Stephen Shankland **posted an interesting article on CNET today**. Here is an exerpt from his article, which you should read in full. He says:

*Clearly, children need some understanding on their own of math, and reliance on a computer has a lot of drawbacks. But computers can also aid those who otherwise would fall by the mathematical wayside, or let people with more advanced abilities bypass drudgery and move on to the challenging material. Graphing calculators can let many students explore curves and functions that realistically they’d more likely ignore if they had to plot them by hand.*

My response to some of the negative comments about his article is:

*Some of you have decided that using technology to handle calculations in mathematics is going to weaken student’s understanding of mathematics. I have to tell you, our student’s understanding of mathematics, and even the vast majority of people’s understanding of what mathematics is pretty bad. Awful. Horrible. I mean, really, really bad.
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*Mathematics is not about calculations. Mathematics is about understanding how our world works through the lens of logical reasoning and pattern forming, and then communicating our understanding of that process to other people.
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*Calculations are a tool in mathematics to understand a process. In my opinion, I want students to understand the processes and ideas that mathematics represents, not the calculations which short-cut that understanding.
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*Here’s an example that Gary Stager suggested to highlight this problem. Ask a typical math teacher to explain to you why "you invert the 2nd fraction and multiply instead" when dividing two fractions works. Ask them to explain the concept behind "inverting and multiplying" two fractions, and you know what, they can’t. They’ve learned a recipe for doing a calculation but have no conceptual understanding of why that rule works, and these are people who are teaching our children about mathematics!
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*We need to move away from the mindset that the most important part of the mathematics curriculum we teach is the rote calculations which can generally be done much faster on a computer, and towards the mindset that students need to be able to formulate problems, decide on appropriate mathematics to use to solve these problems, and then do the calculations on an appropriate device, and finally check that these solutions make sense. These are the steps that Conrad Wolfram and Dan Meyer (in their TED talks) outline as crucial to mathematical understanding, and I completely agree.*

Mathematics education needs to change. Those people who want a "back to basics" approach and get rid of the calculators seem to think that this will improve the mathematics education in our schools. This is flatly not true.

If you ask a random sample of people, they either "weren’t very good at mathematics" and generally hated it, or a very small minority loved it. This opinion spans all age groups and goes back many years, far before the introduction of calculators in schools. If we judge the success of an educational approach by the number of people who enjoy working in a subject, why are so many people who were exposed to that approach before the introduction of calculators hate mathematics so much?

Maybe we need to rethink our approach?

But in the case of the “invert and multiply” rule for dividing fractions, the concept behind it is a proof (not something most elementary math teachers are familiar with). I don’t see how either calculators or computer programs can help kids get at that proof. Real world examples are difficult to come by also, though not impossible. Perhaps I’m missing the point here? I know my own kids enjoy watching mathematical proofs explained on YouTube videos. But these videos feature math teachers explaining the logic behind certain concepts or algorithms in pretty traditional ways.

Most of the theorems and proofs behind fraction rules actually have quite nice visual representations. For example here is a visual representation of the process one goes through when using the invert and multiply rule.

http://mathforum.org/library/drmath/view/62540.html

Many math concepts can be explained without algebraic proofs.

There are some math concepts which don’t have a real world equivalent, and aren’t used in the day to day lives of pretty much anyone except mathematicians, and even then only in their work, not in their mundane lives. I have a simple solution, don’t teach those concepts to kids. Wait until they’ve decided they are going to be a mathematician and teach the concept then. There is plenty of useful and interesting math we can teach which will cover our need to introduce logical reasoning and all the other wonderful things about using math to solve problems. We don’t need to teach the stuff that is really only interesting to a very small number of people.

I agree with you totally. We need to stop teaching math concepts. We need to teach problem solving. We need to provide students with problems that require math to solve them, let them get dirty and wrestle with the problem and guide them as they need it.

If we taught students to be good problem solvers then they would be far better employees of the 21st century. When we teach concepts the students remember them for at most, a semester. Then they forget them because they never use them again. If we teach them to be problem solvers, they can use their problem solving skills almost everyday of their life.

We may need to teach the students some concepts from time to time to solve a problem but that then gives the students purpose for learning the concept.

You light want to check out the book “Problem Solving Strategies – Crossing the River with Dogs” by Ken Johnson and Ted Herr. This resource is a perfect example of how to teach math through problems.

