Sunday, April 13, 2014

Cause and Purpose in Text A neat study in Educational Studies in Mathematics (link) points to a familiar yet disturbing characteristic of elementary mathematics texts.

In the study, samples from eighteen different elementary mathematics texts used in the UK were analyzed. Researchers were interested in how often the texts provided "reasons" for the mathematics they presented—that is, how often the texts explained a mathematical idea (or solicited an explanation from students) in terms of purposes and causes:
There is evidence that the strength and number of cause and purpose connections determine the probability of comprehension and the recall of information read (Britton and Graesser, 1996) and can indicate a teacher's or writer's concern for reasons (Newton and Newton, 2000). Even when writers withhold reasons and provide activities to help children construct them, they cannot assume that this will happen. In books, a concern for reasons, therefore, is often indicated by their presence. Clauses of cause and purpose can, within limits, serve as indicators of this concern (Britton and Graesser, 1996; Newton and Newton, 2000). . . . Clauses are commonly used as units of textual analysis (Weber, 1990). Amongst these clauses, clauses of cause (typically signalled by words like as, because, since) and purpose (typically signalled by in order to, to, so that) were noted.
Having these data, researchers then compiled the "reason-giving" statements into seven different categories based on their "explanatory purpose." The results from the study are shown below. The labels used are my own.

The first four categories (working counterclockwise from the largest section) were considered non-mathematical. Forty percent of the clauses in the sample provided "the purpose of and instructions for games and other activities intended to provide experience of a topic"; 22% provided "reasons in stories, real-world examples and applications and in descriptions of the basis of analogies"; 1.3% provided "the purpose of text in terms of its learning aims and objectives and could be described as metadiscourse"; and another 1.3% of the clauses "justified assertions of a non-mathematical nature." Nearly 65% of "reason-giving" in the texts was non-mathematical.

The next two categories were considered mathematical. Just over 13% of the clauses in the sample provided "the intentions of procedures, operations and algorithms for producing a particular mathematical end"; and just over 9% "attempted to justify [mathematical] assertions (e.g., 'It is a square number because 5 × 5 = 25').

Clauses in the final category (symbols) were considered mathematical or non-mathematical, depending on whether or not the symbols in question were mathematical ones. These clauses provided "the purpose of certain words, units, signs, abbreviations, conventions and non-verbal representations."

This is not to say that writers explain only through clauses of cause and purpose. They may use other devices to the same end and this analysis does not detect them. There is also what the teacher and the child do with the textbook to support understanding, perhaps through practical activity (Entwistle and Smith, 2003). This approach does not detect these directly. The aim of the study, however, is to consider the potential of the children's text to direct a teacher's attention to reasons.
It is important to remember that the results do not tell us that, for example, 40% of the clauses in the sample were instructions. They tell us that 40% of the "reason-giving" clauses were used in instructions. The graph above shows how "reason-giving" statements were used in the textbooks.

Although these results are generally supportive of the conclusions drawn in the study, they also provide further support, especially in light of these values . . .

Clauses of cause ranged from nil to 3.96% of text (using clauses as the unit) with a mean of 0.68% (s.d. 1.08). Clauses of purpose ranged from nil to 8.03% of text with a mean of 4.77% (s.d. 2.08).
. . . for the long-standing contention that contemporary elementary mathematics textbooks are, primarily, classroom management tools.

Newton, D., & Newton, L. (2006). Could Elementary Mathematics Textbooks Help Give Attention to Reasons in the Classroom? Educational Studies in Mathematics, 64 (1), 69-84 DOI: 10.1007/s10649-005-9015-z

Wednesday, January 1, 2014

On Narratives

Idon't think I overstated things any in the very last sentence of this G+ rebuttal:
If we could see as a whole the process of reasoning involved in any mathematics--playing out either in one human mind at one instant or in many human minds across time--it would be clearer to us, I think, that the most noteworthy and valuable part of that process is the awareness of and manipulation of the patterns in the first place. The proof--the deductive proof--while perhaps clever, is just the end product. To praise it as the best part--or worse, to suggest that it represents the entirety of the mathematics present, is a tacit admission of a shallow, spoon-fed understanding of the subject.
At issue here was a clichéd characterization of mathematics as a deductive process. And I think it's important for mathematics educators in particular to understand how and why this characterization is so profoundly and so simply and so powerfully wrong.

