Wednesday, December 17, 2014

Shouldn't Sturgeon's Law Also Apply to Education?

I was complaining to a teacher colleague in a Starbucks about the management where we both used to work—at the time, he had already left and I was still there.

"Everything to them is a 'code red'," I said, borrowing a phrase from yet another teacher colleague at the same place about the same issue. "Everything's important. But this [whatever I was complaining about at the time] is just not important."

He said, "I can't imagine telling my students that something wasn't important."

I don't remember ever being "struck" by something that someone said before that. But I was truly struck by his response—not in a reflexive, emotional way, but a dozen little rememberings happened, a hundred tiny connections, when I heard him say that. It was a perfectly timed, "obvious" truth that managed to pull together a number of pieces of my knowledge and experience and give them their own singular gravity.

Should the General Architecture of Reality . . .

"Ninety percent of everything is crap." This is the phrase most commonly referenced as Sturgeon's Law, formulated by science fiction writer Theodore Sturgeon to emphasize that the volume of lower-quality writing in science fiction was not a characteristic unique to science fiction writing. Indeed, when you substitute "crap" with something a little less bitter, and allow for some variability around 90%, this law is robustly observable:

Sensory Gating We filter out most of the stimuli that pound daily on our senses, attending only to a small percent of what we perceive. This is known as sensory gating. A measure of sensory gating used to help identify subjects with schizophrenia places the reduction in sensory stimuli for those without the illness at around 80%-90%.

Dark Matter and Dark Energy Today it is thought that dark matter and dark energy together make up about 96% of all the mass and energy in the universe, with about 73% going to dark energy and 23% to dark matter. It's far from "crap" (maybe). But this is all stuff that we currently cannot "see, detect, or even comprehend."

Junk DNA Given what we now know, most of the genome is junk, sorry. Even here we don't stray too far at all from Sturgeon's 90%: "at most 10% of the human genome exhibits detectable organism-level function and conversely that at least 90% of the genome consists of junk DNA."

The Pareto Principle Related to Sturgeon's Law is the Pareto Principle, which says that "for many events, roughly 80% of the effects come from 20% of the causes." It has been observed in many natural and social contexts, including wealth distribution and Internet traffic.

. . . Not Be Allowed to Stand Up in Schools?

So, is there a similar 90% in education, in teaching, in schools—however you want to contextualize that? My purpose in writing this post is not to answer yes to that question and then start listing things that are unimportant. Instead, I'd simply like for us to stand back and consider that (1) since education is a part of our natural and social landscape, Sturgeon's Law very likely applies to it in some way, but (2) we are forced to pretend that this is not the case.

Sunday, July 20, 2014

Measuring Misconceptions

When you encounter a multiple-choice problem, you find yourself surrounded by choices with varying levels of influence over you:

If you are knowledgeable about the topic, well rested, and so on, the answer choice with the most influence over you will be the correct one. On the other hand, if you are unprepared, tired, or emotionally distraught (or just don't know), you will probably do less well, not because these characteristics raise the levels of influence of the incorrect answer choices, but rather because they lower the level of influence of the correct answer to match the levels of the incorrect ones. The three incorrect answer choices above, for example, are not more salient when you don't know any German. They are simply as salient as the correct answer, and this--a more even distribution of influence among the choices--is what makes the question difficult.

However, no matter what the question is, one or two of the incorrect answer choices will almost certainly call to you a little more than the others (if the assessment writer has done his job right). The word klar, for example, which is Choice D above, sounds and looks a lot like the English word clear (which is its most immediate translation). And clear bears some semantic resemblance to honest. The word liebe, too, may influence you to choose it simply because it is a bit more well known to English speakers than the other three choices.

Thus, in general, what we expect (and get) among the incorrect answers on multiple-choice tests in a sample is a distribution that is not completely even (all the wrong answer choices have the exact same percentage of responses) and also not completely lopsided (100% of respondents choosing the wrong answer choose the exact same wrong answer).

Misconception Detection

I would argue, however, that lopsided distributions among incorrect answer choices can point to the presence of one or more misconceptions among respondents, especially when--in contrast to the situation outlined above--we can make the further assumption that the population of respondents has been exposed to instruction on the topic.

For example, if we find, in a hypothetical survey of 1,000 fourth graders, that responses to the question 2 + 3 = ? break down as follows--A 1 (1%), B 5 (80%), C 6 (18%), D 10 (1%)--it should raise eyebrows that almost all of the students who answered incorrectly did so in one direction (toward the answer given by multiplication instead of addition).


This very imperfect measure is the one I brought to bear on an analysis of the 2013 results of the State of Texas Assessments of Academic Readiness (STAAR) to see if it could help me identify possible misconceptions, which would lead students to strongly favor one of the incorrect answers among the 3 given for each multiple choice item.

The site linked above has a remarkable amount of information and data related to the assessment. For my analysis, I used the item analysis reports (which give the strand and the percent choosing each answer choice for each question), the student expectations tested, and the actual released tests from 2013:

This graph summarizes answers across Grades 3-8 to 285 multiple choice items in 5 mathematics strands for over 2 million students across Texas. The "misconception strength" was determined for each of the 285 items and then these measures were collected together here in box plots.

