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Navigating through Math

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A couple years ago, I took a trip to Canton, Ohio, home to the illustrious Pro Football Hall of Fame, and home to a group of schools that were using classroom technology in an attempt to boost student performance in math.

Teachers in several Canton schools had set up a system called TI-Navigator, in combination with graphing calculators, both designed by Texas Instruments. The system worked like this:

A math teacher would give a problem to students, who would type answers (such as plotting points on a graph) into their calculators. Their calculators were connected by cords to "hubs," which dangled from a few points in the classroom and which sent signals wirelessly back to the teacher's computer. The teacher then received instant information on how many students -- say, two out of 20, or 18 out of 20 -- had the correct answer. Teachers could then adjust their lessons on the fly, focusing on the problems or concepts that gave students the most trouble.

You sometimes hear these programs referred to as "personal response" or "audience response" systems.

This setup might have made the class look a bit like a hospital ward (with IVs hanging everywhere), but for Canton officials, it was just the right medicine. They saw their students math scores' rise, in some cases dramatically, after using the systems.

But the question remained: Would this technology work as effectively in other math classrooms, among different groups of students?

Now, research by Douglas Owens of Ohio State University suggests the answer is a qualified yes.

Owens is the principal investigator on a four-year, federally funded project that examines the impact of TI Navigator and graphing calculators in math classes. The project, using a research method known as a randomized control trial, is looking at the performance of 1,800 students and 127 teachers from 28 states.

He found that students' math performance, after one year, rose by about 2 points on a 37-point test. It's possible that those gains will increase, as students and teachers become more familiar with the technology, Owens told me, after discussing his research at a conference on June 11 in Washington. The was hosted by the federal Institute of Education Sciences, which is supporting his study.

Read more about his research here http://ccms.osu.edu/.

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The Lure of Technology in Mathematics

Technology is an application of science and science is, in essence, an application of mathematics. A seemingly logical conclusion would be that there is no better place to introduce technology than in the teaching of mathematics. Even current literature in math education attests to this. Clearly, the consensus has been reached that it is technology that will be the saving grace of mathematics education – and the more, the better.

It’s certainly no surprise that this consensus has been reached. Students can now generate graphs literally by the hundreds using graphing calculators or generate table after table of coordinates for these graphs in an Excel spreadsheet. Clearly, this is exactly what is needed to improve math education. Before the introduction of these technological devices, students actually had to generate their own tables of coordinates by manually plugging meaningless numbers into equations that also had no conceptual meaning. And as if that weren’t bad enough, they actually had to manually draw graphs having little, if any, conceptual meaning using these tables of coordinates.

Now, with the availability of computerized spreadsheets and graphing calculators, students can carry out meaningless calculations and construct meaningless graphs far more rapidly and with almost no manual effort at all. With the introduction of technology in math education, students will have the capability to perform hundreds of numerical computations for which they have no conceptual understanding in the time that it used to take them to generate a single meaningless computational result.

Okay, so I’m being facetious, right? Actually no. My point is this. Technology can be a double-edged sword and we have to be particularly careful how we use it in a teaching environment – especially mathematics. Students have been mindlessly carrying out arithmetic calculations and mindlessly drawing graphs in math classes for decades by plugging meaningless numbers into meaningless equations in order to generate meaningless mappings onto meaningless coordinate reference frames and then playing connect-the-dots to construct a meaningless line or curve. With technology, a student will now have the opportunity to do this exact same thing, only this time utilizing an even more mindless protocol – punching numbers into a programmable calculator or a computerized spreadsheet. The technological apparatus will take care of the rest automatically. As if students weren’t already mathematically lazy enough, we now have the technology that will allow them to become even more mathematically lazy. The student’s role is reduced to that of a keypunch operator. Just punch in the numbers and wait for a result that you can neither interpret nor understand.

With graphing calculators, students can now plot graphs without even having to define a coordinate reference frame. This certainly makes mathematics life a lot simpler. Unfortunately, it also completely removes the need for a student to understand how to use a coordinate frame or even what a coordinate reference frame means. After all – If we don’t have to use it, we don’t have to know it. This certainly introduces an improvement in mathematics education. Improve the understanding of mathematics by making it unnecessary to understand as much mathematics. Perhaps if we introduce enough technology, eventually no student will have to understand any mathematics. This would certainly level the playing field and close the achievement gap in math. Every student in every school would now have exactly the same level of knowledge in mathematics – total ignorance.