Simply Wrong

It has to do, in part, with narratives. +Timothy Wilson's essay at Edge provides a nice sketch of social psychological narratives in different contexts. Here's a quote from that piece (emphasis mine):
It's not the objective environment that influences people, but their constructs of the world. You have to get inside people's heads and see the world the way they do. You have to look at the kinds of narratives and stories people tell themselves as to why they're doing what they're doing. What can get people into trouble sometimes in their personal lives, or for more societal problems, is that these stories go wrong. People end up with narratives that are dysfunctional in some way.
Suppose for a moment that what you and I and students (and mathematicians!) consider to be mathematical thinking is defined and shaped not by comparison to any objective ideal or archetype but in large part by the kinds of stories (narratives) we listen to and tell each other and ourselves about mathematical thinking--self-reinforcing narratives which are built out of the thoughts and behaviors that we observe in addition to the words we hear or see about what mathematical thinking is.

If this is the case--and I would argue that it is--then, just as it is with our own or our friends' private narratives, it is not impossible nor even unlikely for these collective, social, public narratives about mathematical thinking to run off the rails, as it were, and gradually stand in opposition to reality. And mathematics as a deductive process is just such a narrative:
Mathematics is not a deductive science--that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
                                                                                  --Paul Richard Halmos

Deductive reasoning is certainly important and plays a key role in all scientific thinking, but the conceptual distance between acknowledging this truth and believing that "mathematics is a deductive process" is one that can be measured in light-years. One need only consider what the mathematics-as-deductive-process narrative excludes from counting as real mathematical activity (such as guessing, play, and entertaining ideas without drawing conclusions) to see that it is not just dysfunctional, but false.

Powerfully and Profoundly Wrong

And dysfunctional it can be. A student who has found it interesting to draw number lines and tick marks to try to continuously halve the interval sizes could not possibly see her activity in this 'deductive' narrative of mathematical thinking. No characters in that story seem to behave as she does or seem to enjoy the things she currently enjoys. So she sees what she does as art or maybe doodling. It is certainly not mathematics.

Educators, parents, and mathematicians themselves, too, can be carried along by this narrative, even when they are rosily confident that they advocate against it. One finds this nested self-delusion wherever adults maintain the conceit that deduction is for grownups (it isn't), that environments which nurture the social acceptability of guessing and failure are for children (they aren't), that deductive reasoning is a Nietzschean or Chardinian (or Guggenheimian?) Superman toward which we direct our students' academic evolutions (it isn't).

Yes, for the sake of children's mathematical narratives, we should stop saying that "mathematics is a deductive process," but we should stop believing it, too.

Narrative Editing

It should come as no surprise that changing a narrative like this is difficult, to say the least. Here's one reason why, courtesy of Philip Tetlock, professor of psychology at the University of Pennsylvania:
The long and the short of the story is that it's very hard for professionals and executives to maintain their status if they can't maintain a certain mystique about their judgment. If they lose that mystique about their judgment, that's profoundly threatening.

Tuesday, September 17, 2013

The Pedagogical Landscape

by Nigel Jones
Neuroscientist Sam Harris argues in The Moral Landscape that debates about morality can be grounded in scientific thinking:
Questions of morality and values must have right and wrong answers that fall within the purview of science (in principle, if not in practice). Consequently, some people and cultures will be right (to a greater or lesser degree), and some will be wrong, with respect to what they deem important in life.
The same is true of teaching (or rather, educating). There are practices that must be better and worse than others, and content that must be objectively better and worse than other content. Some viewpoints are right and some are wrong (to a greater or lesser degree), and some are slightly better or in need of a little improvement. Questions within domains of human interest like education and morality do not have to be simple or clinical or black-and-white in order for their possible answers to be placed on a widely and responsibly endorsed (and interpolated) scale of bad-good-better-best.

