Students may rely on calculators to bypass a more holistic understanding of mathematics, researcher says

Nov 12, 2012

(Phys.org)—Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, says Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research & Development Center. King has proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students published in the British Journal of Educational Technology.

"We really can't assume that calculators are helping students," said King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."

Together with Carol Robinson, coauthor and director of the Education Centre at Loughborough University in England, King examined whether the inherent characteristics of the mathematics questions presented to students facilitated a deep or surface approach to learning. Using a limited sample size, they interviewed 10 second-year undergraduate students enrolled in a competitive engineering program. The students were given a number of mathematical questions related to sine waves—a mathematical function that describes a smooth repetitive oscillation—and were allowed to use calculators to answer them. More than half of the students adopted the option of using the calculators to solve the problem.

"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," said King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."

After completing the problems, the students were interviewed about their process. A student who had used a calculator noted that she struggled with the answer because she couldn't remember the "rules" regarding sine and it was "easier" to use a calculator. In contrast, a student who did not use a calculator was asked why someone might have a problem answering this question. The student said he didn't see a reason for a problem. However, he noted that one may have trouble visualizing a sine wave if he/she is told not to use a calculator.

"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes—especially at the undergraduate level," said King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."

King also suggests that relevant research should be done investigating the correlation between how and why students use to evaluate the types of learning approaches that adopt toward problem solving in mathematics.

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That Guy
5 / 5 (4) Nov 12, 2012
As an engineering tutor, I'm seeing this more and more often. Students will come wanting an equation that fits every situation instead of recognizing that each problem is unique and often can be approached from different directions. They just want to get a number to write down and don't care where it comes from.
verkle
3.7 / 5 (3) Nov 12, 2012
Good article.

I used to tutor Math in college to students of algebra and calculus. A calculator was useless in trying to get them to understand the key points.

Like one of my professors said, you don't have to come up with the right number on a problem to get most of the credit. If your analysis and process were correct, you could get 90% of the credit for that question. I think that is what math is all about.

Horus
1 / 5 (1) Nov 12, 2012
It's a sign that these students are illiterate with the applied mathematics and science subject matters at hand.

During my Mechanical Engineering undergraduate days people has HP 28S and I had my Casio fx 7000 which didn't do all the fancy stuff.

Who finished their Machine Design exam first with the top score? I did. When you know the Calculus, DiffEq, Vector Equations, etc., all you do is reduce down to the correct units and then plug and play into the calculator. Calculators are the last step. It appears these kids don't see patterns in unit conversions and more.

The sooner they do the sooner they can pay attention and understand the subject(s).
ka_
not rated yet Nov 12, 2012
I had a great teacher who always approach class the other way around; he would NEVER show any formula, but instead describe a real world problem where the class essentially would need to come up with a formula to solve it! This does require fundamental math understandings on the teachers part - we would typically end up with far more complicated formulas than those presented in the book, before we then looked at ways to simplify them and thereby find the formulas presented in the book!

The really surprising result of the class with this this teacher was that though we initially we had quite a few students who hated math, at the end of the course all of them overcame that and several even stated they now liked math!

Disclaimer - we had no calculator in our classes so I cant say if this would case a problem, however, I do kind of believe that teachers approach would be effective even when a calculator is present.
dschlink
not rated yet Nov 12, 2012
Absolutely nothing new here. I noticed this problem in graduate school in the late '70s. I was a MBA student and took some engineering classes for fun. Most of the students had zero feel for correct solutions. They'd plug numbers into their calculators and even if they made a large error in data entry, just assume the answer was correct. I had received a bachelors in Engineering Physics just before calculators arrived on the scene and estimating the correct answer was a necessary skill.
ValeriaT
1 / 5 (4) Nov 12, 2012
I do consider the teaching of math as an analogy of teaching assembly language in the courses of computer programming. A dominant majority of textbook assignations can be solved with simulation packages trivially. A dominant majority of practical problems cannot be solved with high-school math at all and these simulation packages must be used anyway.

If so-why to bother students with some math at all? Why not to teach them MathLab or similar simulators directly?
TheGhostofOtto1923
3.5 / 5 (13) Nov 12, 2012
Our brains are calculators but they are not very good ones. This is why we invented whiteboards and spreadsheets to begin with.

Soon enough we will have plug-ins that will do the grunt work for us automatically. And soon after that machines will be doing the conceptualizing, and they will be doing it much better than we ever could.

Lets face it. Humans are a lost paradigm. We will be romping barefoot through the fields like the eloi.
Tesla2
not rated yet Nov 12, 2012
Our brains are calculators but they are not very good ones. This is why we invented whiteboards and spreadsheets to begin with.

Soon enough we will have plug-ins that will do the grunt work for us automatically. And soon after that machines will be doing the conceptualizing, and they will be doing it much better than we ever could.

Lets face it. Humans are a lost paradigm. We will be romping barefoot through the fields like the eloi.