One of the primary problems with using a graphing calculator or computerized spreadsheet to teach math is that it completely disconnects the mathematical function generating a graph from the graph itself. Other than the initial programming of the function into the calculator or spreadsheet, the student doesn’t see or even need to see the form of the function that is generating the graph. Apparently, the current thinking in math education is that the function itself is of little importance. Once programmed into the technological device of choice, its only purpose is to initiate the set of subsequent numerical calculations needed to draw a pretty picture on the display screen. Unfortunately, this is exactly the opposite of what we need to do to improve a student’s conceptual understanding of mathematics. If students are ever going to have a prayer of developing the ability to apply mathematics in understanding the real world, they must be able to visualize the direct connection between the symbolic language of mathematics and the observed behaviors of physical, biological, and economic systems in the real world. Graphs represent this visual link between mathematics and the physical universe.

The behavior of every system in the universe (be it physical, biological, economic, or other) can be described (represented or modeled) symbolically using mathematics. We call this symbolic representation a mathematical function. Mathematical functions are not mysterious, un-interpretable black-boxes and they are certainly not input-output machines as they are so often described in our schools. They are simply symbolic descriptions of what we observe every day in the world around us.

Every mathematical function has a corresponding pictorial representation. We call this visual picture of a mathematical function a graph. This graph is also a visual picture describing the behavior of the system that is represented symbolically by the mathematical function corresponding to this graph.

In teaching graphs in high school math the primary focus has always been on the physical act of drawing of the graph itself – a rather myopic focus in reality. Students draw graphs simply to learn how to draw graphs. Clearly, a graph is more than just lines or curves drawn on some coordinate reference frame whether the coordinate frame is on a sheet of graph paper or a computerized display screen. A graph completes the cycle necessary to develop both a conceptual understanding of mathematics and the ability to use mathematics to understand the universe in which we exist.

mathematical --> graph --> observable
function (symbolic) (visual) (physical)

Anything that eliminates or disrupts this visual link curtails our ability to make the transition from mathematics to the physical world and from the physical world to mathematics. In other words, it disrupts our ability to translate the symbolic language of mathematics into English and English (a verbal description of what we observe in the real world) into mathematics. It disrupts our ability to communicate with the rest of the universe and to understand the information that Nature is attempting to share with us every second of every day.

The problems in math education all stem from the fact that this visual link has never been utilized properly, if at all, in the teaching of mathematics. In the way that math is taught, and has always been taught, the above cycle becomes completely fragmented. Unfortunately, the inappropriate application of technology in mathematics has the potential to disrupt and fragment this cycle to an even greater extent.

Some (perhaps, many) math educators and proponents of technology in the classroom would have us believe that technology is the cure-all for every educational ill in every academic area. No matter what the academic problem, throw enough technology at it and the problem will miraculously disappear. In reality, we often find, down the road, that the problem hadn’t gone away at all – we simply were no longer able to see the problem clearly, because it was hidden under so many layers of technology. Or, perhaps, we simply wished the problem away because we believe so strongly that the use of technology guarantees alleviation of whatever problem is at-hand.

Technology is sometimes a solution in search of a problem to solve. Be wary of such solutions – they often don’t work at all and can actually create more problems than they were originally intended to solve. In contrast, math education is a problem in search of a solution – a solution that works. Eventually, math education will likely make the same discovery that the US manufacturing industry made several decades ago. In its efforts to compete with world-class manufacturers of automobiles and electronics, US companies introduced a plethora of automation technology into their plants. Without question, automation was guaranteed to result in immediate dramatic improvements in both productivity and quality assurance across-the-board. Technology would be the saving grace in US manufacturing’s attempt to compete in the world economy. Unfortunately, this massive injection of technology failed to result in any real improvements in either productivity or quality. As US manufacturers later came to realize, productivity and quality must be designed into the manufacturing process itself – not added post-hoc as a band-aid fix for a terminally ill process.

There is no question that math education is terminally ill. Applying technology post-hoc as a band-aid fix for the problems that currently exist is doomed to failure. Students fail to learn math because the approaches used in teaching mathematics preclude a conceptual understanding of math. Using these same failure-ridden approaches to math education with an external layer of technology tacked on is certain to be a waste of time, effort, and money. Math education is going to have to be completely redesigned from the ground up. It is during this redesign that appropriate solutions (technological or otherwise) to ensure that students develop a conceptual understanding of mathematics will have to be incorporated.