Yet, what I have just stated is seen by most as a fundamental assumption about pedagogical quality that is only 'correct' to the extent that it is collectively condoned: If it is the case that a person does not care about 'pedagogical quality' or if she believes that only she can ever completely and fairly judge her own pedagogical quality, there is no way we can convince her that she is wrong from an objective standpoint.

This is what Harris calls The Persuasion Problem with regard to his arguments about morality. I shall borrow the term, along with the relevant parts of his response:
I believe all of these challenges are the product of philosophical confusion. The simplest way to see this is by analogy to medicine and the mysterious quantity we call 'health.' Let's swap . . . ['pedagogical quality'] for 'health' and see how things look:
Here's how it would look: "If it is the case that a person does not care about health or if she believes that only she can ever completely and fairly judge her own health, there is no way we can convince her that she is wrong from an objective standpoint."
Clearly there are scientific truths to be known about health--and we can fail to know them, to our great detriment. This is a fact. And yet, it is possible for people to deny this fact, or to have perverse and even self-destructive ideas about how to live. Needless to say, it can be fruitless to argue with such people. Does this mean we have a Persuasion Problem with respect to medicine? No. Christian Scientists, homeopaths, voodoo priests, and the legions of the confused don't get to vote on the principles of medicine.
The same goes for education. Of course there are nutty people out there with nutty ideas about how to teach, how to structure schools, what priorities we should have with regard to education, and what counts for quality content (and some of these people and ideas might find themselves in the mainstream on any given occasion--they might be you or me). And of course the issues are complex and amorphous. But there is clearly a quality spectrum in education and good arguments to be made for improvement. Throwing this spectrum into sharp relief and then finding these arguments remain major first steps for a science of education. 

Update (9.18): I missed Ross Douthat's missing the point here. He says that the reason witch doctors and homeopaths are excluded from discussions about medical science is because they're fully on board with medical science. Um . . . okay.

To the extent that there is an indictment to be found in Harris's Persuasion Problem, it is an indictment not of the wackadoodles out there with painfully stupid beliefs, but of the rational majority who either too easily pass judgment on the basis of superstition or too carefully avoid it in order to uphold a comfortable relativism. 

Saturday, August 17, 2013

More on Precision (Rectangles)

An article shared here touches on the notion of precision in mathematics teaching that I have been writing about recently. I'll offer a few thoughts, especially since the article tidily validates a lot of what I've written in my previous posts on the precision principle.

At issue in the article were treatments of 'rectangle' in elementary textbooks, which often define a rectangle (in one way or another) as having "two long sides and two short sides." And comfortingly, I suppose, the same things can be said about this definition that I wrote about the concept of 'line segment'--(1) the idea as presented to young children is remarkably imprecise and (2) it's a different 'version' of the truth that you and I can (weirdly) easily accept (though we shouldn't).

It is less than comforting, however, to see at least the beginning of a predictable rationalization of this situation--one that can be added to the pile:
From a mathematical point of view, there are all kinds of problems with saying that a rectangle has “two long sides and two short sides” (so many that I won’t even attempt to name them). But how bad is this lie? Better yet, how bad is the spirit of this lie? I think it depends on the audience.
Even the first six words of that quote should sound bizarre to us (but they probably don't). What does "from a mathematical point of view" mean here? Does another field of study weigh in on this question? Is there an archaeological point of view on what a rectangle is? In my authoring meetings (for high school mathematics textbooks), I will sometimes hear someone, possibly myself, begin a statement with, "Well, mathematically speaking . . ." Um, what other kind of speaking have we been doing up until now? Have I just spent the last four minutes speaking about exponential functions metaphorically?

