Text Savvy

Are Teaching and Learning Coevolved?

2016-08-20T21:10:00.000-04:00

Just a few pages in to David Didau and Nick Rose's new book What Every Teacher Needs to Know About Psychology, and I've already come across what is, for me, a new thought:

Strauss, Ziv, and Stein (2002) . . . point to the fact that the ability to teach arises spontaneously at an early age without any apparent instruction and that it is common to all human cultures as evidence that it is an innate ability. Essentially, they suggest that despite its complexity, teaching is a natural cognition that evolved alongside our ability to learn.

Or perhaps this is, even for me, an old thought, but just unpopular enough—and for long enough—to seem like a brand new thought. Perhaps after years of exposure to the characterization of teaching as an anti-natural object—a smoky, rusty gearbox of torture techniques designed to break students' wills and control their behavior—I have simply come to accept that it is true, and have forgotten that I had done so.

Strauss, et. al, however, provide some evidence in their research that it is not true. Very young children engage in teaching behavior before formal schooling by relying on a naturally developing ability to understand the minds of others, known as theory of mind (ToM).

Kruger and Tomasello (1996) postulated that defining teaching in terms of its intention—to cause learning, suggests that teaching is linked to theory of mind, i.e., that teaching relies on the human ability to understand the other's mind. Olson and Bruner (1996) also identified theoretical links between theory of mind and teaching. They suggested that teaching is possible only when a lack of knowledge can be recognized and that the goal of teaching then is to enhance the learner's knowledge. Thus, a theory of mind definition of teaching should refer to both the intentionality involved in teaching and the knowledge component, as follows: teaching is an intentional activity that is pursued in order to increase the knowledge (or understanding) of another who lacks knowledge, has partial knowledge or possesses a false belief.

The Experiment

One hundred children were separated into 50 pairs—25 pairs with a mean age of 3.5 and 25 with a mean age of 5.5. Twenty-five of the 50 children in each age group served as test subjects (teachers); the other 25 were learners. The teachers completed three groups of tasks before teaching, the first of which (1) involved two classic false-belief tasks. If you are not familiar with these kinds of tasks, the video at right should serve as a delightfully creepy precis—from what appears to be the late 70s, when every single instructional video on Earth was made. The second and third groups of tasks probed participants' understanding that (2) a knowledge gap between teacher and learner must exist for "teaching" to occur and (3) a false belief about this knowledge gap is possible.

Finally, children participated in the teaching task by teaching the learners how to play a board game. The teacher-children were, naturally, taught how to play the game prior to their own teaching, and they were allowed to play the game with the experimenter until they demonstrated some proficiency. The teacher-learner pair was then left alone, "with no further encouragement or instructions."

The Results

Consistent with the results from prior false-belief studies, there were significant differences between the 3- and 5-year-olds in Tasks (1) and (3) above, both of which relied on false-belief mechanisms. In Task (3), when participants were told, for example, that a teacher thought a child knew how to read when in fact he didn't, 3-year-olds were much more likely to say that the teacher would still teach the child. Five-year-olds, on the other hand, were more likely to recognize the teacher's false belief and say that he or she would not teach the child.

Intriguingly, however, the development of a theory of mind does not seem necessary to either recognizing the need for a special type of discourse called "teaching" or to teaching ability itself—only to a refinement of teaching strategies. Task (2), in which participants were asked, for instance, whether a teacher would teach someone who knew something or someone who didn't, showed no significant differences between 3- and 5-year-olds in the study. But the groups were significantly different in the strategies they employed during teaching.

Three-year-olds have some understanding of teaching. They understand that in order to determine the need for teaching as well as the target learner, there is a need to recognize a difference in knowledge between (at least) two people . . . Recognition of the learner's lack of knowledge seems to be a necessary prerequisite for any attempt to teach. Thus, 3-year-olds who identify a peer who doesn't know [how] to play a game will attempt to teach the peer. However, they will differ from 5-year-olds in their teaching strategies, reflecting the further change in ToM and understanding of teaching that occurs between the ages of 3 and 5 years.

Coevolution of Teaching and Learning

The study here dealt with the innateness of teaching ability and sensibilities but not with the coevolution of teaching and learning, which it mentions at the beginning and then leaves behind.