Technology can be an asset in the high school classroom. What we must keep in mind is that it will be an asset in education, as elsewhere, only if applied effectively. We also must keep in mind that technology may not be required or may only need to be applied sparingly in certain areas of education; i.e., it should only be applied when and where it is needed. For example, contrary to apparent popular belief, its incorporation into mathematics education is not at all necessary for students to gain a significant conceptual understanding of mathematics and an ability to apply mathematics in the real world. The primary purpose of computational tools – such as programmable calculators, computational software, supercomputers, and other technological computing devices – is to reduce the time required to carry out numerical computations and data display in order for scientists, mathematicians, engineers, economists, and business professionals to obtain more results within a much reduced timeframe which allows them much more time to analyze, evaluate, and interpret these numerical results. Actually, this is the only value-added component associated with numerical computation. It is through these analyses and interpretations of numerical data that the behaviors of systems are modeled through simulation and theoretical models tested for consistency with experiment. The key point here is that these users of high performance computational devices already have a conceptual understanding of the mathematical equations describing the modeled systems, the graphical representations of these equations, and the numerical results generated in a computational simulation using these equations. In other words, they already have a complete conceptual understanding of mathematics. The question we must then address in this: In what ways can technology be used most effectively in math when students do not already have a conceptual understanding of mathematics? If our goal is to provide students with a conceptual understanding of mathematics, then technology must be used in ways other than simply to obtain numerical results.

Consequently, we must take great care how we try to incorporate this into math education where students have essentially no conceptual understanding of math whatsoever. Technology can play a role in math education as a tool to assist in the teaching of math. However, technology itself cannot teach a student math. This probably sounds like I am “splitting hairs” here, but a clear distinction is absolutely necessary here. Some parents and teachers, as well as some students, would believe that once technology is incorporated into the classroom, increased understanding will automatically follow. It’s as if it is the technology itself that will now become the mathematics instructor. Unfortunately, this apparent logical progression in thought doesn’t hold.

Any computational device, no matter how technologically advanced it is or how many bells and whistles it possesses, is simply a number generator. There is nothing magical about these devices – they simply spit out numbers. There is certainly nothing inherently educational about these devices – they are simply “number spitter-outers”. It is our conceptual understanding of mathematics that enables us to use these numbers to investigate the behavior of systems in the real world and develop models for these behaviors from the conformational folding of a protein to the dynamics of the atmosphere of Jupiter to the collision of galaxies. Without a conceptual understanding of mathematics underlying these systems, the output from these technologically advanced number generators would be meaningless.

To some extent, technology is simply the new buzzword in education. It sounds impressive and it really looks impressive. But it’s only a tool. Used appropriately, it can be effective in improving education. Unfortunately, if used inappropriately, it can easily become a liability within education.

A good example, in my opinion, of the complete failure of technology in mathematics education is the demonstration problem on drug kinetics included in the NCTM publication Principles & Standards for School Mathematics

First, the use of a spreadsheet to carry out numerical calculations as the initial step in the solution immediately disrupts the visual connection between the behavior of the physical system being observed (absorption of a drug by the body and subsequent elimination of that drug by the kidneys) and a visual representation of this process. A detrimental mistake already on the part of technology.

Second, the form of the equation being programmed into the spreadsheet could easily be mistaken to be a linear equation. In reality, of course, it is not a linear equation – it is a nonlinear, iterative (recursive) equation. It is unfortunate that this is not made immediately obvious since this creates an additional fragmentation of the connection between mathematical function and visual representation of that function. The reason that this is not immediately obvious is the inappropriate use of the variable names NOW and THEN (which is consistently done when using spreadsheets to teach mathematics in high school). Had the equation been written in a form such as

D(t) = 0.4 D(t–1) + 440,

where D(t) is the systemic drug level at time t ,

the recursive nature of the mathematical function describing the behavior of the system we are observing would have been immediately clear. The form of the mathematical function being used in any application must always be specified in a clear and unambiguous manner. Here it is not – another very important point often overlooked in the application of technology in mathematics. When using technology, we apparently no longer have to be concerned whether or not the form of the mathematical function we’re using is specified unambiguously. This clearly demonstrates how the use of technology can easily suppress, if not completely dismiss, the importance of mathematical functions in mathematics.

Third, the graph that is subsequently drawn in section 7.2/part3 entitled Using Graphs, Equations, and Tables to Investigate the Elimination of Medicine from the Body: Graphing the Situation is actually not the initial graph that should have been drawn to represent the behavior that is being observed. In the rush to implement technology, the connection between a verbal description of the behavior being observed and visualization of that behavior has, once again, been disrupted and fragmented.

The primary problem in this example is that the use of technology allows us to clearly see what is happening, but it actually doesn’t allow us to clearly see mathematically why it is happening. This one example should make clear the fact that the application of technology in mathematics education does not necessarily equate to any real improvement in mathematics education.

