Are Calculators Hindering Math Education? Share Your Opinion!

[JonF]

A TI-89 was one of the worst investment I ever made.

 

[AlphaNumeric]

I would say that most of these anecdotes do not make much of a case for withholding information and tools from the students in order to make them learn arithmetic.

When you're 17 and doing 1st order ODEs, you shouldn't be reaching for a calculator to do something like 14*11.

As for my old teacher, comments like "You're being stupid" were not meant as "I think you're stupid for doing that" but more "You're better than that". We were a class of 3 students, the only 3 taking 'Double Maths' in a year of 200, and as such he expected a touch more from each of us. We'd not be scalded for using a calculator to compute the decimal expression for a trig answer, but if someone used a calculator to do a method they were supposed to know the algebraic method, he'd comment, not to put us down but to make us realize it was not worth using a calculator as a crutch in the long run.

I still think of such rationale when I see 1st year students reaching for a calculator to give the decimal expansion for an answer like '2pi', in a maths class. The question (and entire course!) makes no mention of decimals or calculators, but emphasises algebraic expressions but still it's been ground into some students that 'decimals are better' or 'decimals are the proper answer'. Solving x^{2}-2=0 doesn't gives x=\sqrt{2}[/tesx] but "x = 1.414 to 4 sig fig". Fine if you're a physicist or engineering (well, sometimes!) but not as a mathematician.

 

[Crosson]

The machine allows us to expand the scope with which we see mathematics.

Consider the falling scenario:

Teacher: Give me a number.

Student: 6


Teacher: Give me a polynomial.

Student: x^3 +3x -1


Teacher: Give me a matrix.

Student: {{1,4,7},{8,7,0},{4,3,5}}


Teacher: Give me a group.

Student: Z_4 x Z_2

Do you guys see the pattern? All too often students walk away with the unconscious misconception that every mathematical object is built out of a handful of small integers. Its not that they would deny large integer or irrational matrices if they saw them, but that their mind is lacking in examples of these and so they do not think of these things.

I agree that "1.414 with 4 sig figs" is a joke compared to \sqrt(2), but this is why I am in favor of systems that do symbolic math along side numerical math, like mathematica. Mathematica will always return \sqrt(2) or \pi when appropriate, and so it becomes a matter of interest when a approximate decimal is returned.

Without the computer we are confined to a small range of integers. A little time spent with the computer gives us perspective on the smallness of human calculation, a perspective that frees us and empowers us.

I think this debate is rather like an ancient debate that must have occurred over notation. The adoption of arabic numerals vastly expanded the scope of western mathematics, although it is true that with them arithmetic becomes a less mental activity.

When you're 17 and doing 1st order ODEs, you shouldn't be reaching for a calculator to do something like 14*11.

Of course you should, you must have long since mastered a basic facility in arithmetic and could better spend that time learning something new.

For that matter, you should see how the computer generates the analytical solutions for the general class of linear ODEs you are learning about, and the computer will show you how even relatively simple ODEs lead to solutions which are totally out of the realm of human possibility.

 

[morphism]

Are you kidding? If someone asked me for an example of a number, I'm not going to give them pi^(17^(sqrt(3)/2)) * gamma(4.5); unless I'm trying to be a smartass, I'll pick the most natural thing that springs to mind: a natural number. Also, what does this have to do with the topic? Are you somehow implying that calculators help you "think outside the integers"?

And you're saying someone should use a calculator to compute something as trivial as 14*11, when in fact it would take more time and effort to use one for such a thing?!

Sure, computers can be useful, but let's not get carried away...

 

[uart]

I think that calculators and numerical mathematics have their place in mathematics teaching, but I also agree that over-reliance on calculators is a bad thing (and it's also extremely wide-spread these days).

Seriously I’ve got students that will use their calculator to do something like adding 2 to a number. One thing that particularly irks me is when a student reaches for the calculator for something trivial and then to add insult actually gets the wrong answer due to operator error. A typical example might be where a student has an expression like \frac{x+2}{7} and they have to substitute x=9. So they dive straight for the calculator and come up with 9.2857 which they happily write down without a second thought that it can’t be the correct answer.

You’d be amazed at how frequently this type of thing happens when students are over-reliant on the calculator. I suppose the moral of the story is that if you are going to be over reliant on a calculator then at least you’d better learn to use it correctly.