It is an interesting question, however. Discussions in education are increasingly focused on "how students learn," and it seems to be widely accepted that teaching should adjust itself to what we discover about this. But if teaching is as natural a human faculty as learning, then this may be only half the story. How students (naturally) learn might be caused, in part, by how teachers (naturally) teach, and vice versa. And learners perhaps should be asked to adjust to what we learn about how we teach as much as the other way around.

Those seem like new thoughts to me. But they're probably not.

Strauss, S., Ziv, M., & Stein, A. (2002). Teaching as a natural cognition and its relations to preschoolers’ developing theory of mind Cognitive Development, 17 (3-4), 1473-1487 DOI: 10.1016/S0885-2014(02)00128-4

Problem Solving, Instruction: Chicken, Egg

2016-07-30T19:35:00.000-04:00

We've looked before at research which evaluated the merits of different instructional sequences.

In this post, for example, I summarized a research review by Rittle-Johnson that revealed no support for the widespread belief that conceptual instruction must precede procedural instruction in mathematics. The authors of that review went so far as to call the belief (one held and endorsed by the National Council of Teachers of Mathematics) a myth. And another study, summarized in this post, finds little evidence for another very popular notion about instruction—that cognitive conflict of some kind is a necessary prerequisite to learning.

The review we will discuss in this post looks at studies which compared two types of teaching sequences: problem solving followed by instruction (PS-I) and instruction followed by problem solving (I-PS). As far as horserace comparisons, the main takeaway is shown in the table below. Each positive (+) is a study result which showed that PS-I outperformed an I-PS control, each equals sign (=) a result where the two conditions performed the same, and each negative (–) a result where I-PS outperformed PS-I.

Summary of learning outcomes for PS-I vs. I-PS.
Procedural	Conceptual	Transfer
+ = = = = = = = = = – –	+ + + + = = = – –	+ + + + = = = = –

Importantly, 8 of the results reviewed are not represented in the table above. In these results, the review authors suggest, participants in the PS-I conditions were given better learning resources than those in the I-PS conditions. This difference confounded those outcomes (see Greg's post on this) and, unsurprisingly, added 15 plusses, 7 equals, and just 1 negative to the overall picture of PS-I.

Needless to say, when research has more fairly compared PS-I with I-PS, it has concluded that, in general, the sequence doesn't matter all that much, though there are some positive trends on conceptual and transfer assessments for PS-I. Even if we ignore the Procedural column, roughly 55% of the results are equal or negative for PS-I.

Contrasting Cases and Building on Student Solutions

Horserace aside (sort of), an intriguing discussion in this review centers around two of the confounds identified above—those extra benefits provided in some studies to learners in the PS-I conditions. They were (1) using contrasting cases during problem solving and (2) building on student solutions during instruction. Here the authors describe contrasting cases (I've included their example from the paper):

Contrasting cases consist of small sets of data, examples, or strategies presented side-by-side (e.g., Schwartz and Martin 2004; Schwartz and Bransford 1998). These minimal pairs differ in one deep feature at a time ceteris paribus [other things being equal], thereby highlighting the target features. In the example provided in the right column of Table 2, the datasets of player A and player B differ with regard to the range, while other features (e.g., mean, number of data points) are held constant. The next pair of datasets addresses another feature: player B and C have the same mean and range but different distribution of the data points.

There's something funny about this, I have to admit, given the soaring rhetoric one encounters in education about the benefits of "rich" problems and the awkwardness of textbook problems. Although they are confounds in these studies, contrasting cases manage to be helpful to learning in PS-I as sets of (a) small, (b) artificial problems which (c) vary one idea at a time. "Rich" problems, in contrast, do not show the same positive effects.