I wholeheartedly agree, David. Students should take an initial exploratory set of courses in basic numeracy skills in primary school to develop familiarity with the fundamental operations of mathematics. However, once they have observed and experimented with the fundamentals, regardless of whether they mastered the more complex ones or not, they should be free to use technology resources such as calculators, even on standardized tests. Requiring students to continue manually manipulating fractions, performing long division, and pondering other foundational mathematical operations best left to calculators or computers is an archaic and ineffective use of time, talent, and treasure; we stopped using slide rules a while back for a reason.

In this country, we waste considerable amounts of scarce resources (time, young minds, and money) in a senseless devotion to the past. Due to a mathematics curricula imposed mostly by academics, rather than a mixture of academia and industry, we are crippling our nation’s intellectual capacity. It is time these outdated approaches are shown for what they are: loadstones around the neck of our nation.

For those that will lament not doing math the old-fashioned way, each of the procedures for manual mathematical manipulations can be taught in a history of math class, or as a dedicated segment of another class, to illustrate how our “ancestors” used less effective means to perform mathematical operations. Those “skills” are simply prerequisites for manipulating data, a means to an end. However, with their inclusion in today’s mathematics curricula and standardized tests as well the mandate by “old school” math teachers not to rely on technology, these manual operations serve more as a roadblock to students developing higher order mathematical thinking.

Most students get stuck procedurally, or make a minor math error transposing digits, dropping a negative sign or some other lower-order task, and equate that with a lack of mathematical expertise or capability. We have created tens of millions of mathphobic citizens with this approach. In spite of this, we continue to instill fear and loathing of math today. When will we wake up, let go of the past, and embrace our future? When will we push aside the gatekeepers who insist on outdated procedure over unleashing the creative potential of our youth? We do not have to remain on this tragic trajectory. Success is within our grasp, if we embrace in our educational system that which industry recognized over forty years ago: leverage technology to unleash efficiency, creativity, and innovation.

Technological efficiency exponentially improves effectiveness. We can overcome the results of our mistaken insistence on drudgery as the path to mathematical might if we embrace technology and strive for higher-order thinking rather than lower-order procedural prowess. Allow students to leverage their iPhone, iPod, calculator, home computer, or whatever other technology that is accessible to them; and for those without access, make it so. Let them start using technology to assist in problem solving as most in industry do. The time has come. We can no longer afford to constrain our national intellectual capacity due to a well-intentioned, but outdated insistence on doing things as they were done before the Internet age.

The only real debate should be whether to use a conventional or RPN calculator! Or for computers, which OS: Mac OS X, Windows, or Linux?

“Technological efficiency exponentially improves effectiveness. We can overcome the results of our mistaken insistence on drudgery as the path to mathematical might if we embrace technology and strive for higher-order thinking rather than lower-order procedural prowess.”

I love this. As an elementary school educator I cannot agree more. The goal of my class is to get kids to understand the reason behind what they are doing, and for them to conceptualize the math visually and understand the thought process behind it.

http://www.teachingparadox.edublogs.org

A timely post, thanks David.

My colleagues, a Science/Technology consultant and a Cooperative Education consultant, were having a similar discussion with respect to their own painful inter-generational experiences with mathematics education. It’s seems there is hardly a family untouched by “What is the mathematics we need to teach, and how?” My own answer is to use technology MUCH more.

In tech class, we don’t build a deck with hand-saw and hammer. The bigger idea in tech class is not about the saw, it’s about separating materials; it’s not about the hammer, it’s about joining materials.

Using Gary’s example of “invert and multiply” I’d argue that the bigger mathematical idea is reciprocal; division can be thought of as the inverse of multiplication, and every number (NE 0) has a multiplicative inverse. That’s an idea that generalizes to many mathematical forms. The visual illustration of dividing by a fraction is interesting but ultimately one-off and of little practical value, except when used in conjunction with a calculator to justify simple results.

Like good Science Fiction, mathematics is about the ideas.

Hi,

computer is useful for mathematics student.we say computer is not using for only calculation its use lens and logical problem solve.so this is a very important for every student.when we are reading in school there are so many problem create but we do not solve easily way.when we use the computer these problem solve very easily way.So computer is useful for every person.