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Devlin's Right Angle, Part VI

Let me quickly close out the definition of multiplication I was building to in Part V.

I fell far short of my goal of making myself completely clear at the end of that post, so I want to see if I can rescue the clarity of the presentation somewhat—for myself, if no one else.

Anyway, here it is, again:

When you're dealing with operations, like addition and multiplication, it's both different and the same. Different because you can't just throw an operation, pure and unadulterated, at the Induction Principle and ask, "Hey, does it work?" And the reason you can't do that is not because it's a stupid question—it's not. The reason you can't do it is because there is no such thing as an operation, pure and unadulterated. You have to define it first. Same because you still have to deal with the least element (1) and some x + 1 when you're dealing with the natural numbers.

So, let's make the following assumptions about the product of a and b (Note: These are assumptions. And they are not assumptions about "multiplying;" they are assumptions about the result of multiplying. And they are assumptions one would have to make, given the definition of the natural numbers.):

(1) a × 1 = a
(2) a(b + 1) = (ab) + a

Now, here's the freaking crazy part. We assume that (1) and (2) above are true. Then we simply imagine that we already have a jar that contains all the natural numbers, scooped up by the operation of a × b, with b being a natural number. All we have to do in order to show that what we have scooped up is indeed all the natural numbers is to find the number 1 and to find some b + 1.

According to our assumption, then, and (1), the number 1 is in our jar. That is, since we assume that a × b scoops up all natural numbers, and we assume that a × 1 = a, then 1 (substituted for b) is a natural number and must be in our jar.

It is very important to remember that we're not proving anything here, so to speak. As I mentioned in my previous post, we can use the Induction Principle to find out whether a statement like 1 + 2 + 3 + . . . n = n(n + 1) / 2 works for all natural numbers. We check that it works for 1, then we check whether or not it works for some b + 1. If the statement passes those tests, then we know that it works for all natural numbers.

With an operation like multiplication, the same idea applies. However, with multiplication (or addition), we first have to have some guess about how that operation works on 1 and some b + 1. You can see this in (1) and (2) above. The first statement, (1), imagines how the operation applies to the number 1, and (2) imagines how the operation applies to some b + 1. But after we send those statements through the tests of the Induction Principle, the only relevant conclusion that will come out is whether or not they describe an operation that applies to all natural numbers. That's it.

So how do we take those hair-brained statements, imagining some operation called multiplication in the natural numbers, and show that they do indeed work for all natural numbers?

Here's how: First, we just imagine that we're right on the big idea here and that the operation of multiplication (a × b, where both a and b are natural numbers) does indeed apply to all the natural numbers—or, as I said before, the operation of multiplication "scoops up" all natural numbers. Second, we assume that the statements we made in (1) and (2) are true. And these statements tell us how multiplication applies to both the number 1 and some b + 1. Third, we need to show that, given these assumptions, our "multiplication jar," as it were, could contain the number 1 and some b + 1, which would prove that our jar contained all of the natural numbers.

(1) says that a × 1 = a. And since we've assumed that that is true and that a × b scoops up all natural numbers, which we assume are in our jar, then 1 must be a natural number (because we can just substitute b for 1), and, therefore, must be—or, rather, could be—in our jar.

(2) says that a(b + 1) = (ab) + a. Since we've already assumed that a × b scoops up all the natural numbers, obviously (b + 1) can be substituted for b, which means that b + 1 is a natural number and could be in the jar. The only problem is that one must show that a(b + 1) is "uniquely defined," which means that you must show, using axioms of addition already settled upon, that (ab) + a is uniquely defined, which it is, because of the uniqueness of addition. We do, indeed, need to rely upon addition when we define multiplication for the natural numbers.

However, there is no reason to believe that repeated addition is criterial. It should be noted that nowhere--nowhere--in the "definition" above was it necessary to make use of repeated addition to define multiplication. Nowhere.

The Thorn

From what I've seen, what most people get stuck by is the statement in (2) above--a(b + 1) = (ab) + a--what is known as the distributivity of multiplication over addition. So, let's say a = 4, and b = 5. We know that the product of 4 and 5 is 20, right? What the statement above says—if you read it cinemathematically--is that if you want to multiply 4, or a, by the next natural number after 5 (b + 1, or 6), you can multiply 4 and 5 and then just add another 4 [(ab) + a]. In other words, to find 4 × (5 + 1), first find 4 × 5, then just add 4.

Now, of course, this "works" for all natural numbers. And, if you think about it, you could just continue this process repeatedly, using the second part of the definition for multiplication I showed above. In other words, you could find 4 × (5 + 2) by, again, first finding 4 × 5, and then adding 4 and then another 4. The product is 28, bravo!

Hey, you know what we could do? We could basically extend this definition—that is, we can say that "all anyone means when they say" a(b + 1) = (ab) + a is just (a + a + a . . . )btimes! That way, we can say that multiplication is just, by definition, repeated addition, right?

NO. NOPE. NYET. NOPEY-NOPE-NOPE.


Of course, the idea of repeated addition follows very directly and immediately—as a way to calculate multiplication problems—from a definition of multiplication on the natural numbers, but IT. IS. NOT. THE. DEFINITION. OF. MULTIPLICATION.

As Nunes and Bryant (1996!) argued (I'll get the source up soon):

Although there is a conceptual discontinuity between multiplication and addition, there is a procedural connection between these operations. Because multiplication is distributive with respect to addition, repeated addition can be used as a procedure to solve multiplication sums.

What people (including now someone named Adrian with his/her one-person press release to bloggers who write on the subject) constantly refer to is the "procedural connection" while ignoring the "conceptual discontinuity."

Here's another good quote from the same source (again, I'll put it up soon; keep in mind that this is from '01):

The English National Numeracy Strategy (DfEE, 1999, p. 14) suggests that pupils should be taught to understand multiplication as repeated addition. In contrast, the Japanese Association of Mathematical Instruction proposes that "repeated addition is a way to calculate multiplication, not a meaning of it" (Yamonoshita & Matsushita, 1996, p. 291).

Alas, another post shall follow.

Reference:Park, J. & Nunes, T. (2001) "The development of the concept of multiplication." Cognitive Development 16 (2001) 763-773

Part I | Part II | Part III | Part IV | Part V | Part VI

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Dichotomies Are the Reason Daddy Drinks

This happens to me way too often. I stumble across some education "research," read the abstract, print it, and quickly become bored with it.

Then I feel guilty for wasting paper, struggle through the entire piece (PDF), and, usually, find some ideas that seem to be worth further exploration:

There appears to be an inclination within the education community to dichotomise and an associated tendency to (i) ignore the connectedness of the dichotomous categories, and (ii) on occasion, to privilege one category while denigrating the other . . . .

This paper addresses five of these dichotomies: Teaching and Learning; Abstract and Contextualized mathematical activity; Teacher-Centred and Student-Centred classrooms; the teacher's contemporary dilemma: to Tell or Not to Tell; and the related issue for students: to Listen or to Speak. (376)


[The Teaching and Learning dichotomy] is a particularly insidious consequence of the constraints that language (and the English language, in particular) imposes on our theorizing . . . . this is particularly evident in the various translations of Vygotsky, in which the Russian word 'obuchenie' has been represented as either teaching or learning in different translations (Clarke, 2001). The integration of teaching and learning as components of a jointly enacted single activity also occurs in several other languages, including Chinese, Japanese and Dutch. (378)


Differences in the use of abstract and contextualised tasks seem strongly connected to a perceived need in Western classrooms to present mathematics as relevant to students. (379)


In the Swedish classroom, the students demanded that the teacher justify the relevance of what was being taught . . . . Despite the teacher's efforts, students were outspoken in their lack of belief in the relevance of the mathematics they were studying. . . .