And here, some more detail about building on student solutions in instruction. The only note I have here is that it seems worthwhile to point out the obvious: that this confound which also improves learning in PS-I has almost everything to do with the I, and very little to do with the PS:

Another way of highlighting deep features in PS-I is to compare non-canonical student solutions to each other and to the canonical solution during subsequent instruction (e.g., Kapur 2012; Loibl and Rummel 2014a). Explaining why erroneous solutions are incorrect has been found to be beneficial for learning, in some cases even more than explaining correct solutions (Booth et al. 2013). Furthermore, the comparison supports students in detecting differences between their own prior ideas and the canonical solution (Loibl and Rummel 2014a). More precisely, through comparing them to other students’ solution and to the canonical solution, students experience how their solution approaches fail to address one or more important aspects of the problem (e.g., diSessa et al. 1991). This process guides students’ attention to the deep features addressed by the canonical solution (cf. Durkin and Rittle-Johnson 2012).

Loibl, K., Roll, I., & Rummel, N. (2016). Towards a Theory of When and How Problem Solving Followed by Instruction Supports Learning Educational Psychology Review DOI: 10.1007/s10648-016-9379-x

The Appeal to Common Practice

2016-07-28T22:04:00.002-04:00

At any point in a child's life or schooling, he or she presents with a number of things he or she can do and a number—which could be 0—of things he or she knows. We can refer to these collectively as the "knowns." And, of course, the "unknowns" are all those things a child does not know or cannot do at any of the same points. The problem of teaching from the known to the unknown involves making some kind of connection from a student's knowns to a very restricted set of unknowns, which, taken together at any point, form a kind of immediate curriculum.

Now, of course, it is impossible to teach without going from the known to the unknown in some way. On the one hand, a student can't learn anything if s/he has absolutely no knowledge or skills (because then s/he wouldn't exist), and on the other hand, nothing can be described purely in terms of itself. The inevitable connection from known to unknown itself is not at issue. What is at issue is the way this connection is made. What knowns are connected to what unknowns?

The Best "Known"

Over a wide variety of topics, educators will often argue about the quality of the knowns to be connected to specific unknowns. The ongoing debate about whether to teach fractions first or decimals first is an area where this argument pops up, with some making the case that place value is the better "known" to be connected to the unknown of rational numbers (decimals first) while others argue that equal shares is the better known (fractions first). Similarly, one can argue that, for the unknown of improper fractions, proper fractions serve as the best "known," whereas another can argue that, because improper and proper fractions are used in such diverse situations (e.g., "no one says that they have eight fifths dollars"), we must scrap the use of proper fractions as the "known" in introducing improper fractions and come back to the connection later.

While there are certainly substantive reasons that serve as foundations for these arguments, there are also problems that seem almost impossible to duck. One of those is called the appeal to common practice.

Appeal to Common Practice

This is a fallacy. And it works like this: Such and such an action is justified because it is what everyone else is doing or what we've always done. Now, it is pretty rare to see an adult actually commit this fallacy so nakedly. But it does creep up somewhat, um, "un-nakedly." Here's Mark falling into the fallacy with repeated multiplication:

Try to give me a simple definition of exponentiation, which is understandable by a fifth or sixth grader, which doesn't at least start by talking about repeated multiplication. Find me a beginners textbook or teachers class plans that explains exponentiation to kids without at least starting with something like "\(\mathtt{5^2 = 5 \times 5}\), \(\mathtt{5^3 = 5 \times 5 \times 5}\)."

The second of those sentences is pretty clearly the fallacy of appealing to common practice, to the extent that it is used in any way to justify or excuse the teaching of exponentiation as repeated multiplication. But notice what is said in the first sentence: "Try to give me a simple definition of exponentiation, which is understandable by a fifth or sixth grader, which doesn't at least start by talking about repeated multiplication." This, too, is an appeal to common practice, but the practice in this case is not necessarily the teaching of exponentiation as repeated multiplication to fifth or sixth graders but, rather, the teaching of everything before that. The argument is that repeated multiplication is the best "known" because currently the 8 to 10 years of schooling prior to teaching the "unknown" of exponentiation don't prepare students for learning exponentiation any other way (or any better way).

But these circumstances do not make repeated multiplication the best "known," just the most expedient "known." The same goes for repeated addition as a "known" connected to the unknown of multiplication.

All that aside, though, the more general argument is more important: Expedience is not a proper basis for determining quality teaching. Yet, it happens all the time without our noticing it—the appeal to common practice makes it devilishly difficult to discern between expedience and quality.