So, yes, considering our audience is important, and when our audience is there to learn mathematics, it should go without saying that our point of view will be mathematical. And if there are so many imprecisions with our expressions of that point of view that we can't name them all, then maybe our thoughts should turn to how to fix the situation rather than to squeamish theologizing about the "spirit" of our mess-ups. 

If we know that we have no choice but to describe a rectangle as having "two long sides and two short sides" for students in Grade X to be able to recognize it, then what on earth are we doing introducing such baloney in Grade X? Classroom teachers don't have much control over this situation, but publishers and standard-setters are not confined by brick-and-mortar theorizing. They can take the "Shapes" lesson clean out of your 8-year-old's math textbook and plop it down as a "Quadrilaterals" lesson in your 10-year-old's book. Now you've got at least one extra day in 2nd grade to go a little less than a mile wide and a little more than an inch deep on material the students are capable of understanding--and a little less time is wasted in 4th grade explaining why all that bullshit about long sides and short sides is so two years ago.

Saturday, July 20, 2013

The Precision Principle, Part II

f what we're teaching is wrong, then we should either fix it or not teach it.

The not-so-surprising thing about this statement of the precision principle is how easily and readily we agree with it in sentiment while disagreeing in deed. Indeed, when the courage of our conviction is required, it is much easier to find a way to rationalize imprecise teaching than it is to fix precision as an immovable goal and then attend to the inevitable and rewarding challenges such a decision should bring.

Take our line segment example from last time. What is clear to any adult who has a proper understanding of even basic geometry is that a line segment is a 'straight,' one-dimensional, infinitely divisible, cognitive object. There are a number of ways of defining it, even when precision is the goal. So, 'the part of a straight line between two points on the line' may be just as good. Yet, what is often presented to students as meaning 'line segment' is a straight, indivisible 'line' with two points at the ends drawn on paper.

Why? There are a few perhaps useful and certainly easily identifiable rationalizations for this behavior that you can be aware of. Here are two:
  • There's no way Xst/nd/rd/th graders will understand a line segment as a 'cognitive object'.
This one needs a special name--something with 'fallacy'--but only because it is so frequently used, not because it is important or difficult to dismiss. It is pulled off by simply pretending that the person or book attempting to offer the more precise formulation of a mathematical concept has condescendingly intended for you to use his, her, or its exact wording in front of students in the classroom. Perhaps given that this often involves making up a perspective in which there are only two choices--the dirty-elbowed, big-hearted story or the patch-elbowed, heartless exposition--we should use a name we already have.
  • Students need to learn the concrete ideas first and then gradually move to the abstract.
In his 1993 paper, Registres de représentations sémiotique et fonctionnement cognitif de la pensée, Raymond Duval best expresses the important antagonism between concreteness and the goal of improved precision in mathematics education. (Note: semiotic representations in this context are words, images, manipulatives, etc., that function to represent concepts.)
On the one hand, the learning of mathematical objects cannot be other than a conceptual learning and, on the other hand, it is only by means of semiotic representations that an activity on mathematical objects becomes possible. This paradox can constitute a real vicious circle for learning. In what way can subjects in their phase of learning avoid mistaking mathematical objects for their semiotic representations if they can relate only to semiotic representations?
Using our line segment example, we can edit Duval's question above to read
In what way can students in their phase of learning avoid mistaking the precise concept of a line segment for a picture or description of one if all they could possibly have is the picture or description?
An answer in line with the precision principle would include, first of all, the notion that there is no pinnacle of precision for any concept. The 'mathematical objects' Duval refers to do not exist outside of our reasoning about them, so to 'mistake' them for something else is all we can ever do. This does not mean, however, that anything goes. There are perspectives on mathematical concepts that are better than others, and, importantly, there are perspectives not worth using at all.

