Precision, Clarity, Order, Cohesion
To quote myself from last time:
Well, I'd like to offer just one example here (with a few hypothetical details) that might demonstrate to readers what such a conversation would look like--and why such a conversation would be important.
But before I do that, I think I should do something to try to convince skeptical readers that such an exercise is not an alien way of thinking, not a new-fangled concoction from some no-name blogger looking to air yet another opinion about education. To do that, I will turn briefly to the law as an analogy.
These distances (among others) between the facts (and contexts) before a court and the events (and contexts) to which they relate serve collectively to separate the thinking of actors in legal proceedings and other people, because the distances describe different inputs to that thinking.
Given these distances and differences in "thinking," one would be right to wonder why everyday folks grant any legitimacy at all to the courts of law that function to settle their disputes. The answer, of course, in part, is that both citizens and courts operate under a set of general guidelines, or principles. Citizens agree that these principles apply to them, and courts work to interpret and apply them to settle disputes.
Teachers would agree, I imagine, that the content of their teaching should be presented accurately (precision), such that it is understandable to their students (clarity). And I think there would be little disagreement that the elements of their instruction should be arranged so that students can see the relatedness of ideas (packaging, or cohesion) while also moving in steps from known to unknown (order). In other words, teachers would have little trouble considering that these principles apply to the instruction they present. [The descriptions I included for the principles above are just my own interpretations. There are certainly others.]
It is also possible, in my view, for a diverse and intelligent group of people--at some temporal and informational distance from the activities of a classroom--to usefully interpret these principles as they apply to classroom instruction without requiring the full range of inputs that are available to a classroom teacher's thinking.
This brings me to the example I mentioned above of the kind of conversation that can take place regarding mathematics instruction--a conversation that can be principles-based and, while "disconnected" (temporally and informationally) from the classroom, can be practical and useful.
Consistent with the standards and the specific observations and suggestions in the widely referenced publications listed above, textbook lessons typically introduce students to the idea that the area of a non-rectangular parallelogram can be described by the same formula as that used to describe the area of a rectangle with the same base and height, using one or more illustrations like the one below, accompanied by written instruction to help students understand the illustration:
Generally, the purpose of this written instruction is to clarify for students that (a) a part of the non-rectangular parallelogram was simply removed and then re-attached somewhere else on the figure--a transposition that does not change the area of the figure, and (b) the resulting figure is a rectangle whose height and base are the same as that of the non-rectangular parallelogram.
Let's start with a very simple idea: rectangles and squares are special kinds of parallelograms. Deriving or developing a formula to describe the area of certain non-rectangular parallelograms based on what is taught about the area of rectangles--as is often done in this instruction--is a bit like arguing that the numbers below are all integers because you can count up from them or count down from them and reach the integer 9.
3, 1, 11, 45
It turns out in both cases that what is stated is true--non-rectangular parallelograms and rectangles share the same area formula, and the numbers above, along with the number 9, are all integers. And it so happens that in both of these cases, the reasons are difficult to argue with as well--transforming a non-rectangular parallelogram into a rectangle without gaining or losing area is certainly a convincing demonstration, and so long as one construes "counting" as being restricted to integers, the reason given above for the "integer-ness" of 3, 1, 11, and 45 is satisfactory, albeit sloppy.
But, still, those reasons are wrong. The "integer-ness" of integers has nothing to do with the number 9, and the reason the area of a parallelogram can be described by the formula A = bh or A = lw has nothing to do with rectangles and everything to do with the fact that parallel lines are at all corresponding points the same distance apart:
This idea--this more precise idea--awaits actual fleshing out in lessons, where the other principles (clarity, order, and cohesion), along with precision again, must be considered. Note that the new arrangement above is suggestive of reworked orders and packagings of content. If, for example, we are to keep this topic (parallelogram area) where it typically falls in a mathematics curriculum, then we would need to move up discussions about parallel lines from where they typically fall (order change), and such discussions could no longer be considered in isolation (cohesion change)--two extremely small examples of the enormous ground mathematicians, teachers, education specialists, etc., could cover in the kinds of discussions I recommend in this article.
And I'd like to think I'm a little bit closer now to figuring out what such a methodology could look like.
To each writing and editing task, I bring four important resources: (1) my own knowledge about the lesson that I am writing, (2) a list of related lessons that have likely already been presented, (3) a list of related lessons that will likely be presented in the future, and, most importantly, (4) a memory not only for the content of each of the lessons mentioned above, but also a memory of all the different ways those lessons have been presented.