Interleaving Study Is Not Interleaving Learning

2016-07-18T20:29:00.000-04:00

The study I briefly discuss here is closely related to one I wrote up on interleaving just a few months ago (no surprise, since the two studies share an author). You can find a free link to the article referenced in that post here.

In the latest research, the authors found that a blocked schedule (presenting examples from one category at a time) outperformed an interleaved schedule (interspersing examples from all the categories) for category learning when the examples to be classified were more highly discriminable. This result was consistent across the two experiments in the study (p = 0.055 and p = 0.04). Importantly, however, although interleaving was a better strategy for learning categories of lower discriminability, the effects across the experiments were much weaker (p = 0.2 and p = 0.08). Blocking had either a significant or close to significant effect, whereas interleaving didn't get nearly as close (if you like p-values, anyway).

The Study

The participants in the first experiment of the study (we'll only focus on that one here) were quite a bit older than I'm used to reading about in education studies: between 19 and 57 years old, with a mean age of 30. (This is similar to the previous study.) Subjects were divided into two groups, one of which was presented with images like these:

These images represent the four presented categories: long, steep; long, flat; short, steep; and short, flat. One subset of participants in this group was exposed to 64 of these images in a blocked schedule (16 from each category at a time) while the other was presented with the images in an interleaved schedule. Each example was appropriately labeled with a category letter (A, B, C, or D). After this initial exposure, subjects were then given a test on the same set of 64 images, randomly ordered, requiring them to assign the image to the appropriate category.

The other group of participants was presented with similar images (line segments) and in blocked or interleaved subsets. However, for this group the images were rotated 45 degrees. According to the researchers, this created a distinction between the groups in which the first was learning verbalizable, highly discriminable categories ("long and flat," etc.) whereas the second was learning categories difficult to express in words—categories of low discriminability.

Discussion, Questions, Connections

As mentioned above, the blocked arrangement of the examples produced a learning benefit for categories of higher discriminability when compared with interleaving. The same cannot be said for interleaving examples in the low-discriminability sequences, although the benefits for interleaving in these sequences were headed in the positive direction. So we are left to wonder about the positive effects of blocking here.

The authors suggest an answer by mentioning some data they are collecting in a separate pilot study: blocking makes it easier for learners to disregard irrelevant information.

We compared learning under a third study schedule (n = 26) in which the relevant dimensions were interleaved, but the irrelevant ones were blocked (i.e., this schedule was blocked-by-irrelevant-dimensions, as opposed to blocked-by-category), which was designed to draw learners’ attention to noticing what dimensions were relevant or irrelevant. On both the classification test and a test in which participants had to identify the relevant and irrelevant dimensions, this new blocked-by-irrelevant-dimensions condition yielded performance at a level comparable to the blocked condition and marginally better compared to the interleaved condition.

Therefore, although we initially hypothesized that participants, when studying one category at a time, are better able to compare exemplars from the same category and to generate and test their hypotheses as to the dimensions [that] define category membership (and this may still be true, particularly for Experiment 1), these pilot data suggest that with the addition of irrelevant dimensions . . . the blocking benefit is perhaps more likely driven by the fact that it allows participants to more easily identify and disregard the irrelevant dimensions.

This strikes me as making a good deal of sense. And it points to something that I may have previously confused: interleaving study examples is different from interleaving initial learning examples. When students are first learning something, blocking may be better; after acquisition, interleaving may benefit learners more.

We have a tendency, in my opinion, to ignore acquisition in learning. I'm not sure where this comes from. Perhaps it is believed that if we are justified in rejecting tabula rasa, we are safe to assume there are absolutely no rasas on any kid's tabula. At any rate, it's worth being clear about where in the learning process interleaving is beneficial—and where it may not be.

Postscript: It’s not unusual to believe, about a child’s cognitive subjectivity, that it is like a large glop of amorphous dough and that instruction or experience acts like a cookie cutter, shaping the child’s mind according to pre-made patterns and discarding the bits that don't fit.

But these results could suggest something different—that prior to learning, the world is a hundred trillion things that must be connected together, not a stew of sensation that must be partitioned into socially valuable units.