By contrast, in the classroom in Shanghai, mathematics tasks tended to be very abstract in character and the teacher made no effort to demonstrate or argue for the real world applicability of the mathematics being studied. . . . However, in the post-lesson interviews, the Chinese students consistently expressed strong beliefs in the utility of mathematics in general and in relation to the specific mathematics they were studying [ed: There is a bit of apples-and-oranges at work here, so be warned. "Demanding" relevance of what you're studying and "believing" that what you're studying has relevance are two completely different things.]. . . . . Svan has christened this the "Expanded Relevance Paradox" (Svan & Clarke, in preparation) and means, by this term, to refer to the paradoxical character of application-oriented mathematics teaching associated with subjective irrelevance and pure mathematics-oriented mathematics teaching associated with subjective relevance . . . . Mathematical tasks are a constituent element of the social activity in which students engage. Attempts to increase the 'relevance' of these tasks through a figurative contextualisation may be counter-productive if these efforts are perceived by students to be artificial and are interpreted as reifying the very distinctions they seek to dissolve. (380)

This one was really good. Hell, it's been "good" for a long time. (emphasis mine):

One common interpretation of the constructivist manifesto (i.e., that "knowledge is the result of a learner's activity rather than of the passive reception of information or instruction," von Glasersfeld, 1991, p. xiv) has been that it became no longer legitimate for teachers to "tell" students anything. This position is not a logical consequence of adherence to constructivist learning theory, which suggests that students inevitably construct their own mathematics, whatever the classroom situation (Cobb, 1995). However, Telling or Not-Telling have been constructed oppositionally with such success that publications on contemporary pedagogy, . . . while usefully discussing many pedagogical strategies, see no need to address any strategies that might be construed as analogous to "telling" and even articles that purport to address the issue (such as Chazan and Ball, 1999) offer teachers little insight into how (and, as importantly, when) their mathematical knowledge might be articulated explicitly to the benefit of their students.


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Wu and I

Allison pointed me to some notes from a presentation by Hung-Hsi Wu at NCTM's 2007 meeting, and I've been meaning to write about them.

I gave a presentation at that conference, so during Wu's talk I was, if I remember correctly, holed up in my hotel room frantically reviewing my notes—a shame, considering that what appears to be the central point of Wu's presentation was, with fewer differences than similarities, the idea that I tried to flesh out for textbooks when I first started writing this blog three years ago. (A further shame is that many of my posts which directly addressed that idea are gone forever.)

Wu's idea below, minus the brilliant parallel to engineering, is essentially what I have referred to as the Boundary Principle (emphasis in original):

Engineering cannot cater to any human need no mater [sic] how scientifically absurd, any more than it should produce any product that is useless though scientifically correct.

Engineering must mediate between two extremes:

(1) inviolable scientific principles.
(2) user-friendliness of the final product.

Mathematics education, as Wu explains, is constrained by the same impossibility on the one hand and impracticality on the other. That is, it is not possible for mathematics education to serve every student desire, nor practical for it to attempt to do so. Rather, mathematics education (what Wu refers to as "mathematical engineering") must strike a principled compromise between its often conflicting fealties to both mathematics students and to mathematics itself.

Wu goes on to list five "inviolable scientific principles" for mathematical engineering, four of which fit reasonably well under my three for textbooks—accuracy, coherence, and language:

Precision: Mathematical statements are clear and unambiguous. At any moment, it is clear what is known and what is not known.

Definitions: Bedrock of mathematical structure (no definitions, no mathematics).

Reasoning: Lifeblood of mathematics; core of problem solving.

Coherence: Every concept and skill builds on previous knowledge and is part of an unfolding story.

Purposefulness: Mathematics is goal-oriented. It solves specific problems.

I would disagree that the last of these rises to the importance of being a principle, and Wu's contention that the damaging split (visit the links in Allison's post [linked above] for more details) is between mathematicians on the one hand and mathematics educators on the other is, I think, a bit simplistic. Nevertheless, I was heartened to see so many similarities between Wu's ideas and my own.

And here's just a fantastic section from Wu's article "How mathematicians can contribute to K-12 mathematics education" (again, see Allison), which, to my mind, connects with some ideas here, here, here, definitely here, and certainly here and here:

At present, what most children get from their classroom instruction on fractions is a fragmented picture of a fraction with all these different "personalities" lurking around and coming forward seemingly randomly. What a large part of this research does is to address this fragmentation by emphasizing the cognitive connections between these "personalities". It does so by helping children construct their intuitive knowledge of the different "personalities" of a fraction through the use of problems, hands-on activities, and contextual presentations.

This is a good first step, and yet, if we think through students' mathematical needs beyond grade 7, then we may come to the conclusion that establishing cognitive connections does not go far enough. What students need is an unambiguous definition of a fraction which tells them what a fraction really is. They also need to be exposed to direct, mathematical, connections between this definition and the other "personalities" of a fraction. They have to learn that mathematics is simple and understandable, in the sense that if they can hold onto one clear meaning of a fraction and can reason for themselves, then they can learn all about fractions without ever being surprised by any of these other "personalities".

Update: I promised to post this great quote from Allison on the Wu presentation/paper. It's a fantastic quote, and it's like a bottom line for people who read this post (or read this blog) and say, "Harumphh, he's so abstract":

The point is to simplify without lying, misleading, or otherwise undermining future growth, so that the presentation true for the 4th grader is still true for the 12th, but the 12th can handle more, and can see the connections to the stuff already in his mind. Mastery is then possible.

Update II: One reason I like Allison's statement so much--aside from the fact that it's pretty much spot on in interpretation--is that she uses the word lying. Exactly. I can go into detail later, but that's exactly what a lot of mathematics education is today--straight-up lying to kids' faces.

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Historical Phenomenology of Mathematics

One of these days, I'll have to pick up the book. For now, I'll write about the ideas, using incomplete information.

First, a definition from the man himself, the late Hans Freudenthal. This is his description of historical phenomenology as it concerns mathematics:

Our mathematical concepts, structures, ideas have been invented as tools to organise the phenomena of the physical, social and mental world. Phenomenology of a mathematical concept, structure, or idea means describing it in its relation to the phenomena for which it was created.

In a previous post, I shared an example of what seems to be an early conception of the median, quoted in a paper by Bakker and Gravemeijer titled An Historical Phenomenology of Mean and Median. The example comes from a book by Edward Wright, cartographer and mathematician, circa 1599:

Neither if there be disagreement betwixt observations, are they all by and by to be rejected; but as when many arrows are shot at a mark, and the mark afterwards away, he may be thought to work according to reason, who to find out the place where the mark stood, shall seek out the middle place amongst all the arrows: so amongst many different observations, the middlemost is likest to come nearest the truth.