Secondly, no knowledge--about mathematics or anything--comes to us except through our senses, so to say that students must learn concrete ideas before abstract ones is to say nothing at all, really--or, at best, it is to say something so self-evidently true as to not be worth saying at all. The most obtuse and insular triumphs of your philosophical and mathematical reasoning are built on foundations of concrete inputs.

So, the concrete-before-abstract non-response gives up too easily. Just because we focus on providing concrete experiences of mathematics to younger students does not give us license to lie to them.

I like to think about what could happen if we really took a principled stance as a mathematics education community to maximize the accuracy of information environments. Far from ending debate, it would create more--debate that at least would be centered around a truly commonly held ideal.

Is it possible to teach geometric concepts accurately to young students? Good research question. What are better ways to handle Duval's supposed paradox of trying to teach the intangible through the tangible with regard to geometry (clarity principle!)? Or, where can we move this teaching (order principle!) in the K-12 sequence so that students are able to appropriate the concepts in full? How can we better structure our teaching leading up to these concepts so that students are better prepared to learn them? What other issues come into view as a result?

See how it works?

Saturday, June 1, 2013

The Precision Principle

How many line segments are shown in this diagram?

  • 2
  • 3
  • 4
  • none of the above

The correct answer, of course (or normative answer, if you prefer), is "none of the above." There are an infinite number of line segments in that representation. Sure, some of the line segments' endpoints are represented as black dots in the picture, and all but one of these points is labeled, but none of that information is relevant. Drawing black dots and writing letters are not ways of calling points into existence; they're ways to help people visualize and name what is already there.

Still, despite knowing the correct answer, you and I can easily articulate (or at least understand) rationales for choosing each of the incorrect answers:
Choice (a): Students count only line segments AD and DC, because their endpoints are drawn and labeled and neither of them includes other line segments.

Choice (b): Most likely, students count only those line segments that do not contain other line segments and have drawn endpoints.

Choice (c): Students count all of the line segments with drawn endpoints.
Why can we so easily figure out the logics that lead to the incorrect answers? It seems like a silly question, but I mean it to be a serious one. At some level, this should be a bizarre ability, shouldn't it? Imagine seeing a person slam their body into an outside wall and look around confusedly, over and over again. Would even one logical rationale (much less three of them) for this non-normative behavior come quickly to you? For example, would you say or think, almost without hesitation, "Ah, poor guy, he has probably forgotten that he can't walk through walls." Yet, we can quickly and easily construct rationales for missing the geometry question (even when we are in charge of instructing students correctly!)--not dismissive rationales as we might be wont to assign to the wall-walker, like, "they were just goofing around" or "they're crazy," but real, logical mappings of what produced the incorrect answers. Why?

The answer is that we can easily switch back and forth between different "versions" of the truth. We can think of and talk about line segments as straight, indivisible "lines" with little dots at each end drawn on paper. Or we can think of and talk about line segments as infinitely divisible, "straight," one-dimensional cognitive objects with finite length. And of course we can mix and match the versions as well. (Even a quick Google search for "line segment" will give you both versions. Compare the images of line segments presented next to the definitions.)

But why should there be two versions of the truth when one is clearly more accurate (more precise)? What I would suggest is that most of the answers educators provide to this question are bad ones. And in place of those bad answers I would suggest the Precision Principle: accuracy within information environments should be maximized. I'll talk more about both of these answers in a future post.

Wednesday, May 22, 2013

The Art of Clarity

ne of my favorite examples illustrating the necessity and importance of organizing information environments is from 2001, in California.

Composite of Ankrom at work.
In broad daylight, artist Richard Ankrom, posing as a Caltrans worker, installed a homemade road sign above the 110 freeway in Los Angeles to inform travelers of the upcoming 5 North Interchange. The sign--which was designed by Ankrom perfectly to specifications--remained for eight years until it was replaced in November of 2009. Luckily, as of this writing, Google Maps street view still shows Ankrom's handiwork--just 9 months before it was taken down.

