Just a thought or two on this whole calculator business. Mainly, I would like to point to the calculator discussion as a good example of the harmful effects of the so-called math wars. One person noted the irresponsible hyperbole and pure misrepresentation in the anti-calculator article. I agree that this was despicable and has no place in what should be an intelligently conducted discussion. But the worst part about it is that when someone uses this kind of attack, they almost completely destroy the hope that their valid points might see the light of day. It takes much more discipline than the average reader uses to look to see if there is anything useful in something that is otherwise filled with reactionary, inflammatory, and unjustified baloney.
I think there is a critical and valid issue being raised, or maybe just alluded to, and that is that "inappropriate" use of calculators could possibly be detrimental to learning. Of course, much of the whole debate is really about defining what "appropriate use" is, but well-intentioned reformers often fail to think about questions like "what if this curriculum gets into the hands of a teacher who doesn't understand math, or doesn't want to think, or is just coasting to retirement?". Whether we like the idea or not, it is quite possible that children will be better served by a curriculum that many of us would see as sickenly rote-based and boring, if the teacher presenting it doesn't understand math very well. We often dismiss this idea without careful consideration by saying that we'll do "teacher training" to make sure that the teachers know how to use it. But that's not a trivial problem, given that the teacher in question may have been "hating" math for 40 years. Are you going to turn that around in a 3-week summer program?
I'm not saying that I know the answer to any of these questions, only that the polarizing effect of nasty, immature debate (on either side of the conflict) tends to keep us from carefully investigating the middle ground. Which is not guaranteed to be where the best course is--but we are guaranteed not to see what it has to offer if we don't look.
I am commenting on the comments to article on calculators Jerry posted 9/29.
One responder to the article said:
>I'm in complete agreement.
>My basic position, stated approximately 3 zillion times (a number not >accessible even on the most powerful calculator) is that students should >be given calculator licenses the same way that we give driver's licenses. >Only students demonstrably competent in arithmetic should be allowed to >have them.
I am not in complete agreement. I am not in complete disagreement.
It has to do with my basic philosophy of what it means to learn mathematics. For example, I want students to understand scientific notation. Thus I want them to understand the effect of multiplying a power of ten on any number.
Consider 7.09 x 10^3 I could just tell them to memorize "move the decimal as many places to the right based on the value of the exponent."
But, I would rather present it in a way that they will come to this conclusion on their own. Thus, I see calculators as a pedagogical tool for understanding the mathematics.
Consider the following set of problems provided by the text:
7.09 x 10^0 7.09 x 10^1 7.09 x 10^2 7.09 x 10^3 7.09 x 10^4 7.09 x 10^5
Next I have them compute each answer and list them beside each problem.
Consider the time it might take to do each calculation pencil and paper (using no shortcuts). It is too time consuming. We usually just tell them "the rule" and move on.
BUT, after I teach them how to use their X^y keys and their "constant" keys and "memory" keys, they actual enjoy the challenge of mastering the calculator and keys they heretofore could not use.
We next review their answers
7.09 x 10^0 = 7.09 7.09 x 10^1 = 70.9 7.09 x 10^2 = 709 7.09 x 10^3 = 7,090 7.09 x 10^4 = 70,900 7.09 x 10^5 = 709,000
and note how the decimal "moves" to the right with each answer. (I call it the "vertical" pattern.) We next consider the relationship of where the decimal was in the factor and where it moves to in each product. (I call it the "horizontal" pattern). This horizontal pattern is discovered by the teacher rather than decreed by the teacher. It is something the students conclude themselves and the teacher facilitates. It is empowering.
It was all facilitated by a tool (the calculator) that made data generation quick and easy, thus providing more time for analysis of patterns. I would have never consider approaching this topic (and many others) in this manner with technology to facilitate the data analysis.
I agree with Tony (Ralston). This business of elementary children not using calculators because they have to learn the basic facts is getting a lot of flak that is unnecessary. Good teachers know that children need to learn their basic facts for the four basic operations of mathematics. I have been across the state of Kentucky in Sept. and last year and have never talked with a teacher who let their elementary students use calculators to solve basic arithmetic. Most teachers still do timed tests. However, elementary teachers do let their students use calculators for higher order thinking with problem solving. It is in this way that students are grasping higher levels of thinking. Children are also being taught to estimate so they will know if the answer the calculator gives them is a reasonable answer. If California is having a problem with elementary students using calculators for basic arithmetic then they need to beef up their monitoring of what is going on in elementary classrooms, not say that the children should not be using them at all. Maybe their teachers need to learn when it is appropriate for elementary children to use calculators and when it is not appropriate. My final comment is that children can not learn the basic facts all on their own or just when they are in school. It takes parents or someone working with them outside the classroom so they don't forget them. How many times have we all heard a teacher say at the beginning of the year that "this class does not know their facts. didn't they learn them last year?" Yes teachers, most of them learned them the year before, regardless of the grade level; however, over the summer these were not reinforced. Therefore they have been forgotten. My suggestion is to have the students use these basic facts at the beginning of the year in some meaningful way, not just drill, drill, drill again and again. Doing 50 problems of the same nature is not going to necessarily help them with their basic facts and knowing how to use them.
From reading the Readers' Digest by David Galernter, I see no evidence of a background in TEACHING or MATHEMATICS TEACHING by Mike McKeown & David Gelernter.
I feel concerned when academics generalise about how calculators are used, after acquiring the briefest of snapshots from some classroom.
In my experience as a teacher and advisory mathematics teacher, is there is a wealth of innovative approaches and strategies developed by classroom teachers that engage students in rich activities. Many of these activities create conditions where students have to confront their understanding of mathematical concepts, and are encouraged to take risks in a supportive environment. I sense that Mike McKeown & David Gelernter have a very Newtionian construct of what mathematics and mathematics learning is.
Many Australian researchers and teachers (Paul Swan & Alistair McIntosh, both from Edith Cowan University in Western Australia) have captured many wonderful ideas from the classroom, that involve the use of calculators as an instructional tool.
Paul has worked with a new calculator that has the ability to have some keys temporarily disabled, which allows for further innovation by the classroom teachers.
I implore readers of this list server to be more discerning of what they read. Readers' Digest does have an important role in our society (probably the single biggest factor in encouraging me to read), but it is not exactly the pinnacle of journalism...more like right wing propaganda, once readers view it at a meta-level.
Australia is currently in the grip of similar influence with the rise of the One Nation Party, spewing out propaganda in such volume and intensity that the masses who are 'radiating averageness' accept her ideas without question. In a similar vein, Readers' Digest must be viewed similarily, including its flawed arguement about calculators.
Sure, many calculators are misused. But the innovation and wisdom of classroom teachers should be valued and encouraged.