For a more accessible example of "Wright's median," one can think of a guess-the-number contest at a fair in which participants are invited to publicly guess (write down) the number of, say, jellybeans in a jar to win a prize. Here, each guess is like an arrow, and the exact number of jellybeans like the missing bullseye. Assuming that every participant had access to the same information (they all saw the same jar of jellybeans) and that the distribution of their "errors" was relatively normal, the median (or mean) of all the guesses would likely fall close to the exact number of jellybeans in the jar:

Treynor, former editor of Financial Analysts Journal, told us that when he taught finance, he would pass a jar of beans among his students and have them guess the number. As I wrote: "The guesses would vary wildly, but always, when the number guessed in total was divided by the number of students guessing, the result was within 3% of the correct number, he said. As there were 52 of us assembled, and a bowl of peppermint candies on the table, we tried the experiment. A low guess of 32 was recorded, a high of 71. The median guess was 46, the mean was 45. The correct total was 46, a number only one of the 52 had guessed."

More later. Bye!

Reference: Bakker & Gravemeijer. 2006. An Historical Phenomenology of Mean and Median Educational Studies in Mathematics. 62: 149-168.

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The Wright Median

Historical and didactical phenomenology are both mouthfuls. Interesting, though. More on these later.

The paragraph below seems to refer to the concept of the median, and was written by Edward Wright, cartographer and mathematician, in about 1599:

Neither if there be disagreement betwixt observations, are they all by and by to be rejected; but as when many arrows are shot at a mark, and the mark afterwards away, he may be thought to work according to reason, who to find out the place where the mark stood, shall seek out the middle place amongst all the arrows: so amongst many different observations, the middlemost is likest to come nearest the truth. (Eisenhart, 1974, p. 52, spelling modernized)


Reference: Eisenhart, C.: 1974, 'The development of the concept of the best mean of a set of measurements from antiquity to the present day', 1971 ASA Presidential Address. Unpublished manuscript.

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Relojs and Recontextualizing

I just finished reading a nice paper addressing the relevance of Cummins's (1979) BICS/CALP distinction with regard to teaching ELLs.

From the introduction:

Drawing a line between something considered to be BICS [basic interpersonal communication skills] (seen as of little use in the classroom), and something considered to be CALP [cognitive academic language proficiency] (seen as prerequisite to academic success), can easily work against the fundamentally creative endeavor of this kind of learning. This article . . . offers a framework rooted in recontextualization (Dyson, 1999, 2003) that can better help us think about teaching ELLs.

Of course, you need to read the paper and do some of your own recontextualizing to figure out what that paragraph above is saying. I link to it partly because I like the ideas in it (for a specific purpose I have in mind), but mostly because I liked this part of a story the author relates about a young ELL named Adriana:

After a few weeks in which Adriana appeared to be making great progress, Mr. Vélez was suddenly puzzled by a new kind of error she began making. When she encountered a t in a word—a letter with which she was familiar and had seemed to have under control—she made the /r/ sound. For several lessons, he was flummoxed by this development: he would discuss it with her, even point out the r in her name, but invariably she would read t with a /r/ sound the next time around. But then he noticed that she was looking up at the English sound-spelling cards posted on the wall, where a picture of a timer accompanied the letter t. That is, to Mr. Vélez it was a timer; to Adriana—when he asked her—it was a reloj (clock), not unlike the one in her picture dictionary. Adriana was recontextualizing—figuring out what sound the letter t needed to make on the basis of her sense of how to use letter-picture resources.

Maybe Adriana knew the concept of timer in Spanish . . . but it is quite possible that she did not, and was substituting a familiar concept instead. But it was not this "lack" [of CALP] that was holding her back in reading. And it is highly unlikely that a discussion of the meaning of timer, on its own (as decontextualized language), would have helped much.


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Different Strokes for Different Folks

Noted without comment . . .

According to conflict theorists, the hidden curriculum is the transmission of cultural values and attitudes, such as conformity and obedience to authority, through implied demands found in rules, routines, and regulations of schools (Snyder, 1971).

Although students from all social classes are subjected to the hidden curriculum, working-class and poverty-level students may be affected the most adversely (Polakow, 1993; Ballantine, 2001; Oakes and Lipton, 2003). When teachers from middle- and upper-middle-class backgrounds instruct students from working- and lower-income families, the teachers often have a more structured classroom and a more controlling environment for students. These teachers may also have lower expectations for students' academic achievements. For example, one study of five elementary schools in different communities found significant differences in how knowledge was transmitted to students even though the general curriculum of the school was organized similarly (Anyon, 1980, 1997). Schools for working-class students emphasize procedures and rote memorization without much decision making, choice, or explanation of why something is done a particular way. Schools for middle-class students stress the processes (such as figuring and decision making) involved in getting the right answer. Schools for affluent students focus on creative activities in which students express their own ideas and apply them to the subject under consideration. Schools for students from elite families work to develop students' analytical powers and critical thinking skills, applying abstract principles to problem solving.

Through the hidden curriculum, schools make working-class and poverty-level students aware that they will be expected to take orders from others, arrive at work punctually, follow bureaucratic rules, and experience high levels of boredom without complaining (Ballantine, 2001). Over time, these students may be disqualified from higher education and barred from obtaining the credentials necessary for well-paid occupations and professions (Bowles and Gintis, 1976).



Reference: Kendall, D.E. Sociology in Our Times: The Essentials, Fifth Edition. Thomson Wadsworth Publishers. (p. 385)

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The Wason Selection Task, Finale

My 3-year-old can make free-throws. My proof? When I put her on my shoulders close to the basket, she makes it almost every time.

In a sense, this is what Margolis is saying about the Wason Selection Task and the reduced-array selection task. People aren't bad at modus tollens reasoning, because when we alter the wording in the task and/or reduce the number of possible answers, a "clear majority" of people respond correctly:

The most essential aspect of the puzzle does not depend on a presumption that people have difficulty in consciously grasping the underlying abstract logical relations. As it happens, people do have exactly that difficulty. Few readers aside from those who routinely have occasion for using formal logic (because they teach it, for example) are likely to have "gotten" [it] at first glance, without thinking it over a bit . . . But that is not at all the important point of the Wason material.

Rather the important point is this. If the two pairs of tests each involve questions that are exact logical equivalents, as they do, then why should we find one easy and the other hard?

I'd like to answer Margolis's question with an analogy. Suppose Dick and Jane are traveling together in a foreign country. In that country, Language X is spoken. Language X is Dick's first language, and he has been speaking it all his life. It is not Jane's first language, but she learned to speak and understand it in school very well, and she has been conversing in the language for years with friends that also learned the language.

Scenario 1: Dick and Jane travel to a region of the country where a rather strange dialect of Language X—called Low Language X—is spoken. Even though Dick has never encountered the dialect before, he has only a little trouble understanding it. Jane, on the other hand, has considerable difficulty with it. In fact, she only understands about 10% of what is said.

Scenario 2: Next, Dick and Jane travel to a region of the country where a dialect called High Language X is spoken. Both Dick and Jane can understand everything perfectly. In this region, Jane can understand a "clear majority" of the utterances.

These two scenarios have an underlying "logical equivalence" in that, in both situations, the same language is spoken. Is it reasonable, then, to argue that Jane's hypothetical lack of knowledge, as a non-native speaker, of the "underlying abstract logical relations" of the language is irrelevant in explaining her very poor understanding of Low Language X?

Of course not. But if we map Margolis onto this analogy, the answer would be yes. And he gets to this yes through two awkward assumptions, again mapped onto the analogy: (1) Jane should be able to understand Language X (in all its forms) by virtue of her considerable everyday experience with the language:

[The inability to handle modus tollens] has always warranted more suspicion than it has received, since anyone who listens to their children will hear them quite readily make what are functional equivalents of modus tollens inferences. And not very surprisingly, since the world provides us with endless occasions to make such inferences. (If I picked my keys off the desk, they would now be in my pocket. My keys are not in my pocket. So they are probably on my desk.)