This may be why blocking could work well for newly learned categories and for so-called highly discriminable categories: because what is new to us is highly discriminable—separate, without precedent, meaningless.

Image credit: Danny Nicholson

Noh, S., Yan, V., Bjork, R., & Maddox, W. (2016). Optimal sequencing during category learning: Testing a dual-learning systems perspective Cognition, 155, 23-29 DOI: 10.1016/j.cognition.2016.06.007

The Problem of Stipulation

2016-06-30T22:09:00.000-04:00

I think it would surprise people to read Engelmann and Carnine's Theory of Instruction. (The previous link is to the 2016 edition of the book on Amazon, but you can find the same book for free here.)

Tangled noticeably within its characteristic atomism and obsession with the naïve learner are a number of ideas that seem downright progressive—a label never ever attached to Engelmann or his work. One of these ideas in particular is worth mentioning—what the authors call the problem of stipulation.

The diagram at left shows a teaching sequence described in the book, in all its banal, robotic glory.

To be fair, it's much harder to roll your eyes at it (or at least it should be) when you consider the audience for whom it is intended—usually special education students.

Anyway, the sequence features a table and a chalkboard eraser in various positions relative to the table. And the intention is to teach the concept of suspended, allowing learners to infer the meaning of the concept while simultaneously preventing them from learning misrules.

Stipulation occurs when the learner is repeatedly shown a limited range of positive variation. If the presentation shows suspended only with respect to an eraser and table, the learner may conclude that the concept suspended is limited to the eraser and the table.

You may know about the problem of stipulation in mathematics education as the (very real) problem of math zombies (or maybe Einstellung)—which is, to my mind anyway, the sine qua non of anti-explanation pedagogical reform efforts.

But of course prescribing doses of teacher silence is only one way to deal with the symptoms of stipulation. Engelmann and Carnine have another.

To counteract stipulation, additional examples must follow the initial-teaching sequence. Following the learner's successful performance with the sequence that teaches slanted, for instance, the learner would be shown that slanted applies to hills, streets, floors, walls, and other objects. Following the presentation of suspended, the learner would be systematically exposed to a variety of suspended objects.

Stipulation of the type that occurs in initial-teaching sequences is not serious if additional examples are presented immediately after the initial teaching sequence has been presented. The longer the learner deals only with the examples shown in the original setup, the greater the probability that the learner will learn the stipulation.

It's a shame, I think, that more educators are not exposed to this work in college. It's a shame, too, that Engelmann's own commercial work has come to represent the theory in full flower—unjustly, I believe. Theory of Instruction could be much more than what it is sold to be, to antagonists and protagonists alike.

Update (07.13): It's worth mentioning that, insofar as the problem of stipulation can be characterized as a student's dependence on teacher input for thinking, complexity and confusion can be even more successful at creating cognitive dependence than monotony and hyper-simplicity. When students—or adults—are in a constant state of confusion, they may learn, incidentally, that the world is unpredictable and incommensurate with their own attempts at understanding. In such cases, even the unfounded opinions of authority figures will provide relief.

Audio Postscript

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Growing Up Too Fast or Too Slowly?

2016-06-19T01:02:00.001-04:00

If someone were to ask me where this post came from, I would be inclined to answer "everywhere." But I can offer two or three of the connections explicitly here. My thinking of late was simple enough: could variable levels of patience with children's developmental timelines influence one's traditionalist or progressivist orientation to education?

First, I was remembering an interaction I had with a former colleague last year¹. We were talking about some math activities we were designing for third graders. What I recall from the exchange is that I was interested in being a little more helpful with the information we provided, suggesting that we had time to fade away our help (the program we were working on was serving children in Grades 3–8). She, on the other hand, argued that "students can't wait forever" to be expected to reason about mathematics. (I've been on her side of the argument many many times too, though never with a time-is-running-out rationale.)

Second, I came across, quite unexpectedly, some related, illuminating thoughts in a book called Algorithms to Live By: The Computer Science of Human Decisions. The authors attempt to provide some real-world relevance to the tradeoff between exploration and exploitation—a longstanding tension in computer science between "gathering information and . . . using the information you have to get a known good result."