Even if intelligent and/or experienced discussants were to restrict themselves to a specific lesson at a specific grade level and bring only those four resources mentioned above into a debate, one could write ahem, a lot, about--and education could gain a lot from--the ensuing conversation.
Well, I'd like to offer just one example here (with a few hypothetical details) that might demonstrate to readers what such a conversation would look like--and why such a conversation would be important.
But before I do that, I think I should do something to try to convince skeptical readers that such an exercise is not an alien way of thinking, not a new-fangled concoction from some no-name blogger looking to air yet another opinion about education. To do that, I will turn briefly to the law as an analogy.
The Law as Analogy
When a case is brought before a court, many, if not all, of the facts of the case under consideration are related to events in the past. That is, there is generally a temporal distance between the facts considered by a court and the events to which those facts relate. What's more, there is an informational distance between the actual events and the facts considered. Certain aspects of the events are submitted for consideration while others are not, and certain aspects of the events are deemed irrelevant while others are relevant. Courts may also find other areas of law not mentioned in a dispute both relevant and applicable.These distances (among others) between the facts (and contexts) before a court and the events (and contexts) to which they relate serve collectively to separate the thinking of actors in legal proceedings and other people, because the distances describe different inputs to that thinking.
Given these distances and differences in "thinking," one would be right to wonder why everyday folks grant any legitimacy at all to the courts of law that function to settle their disputes. The answer, of course, in part, is that both citizens and courts operate under a set of general guidelines, or principles. Citizens agree that these principles apply to them, and courts work to interpret and apply them to settle disputes.
Back to Education
What I have in the past proposed (and will do so again here) are principles for mathematics instruction which can be easily accepted as applicable to daily classroom teaching and interpretable at some temporal and informational distance from that teaching. Here they are again (ignore the Distributor/Producer and Consumer labels):Teachers would agree, I imagine, that the content of their teaching should be presented accurately (precision), such that it is understandable to their students (clarity). And I think there would be little disagreement that the elements of their instruction should be arranged so that students can see the relatedness of ideas (packaging, or cohesion) while also moving in steps from known to unknown (order). In other words, teachers would have little trouble considering that these principles apply to the instruction they present. [The descriptions I included for the principles above are just my own interpretations. There are certainly others.]
It is also possible, in my view, for a diverse and intelligent group of people--at some temporal and informational distance from the activities of a classroom--to usefully interpret these principles as they apply to classroom instruction without requiring the full range of inputs that are available to a classroom teacher's thinking.
This brings me to the example I mentioned above of the kind of conversation that can take place regarding mathematics instruction--a conversation that can be principles-based and, while "disconnected" (temporally and informationally) from the classroom, can be practical and useful.
Area of a Parallelogram -- The Facts
A very common way of teaching students the reasoning behind the formula for the area of a non-rectangular parallelogram is to demonstrate how to take apart such a parallelogram and turn it into a rectangle. Some states' elementary and middle-school mathematics standards are fairly explicit about this "way" of teaching. Here are two examples (emphases mine):Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram). --California, 1997, Grade 5 (PDF)
Derive and apply formulas for areas of parallelograms, triangles, and trapezoids from the area of a rectangle. --Florida, 2007, Grade 5The NCTM's 2000 publication Principles and Standards for School Mathematics (PSSM) also endorses this method as consistent with stimulating students' understanding of, and investigation into, area (emphasis mine):
One particularly accessible and rich domain for such investigation is areas of parallelograms, triangles, and trapezoids. Students can develop formulas for these shapes using what they have previously learned about how to find the area of a rectangle, along with an understanding that decomposing a shape and rearranging its component parts without overlapping does not affect the area of the shape. --PSSM, 2000, p. 244And, from the third edition of this well-known book by Van de Walle, we have the following:
Parallelograms that are not rectangles can be transformed into rectangles having the same area. The new rectangle has the same height and two sides the same as the original parallelogram. Students should explore these relationships on grid paper, on geoboards, or by cutting paper models, and should be quite convinced that the areas are the same and that such reassembly can always be done with any parallelogram. As a result, the area of a parallelogram is base times height, just as for rectangles. (p. 334)(I should note that, of the four snippets I present above, only Van de Walle's even makes an attempt at a consistent distinction between non-rectangular parallelograms ["parallelograms that are not rectangles"] and rectangles. But then even in the Van de Walle snippet, this distinction breaks down in the last sentence. Already not a good sign if we have precision in mind.)