And (2) the two scenarios have the same underlying features, so the different surface features make no difference:

If the two pairs of tests each involve questions that are exact logical equivalents, as they do, then why should we find one easy and the other hard?

Reading these assumptions in reverse and in context makes more sense: There is no "logical" difference between understanding High Language X and Low Language X. Therefore, Jane's everyday experience with High Language X should transfer to Low Language X.

These assumptions obviously ignore a central fact: Since Jane does not have knowledge of the "underlying abstract logical relations" of the language, then the "logical equivalence" between the two scenarios is NOT AN EQUIVALENCE AT ALL, to her.

Or, to put this back into the original context of reasoning ability: Since people do not, in general, have knowledge of formal reasoning rules, the "logical equivalence" betweeen the Wason Selection Task and the reduced-array selection task (or the "drinking" version) is NOT AN EQUIVALENCE AT ALL, to people.

The reason why we fail the WST in large numbers but do well on the RAST is because the two tasks are not the same to us. We don't have deep understanding of the abstract logical relations that can transfer between the two tasks. At the very least, that is a possible explanation. Thus, we can't simply dismiss people's knowledge of and access to formal reasoning ability in comparing the results from the WST and the RAST.

Back to Framing

The same two assumptions I mentioned above influence a lot of thinking in elementary mathematics education: (1) children should be able to understand mathematics (in all its forms) by virtue of their considerable everyday experience with it, and (2) both "everyday" mathematics and formal mathematics have the same underlying features (logical equivalence), so the surface features make no difference. As a result of these assumptions, we feel more and more comfortable discounting the relevance of formal mathematical knowledge, access to that knowledge, and correct application of that knowledge in favor of sanctifying students' gut intuitions and misperceptions.

Keith Devlin, though he is by far much more reasonable in many of his other writings, uses the two assumptions above to make his "birds do math" argument:

When we humans try to emulate the navigational feats of lobsters or migrating birds, we have to resort to mathematics. In human terms, those creatures have built-in mathematical ability: they have brains that have evolved to carry out the trigonometrical calculations necessary to determine north from the position of the sun or to set a course based on a knowledge of where the North Pole lies. They are, in short, natural born mathematicians.

Obviously, though, we are not birds or lobsters. If it is indeed the case that our species does not have much in the way of built-in mathematical ability, this is actually a good thing. It means that we can continually improve our ideas, our understanding of and power over our world and our universe. We can erase and start over. We can imagine ourselves beyond what we are capable of today. What it also means is that we are in charge of giving ourselves this ability. It can not be drawn out of us. It doesn't happen by osmosis.

The same can be said for reasoning ability. That we are not hard-wired to reason in formal, abstract ways is a cause for celebration. It means that we can recognize when it is not appropriate and apply it even when it is counterintuitive to do so. But, again, it also means that we have to teach it to ourselves.

In Conclusion

The conclusions I have been building to in this series on the Wason Selection Task are quite obviously simply extensions of those I presented in my framing argument. I have tried to show here, however, that a subtle and powerful excuse that leads to poor teaching (or no teaching)--that students are inherently mathematical--has a sibling in cognitive science--that people are inherently rational. It is my somewhat educated opinion that removing this excuse would be an important first step in making mathematics fluency for all happen.

In his book, Margolis quotes Nisbett and Ross (1980) in reference to people's general inability to handle formal reasoning: "If we're so dumb, then how did we get to the moon?" I like Tom Hanks' (as Jim Lovell) statement as an appropriate response to this question:

From now on, we live in a world where man has walked on the moon. And it's not a miracle, we just decided to go.

Indeed, if we ever see the day when all of our students are fluent in the language of mathematics, it will not be because of any miracle curriculum or teaching method or law. We will just have decided to make it happen.

Wason Task: Part I | Part II | Part III | Part IV | Part V | Part VI

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The Wason Selection Task, Part V

Margolis's claim is that his explanation does away with the longstanding idea that people are bad at modus tollens reasoning on the WST.

I’ll do my best to show you how he gets there. I'll set off what I understand as the two parts of his argument in blockquotes (the second part of his argument shown below actually is a quote, though I've partitioned it).

The First Part of the Argument

(1) Subjects perform very poorly on the 4-card version of the Wason Selection Task, but a "clear majority" finds the correct answer to the reduced-array selection task. These results are produced by a combination of two factors: (a) subjects are misunderstanding the cards as categories in both tasks and (b) some subjects interpret the "if" in the task as a simple "if" [the most common interpretation] and others interpret it as "if and only if."

(2) The misunderstandings in the 4-card version yield mostly "incorrect,” though “logical,” answers, whereas the same misunderstandings produce "correct and logical” answers in the 2-card version.

(3) For the 4-card version, there are in fact several logical responses, given that subjects misunderstand the cards as categories. For the "if" reading, either P or not-Q would be logical. For the "if and only if" reading, there are four logical responses: P and Q, P and not-P, Q and not-Q, not-P and not-Q.

(4) The correct answer, P and not-Q, is not a logical answer, given the misreading.

With the first part of his argument, Margolis shows us that there is an explanation that both fits the results and does not require us to postulate a general inability to handle modus tollens. In fact, the correct answer in the reduced-array selection task, which a "clear majority" of subjects choose, does seem to require modus tollens reasoning.

However, the most common responses to the 4-card task (either P alone [the “if” reading”] or P and Q [the “if and only if” reading]) involve modus ponens reasoning, not modus tollens. So what remains to be shown is that people can handle modus tollens in the 4-card task. (Margolis might likely disagree that this second part is necessary.)

The Second Part of the Argument

[1] The [not-P, not-Q] response [ed.: a response involving modus tollens] is in fact another correct response to the illusory reading of the cards as categories.

[2] That [this response] is almost never seen shows the effect of the salience of [the P and Q cards] in the rule.

[3] But the wording of the [instruction] has a recency advantage over the wording in the rule. This turns out to be so strong that the predominant response is reversed by reversing the order in which the two clauses in the [instruction] are presented. . . . The exceptionally heavy [P and Q] responses are elicited by making the task read: "Circle two cards to turn over to check whether the rule has been violated." But [P and Q] as the dominant response switches to [not-P and not-Q] when the instruction is turned around to read: "Figure out which two cards could violate the rule, and circle them."

I have a few problems with Margolis's conclusions. First, even if the categories explanation were the only one possible to explain subjects' performance on the 4-card task, it is not required to explain subjects' improved performance on the RAST. Indeed, Margolis himself says that it seems as though subjects are still interpreting the cards as categories in the 2-card version. This, of course, is not proof that they are. Other explanations should be considered.

For example, it should be noted that there are a total of 15 possible responses to the 4-card task. Subjects may choose a single card (4 possibilities), a pair of cards (6 possibilities), three cards (4 possibilities), or all the cards (1 possibility). In the reduced-array selection task, there are only 3 possible responses: each individual card or both cards. If subjects were guessing perfectly randomly from all available possibilities, the correct answer in the 4-card task would theoretically show up about 6.7% of the time. In the RAST, that jumps to about 33.3% of the time—nearly a 400% increase in correct answers. Even when we discount the three- and four-card possibilities in the 4-card version (without similarly dismissing the possibility of selecting both cards in the 2-card version), the theoretical number of correct answers in the 2-card version is more than triple that in the 4-card version.