The reason he did it?
Caltrans had never put in any signage for the interchange and the spot was legendarily confusing to motorists.
And also, art. But it should be important for us to see these as one and the same reason. What Akrom accomplished was a reorganization of a dysfunctional information environment--an art form unto itself.

Of course, clarity is the operative principle under which Ankrom labored. One can imagine--in the sign's absence--the rightmost lanes of this stretch of freeway unnecessarily packed and slow as drivers heading to the 5 North quite rationally chose the 5 South as the safest bet. And then just 2 miles farther on, more delay perhaps, as travelers found themselves trying to move across lanes of traffic in too short a distance. Ankrom corrected this environment to make important information clear and present to his "students."

The equally valid inverse of this principle was put well by Dewey in Democracy in Education:
It is the business of the school environment to eliminate, so far as possible, the unworthy features of the existing environment from influence upon mental habitudes.
Indeed, the following, from the same tome, should disabuse even the most zealous reformer of the notion that Dewey found expert control over information environments abhorrent. On the contrary,
The subject matter of education consists primarily of the meanings which supply content to existing social life. The continuity of social life means that many of these meanings are contributed to present activity by past collective experience. As social life grows more complex, these factors increase in number and import. There is need of special selection, formulation, and organization in order that they may be adequately transmitted to the new generation.

Saturday, May 11, 2013

Bongo-Bongoism in Education

In time I will argue--successfully, I hope--that the no doubt confusing philosophizing going on in my previous post on this topic was neither a hypocritical misapplication of good principles of information organization nor a faithful application of bad principles.

For now, though, the confusing post remains, and I shall do my best in this one and the next few to try to make my thoughts clearer while expanding on them a bit from what I hope to be a firmer foundation.

Because of (among other things) education's enormous social and institutional complexity, even reasonable, mainstream discussions within mathematics education tend to suffer from a kind of bongo-bongoism, "the venerable but ultimately sterile anthropological practice of countering every generalization with an exception located somewhere at some time":
When a generalization is tentatively advanced, it is rejected out of court by any fieldworkers who can say: 'This is all very well, but it doesn't apply to the Bongo-Bongo.'*
This is not so much a bug as it is a feature of the discourse among educators who directly and daily influence the learning of students. After all, these are people who traffic in exceptions rather than rules--for whom bongo-bongoism is a necessary, valuable, critical orientation toward "outside" ideas (from research, standards, professional development gurus, and textbooks) that often miss the trees for the forest (or, rather, the students for the statistics).

Yet it is one thing to be naturally suspicious of generalization and quite another to treat it as a thought crime. On this view, Eratosthenes's investigations concerning the circumference of the Earth should never have begun given that they were based on wild assumptions about the shape of the planet, the direction of the Sun's rays, and the locations of cities in Egypt. Better to wait for a 25,000-mile-long tape measure. Darwin's belief-shattering ideas, too, are ridiculous in this light. Were you there? is a question indoctrinated children are taught to ask of those who present the evidence for the theory of evolution by natural selection. This is much worse than missing the forest for the trees. It is missing the forest for the missing trees.

Mathematics education desperately needs a good generalization--one that, at the very least, can keep its denizens somewhere between a healthy bongo-bongoism and a destructive solipsism.

Saturday, May 4, 2013

Teaching as Shaping and Organizing an Information Environment

The principles of precision, clarity, order, and cohesion are clearly applicable to situations involving intentions to communicate carefully.

Thus, one way I can attempt to argue for the extension of these principles to mathematics education is to outline the similarities between careful communication on the one hand and mathematics education on the other. To borrow from the syntactical structure of the Transitive Property: if these principles apply to careful communication, and careful communication is what mathematics education is all about, then the principles should apply to mathematics education as well.

