Jessica Furr

EMT 468

October 25, 1996

Any discussion of classroom philosophy and practice must first address two questions: What do I want to teach my students? And what does society expect them to learn? Every subject has a unique contribution to the overall development of citizens. English produces vital communication skills. Social studies teach us to view the world and our place in it contextually. Science teaches students to be curious about their environment and to seek understanding through experimentation and inductive reasoning. And mathematics, as part of this collaborative effort, should teach us to understand phenomena through structural analysis and deductive reasoning with the tools of symbolic manipulation.

Each of these purposes is unique as a perspective and an approach to problem-solving. Together they equip citizens to be aware, successful, and adaptable in our society. The ability to effectively utilize each approach obviously requires a person to have at hand a standard body of knowledge and proficiency in basic, necessary skills, like reading, writing, and computation. But a student who is only taught a collection of facts and skills and isn't taught how to use them in problem-solving is just as handicapped as the kid who sees the best way to approach a problem, but who can't do anything when he gets there. A good curriculum must teach understanding and knowledge.

So why is there an implicit expectation that the role of the teacher is to dispense knowledge and supervise skill acquisition? I believe there are two reasons why this has occurred. One, facts and skills are easier to observe and quantify, and are therefore a more accessible standard by which teacher performance can be measured. And because there exists an unsupported assumption that if a student can recite facts and perform on demand, then he must necessarily understand the contextual relevance of his knowledge. In mathematics, at least, we know this is far from the truth.

How does this discussion relate to formulating a position on technology in the classroom? The continuum of choices between exploiting the full potential of instructional technology or ignoring such tools is equivalent to the spectrum of teaching for understanding or for calculational proficiency. In seconds, computers and calculators do computation that could take students hours. These hours are the ones that can be spent analyzing data, equations, functions, and systems. Teachers only have a limited amount of time, and they can either drill algorithms, or they can take that time to construct knowledge. The argument can also be made at this point that there may be no need for drill once understanding is achieved. The premise of the following discussion is that I am convinced as a teacher to strive towards teaching for understanding. The compilation of research and policy statements presented here is in defense of the effectiveness of technology in the classroom towards this goal.

There is a legitimate argument to be made against the relevance of technology in the classroom. The history of instructional technology and educational reform is long and less than impressive. It is important to remember that innovation is not new to education. A speech given at Heidelberg University around 1500 shows the hopes generated by the invention of the printing press.

"...each of our pupils will be able to have for himself the wisdom and scholarship of our great teachers, past and present. Consider its importance to our scholars, what it will mean to our young pedagogues and the manner in which they shall execute their duties. Pedagogy, gentleman, shall be revolutionized . . . No longer will our teachers be mere dispensers of knowledge. Our teachers will become implementers and facilitators of scholarship."

Some inventions, like the slate tablet, pencil and paper, and calculators, were valued for allowing students to write down their work and to keep it for later use, setting them free from having to memorize everything and doing only those calculations which could be handled mentally. Other innovations, like the printing press, photographs, audio recorders, and movies, carried with them an expectation that they could provide individualized instruction, provide for greater equity in education, and perhaps even replaced the need for teachers. Unfortunately, none of these innovations lived up to the revolutionary expectations they created.

Computers are unfortunately most commonly understood in terms of this last category, i.e. a more sophisticated form of presenting traditional knowledge. Software for drill is merely a glorified worksheet. Even Intelligent Tutoring Systems, which are more sophisticated forms of practice software, still present rather contrived problem-solving situations adapted to a student's ability level. Computers used in this way are part of the status quo and, like past experiments, will have very little impact on improving students' learning in mathematics. However, there is a characteristic difference in the opportunities presented by current technological advances as opposed to those of the past. Today, the powerful abilities of computers to model realistic situations is opening up a new world of problems that could hardly be explored before, even by experts. Today's computers are not only new tools for problem-solving and communication, but are also the source of inevitable curriculum changes.

The present effect of the growing body of new mathematical knowledge and the new opportunities for solving different kinds of problems, is a shift in the type of learning emphasized, more than a shift in particular content. The NCTM Standards explain that "new technology not only has made calculations and graphing easier, it has changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them," The effect of technology on society at large also has implications for the student.

"Basic skills today and in the future mean far more than computational proficiency... the calculator renders obsolete much of the complex paper-and-pencil proficiency traditionally emphasized in mathematics courses. Topics such as geometry, probability, statistics, and algebra have become increasingly more important and accessible to students through technology."

While the same prerequisite facts and skills may continue to be necessary, the overall goal of learning math is evolving as our society requires increasingly higher-order thinking skills from a greater portion of the population. Here we begin to see a mutual support between technology and instruction for understanding. The opportunities created by computers justify the need for more honest mathematical studies while choosing to teach the structures of math requires the assistance of computers.

Another legitimate argument among those who hesitate to integrate technology into the classroom is that students will become dependent on machines and not develop any personal proficiency in math. Representatives of the National Council of Teachers of Mathematics respond strongly to this evaluation.