Of course this doesn't fully explain the "clear majority" of correct responses to the 2-card version, but it should cast some doubt on the idea that a category misreading explains all of the improvement.

Second, the reduced-array selection task ostensibly tests subjects using only the “difficult” cards—the one they choose incorrectly most often (Q) and the one they incorrectly almost never choose (not-Q). Yet, the former is not often chosen by itself in the 4-card version. The Q response is most often paired with the P response. Thus, it may be that there is only one “difficult” card in the RAST—the not-Q card. The fact that subjects choose this card in the RAST may be explained by using Margolis’s own words:

Other than for rhetorical purposes, we do not ask questions with obvious answers.

These objections, however, are not the most important disagreement I have with Margolis. For that, I'll require another post.

Wason Task: Part I | Part II | Part III | Part IV | Part V | Part VI

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Good Intuitions

In order to respond to Margolis's ideas regarding the Wason Selection Task, I thought it appropriate that I find the horse's mouth, as it were.

So I got the book for Father's Day. And I just want to share one little section to show you where Margolis is coming from.

Margolis leads into a discussion of the Wason Selection Task by likening it to two other famous experiments: the "Linda" problem [wiki] and the "taxi" problem.

All three experiments share the following basic character: for each, some elementary inference or intuition seems beyond the competence of a very large fraction of intelligent subjects . . . For Wason's selection problem the unavailable intuition is an elementary logical inference: "If p then q" implies "If not-q, then not-p" (modus tollens). . . .

If it were true that these experiments showed what they have so often been claimed to show, we would have a very deep puzzle. For the least controversial notion in psychology is that some version of the law of effect [wiki] is essential to an account of performance (Dennett 1983). On a plausible theory of human judgment, normal human beings should exhibit good intuitions about pervasive features of their experience.

Seriously? The law of effect?

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The Wason Selection Task, Part IV

Yes, my friends, it's sad but true. I must here begin to wrap up my Wason Selection Task (WST) series. Before I start, however, let's recap.

In the first post in the series, I introduced the formal version of the WST:

wason4

Each card has a letter on one side and a number on the other, but you can see only one of these for each card. Here is a rule: "every card that has a D on one side has a 3 on the other." Your task is to select all those cards, but only those cards, which you would have to turn over in order to discover whether or not the rule has been violated.

Research has consistently found that only around 10% of the general population finds the correct answer to this task (D and 7). And Inglis and Simpson (2004) (referenced in Part I) found that less than half of the mathematics academic staff in their study came up with the correct answer. Most people either choose the D card alone or the D and 3 cards.

In the second part of the series, I mentioned that the WST is considered to be a conditional reasoning task, and I analyzed it according to the structure of conditional reasoning arguments. In short, a basic conditional reasoning argument starts with an "if P, then Q" statement, which is followed by a statement about the truth or falsity of P or Q. The final statement is the conclusion. In the WST, the rule given is considered to be the "if P, then Q" statement, and each of the four cards is one of the four possible second statements (P is true, P is not true, Q is true, Q is not true). However, as I noted briefly in my second post, the WST is not as straightforward as a typical conditional reasoning task—a fact which, while likely worthy of analysis, is mostly irrelevant to and beyond the scope of this series.

Finally, in Part III, I briefly described some explanations that have been given for why subjects fail the WST in large numbers. These include the matching bias and the confirmation bias.

In this post, I'd like to do one thing: review Margolis's explanation for subjects' failures to find the correct answer to the WST. In the following post, I will (1) finish up Margolis and take a look at Myrtle's "birds do math" idea, and (2) connect some conclusions regarding the WST with some thoughts about math education.

Margolis's "Categories" Explanation

The cards most often chosen by subjects in the WST are the D card alone (P) or the D and 3 cards (P and Q), whereas the correct answer is to choose the D and 7 cards (P and not Q). In the language of conditional reasoning, a majority of subjects correctly identify modus ponens (D) as valid but incorrectly identify affirming the consequent (3) as valid. And while they hardly ever incorrectly identify denying the antecedent (K) as valid, they almost always never correctly identify modus tollens (7) as valid.

Margolis uses this fact to begin his explanation and to introduce what is known as the reduced-array selection task (RAST):

So there seem to be two easy cards: [D], which is rarely missed, and [K], which is rarely chosen; and two hard cards: "3" and "7", which supply nearly all the errors. Overall, about 90% of subjects in fact do make errors. So what will happen if subjects are shown only what Wason called a "reduced array". Delete the two easy cards, and have subjects judge only the two hard cards. One might suppose, since essentially all errors are caused in relation to the hard cards, that subjects will continue to do badly.

But they don't! If this test is run on a group of reasonable size (say a class), those asked to respond to the 4-card version will typically return the usual 10% correct responses. But those given the reduced array will return a clear majority of correct responses! What can possibly account for this large improvement, related to merely removing the two cards that are ordinarily judged correctly anyway?

Margolis reasons that if subjects have difficulty with affirming the consequent (3) and modus tollens (7) in the 4-card version, then they should continue to have problems in the reduced-array selection task, or RAST. But since they don't, subjects' difficulty with these forms of arguments cannot explain their improvement in the RAST. Something else must:

This odd, even bizarre, improvement can be explained if subjects are seeing the cards not as particular cards but as indicating categories of cards. If explicitly asked, subjects understand the intended meaning of the question. But their responses make logical sense only with respect to a drastic misreading of the question. The question is misread as being about which categories of cards should be examined; for example, any cards with a "D" on either side; rather than about the particular card shown with a "D" on its upside.

The category idea is difficult to grasp, so let me see if I can make it clearer, using a very strange analogy:

Imagine you are in a room with a number of boxes. You are told that in each box is a dog or some other animal, and each box has a 3 or some other number written on the back of it. Here is a rule: If it is a dog, then its box reads "3." How would you proceed to identify if the rule has been violated?

In this situation (given the more common "if, then" reading rather than the "if and only if" reading), you can simply check all the dogs' boxes (D) for 3's. There is no need to check all the boxes with numbers other than 3 written on them (7), because, having already checked the category of dogs—i.e., all the dogs—you can tell whether or not the rule has been violated. According to Margolis, in the WST, this kind of thinking leads subjects to choose the D alone, and the "if and only if" reading of the rule, along with the matching bias, leads them to choose D and 3.

However, in our scenario, if you were able to check only the numbers on the boxes, then the only correct response—even when reasoning with categories (and, again, given the more common "if, then" reading)—would be to check all the boxes with numbers other than 3 written on them (7). This, says Margolis, explains why subjects make the correct response in the RAST—given the two "hard" cards, 3 and 7:

Subjects still seem to be misinterpreting the cards as categories. But with the reduced array, the only available correct response for the "category" reading is also the correct response for the intended reading! [D] is still salient in the question, but since it is no longer available, subjects must pick the "7".


Wason Task: Part I | Part II | Part III | Part IV | Part V | Part VI

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The Wason Selection Task, Part III

I stated in Part II that evidence suggests that people do well on conditional reasoning tasks but not on the Wason Selection Task.

For example, in response to Margolis's suggestion that subjects perform poorly on the Wason Selection Task (WST) because they misinterpret the rule, Prudkov writes

The formal wording of the task is unusual but very clear; most people in an industrial society are able to solve similar problems and are familiar with the reuisite [sic] logical rules. Nevertheless, it is very difficult to find the right solution to the task (Johnson-Laird & Wason 1977). . . . Why do the subjects misinterpret the task if their knowledge and skills allow the correct interpretation to be constructed? (Margolis's version confirms this.)