 

[mathwonk]

doodlebob and others are stating correctly that some teachers can make a calculator based program fun. nonetheless their students will never learn any ideas until they lay down the calculator and start thinking about what has been displayed there.

the stimulation from the calculator needs complementing by actually thinking. and thinking is not necessarily torture. we are thinking beings after all.

fun is good motivation for work, but it is not necessarily useful work.

being asked to think, should not be equated with torture.

for extremely weak and frightened students, afraid to begin to think or imagine, calculators may help take away some of the fear. but at some point they must be abandoned for real conceptual work, or there is no gain.

for very strong students, already in possession of computational skills and conceptual understanding, a calculator does little harm, but has little value either except for experimenting in cases of great computational complexity.

For the great majority of students who are able to begin to learn concepts, but need practice both in computing and in reasoning, calculators make this process harder, and are actually harmful.

thus calculators may help motivate beginners to learn, and can be used valuably by those who have already learned, but for those who are in process of learning unless used with great care, they are useless and even detrimental.

this is the lesson of a lifetime of teaching.

 

[Trail_Builder]

im only high school level student and always use a calculator if allowed (within reason, like i know most easy powers and stuff), and find i make silly mistakes trying to work without it.

though i do try...

o well, whatever floats you boat

 

[mathwonk]

you are the perfect illustration of someone who is beign harmed by a calculator, since you are not learning to stop making silly mistakes.

there is a reaSON FOR SILLY MISTAKES, LIKE misunderstanding basic ideas.

unless you change this, in a few years you may well be in college wondering why you are failing calculus, or abstract algebra.

 

[Crosson]

there is a reaSON FOR SILLY MISTAKES, LIKE misunderstanding basic ideas.

Silly mistakes come from lack of concentration, it is the fact that they don't come from real misunderstanding that makes them "silly".

If the complaint is that students make errors in the use of the machine, how does this argue against teaching the use of the machine in the classroom?

 

[mathwonk]

i recommend locking this thread as it is potentially endless. everyone, especially those with no experience, has an opinion.

 

[JasonRox]

Silly mistakes come from lack of concentration, it is the fact that they don't come from real misunderstanding that makes them "silly".

I disagree.

You shouldn't have to concentrate much for small things, and I believe that's what he's saying.

 

[Gib Z]

Eventually the thread will die, just you wait mathwonk.

I have little experience compared to that of mathwonks, but I still think calculators disrupt the theory of math. People in my school know Newtons method just fine at my school, but if you told them to square root something by hand they would stare blankly at you.

The sqrt button of the god damn calculator stops them from realizing they can use what they know rather than use a calculator!

 

[arildno]

Silly mistakes come from lack of concentration, it is the fact that they don't come from real misunderstanding that makes them "silly".

Partly true, but wholly misunderstood.
"Concentration" is not an automatic, inherent ability in everyone, the skill to concentrate and not let yourself be distracted must be meticulously, and painstakingly, built up. In particular, close attention to what you write AND HOW YOU GET THERE is a crucial feature about the concentrated, working mind that requires an on-going understanding of intermediate steps that the short-cuts in calculator use inhibits, by way of concealing these steps within the machinery.

Only those students that are good at maths ought to be allowed to use a calculator, everyone else should be denied that "aid", for their own good.

 

[chaoseverlasting]

Calculators inhibit your ability to learn techniques that would enable you to do those calculations quickly. I am a student and I try not to use calculators at all. For one thing, they arent allowed in my exams, where the time I spend on calculations is very valuable. Therefore, I must be quick and accurate when I calculate something. If I am dependent on a calculator, I won't ever be able to do that. Like 14*11 =140+14=154.

 

[mathwonk]

the only time i have found a calculator useful was in teaching graduate algebra once, and trying to carry out examples of some of the proofs of the fundamental theorem of symmetric functions, that all symmetric functions are polynomials in the elementary ones.

the proofs consisted of very impractical algorithms for transforming a symmetric function into a polynomial in the eleme ones, but these transformations took forever and were very hard to complete without error.

thus even finding the basic expressions for the discriminant of a cubic polynomial was essentilly undoable for me. , i.e. if r,s,t, are the roots of a cubic, the discriminant is a formula for the symmetric product
(r-s)^2 (r-t)^2 (s-t)^2 in terms of the coefficients of the polynomial, namely in terms of r+s+t, rst, and rs+rt+st. Try it.

I this case only, I used mathematica to cary out the operations. But then i learned a better way, using the resultant, to do it by hand. Its all in my webnotes.