The good news here is that the grade-level averages for "misconception strength" drop from Grade 3 to Grade 8. (This is shown in the small inset line graph in the grouped box plot above.) Surprising--to me at least--though, is that there is no strong trend at all among the strands across grade levels. There is a slight decline across the grade levels for each strand except for Measurement, which shows a very slight increase, but none of these changes is noteworthy.

The Items

We like to focus on means and medians when analyzing data, but here the extremes and outliers are probably more interesting. (Outliers in the plot above are points above or below the box plots.) So, let's take a look at some of the items which contained a strongly influential incorrect answer choice--enough to suck in a fair number of students who had ostensibly been prepared to avoid such influence.

Edges Are Not Points, Grade 3
(A: 66%*, B: 30%, C: 1%, D: 1%; MS = 0.42721)

The asterisk above shows the correct answer choice. The percent of students who chose each answer is given along with our "misconception strength" measure (MS = 0.42721). Students here seemed to be carrying the misconception that "edges" are the outer points of figures (like they are in the real world).

Primes > 2 Are Odd but Not Vice Versa, Grade 5
(F: 59%*, G: 7%, H: 3%, J: 30%; MS = 0.29744)

Answer Choice J is exactly what you'd get if you crossed out all the even numbers and counted up what was left. This may also point to an issue with basic multiplication facts. I can imagine many students thinking of 45 and 49 as prime simply because they can't remember their factors.

Forget About Mirrors, Grade 8
(A: 5%, B: 4%, C: 38%, D: 52%*; MS = 0.33611)

Thirty-eight percent of students could not live with the x-axis being the line of reflection, so they moved it up 2 units in their minds.

Not Everything Is Left-to-Right, Grade 4
(F: 6%, G: 5%, H: 48%*, J: 41%; MS = 0.32192)

It seems that what happened here was that 41% of students (that's over 143,000 students, by the way) read the table from left to right, got the correct expression, and then didn't notice that the answer choices all read out the table from right to left.

Symbols Are Important Too, Grade 7
(F: 3%, G: 46%, H: 45%*, J: 5%; MS = 0.36696)

They knew that the answer was 4, and they didn't care about the symbolic gobbledy-gook around it.

Stop Being Fooled by Position, Grade 7
(A: 2%, B: 71%*, C: 25%, D: 2%; MS = 0.37387)

Students used the positions of the triangles to determine correspondence rather than corresponding angle measures. A well known misconception.

Superficial Mapping, Grade 7
(A: 8%, B: 58%, C: 33%*, D: 2%; MS = 0.36918)

Here we have a relatively rare case of a significant majority of students choosing one wrong answer (compared with those who chose the correct answer). The information in the diagram in B maps superficially to the bullet points in the question, and that was enough for most students.

Division Is Unnatural, Grade 5
(F: 21%, G: 75%*, H: 3%, J: 1%; MS = 0.35975)

A large majority answered this correctly, but 21% still wanted to multiply instead of divide.

More Left-to-Right Fixedness, Grade 6
(F: 35%, G: 2%, H: 7%, J: 56%*; MS = 0.33005)

Order of operations difficulties appear frequently in the STAAR results. The misconception scores for this one and the other order of operations questions are moderated by the apparent difficulty students have with these questions (44% answered this one incorrectly, which is 155,000 sixth graders). Given some of the difficulties grownups have with the order of operations, it is no surprise to see the same problems reflected in the next generation.

Why Are Misconceptions Important?

We often look at misconceptions as the products of "doing things wrong." They are, after all, associated in our minds with wrong answers as opposed to right ones. But I prefer to see misconceptions as the complete opposite--they are the products of "doing things right" when it comes to teaching, even though those local "right"s are situated inside global "wrong"s.

It's true, for example, that the "answer" to 6 + 3 = ? is 9 and the answer to 2x + 3 = 9 is x = 3. If you repeat problems like this hundreds of times in classrooms over the course of a student's elementary and middle school education, you are doing something "right" hundreds of times, so long as your focus is on the discrete set of problems. Taken as a whole, however, you have developed the misconception that math problems are to be analyzed from left to right, always or almost always. Or perhaps more accurately, you have developed a notion of "normal" math and "weird" math, with left-to-right-isms belonging to the normal category.

Monday, June 2, 2014

Understanding Understanding

A solution to a math problem is obviously different from the reasoning used to arrive at that solution.

Less obvious but equally true is that our reasoning about a math problem is different from the understanding in which that reasoning is embedded. To tease out the distinction between solutions, reasoning, and understanding, I like to think about brainteasers like this well known brainteaser, posted by +Khan Academy.

The solution is 10, just to get that out of the way. Ten light bulbs are on after 100 people go through flipping light switches. But the description below is also a decent solution to the problem, and uses flawless reasoning:

Write down the counting numbers from 1 to 100. Draw O's above each of the 100 numbers to show that they are "on." Then draw X's above the multiples of 2 to show that they are "off." Continue in this way, alternating between O's and X's for each of the other n numbers up to and including 100. The result is that 10 of the numbers from 1 to 100 show a final O at the top of their column, indicating that 10 of the light bulbs are on when the process is complete.