One of the curious things about human beings, which any developmental psychologist aspires to understand and explain, is that we take years to become competent and autonomous. Caribou and gazelles must be prepared to run from predators the day they're born, but humans take more than a year to make their first steps.

Alison Gopnik, professor of develpmental psychology at UC Berkeley . . . has an explanation for why human beings have such an extended period of dependence: "it gives you a developmental way of solving the exploration/exploitation tradeoff . . . Childhood gives you a period in which you can just explore possibilities, and you don't have to worry about payoffs because payoffs are being taken care of by the mamas and the papas and the grandmas and the babysitters . . . .

If you look at the history of the way that people have thought about children, they have typically argued that children are cognitively deficient in various ways—because if you look at their exploit capacities, they look terrible. They can't tie their shoes, they're not good at long-term planning, they're not good at focused attention."

[But] our intuitions about rationality are too often informed by exploitation rather than exploration.

Third and finally—and again, entirely and bizarrely unexpectedly—was this, which rounds out our trip from left of center to neutral to right of center (admittedly leaving the left of center unfairly underrepresented):

Treat children like children, treat grown-ups like grown-ups. An 11-year old doesn't need to teach himself, and shouldn't. A 22-year old does need to teach himself and must. And the best way to become a self-teaching 22-year old is to have teachers and parents who directly teach you when you're 11. People have known this for hundreds of years--thousands of years--and yet our public schools have somehow forgotten.

Although I'm more inclined to favor early and protracted exploration (input, learning) followed by later exploitation (performance, output) at virtually every scale in education—something that I have undoubtedly been unable to conceal even in this post—I don't intend in this writing to pass judgment one way or another²; only to suggest that where one finds oneself on the exploration-exploitation continuum is likely predictive of where one finds oneself on the traditionalist-progressivist continuum. Respectively.

Image credit: Ken Munson Photography

As far as education goes, I would consider myself right of center, and I would say my colleague was left of left of center. But that might be just what I would say. To her, she may have been either neutral or left of center, and I was right of right of center. That's how this game often works when you're not thinking hard about it: you're always neutral, and the other guy is the extremist.

I won't even mention that, given certain assumptions, the optimum balance (according to something called the Gittins Index) between exploration and exploitation is not 50-50; more like 70-30. : )

Gradient Descent

2016-05-31T19:55:00.000-04:00

In this post, we looked at the perceptron, a machine-learning algorithm that is able to "learn" the distinction between two linearly separable classes (categories of objects whose data points can be separated by a line—or, with multi-dimensional categories, by a hyperplane).

The data shown below resemble the apples and oranges data we used last time. There are two classes, or categories—in this case, the two categories are the setosa and versicolor species of the Iris flower. And each species has two dimensions (or features): sepal length and petal length. In our hypothetical apple-orange data, the two dimensions were weight and number of seeds.

Using just two dimensions allows us to plot each instance in the training data on a coordinate plane as above and draw some of the prediction lines as the program cycles through the data. For the above data, a solution is found after about 5–6 "epochs" (cycles of \(\mathtt{n}\) runs through the data). This solution is represented by the blue dashed line.

This process is a bit clunky. The coefficients, or weights \(\mathtt{w}\) are updated using the learning rate \(\mathtt{\eta}\) by \(\mathtt{\Delta w_{1,2} = \pm 2\eta w_k}\) and \(\mathtt{\Delta w_0 = \pm 2\eta}\). This process, though it always converges to a solution so long as there is one, tends to jolt the prediction line back and forth a bit abruptly.

Gradually Reducing Error

With gradient descent, we can gradually reduce the error in the prediction. Sometimes. We do this by making use of the sum of squared errors function—a quadratic function (parabola) that has a minimum: \[\mathtt{\frac{1}{2}\sum(y - wx)^2}\]

This formula shows the target vector \(\mathtt{y}\) (the collection of target values of 1 and -1) minus the input vector—the linear combinations of weights and dimensions for each object in the data, \(\mathtt{w^ix^i}\). The components of the difference vector are squared and summed, and the result is divided by 2, which gives us a scalar value that places us somewhere on the parabola. We don't use this "cost" value except to keep track of the cost to see if it reduces over cycles.