"Contrary to the fears of many, the availability of calculators and computers has expanded students' capability of performing calculations. There is no evidence to suggest that the availability of calculations makes students dependent on them for simple calculations. Students should be able to decide when they need to calculate and whether they require an exact or approximate answer. The should be able to select and use the most appropriate tool. Students should have a balanced approach to calculation, be able to choose appropriate procedures, find answers, and judge the validity of those answers."

In fact, there is a growing number of studies which support the effectiveness of technology in increasing student understanding of mathematical concepts. This is, in part, because writers of educational software have begun to respond to the evidence that learning is not only about looking, listening and practicing.

"There is now a substantial body of research in instructional psychology showing that learning is an active and constructive process. Learners are not passive receptacles of information, but they actively construct their knowledge and skills through interaction with the environment and through reorganization of their prior mental structures.

It is understandable then, that the most effective innovations are a new generation of educational software which present information visually in two and three dimensions and allow exploration of geometric and functional relationships. These programs help students develop spatial understanding, reveal immediate relationships in changing variables, and build bridges to higher-order mathematical thinking.

The great challenge inherent in this shift of pedagogical purposes to teaching concepts and structures, is that the same old questions and textbook activities are no longer sufficient to make full use of the computer's capabilities for enhancing learning. "Designing a problem in a computer based environment requires a new analysis of mathematical objects, operations and feedback. This explains perhaps why teachers can be reluctant to use computers in their classrooms."

"Implementing the embedded view of educational computing outlined in this paper, which is based on a social constructivist conception of learning and oriented toward higher-order objectives of mathematics education, entails a total modification of the teaching-learning environment involving changes in children's learning activities, in the role of the teacher, in the social interaction patters in the classroom, etc."

Such changes are necessary, however, to meet the new expectations for teaching mathematics. The first four NCTM standards for every grade level emphasize the purposes of school mathematics for making connections, communicating, solving problems, and reasoning. There are some educators who believe that The Standards didn't go far enough in emphasizing the necessity of technology in reaching these goals.

"Adds Kaput: 'You can't really achieve what the Standards suggest without technology.' ...Andee Rubin agrees. Teaching next-generation math without technology "would be like trying to do a lab science without a microscope," she says. "Our focus is to see math as a science, an investigative, exploratory subject. And we need the [technology] tools to do that."

One of the main factors for developing problem-solving skills is that students work on relevant, real-world problems. The difficulty with "real" problems, and the reason they have been so rarely used in the past, is because the data from such situations rarely comes in nice whole numbers which allow students to easily compute and analyze the situation. Now, computers can do the tedious computation and give students the time to analyze and reflect upon the problem or concept.

Another relevant aspect of problem-solving is also an important factor in making connections, namely that students understand the relationship between various forms of representation of a single concept. "Students who are able to apply and translate among different representations of the same problem situation or of the same mathematical concept will have at once a powerful, flexible set of tools for solving problems and a deeper appreciation of the consistency and beauty of mathematics"

All of this makes more math accessible to more students. But some still argue that students cannot grasp such higher-order concepts before mastering computational skills. Bruner, a cognitive psychologist, argued this point over a decade ago.

"Learning is most often figuring out how to use what is already known in order to go beyond what is currently being thought - and this involves knowing something structural about what is being contemplated - how it is put together. 'Knowing how something is put together is worth a 1,000 facts about it. It permits you to go beyond it.'... ' A curriculum ought to be built around the great issues, principles, and values that a society sees as worthy of continual relevance for its members, 'If understanding of number, measure, and probability is judged crucial in the pursuit of science, then instruction in these subjects should begin as intellectually honestly and as early as possible in a manner consistent with the child's forms of thought.'"

As a closing discussion, I must discuss the specific present challenges inherent in integrating technology into the classroom. New types of knowledge will require teachers to adjust both their own understanding of math and of pedagogy. A great deal of hard change lies ahead if the potential of technology as we understand it is to be realized.

"To realize all of these potentials will take more than developing and validating educational hardware and software. Engrained behavioral patterns that tend to make human beings resistant to change must be overcome. The application of technology will entail a significant alteration of the traditional teaching-learning environment. For technology to succeed in achieving its promise, a majority of Americans must become convinced that the new way is the better way - and this may be the most difficult task of all."

At the heart of educational reform is a change of attitude to accept the new ways, to believe in their inherent value. Second is a willingness to contribute to the reform movement which needs new curriculum, new software, new problems to pose, and new techniques for teaching and classroom management. All of this means a lot of work, and understandably, not everyone is ready to jump in with both feet. I've often heard people say "Using computers isn't worth the risk." I would respond to such a teacher by suggesting that she reflect upon her own limitations and her responsibility to the students and ask: "Am I worth the risk?"

It all comes down to two questions: What do you believe students most need to know in order to be prepared in life? And two, do you care about teaching enough to do the extra work? My answer is that students need to learn understanding and I care deeply about helping them achieve that understanding. Therefore, I must develop a philosophy and practice in my classroom that takes full advantage of available technology.

Bibliography

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