And Handley (PDF) cites these sources in describing the contrast in performance between other conditional reasoning tasks and the WST:

There is an apparent anomaly in the reasoning literature: In relation to people’s performance on simple conditional inference tasks, performance on abstract versions of Wason’s selection task is particularly poor when tested against the criteria of formal logic (for a review, see Evans, Newstead, & Byrne, 1993; Manktelow, 1999).

Now, of course, what these two writers are describing is only the relative performance differences between participants in the WST and those in other conditional reasoning tasks. It is not at all evident that the general population performs perfectly with any kind of logical reasoning, much less conditional reasoning. Inglis and Simpson (PDF), for example, write

It has been found that many people respond in an apparently irrational, non-normative fashion when given straightforward logical reasoning tasks. For example, experimenters have found that people are much more likely to endorse logical arguments as valid if the conclusions are believable. Conversely, it is much harder to correctly evaluate logically valid arguments when the conclusion is unbelievable (Evans, Barston & Pollard, 1983).

And Thompson and Byrne (PDF) specifically single out simple conditional reasoning as being problematic as well:

For the modus ponens inference (MP), reasoners are given the true antecedent (TA), Sarah went to Moose Jaw, and they are asked to judge the validity of the true consequent (TC), Tom went to Medicine Hat. For the modus tollens inference (MT), reasoners are given the false consequent (FC), Tom didn’t go to Medicine Hat, and are asked to judge the validity of the false antecedent (FA), Sarah didn’t go to Moose Jaw. These two inferences are valid, regardless of whether one interprets the conditional as an implication relation (the antecedent is sufficient but not necessary for the consequent) or an equivalence biconditional relation (the antecedent is sufficient and necessary for the consequent). The findings indicate that reasoners tend to make the MP inference readily but that the MT inference is more difficult and many individuals conclude erroneously that nothing follows (see Evans et al., 1993).

Yet, even though subjects don't always perform optimally on any kind of logical reasoning task, they perform extraordinarily poorly on the WST. Why is this the case?

There are a number of explanations. One of the simplest is something known as the matching bias. The matching bias is understood as the tendency to choose items in the WST that match the names given in the rule. Thus, since the rule given in the formal task I presented in Parts I and II reads "every card that has a D on one side has a 3 on the other," participants are more likely to choose the D and 3 cards. Similarly, if I changed the rule in the "drinking" version of the task from "every person that has an alcoholic drink is of legal age (21)" to "every person that is drinking vodka is 29," participants, falling prey to the matching bias, would be more likely to choose the Vodka and 29 cards.

Another explanation is something known as the confirmation bias (wiki). In essence, this bias, as applied to the WST, would lead subjects to look for ways to confirm the rule, rather than finding ways in which the rule could be violated. Thus, in the formal task, subjects choose the D card to hypothetically confirm that a 3 is on the other side and the 3 card to see whether or not there is a D on the other side.

And Margolis, in his "misinterpretation" explanation, provides the juicy quote that will take us into the next post. I'm putting the funny stuff in red:

The cognitive illusion comes at the stage of interpreting the task, not from the inability to handle modus tollens that is the usual explanation. That claimed inability has always warranted more suspicion than it has received, since anyone who listens to their children will hear them quite readily make what are functional equivalents of modus tollens inferences. And not very surprisingly, since the world provides us with endless occasions to make such inferences. (If I picked my keys off the desk, they would now be in my pocket. My keys are not in my pocket. So they are probably on my desk.)


Wason Task: Part I | Part II | Part III | Part IV | Part V | Part VI

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The Wason Selection Task, Part II

Before we sink our teeth even deeper into the Wason Selection Task, we should start by briefly reviewing conditional reasoning arguments.

Conditional reasoning generally starts with the statement "if P is true, then Q is true" or "if P, then Q" (P &rarr Q). For example, the statement "If this tree is a spruce (P), then it has needles (Q)" is a conditional statement.

There are four conditional reasoning arguments that apply to the Wason Selection Task—two of them valid and two of them invalid. Each of these introduces a different second statement. Thus, each can be identified according to the statement introduced right after the "if P, then Q" statement.

Modus Ponens (P is true): This argument proceeds as follows: If P is true, then Q is true. P is indeed true. Therefore, Q is true. This is a valid form of reasoning. Example: If this tree is a spruce (P), then it has needles (Q). This tree is indeed a spruce (P). Therefore, this tree has needles (Q).

Denying the Antecedent (P is not true): This is a fallacy and proceeds as follows: If P is true, then Q is true. P is not true. Therefore, Q is not true. Example: If this tree is a spruce (P), then it has needles (Q). This tree is not a spruce (not P). Therefore, this tree does not have needles (not Q).

Affirming the Consequent (Q is true): This is also a fallacy and proceeds as follows: If P is true, then Q is true. Q is indeed true. Therefore, P is true. Example: If this tree is a spruce (P), then it has needles (Q). This tree indeed has needles (Q). Therefore, this tree is a spruce (Q).

Modus Tollens (Q is not true): This argument proceeds as follows: If P is true, then Q is true. Q is not true. Therefore, P is not true. This is a valid form of reasoning. Example: If this tree is a spruce (P), then it has needles (Q). This tree does not have needles (not Q). Therefore, this tree is not a spruce (not P).

It is important to note that the terms valid and invalid used to describe these arguments tell us nothing about the correctness of their conclusions. For example, each of these lines of reasoning is logically invalid . . .

Affirming the Consequent: If today is June 1, then tomorrow is June 2. Tomorrow is indeed June 2. Therefore, today is June 1.

Denying the Antecedent: If today is June 1, then tomorrow is June 2. Today is not June 1. Therefore, tomorrow is not June 2.

. . . even though they are undeniably correct. Formal logic does not concern itself with the contents of arguments, only their form.

The rule given in any Wason Selection Task is considered to be a statement of the form "if P, then Q." So, the rule "every person that has an alcoholic drink is of legal age (21)" that I included in my previous post might be recast as "if alcoholic drink (P), then legal age (Q). Similarly, in the more formal version below, the rule "every card that has a D on one side has a 3 on the other" might be recast as "if D (P), then 3 (Q)."

Accordingly, each of the four answer choices in a Wason Selection Task is seen as the second statement in a conditional reasoning argument—either a statement about P (i.e., P is true [P] or P is not true [&sim P]) or a statement about Q (i.e., Q is true [Q] or Q is not true [&sim Q]).

In the "drinking" version of the selection task, for example, statements about drink type are the Ps and statements about age are the Qs:

wason3

The Ps and Qs for the more formal task would be assigned this way:

wason4

(Notice, by the way, how brilliant I was to include the html code for the tilda, "& sim," in the art, expecting it to be translated. Not one of my prouder blogging moments.)

Though the Wason Selection Task seems to be structured as a conditional reasoning task, it is not that straightforward. Participants are not asked to determine whether or not a conclusion is derived logically from statements. If that were the case, you would be shown the cards, given the "if P, then Q" statement, and simply asked, for each card, Is there a 3 on the other side? or Is there a D on the other side?

There is some evidence to suggest that people do fairly well on conditional reasoning tasks, but very poorly on the Wason Selection Task. I'll take up some interesting arguments as to why this is in Part III.