To get a sense of what understanding is, though, you have to really feel how uncomfortable you would be with the prospect of everyone in the world accepting this long-form answer from their students and children as perfectly correct and normal.

That discomfort is what it feels like when something scrapes against your understanding,an understanding of what mathematics is supposed to look like and be like.

This is the ineffable irritation many people feel when they encounter things like the homework at the right. Frustrated Parent's understanding is a constellation of ideas about mathematics--that it should be high stakes and "real world" ("In the real world...", "termination") that it is a dry, workplace calculation tool ("get the answer correct"), that it is wound tight and enigmatic ("simplification is valued over complication") and should not take a long time ("under 5 seconds").

A person's understanding of math is often not something they can explain in words--or even have thought about in words. At best, it is often not something that a person has really examined thoroughly. But it's worth examining, without feeling like you need to make a decision about it.

There's nothing wrong with a little introspection to try to understand your own understanding. It may help you empathize with students a little more.

P.S.: If you've read this far and you're still curious about the real mathematical reasoning that gets you to the solution of 10 above, you're welcome.

Sunday, April 27, 2014

Failure of Further Learning

The central idea in the paper I discuss here is the "failure-of-further-learning" effect--an effect documented since the 1930s.

Kay (1955), for example, observed that after initially 'learning' and recalling text passages read to them, subjects did not learn anything new from further exposure to the passages. Rather, the subjects' initial recall of the material seemed practically impervious to change:

"[Participants] were repeating the same correct items, making the same errors, and omitting the same items 1 week after another … The initial reproduction had established the content of the passage; as it was laid down then so, with only slight modifications, was it held 6 weeks and ultimately 6 months later."

The predicament summarized in the quote above is replicated thousands of times every year, as many students (to give just one example) make the practically inevitable mistake of thinking of the sum of two fractions as the sum of the numerators over the sum of the denominators (PDF). (Why shouldn't they make this mistake? By the time students encounter fractions, addition of "numbers" (i.e., counting numbers) has been "laid down" for all they know. Their initial success has almost paradoxically limited their future potential for learning.)

So, what causes the failure-of-further-learning (FOFL) effect and how can it be mediated? Fritz, et. al conducted four experiments with the goal of shedding light on these questions. Below are descriptions of the experiments from their abstract.

Experiments 1 and 2 attempted to circumvent the effect by varying the activities of participants and requiring interactive exploration. In both experiments, recall after four weekly sessions showed little benefit beyond performance on the first recall. Experiment 3 interfered with the formation of an immediate situation model by introducing passages that were hard to comprehend without a title. Performance improved substantially across four sessions when titles were not supplied, but the standard effect was replicated when titles were given. Experiment 4 made verbatim memories available by incorporating all re-presentations and tests into one session; as predicted, recall improved over successive tests.

I'd like to focus in this post on the details of Experiment 3, with a follow-up post later discussing some of the implications of the research as a whole.

Experiment 3

We hypothesized that the failure to improve recall with successive encounters with the passages was a function of having formed a situation model based on understanding the passage in the initial reading.

Thus, if the situation model (the gist of the passage) frustrates future learning of the same material, then it follows that making it difficult for learners to get the gist in the first place should allow for larger increases in future learning.

So Fritz, et. al, gave participants both titled and untitled text passages to learn. Withholding titles made it difficult for subjects to form a situation model (schema) of the passage—a trick borrowed from well known research by Bransford and Johnson (1972). Here is an example passage:

The procedure is actually quite simple. First you arrange things into different groups. Of course, one pile may be sufficient depending on how much there is to do. If you have to go somewhere else due to lack of facilities that is the next step, otherwise you are pretty well set. It is important not to do too many things. That is, it is better to do too few things than too many. In the short run this may not seem easy but complications can easily arise. A mistake can be expensive as well. At first the whole procedure will seem complicated. Soon, however, it will become just another facet of life.

Think of reading this same passage but with the large title "Washing Clothes" above it to get a sense of how removing the title inhibits your gist-making process (and, conversely, how including the title might allow the gist to muddy your memory of the actual details of the information in the text).

As the researchers predicted, inhibiting the formation of the gist of the passage deleted the FOFL effect (while the effect was still present in the titled-passage condition):

The linear improvement for the untitled-passage condition was significant while that for the titled-passage condition was not. Yet, it is worth noting some of the obvious: (1) At no point over the 4 weeks did the gist-less participants (untitled condition) perform better (in verbatim recall) than those who were allowed to form gists of the passages (titled condition), and (2) the removal of the FOFL effect for the untitled-passage condition was purchased at a significant cost—participants in this condition began with about half as much memory for the passages than did their titled-passage counterparts.

Fritz, C., Morris, P., Reid, B., Aghdassi, R., & Naven, C. (2013). Failure of further learning: Activities, structure, and meaning British Journal of Psychology DOI: 10.1111/bjop.12060

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. 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

I don'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.*

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

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

If 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 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?

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.