And getting eaten by those who do the thinking too...
doubledodge
not rated yet Nov 13, 2012
Anyone remember the slide rule? I still have several and although it may be faster and more accurate to use a calculator, the slide rule certainly gave one a better feel for the numbers and of course you had to take care of your powers of 10 yourself. With regular use you could almost visualise the result in your head and that certainly helps me to see when I have pressed the wrong key on my calculator even now.

So BRING BACK THE SLIDE RULE!
antialias_physorg
5 / 5 (1) Nov 13, 2012
I used to tutor Math in college to students of algebra and calculus. A calculator was useless in trying to get them to understand the key points.

That mirrors my experiences as a tutor. While calculators free them from the nitty-gritty stuff it is just that nitty-gritty stuff that you need to have a solid foundation in. Understanding builds on solid foundations.

If you're unsure about the basics without your electronic crutch then you will not trust your own understanding of anything further up. And that insecurity translates into an inability to efficiently and effectively employ that understanding to solve new problems (or combine knowledge to find new ways of solving problems - e.g. in physics)

If your analysis and process were correct, you could get 90% of the credit

Yes. The numbers don't really matter. Because once you hit real life you WILL use computers to solve your math problems. (Almost no real problem can be solved analytically).
thingsmith
not rated yet Nov 13, 2012
I do consider the teaching of math as an analogy of teaching assembly language in the courses of computer programming. A dominant majority of textbook assignations can be solved with simulation packages trivially. A dominant majority of practical problems cannot be solved with high-school math at all and these simulation packages must be used anyway.


Do you really want to carry the math simulator when you go shopping? Imagine the extra time required to determine whether the 8, 16, or 24 jar of peanut butter is cheaper and more this to everything else you want to buy.(FYI the biggest is not always cheapest.) Of course if you are wealthy it does not matter.

Clearly you view on programming is to always buy more bigger and more hardware. Sorry, this does not work in the real world. Programmers need to optimize their code and optimizing compilers do not alway work.
alfie_null
5 / 5 (4) Nov 13, 2012
If so-why to bother students with some math at all? Why not to teach them MathLab or similar simulators directly?

I have this silly little quirk. I just feel more comfortable with the idea that (for instance) the engineer who designed the bridge I use when driving to work has a good understanding of how numbers and equations work, which ones to use to solve which problems, and is not just blindly plugging data into a computer program.
antialias_physorg
5 / 5 (1) Nov 13, 2012
I have this silly little quirk.

Me too. Especially when it gets down to doing any kind of serious analysis you will get into the area of statistics and simulations.
If you have no real grasp of the intricacies of statistics (or the underlying principles and limitations of the simulation) then the likelyhood that you will get misleading results is enormous (as can be witnessed by many semi-informed statements in the comment sections on physorg, BTW).

'Blindly plugging in data' will just lead to GIGO (garbage in, garbage out)
hb_
not rated yet Nov 13, 2012
@natello

The problem is, that when the students use only simulation tools, they do not acquire an understanding of the underlying math and processes. You may beleive that the students - freed of the tedious calculating - would have more time to grasp the core concepts. All empirical data, however, indicates the opposite.

When someone encounters a problem in the real world, knowing if the basic assumptions and if the results are reasonable is really valuable. Even routine simulations can have incorrect data at the input, and this is why judging if the result is reasonable or not is necessary for any true engineer.
ValeriaT
1 / 5 (1) Nov 13, 2012
The problem is, that when the students use only simulation tools, they do not acquire an understanding of the underlying math and processes
Indeed. And they can solve the real-life problems in addition! Whereas in contemporary version of educational system they cannot do both math, both real-life simulations well. What such learning is about, after then?

In the same way, the programmers aren't required to understand the assembler and machine code programming. Why they should understand it? Will it help them in solving of real life problems better?

Not to say, I've persistent problem even with professional physicists: they do understand math (sometimes) - but they (usually) don't understand the physics. They cannot imagine, what it is possible in it and what not. I know, where their problem actually is - they cannot imagine even the Schrodinger wave solution. They did never see, what the quantum mechanics or relativity really predict. Not to say about experimental physics.
ValeriaT
1.5 / 5 (2) Nov 13, 2012
The problem is, that when the students use only simulation tools, they do not acquire an understanding of the underlying math and processes.
This is correct - but who of them actually did see, how the differential or integral or another functional is working? They should understand the principle of the calculus - but should they spend their most intellectually productive time with memorizing of algorithms, which most of programs already handle a way better?

That is to say, even the courses of math should be more illustrative and interactive. It will support the creativity of thinking and multidimensional imagination. I've no problem if the students at the secondary school level become familiar with quantum mechanics. But they should learn about it in interactive way. Unfortunately, we have no good simulators for it developed yet.
drhoo
not rated yet Nov 13, 2012
"Mathematics is primarily a tool for contemplation not calculation"

Nassim Taleb
Eikka
1 / 5 (1) Nov 17, 2012
Imagine the extra time required to determine whether the 8, 16, or 24 jar of peanut butter is cheaper


That's why you look for the unit price on the price tags.

That's an example of a trivial problem that you could solve easily in your head, but in reality you aren't going to calculate if 2 for $3.99 is cheaper than 1 for $1.99 to save one cent.