It’s really too bad that technology wasn’t used more effectively in solving this problem. This actually is a very good problem – much better than those typically found in a math textbook. The faux-pas here was that we were too focused on obtaining numerical results and, consequently, applied technology prematurely to solve this problem. This demonstration problem is an example of an analysis of experimental data. If you were working, say, in a clinical laboratory after graduation and you were given this same problem, the immediate application of technology to solve this problem would certainly be warranted. The reason: Your only interest would be in obtaining an accurate numerical result to pass along to the physician.

But our goal in math education cannot simply be to obtain numerical results, particularly if it inhibits in any way the student being able to gain a complete conceptual understanding of a problem. A simple thought experiment can clearly demonstrate why relying too much on technology in math education may not allow a student to gain a complete conceptual understanding of mathematics, which must be our ultimate goal in math education.

Suppose a student is interested in modeling the behavior of some system that he or she is observing in order to gain an understanding of this system’s behavior. But let’s also say that the observer has little or no experimental (numerical) data. The only information that the student has is a verbal description of this behavior gained solely through observation.

For example, let’s say that the only information that the student has in the drug elimination problem above is that the patient takes some prescribed dosage of medication every 8 hours and that at the end of 8 hours, the amount of drug eliminated from the body through the kidneys is some constant fraction of the amount of drug present in the system immediately after each dosage of medication is taken.

Here we clearly have insufficient numerical data to plug into a computer spreadsheet or any other computer program. It’s exactly the same problem as we had previously except that there are no numbers to compute. Can this problem be solved using the same technology devices used in solving the initial problem where we were given numerical data? The answer is: It can’t. Technology is absolutely of no value in solving this problem. Yet, the two problems are exactly the same problem. After all, changing the numbers doesn’t change the problem and here we’ve changed the numbers by eliminating them entirely.

It should now be clear that there is a limit to the application of technology in math education.

So let’s say that a given student has been taught to solve the initial problem using technology. Unfortunately for the student, this same technology cannot be used to solve the latter problem. It’s likely that this same student is not going to have any idea how to even approach solving this problem now that the numerical data has been removed from the problem.

What we find is that the student has learned how to solve the problem by using technology but is unable to solve the exact same problem when technology cannot be used. The questions that now beg to be answered are these:

Did the use of technology in solving this problem teach the student any conceptual understanding of mathematics or did it only teach the student how to use a spreadsheet to compute some numerical results?

How will technology be used to give students an understanding of mathematics through observation when there are no numbers to compute and no numerical results to obtain?

My assessment would be that the student had not actually gained any conceptual understanding of mathematics at all through the use of technology.

There is no question that there is a place for technology in math education. Published results from research in math education have been reported clearly demonstrating statistically significant improvements in math achievement when technology is used in the classroom. So far, I am not all that impressed with the levels of improvement being reported. Of course, these are all statistical studies and, without looking at the actual raw data, we really don’t know if all of these studies are even valid, particularly since studies of this type are quite subject to unintentional selection bias. Remember: don’t believe everything you read just because you read it.

What we may discover is that one of the greatest benefits derived from the introduction of technology into math education is its effect as a marketing strategy. Given that so many students in high school are obsessed with computer games, the appeal of having technology available in the classroom may generate an interest in mathematics within a larger segment of the student population than ever before possible. If so, perhaps the inclusion of technology in math education will turn out to be a successful step forward – assuming that the real benefits that are realized actually outweigh the costs incurred.

Definitely, technology will likely find increased application in math education in the future. The exact role it will play and the extent to which it can be applied is uncertain, at least in my mind. If it is incorporated in an effective manner as an integral component in a complete redesign of secondary math education, its role may be significant. Only time will tell if it will turn out to be the panacea that many in math education believe it will be. Until then, we need to keep the application of technology in math education in perspective. If it is applied effectively and only when and where it is needed, it is likely to be of some benefit in helping to improve student achievement in mathematics. In contrast, if we see indications that it is being over-incorporated in situations where its use or effectiveness is clearly questionable then we must ask if the primary concern is improvement of education or simply an exploitation of a new and potentially profitable market for technology.

Perhaps it appears that I am being much too over-critical of technology in math education. And you may find my comments to be even more surprising given that I am actually a daily user of high performance computing technology in scientific research. However, the concerns that I have voiced and the questions that I have posed here are concerns and questions that we all should have if our vision is to ensure real improvement in mathematics education in our schools.

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