reason.
I am asking myself your original question every day in my job. I used to be a no-calculator-till-you've-mastered-it kind of person, because that's the way I was taught. Using calculators was seriously frowned on when I originally learned this stuff, but I am not convinced that learning it one way is preferable to another. It seems to make more sense to push problem-solving with the technology of the day than hold "the way I was taught" as sacred. Not to be That Guy, but how many of us really learned what a square root was, or what division meant, by correctly using some by-hand algorithm? Moreover, if you want to know the value of log 3, what do you use to get it?
I think we're seeing a big change coming in how math education is treated in light of evolving technologies. I think your question is becoming more important. While I wasn't totally inspired with the way this talk approached the topic, I am excited to see what happens with the ensuing discussion.
posted by klausman at 4:36 PM on July 3, 2011 [1 favorite]
Net Prophet: Good question! That's why I ask students to explain any assumptions they make. If someone just decided that the ball was tossed down the stairs, (and maybe also assumed something interesting happened at the bottom), then I'd call it good and grade their (b)-(d) based on their (a). That sort of thing actually happens a lot. It may make the grading a little tougher but it's also more interesting to read.
posted by monkeymadness at 9:31 AM on July 4, 2011
posted by monkeymadness at 9:31 AM on July 4, 2011
Response by poster: Thanks to everyone. If anaelith could explain the process of simplification, it would be even more valuable. I could follow a good portion of it but not everything. I was going to mefimail you but your mefimail is unfortunately disabled.
I decided to do it this way: do the problem by hand, and if your answer is one of the multiple choice answers, then fine. If not, use a calculator to check. If it's still not one of the answers, then choose "None of these" (the fifth choice). Another factor is that my son is only ten years old and I don't want this to be traumatizing (just very challenging!).
posted by strangeguitars at 6:31 AM on July 5, 2011
I decided to do it this way: do the problem by hand, and if your answer is one of the multiple choice answers, then fine. If not, use a calculator to check. If it's still not one of the answers, then choose "None of these" (the fifth choice). Another factor is that my son is only ten years old and I don't want this to be traumatizing (just very challenging!).
posted by strangeguitars at 6:31 AM on July 5, 2011
- Thought: Decimals! Oh no! Solution: Convert decimals to fractions. Example: 0.55 = 55/100
- T: Unsimplified fractions! Oh no! S: Simplify fractions. E: 55/100 = 11/20
- T: Powers of fractions. S: Re-write powers of fractions as a power of the numerator over a power of the denominator. E: (11/20)^4 = (11^4)/(20^4) Bonus side note:
- T: Multiplication of fractions:
S: Multiply straight across. You could do some simplification here, but with the powers floating around I prefer to do it after the next step. - T: Numbers are too big.
S: Factor all numbers into primes.
E: 20 = 2*2*5, so 20^4 = ((2^2)*5)^4 = ((2^2)^4)*(5^4) = (2^8)*(5^4) - T: Fractions are not simplified.
S: Cancel out powers.
E: (2^2)/(2*(2^8)) = 1/(2^7) - T: Two fractions being added.
S: Find a common denominator.
E: Multiply right addend by 1 in the form of ((2^3)*(5^4))/((2^3)*(5^4)). Note that having everything factored makes finding the least common denominator easy. - T: Two fractions with same denominator being added.
S: Do the addition.
E: (a/b)+(c/b) = (a+c)/b - T: Order of operations, must do exponents before addition. But, powers are too big and awkward.
S: Combine twos and fives into tens.
E: (2^3)*(5^5) = (2^3)*(5^3)*(5^2) = (10^3)*(5^2) - T: Order of operations (still).
S: Do the multiplication, then the addition. - T: Dividing by a really large number is awkward.
S: Divide by 10000 first, then divide by 56 OR divide by 56 and then divide by 10000.
a a a a^n
(a/b)^n = - * - * ... * - = ---
b b b b^n
Ten is a good age, I was worried that your son was older, which would probably be too late. Far too many students are unsure when it comes to this stuff because it's practically still new to them and not yet an ingrained habit.
posted by anaelith at 8:13 AM on July 5, 2011
Response by poster: You are really a god/ess (I have no idea what your gender is).
3. I didn't know that, and I didn't see it in Wikipedia. Very cool. I guess it makes sense because (a^n)*(b^n) = (a*b)^n, right?
I don't understand #5 at all. I do, however, understand how to factor.
I don't understand #6.
Don't understand #7.
Wouldn't have come up with #9. Pretty impressive. :)
posted by strangeguitars at 8:50 AM on July 5, 2011
3. I didn't know that, and I didn't see it in Wikipedia. Very cool. I guess it makes sense because (a^n)*(b^n) = (a*b)^n, right?
I don't understand #5 at all. I do, however, understand how to factor.
I don't understand #6.
Don't understand #7.
Wouldn't have come up with #9. Pretty impressive. :)
posted by strangeguitars at 8:50 AM on July 5, 2011
Number 6: When dividing two exponents with the same base, you subtract the exponents. In this case we have 2^2 in the numerator and 2^9 in the denominator, so the answer is 2^(2-9) = 2^-7 = 1/2^7
Number 7: you're trying to get the denominators of the numbers you're adding to be the same, so that you can add the numerators. To do this, you multiple both top and bottom by the same thing (so you're really multiplying by 1, but in a simpler form).
Simpler example: 2/3 + 3/4
Multiply 2/3 x 4/4 = 8/12 and 3/4 x 3/3 = 9/12. In both cases you've multiplied each number by 1, but in a particularly useful form.
Now 8/12 + 9/12 = 17/12
posted by Dr.Enormous at 2:44 PM on July 5, 2011
Number 7: you're trying to get the denominators of the numbers you're adding to be the same, so that you can add the numerators. To do this, you multiple both top and bottom by the same thing (so you're really multiplying by 1, but in a simpler form).
Simpler example: 2/3 + 3/4
Multiply 2/3 x 4/4 = 8/12 and 3/4 x 3/3 = 9/12. In both cases you've multiplied each number by 1, but in a particularly useful form.
Now 8/12 + 9/12 = 17/12
posted by Dr.Enormous at 2:44 PM on July 5, 2011
Oh yeah, number 5: You can distribute exponents:
(A*B)^2 = (A^2)*(B^2)
Essentially, this says that squaring the product of two numbers is the same as squaring each, then multiplying them. In this case, you're breaking down 20^4. Since 20 = 2*2*5, then substitute that in for 20:
204 = (2*2*5)4
But 2*2 is the same as 22, so we now have (22*5)4 .
Distribute the exponent, and you have (22)4* 54
When you have an power of some number raised to a power, you multiply the two exponents, so (22)4 becomes simply 28, and the whole expression is now 28*54
posted by Dr.Enormous at 2:54 PM on July 5, 2011
(A*B)^2 = (A^2)*(B^2)
Essentially, this says that squaring the product of two numbers is the same as squaring each, then multiplying them. In this case, you're breaking down 20^4. Since 20 = 2*2*5, then substitute that in for 20:
204 = (2*2*5)4
But 2*2 is the same as 22, so we now have (22*5)4 .
Distribute the exponent, and you have (22)4* 54
When you have an power of some number raised to a power, you multiply the two exponents, so (22)4 becomes simply 28, and the whole expression is now 28*54
posted by Dr.Enormous at 2:54 PM on July 5, 2011
This thread is closed to new comments.
Also, calculators should be allowed when teaching order of operations, because that way you know which kids actually get it and which kids don't/are cheating. (Since all but the fancy graphing calculators will ignore order of operations rules.)
posted by phunniemee at 4:41 AM on July 3, 2011 [1 favorite]