 

[mathwonk]

by the way, just because i have decades of experience does not mean your experience may not differ from mine, even if i often disallow this possibility. that's what makes it hard to tell just which advice to take from your elders.

 

[D H]

im only high school level student and always use a calculator if allowed (within reason, like i know most easy powers and stuff), and find i make silly mistakes trying to work without it.

A calculator is no guarantee against silly mistakes. If you can't do simple problems in your head or on paper, how are you to know the answer on the calculator is correct? Over-reliance on calculators is lazy and stupid.

Keep in mind that when calculators are not allowed on a test, the test questions are usually designed to yield nice, neat answers. Moreover, the instructor would much rather see an answer like (\sqrt5-1)\pi[/tex] than 3.88322207745093 because the former shows that the student was thinking while the latter shows the student was punching buttons. This is particularly so if the answer was supposed to be (\sqrt5-2)\pi[/tex].<br /> <br /> How many of you have given $20.50 for an item that costs $5.32 and received a stymied look from the cashier followed by $5.18 in change? I asked for the manager when the cashier would not relent ("I gave you a twenty and change. I should get 15 and change back." "Well, the cash register told me to give you $5.18."). The manager quickly saw a twenty at the top of the 10 dollar bill bin and fixed the problem.

 

[JasonRox]

How many of you have given $20.50 for an item that costs $5.32 and received a stymied look from the cashier followed by $5.18 in change? I asked for the manager when the cashier would not relent ("I gave you a twenty and change. I should get 15 and change back." "Well, the cash register told me to give you $5.18."). The manager quickly saw a twenty at the top of the 10 dollar bill bin and fixed the problem.

Sounds like the cashier has mistaken your $20 for a $10. Not a mathematical mistake.

 

[D H]

She made several mistakes. First, she put the twenty in with the tens. Second, her fingers by-passed her brain and automatically entered $10 because that is where her fingers put my bill. Third, she refused to believe her cash register could be mistaken. Fourth, she did not double-check the subtraction. She knew I gave her a twenty and still insisted the amount on the register was correct.

 

[JasonRox]

She made several mistakes. First, she put the twenty in with the tens. Second, her fingers by-passed her brain and automatically entered $10 because that is where her fingers put my bill. Third, she refused to believe her cash register could be mistaken. Fourth, she did not double-check the subtraction. She knew I gave her a twenty and still insisted the amount on the register was correct.

The first mistake led to all the others. Not a mathematical error here.

How do you know she knew you gave her a $20? You're not her.

 

[D H]

I told her I gave her one. She agreed that I did. I even said as much in my first post on this subject:

"I gave you a twenty and change. I should get 15 and change back." "Well, the cash register told me to give you $5.18."

You are missing the point. It is quite common for people with any mathematical sense to pay a bill some paper bills and a bit of change. (Who wants a pocket full of nearly worthless coins?) It is also quite common for cashiers to be completely stymied by such an action. They let the register do the work for them.

 

[symbolipoint]

HERE is an idea about "Calculators in Education":

Make a specific dedicated board for Calculators in Education. As a main topic board, the topic and discussions will be easier to find.

 

[symbolipoint]

From arildno:

Only those students that are good at maths ought to be allowed to use a calculator, everyone else should be denied that "aid", for their own good.

That was exactly the method many years ago, at least in some institutions. Students would study and hopefully learn arithmetic and mathematical ideas at the basic level. When they entered into the higher college preparatory courses, they would be allowed to use scientific calculators. The use of the calculators was not used as substitute for understanding nor for skills. STUDENTS STILL WERE REQUIRED TO SHOW THEIR STEPS AND SOLUTION PROCESSES.

 

[drpizza]

I've looked back at some of the questions I had about calculator use. It's beneficial to see the same students in two different settings - math and physics. I can't recall ever seeing, for example, a quadratic equation in a math textbook that didn't have integer coefficients. In physics, the students might run into 0 = 15.8 + 4.8t + .5(9.81)t^2 and not even recognize that it's a quadratic equation. Maybe, if we're going to give them calculators, we also give students more realistic problems with "not so nice" numbers to work with, rather than the contrived problems such as "Bob is 5 years younger than Steve..."

 

[arildno]

Of course, intelligently used, the calculator can be a powerful aid.
This is when the actual calculations become excessively tedious, and when
we are primarily interested in seeing how the student develops a general strategy for solving a problem, in which such calculations are merely annoying sub-problems.

 

[Crosson]

in which such calculations are merely annoying sub-problems.