Wason Task: Part I | Part II | Part III | Part IV | Part V | Part VI

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The Wason Selection Task, Part I

The Pope, a nun, Kermit the Frog, and Bruce Lee are all sitting at a bar. Well, actually it's just four people, represented by the cards below.

wason1

Each person has an age and a drink type, but you can see only one of these for each person. Here is a rule: "every person that has an alcoholic drink is of legal age (21)." Your task is to select all those people, but only those people, that you would have to check in order to discover whether or not the rule has been violated.

Most people have little trouble picking the correct answer above. But, "across a wide range of published literature only around 10% of the general population" finds the correct answer to the infamous Wason selection task shown below:

wason4

Each card has a letter on one side and a number on the other, but you can see only one of these for each card. Here is a rule: "every card that has a D on one side has a 3 on the other." Your task is to select all those cards, but only those cards, which you would have to turn over in order to discover whether or not the rule has been violated.

In fact, Matthew Inglis and Adrian Simpson (2004) found that mathematics undergraduates as well as mathematics academic staff, though performing significantly better than history undergraduates, performed unexpectedly poorly on the task, with only 29% of math undergrads and a shocking 43% of staff finding the correct answer.

All of this leads to some interesting ideas, which I'll get to in a future post.

Reference: Inglis, M. & Simpson, A. Mathematicians and the Selection Task. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 2004. (3) 89-96.

Wason Task: Part I | Part II | Part III | Part IV | Part V | Part VI

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Consonance and Conflict

Okay, I lied. I'll get to that framing thing soon. And, of course, I have to come back to the skills-based thing also.

But allow me in the meantime to post a nice quote from Uri Leron, which is not altogether unrelated to the framing issue. And it certainly relates to the cinemathematics present in mathematics education today:

According to the algebraic image of functions, an operation is acting on an object. The agent performing the operation takes an object and does something to it. For example, a child playing with a toy may move it, squeeze it, or color it. The object before the action is the input and the object after the action is the output. The operation is thus transforming the input into the output. The proposed origin of the algebraic image of functions is the child's experience of acting on objects in the physical world. . . . Inherent to this image is the experience that an operation changes its input—after all, that's why we engage it in the first place: you move something to change its place, squeeze it to change its shape, color it to change its look.

But this is not what happens in modern mathematics or in functional programming. In the modern formalism of functions, nothing really changes! The function is a "mapping between two fixed sets" or even, in its most extreme form, a set of ordered pairs. As is the universal trend in modern mathematics, an algebraic formalism has been adopted that completely suppresses the images of process, time, and change.


REFERENCE: Leron, Uri. Mathematical Thinking and Human Nature: Consonance and Conflict. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 2004. (3) 217-224.

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Hands-On, Brains-Off

Methodologies and philosophies that facilitate classroom management are almost always preferred over those that facilitate actual learning.

That's not to say that classroom management and learning can't coexist, nor is it to say that most practitioners don't look to improve learning in their students. However, given the choice between a programmatic methodology or philosophy and a pedagogical one, educators seem almost always to be more attracted to the former. The environments in which they work make it nearly impossible not to have such preferences.

One such programmatic methodology—although it has pedagogical potential—is the nefarious hands-on activity, which is so beloved by elementary math teachers that entire programs can be rejected for not including it as a feature. Is it because educators see hands-on activities as vital to the academic success of students? Nope:

Patricia S. Moyer-Packenham, a researcher from George Mason University, in Fairfax, Va., interviewed and observed 10 middle-grades teachers using manipulatives to teach math. In a paper published in 2001, she noted that many of the teachers saw the classroom toys as a "fun" reward for students, rather than as a way to enhance their learning.

This is a quite shocking, though not entirely unexpected, admission from even such a small sample of teachers. But it is consistent with my experience:

Most hands-on activities aren't written to help kids "move from the concrete to the abstract." They are written to keep kids firmly in the concrete, playing with fraction strips; counters; ones, tens, and hundreds blocks; etc. . . Hands-on activities are preferred because they are seen as effective classroom management tools.

In this article from Education Week, which I also linked to above, we find research that calls into serious question the current use of manipulatives in mathematics classrooms:

Researchers found that children taught to do two-digit subtraction by the traditional written method performed just as well as children who used a commercially available set of manipulatives made up of individual blocks that could be interlocked to form units of 10.

Later on, though, the children who used the toys had trouble transferring their knowledge to paper-and-pencil representations. Mr. Uttal and his colleagues also found that the hands-on lessons took three times as long as the traditional teaching methods.

Three times as long to teach kids to have a somewhat inflexible knowledge of two-digit subtraction. Certainly not a glowing endorsement of hands-on activities. Yet, the problem isn't in the manipulatives themselves (though that can't be discounted when you're talking about mathematics); it's in how they are used in instruction:

One problem is that children . . . sometimes fail to grasp the symbolic value of the objects they're using, according to a panel of experts who presented research on the topic during a national meeting of the Society for Research in Child Development held in Boston last month.

Students might correctly perform the classroom procedure, connecting 10 blocks here, for instance, or taking away blocks from another pile, without thinking about what the objects are meant to represent. Younger children, in particular, also can get lost in play with the toys or become distracted by superficial features of the toys, such as realistic details or bright colors, that have nothing to do with the academic concept being taught.

And textbook publishers, though they spend a lot of time washing their hands of what actually goes on in classrooms, also share some of the blame for this.

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Routine, Interactive, and Skills-Based

The three characteristics of natural parent involvement I derived in large part from Vygotsky's social development theory.

In essence, Vygotsky's theory is a descriptive theory of cognitive development—one which assigns a central role to the social environment and to culture:

Every function in the child's cultural development appears twice: first, on the social level, and later on the individual level; first, between people (interpsychological), and then inside the child (intrapsychological). This applies equally to voluntary attention, to logical memory, and to the formulation of concepts. All the higher functions originate as actual relations between human individuals.

Vygotsky saw cognitive development as proceeding from the social to the individual through shared activity mediated by cultural signs:

Symbolism and the conventionality of signs were perceived by Vygotsky as important characteristics of human activity that are imposed on an individual’s behavior, shaping it and reconstructing it along the lines of the sociocultural matrix. The concept of activity thus was perceived as an actualization of culture in individual behavior, embodied in the symbolic function of gesture, play, and speech systems.

A classic example of the merits of social development theory is that of the crying baby learning through the mediation of its parents' behavior the intentionality of its own behavior. Volition itself begins first as an interpsychological (or intermental) object and gradually becomes an intrapsychological (or intramental) object:

Initially, an infant's cries are not intended by the infant to be a form of communication, their existence is simply an undirected expression. When they cry, we act on their behalf, giving meaning to their communication. She or he can communicate only through their relationship with us. This is an example of intermental ability. At a certain point in the infant's development, this changes, with the infant's behaviour becoming intentional. When an infant is able to use crying instrumentally, that is as an intentional act of communication, the ability demonstrated is intramental.

One ready implication of this idea is that, assuming that natural cognitive development in children can be characterized as moving from the intermental to the intramental, we would expect parents to be naturally capable of facilitating this process. Some research has shown that not only is this the case, but that this process of development and facilitation occurs throughout a child's life.