Reaching this kind of abstraction about mathematics education--that it is, at bottom, about careful communication (or, rather, a careful shaping and organizing of an information environment)--seems either impossible or inadvisable given its enormous institutional complexity. There are standards, unions, professional organizations, research, students, publishers, national news reports, technological innovations, blogs, assessments, philosophies, markets, buzzwords, and on and on. Only an inept or mendacious observer could find just one story or one solution among all of this disparity.

Yet, why not make use of mathematical thinking to analyze mathematics education? Mathematics derives its power as a cognitive tool because of abstraction. It tosses away pesky and unhelpful and irregular incidentals that stand in the way of useful generalizations about the world. We can at least try to do the same.

A good way to start, I think, is to ask, What is teaching? What image or images come to mind when you read or hear the word teaching?
  • a person giving another person directions to a location?
  • someone reading a newspaper?
  • a group of finches from the Galapagos Islands?
  • a painting?
I would expect that none of the above would be your answer. For me, the strongest image conjured by the word teaching is a classroom with a teacher and students, doing . . . something. I can't presume to think that this is the image everyone would have, but I would expect it to be the number one answer on Family Feud.

In truth, though, the only difference between the image of a classroom with a teacher and students and, say, that of a person giving another person traveling directions is the content of the message being delivered. In both situations there is a sender, a receiver, and an educational message.

What about the image of a person reading a newspaper? Is this teaching? Here again we have a sender, a receiver, and an educational message, though the sender is not present, may be more than one person, and has had his or her educational intentions filtered through a number of limitations outside of his or her control. With the exception of the presence or absence of the sender, this description could just as well describe any school teaching environment.

To consider the painting and the Galapagos finches as examples of teaching is to move much further along the abstraction ladder. Whether or not either have identifiable or even extant senders or receivers is irrelevant. Both represent information environments that have been carefully organized--in the case of the former, by a perhaps long-dead painter; in the case of the latter, by the learner himself. A lesson plan that sits on a desk (or in the cold of space) for possibly all of eternity is still an example of teaching, and the cosmos itself, though lacking a "sender" altogether as far as we can tell, is a teacher to the extent that a creature can organize and shape its informational content.

A line of inquiry seems naturally to follow the description of teaching as a careful shaping and organizing of an information environment: are there criteria we can apply to this process to maximize its efficacy? This is the question the principles of precision, clarity, order, and cohesion attempt to answer in the affirmative.

Monday, April 15, 2013

Precision, Clarity, Order, and Cohesion

It would be best for me, I think, to introduce the four principles I will talk about--precision, clarity, order, and cohesion--with a mundane example followed by a question.

So, let's start with this nonsense, which is a jumble of something that is not nonsense:
n u p i k u e o c x e e g c z t e t o g d i r n o h g

First, I need to put these bits of information in the correct order. For people with background knowledge in reading English text, this re-ordering will help out quite a bit, although there will still be confusion. For those without this background knowledge, the text below is likely just as confusing as that above:
x e n u z i c k e d t h e p u c e t o g o r i g o n g

So, now we need to package this information. Bits that belong together should appear together, and bits that belong apart should be separated from each other. That is, we correct the information to give it cohesion:
Xenu zicked the puce to gorigong.

Note that placing a capital letter at the beginning of the sentence and a period at the end also helps to give the information more cohesion--we clearly identify the sentence as a separate 'chunk' of information, a complete thought.

For clarity, we at minimum should probably define 'zicked' and 'puce' and 'to gorigong." (Or did you think that 'gorigong' was a name?) But note that addressing clarity here requires us to revisit order and cohesion also: Should we define the terms before using them in instructional sentences or after? Should we define them all together or separately? Etc.

Finally, to apply the precision principle here simply requires us to ask if the resulting clear, cohesive, and well ordered information is actually true--although, as we will see, this is only a starting question.

With those basic concepts introduced, I ask the question: In what way can attentiveness to these principles shine light on how to teach mathematics?