Even worse, many students use these tedious calculations to convince themselves that they are "working hard" and that they "deserve" a good grade.

In my opinion this is the biggest problem with doing arithmetic, algebra and calculus by hand: it distracts the students from the the difficult task of applying the concepts effectively in a variety of situations.

As a physics tutor, I explain to intro students that a good way to study is to look at every problem in the section, but only to the point of understanding the application of the concepts. Some students cannot receive this advice, they believe in doing the complete problem everytime including algebra, arithmetic and physical units, which is absurd to do for practice sake beyond the first few weeks of the first semester!

 

[mathwonk]

dr.pizza, your post reminds me of the students in my calculus class who think that the smallest positive real number is 1, and that the linearity properties of the integral or derivative, only apply to integer multiples. I.e. basically they think all real numbers are integers, and why not, if all the problems we give them involve only integers!And Crosson, it takes a very long time, indeed years rather than weeks, doing "annoying" calculations before one can safely ignore them as distractions. Until that time one should call them "enlightening" calculations!

You may be right about your students' needs, but my own students are not like you and me, they did not do so many computations in lower school, and have almost no intuition for them. They have used calculators so long they never did acquire any skill or knowledge of basic operations.

E.g. many if not most of my students do not even realize that the reciprocal of a large positive number is a small number. In that situation, calculators have become monsters. I invite you to try this experiment on your own scholars. See if your assumptions about their knowledge are perhaps over optimistic.

 

[JasonRox]

You may be right about your students' needs, but my own students are not like you and me, they did not do so many computations in lower school, and have almost no intuition for them. They have used calculators so long they never did acquire any skill or knowledge of basic operations.

E.g. many if not most of my students do not even realize that the reciprocal of a large positive number is a small number. In that situation, calculators have become monsters. I invite you to try this experiment on your own scholars. See if your assumptions about their knowledge are perhaps over optimistic.

I never had that experience in school (I skipped all the time). If you understand what you're doing, you should understand what solutions look like and stuff.

In the Calculus course that I TA in, I get students asking for help on assignments or to look at there work. I never look at the assignments, so when they show me their work or problem it's the first time I seen it. After I see the problem and then their solution without even working it out, because of understanding, I can already decide whether the solution makes any sense at all. If it doesn't make sense, I tell them and see how they did and throw a few hints on what they should work on and what not.

For the reasons above, this is why I choose courses like Calculus and Linear Algebra to TA in. It's because I understand them. I don't like getting nervous when somebody asks me a question and I have to pull out the textbook, like other TA's might have to do. I hate that. The only time I use the textbook is to check where they are so I don't show them knew stuff. Even for examples, if you understand it, you can create your own without too much difficulty. So, I don't understand why professors always use the examples given in the textbook it completely defeats the purpose or having the examples in the first place.

Anyways, I also have my weak spots, even in Calculus (I just don't tell anyone until I got it figured out. ).

In the end, understanding is key. Calculators won't get you there. Because you can plug in any information into your calculator but you'll never intuitively know whether the answer makes sense. You just have to hope that you did the steps correctly and inputted it in your calculator correctly.

Note: It's like that one time in Differential Equations where I pointed out two different ways to prove/solve a problem on the board. The professor dismissed both of mine, and went out with "easier" methods the other students came up with. As it turns out, he tried mine after the others, and it was done in 2 steps. He didn't bother trying the other one because he said it would be over their heads although it was the easiest method of all and could be done in one step. What were the concepts? The first was just finding the derivative of the equation and then using basic Linear Algebra, and the other method was using Taylor Series. Two things that you should know when in Differential Equations. Sad.

People wonder why they do bad in courses like Linear Algebra and Abstract Algebra. Maybe it's because a calculator won't tell you if a set with so-and-so operations is a vector space or a group!

 

[uart]

E.g. many if not most of my students do not even realize that the reciprocal of a large positive number is a small number. In that situation, calculators have become monsters.

Yeah I've also noticed that. It definitely shows a big lack of mathematical intuition.

Another case I came across recently is in the teaching of polynomial division. When I point out the obvious similarity in the procedure with that of "normal" long division all I get is a lot of puzzled faces. The reason of course is that none of the students have ever done a long division by hand.

 

[mathwonk]

I think you have put your finger on the reason polynomials and abstract algebra is so hard for todays students - they have no real life experience doing the calculations that are being abstracted.

so calculators have made the jump from elementary to abstract math almost impossible, by taking away the bridge that used to link the two.