In her book Apprenticeship in Thinking, Barbara Rogoff provides many examples of this natural ability, including this one, which I have posted before:

To maintain understanding of the message, mothers label penguins "penguins" rather than "birds" until children have established the bird prototype, at which time mothers begin remarking that "penguins are birds." They appear to protect the process of forming prototypes by not distinguishing the specific examplar birds until children have a well-established set of prototypes and have labels for atypical birds. While these maternal adjustments may be useful for children's concept acquisition, they also reflect adherence to principles of communication (e.g., Clark & Haviland, 1977): that a speaker be sensitive to the perspective and knowledge of the listener, and that conversation focus on what is deserving of comment from the joint perspective of speaker and listener.

And I can provide at least one example from my own life, which I'm sure many other parents can relate to. Before about the age of 2 or 3, children rely on their parents' interpretation of an injury event in order to learn how to respond to it themselves. When children fall and injure themselves just slightly, they often simply become quiet. Parents instinctively respond to events like these by temporarily resisting the urge to attend to a child's injury in order to actively play down the seriousness of the event. This behavior can sometimes be comical. On more than one occasion, my wife and I have both found ourselves dancing around and singing in the presence of an injured child (slightly injured, mind you), so great is the need to interpret for him or her that the injury isn't all that bad. And, indeed, research tells us that this kind of parent behavior is capable of influencing children's subjective experiences of pain.

Assuming, then, that cognitive development proceeds (in settings outside of school) from the social to the individual through activity and that parents are naturally capable of facilitating this process, how might schools structure parent involvement programs (specifically, academic parent involvement programs, which have the greatest impact on achievement) in order to take advantage of what is already going on at home? Here are three things I would suggest all parent involvement should be:

Routine

Parent involvement should be routine. And by routine, I mean two things.

First, it should be fairly consistent and regular. Once or twice a week, if not three. If we forget about school for a moment and look at how parents are naturally involved in their children's cognitive development, we see that it is an everyday process. Children learn in environments that are consistent and regular. (If those environments are consistently and regularly chaotic, children learn from that too.)

Second, parent involvement should be non-instructional. This is because cognitive development (outside of school) proceeds naturally through activity, not through school-like instruction. Thus, it is awkward for parents to "teach" a measurement concept, but not awkward for them to read and discuss a story that presents measurement concepts. It is awkward for parents to "teach" their children about integers, but not awkward for them to help their children solve a word problem about integers. It is awkward for parents to "teach" their children multiplication and division, but not awkward for them to help their child play a game involving multiplication and division. This is not to say that parents can't teach their children the way teachers do; it's to say that they don't naturally do so. School and home are different social institutions, and teachers and parents have different social roles and ways of operating.

Interactive

This one's pretty easy. Since cognitive development proceeds from the social to the individual through shared activity, it makes sense that effective parent involvement be interactive. Both child and parent should be able to be comfortably involved. This means that not only should it be relatively quick and easy for parents, but it should also focus on material that the student has already learned, so that the student can be actively and equally involved.

Skills-Based

The relationship between this characteristic and Vygotsky's theory is a little hard to explain. I'll have to come back to it in another post. In the meantime, you can chew on this reason for why parent involvement should be skills-based:

Many observers have pointed to the NCTM's 1989 Curriculum and Evaluation Standards as a document responsible for dramatically changing the way math is taught in the United States—away from traditional, algorithm-centered instruction toward more conceptual understanding and exploration of deep mathematical ideas.

Okay. Let's assume that that change occured overnight. So, every five-year-old starting Kindergarten in 1989 learned this "new math" up through the 8th grade. Let's further assume that all those children had children at the age of 25 (the average age as of 2004). Then, five years later, they became parents of Kindergartners who are now using the "new" math.

The year that those children would enter Kindergarten: 2014.

If "new" math took hold everywhere overnight in 1989 (which of course it didn't, and in some places it still hasn't), we would have to wait until 2014 to begin to see parents of school-aged children who had a strong familiarity with "conceptual understanding," "explain your answer," "draw a model," "guess and check," etc.

And so the gap between parents and schools remains.

REFERENCES: Daniels, H. An Introduction to Vygotsky. Routledge. New York. 1996; Rogoff, B. Apprenticeship in Thinking. New York: 1990; Gleason, M. Parents' Assistance of Their Children's Scientific Reasoning. Cognition and Instruction. 17(4), 1999.

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The Point

The responses I received to my vocabulary comment were not contradicted by the research I presented . In fact, they could be supported by it.

And there are other reasons to call out vocabulary in a text. Here's Robert with one of them:

If my calculus students encountered a problem involving relative maximum values of a function, the first thing they need is to be fluent with the meaning of the term "relative maximum." If they're not, then they need to go look it up--and the cues show them where to look (by flagging the point at which the term is first introduced).

I agree that highlighting important vocabulary words helps signal their importance and can help with navigating a search for their meaning. But, to me, the idea should be to also help students remember the meaning of the words. In truth, publishers don't highlight vocabulary words because doing so facilitates recall of those words or because it signals their importance or because it helps with textbook navigation. They highlight vocabulary words because it signals to customers that the publisher cares about vocabulary. The point is that if we truly cared about learning, we could do more with text to facilitate learning.

Consider the following series of experiments conducted by Bruce Britton (whose work we have looked at before) and Robert Sorrells out of the University of Georgia. They began with questions that are rarely, if ever, asked in educational publishing:

Our development of the method began when we encountered Colomb and Williams' (1987) claim that all expository texts have a point, which is a central tenet around which the text is organized. From this idea, our three hypotheses emerged. First, do expert readers generally agree about the point of a text, whereas less skilled readers differ in their judgments of the point? Second, do readers who have the point of a text signaled to them recall more of that text than readers who do not have the point signaled? Finally, do readers who have the point signaled to them construct a different structure of the text, incorporating more of the important information than those who do not? If these hypotheses are supported, they suggest that identifying the point of a text and informing readers that it is the point would improve the learnability of that text.


Experiment 1

To answer the first question, the researchers assembled five faculty with Ph.D.s in English as the "expert readers" and 12 introductory psychology students as the "less skilled readers." Each individual was given three brief expository texts, without titles, and asked to simply underline the point of each after a careful read. The texts were provided by Colomb, who identified the point of each to the researchers (point units) along with the "themes related to the point." The "themes" (theme units) were ideas that were not as informative as the point, but were more important than other ideas (other idea units) in the texts. Each text was a different length, and "the point" in each text occurred in a different location.

The researchers found that the "expert readers" identified 86.4% of the points, whereas the "less skilled readers" succeeded only in identifying 49.5% of the points, a significant difference (p < 0.05). Less skilled readers also showed greater variance in their individual results.

The importance of this result is that it points to the possibility of experts signaling the point for less skilled readers. The next two experiments focus on whether or not such signaling can improve recall and/or comprehension. In these experiments, Britton and Sorrells set out to test this general hypothesis:

If the point of a text telegraphs the information in that text, as Colomb and Williams suggested, then cuing the point should affect the recall of other information.


Experiment 2

This experiment was conducted to answer the second and third of the researchers' specific questions:

Do readers who have the point of a text signaled to them recall more of that text than readers who do not have the point signaled? [And], do readers who have the point signaled to them construct a different structure of the text, incorporating more of the important information than those who do not?

Each of twenty-nine introductory psychology students was given the three texts used in Experiment 1. (Recall that each text was broken down by the researchers into "point units," "theme units," and "other idea units.") Each participant was asked to read the text and write down what he or she remembered (free recall). For participants in the experimental group, the point of each text was underlined, and written instructions identified the underlined sentence as the point of the text. Participants in the control group received the texts without modification of any kind.