Consistent with the standards and the specific observations and suggestions in the widely referenced publications listed above, textbook lessons typically introduce students to the idea that the area of a non-rectangular parallelogram can be described by the same formula as that used to describe the area of a rectangle with the same base and height, using one or more illustrations like the one below, accompanied by written instruction to help students understand the illustration:
Generally, the purpose of this written instruction is to clarify for students that (a) a part of the non-rectangular parallelogram was simply removed and then re-attached somewhere else on the figure--a transposition that does not change the area of the figure, and (b) the resulting figure is a rectangle whose height and base are the same as that of the non-rectangular parallelogram.
The Analysis
It's worth taking a moment--before proceeding to an analysis of the above instruction--to refresh one's perspective with the words of the late Isaac Asimov; these from an article titled The Relativity of Wrong.The basic trouble, you see, is that people think that "right" and "wrong" are absolute; that everything that isn't perfectly and completely right is totally and equally wrong. However, I don't think that's so. It seems to me that right and wrong are fuzzy concepts. . . .What I will want to do first in the next few paragraphs is argue that the instruction referenced above is wrong. And what I mean by wrong, at least at the very beginning of my argument, is not that everything about the instruction is "all wrong, terribly wrong, absolutely," but that it is "wrong enough to be discarded" in favor of better instruction.
Nowadays, of course, we are taught that the flat-earth theory is wrong; that it is all wrong, terribly wrong, absolutely. But it isn't. The curvature of the earth is nearly 0 per mile [0.000126 per mile], so that although the flat-earth theory is wrong, it happens to be nearly right. That's why the theory lasted so long. . . .
[However], although the flat-earth theory is only slightly wrong and is a credit to its inventors, all things considered, it is wrong enough to be discarded in favor of the spherical-earth theory.
And yet is the earth a sphere?
No, it is not a sphere; not in the strict mathematical sense. . . . Actual measurements of the curvature of the earth were carried out in the eighteenth century and Newton was proved correct.
The earth has an equatorial bulge, in other words. It is flattened at the poles. It is an "oblate spheroid" rather than a sphere. . . .
Even the oblate-spheroidal notion of the earth is wrong, strictly speaking. In 1958, when the satellite Vanguard I was put into orbit about the earth, it was able to measure the local gravitational pull of the earth--and therefore its shape--with unprecedented precision. It turned out that the equatorial bulge south of the equator was slightly bulgier than the bulge north of the equator . . .
There seemed no other way of describing this than by saying the earth was pear-shaped, and at once many people decided that the earth was nothing like a sphere but was shaped like a Bartlett pear dangling in space. Actually, the pearlike deviation from oblate-spheroid perfect was a matter of yards rather than miles, and the adjustment of curvature was in the millionths of an inch per mile.
Let's start with a very simple idea: rectangles and squares are special kinds of parallelograms. Deriving or developing a formula to describe the area of certain non-rectangular parallelograms based on what is taught about the area of rectangles--as is often done in this instruction--is a bit like arguing that the numbers below are all integers because you can count up from them or count down from them and reach the integer 9.
It turns out in both cases that what is stated is true--non-rectangular parallelograms and rectangles share the same area formula, and the numbers above, along with the number 9, are all integers. And it so happens that in both of these cases, the reasons are difficult to argue with as well--transforming a non-rectangular parallelogram into a rectangle without gaining or losing area is certainly a convincing demonstration, and so long as one construes "counting" as being restricted to integers, the reason given above for the "integer-ness" of 3, 1, 11, and 45 is satisfactory, albeit sloppy.
But, still, those reasons are wrong. The "integer-ness" of integers has nothing to do with the number 9, and the reason the area of a parallelogram can be described by the formula A = bh or A = lw has nothing to do with rectangles and everything to do with the fact that parallel lines are at all corresponding points the same distance apart:
This idea--this more precise idea--awaits actual fleshing out in lessons, where the other principles (clarity, order, and cohesion), along with precision again, must be considered. Note that the new arrangement above is suggestive of reworked orders and packagings of content. If, for example, we are to keep this topic (parallelogram area) where it typically falls in a mathematics curriculum, then we would need to move up discussions about parallel lines from where they typically fall (order change), and such discussions could no longer be considered in isolation (cohesion change)--two extremely small examples of the enormous ground mathematicians, teachers, education specialists, etc., could cover in the kinds of discussions I recommend in this article.
Conclusion
I have been short on posts in 2009, but I have had one main idea--the method of education. What I tried to argue, beginning with the Garfunkel's Syndrome series and ending, for the most part, with this post (though these ideas go back a long way), was that part of the answer for just some of the troubles of education--specifically mathematics education--could be found in a methodology for approaching its questions.And I'd like to think I'm a little bit closer now to figuring out what such a methodology could look like.
Labels: education, mathematics, textbooks