Okay, so we don't know what side of the parabola we're on. In that case, we look at the opposite of the gradient of the curve with respect to the weights, or the opposite of the partial derivative of the curve with respect to the weights: \[\mathtt{-\frac{\partial}{\partial w}(y - wx)^2 = -\frac{\partial}{\partial u}(u^2)\frac{\partial}{\partial w}(y - wx) = -2u \cdot -x = -2(y - wx)(-x)}\]

Multiply this result by \(\mathtt{\frac{1}{2}}\), plug that summation back in, and we get a gradient of \(\mathtt{\sum (y - wx)(x)}\). Finally, multiply by the learning rate \(\mathtt{\eta}\) to get the change to each weight: \[\mathtt{\eta\sum (y - wx)(x)}\]

An Example

Let's take a look at a small example. I'll use data for just 10 flowers in the Iris data set. All of these belong to the setosa species of the Iris flower. I'll use a learning rate of \(\mathtt{\eta = 0.0001}\).

Sepal Length (cm)	Petal Length (cm)
5.1	1.4
4.9	1.4
4.7	1.3
4.6	1.5
5	1.4
5.4	1.7
4.6	1.4
5	1.5
4.4	1.4
4.9	1.5

In that case, all of these instances have target values of \(\mathtt{-1}\), which don't change as the data cycle through. Our \(\mathtt{y}\) vector, then, is [\(\mathtt{-1, -1, -1, -1, -1, -1, -1, -1, -1, -1}\)], and our starting weights are [0, 0, 0]. The first weight here is the "intercept" weight, or bias weight, which gets updated differently from the others.

Our input vector, \(\mathtt{w^{T}x}\), is the combination sepal length × weight 1 + petal length × weight 2 for each object in the data. At the start, then, our input vector is a zero vector: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. The difference vector, \(\mathtt{y - w^{T}x}\), is, in this case, equal to the y vector: just a collection of ten negative 1s.

The bias weight, \(\mathtt{w_0}\), is updated by learning rate × the sum of the components of the difference vector, or \(\mathtt{\eta\sum(y - w^{T}x)}\). This gives us \(\mathtt{w_0 = w_0 + 0.0001 \times -10 = -0.001}\).

The \(\mathtt{w_1}\) weight is updated as 0.0001(5.1 × -1 + 4.9 × -1 + 4.7 × -1 + 4.6 × -1 + 5 × -1 + 5.4 × -1 + 4.6 × -1 + 5 × -1 + 4.4 × -1 + 4.9 × -1) and the \(\mathtt{w_2}\) weight is updated as 0.0001(1.4 × -1 + 1.4 × -1 + 1.3 × -1 + 1.5 × -1 + 1.4 × -1 + 1.7 × -1 + 1.4 × -1 + 1.5 × -1 + 1.4 × -1 + 1.5 × -1).

Through all that gobbledygook, our new weights are \(\mathtt{-0.00486}\) and \(\mathtt{-0.00145}\) with a bias weight of \(\mathtt{-0.001}\). You can see below how different the gradient descent process is from the perceptron model. This model in particular doesn't have to stop when it finds a line that separates the categories, since the minimum may not be reached even when the program can classify the categories precisely.

Enhance the Salience of Relevant Variables

2016-05-27T12:10:00.000-04:00

This study has some connections to things I've written up here—in particular, Hallowell, et. al on how the salience of characteristics of solids affects the responses of young children to certain visuo-spatial tasks. When these salient characteristics are irrelevant to the task, inhibitory processes can be measured, mentioned in studies here and again here.

The study we will look at in this post explored whether increasing the salience of the relevant variable in tasks with both relevant and irrelevant variables of competing significance would improve students' performance. In particular, researchers looked at whether increasing the salience of perimeter would help students with responses to tasks where area was irrelevant but salient.

Three groups of shape-pair categories were used, along with two levels of complexity:

In the Congruent condition, the shape with the greater area also has the greater perimeter. In the Incongruent Inverse condition, the shape with the smaller area has the greater perimeter. And in the Incongruent Equal condition, the areas differ, but the perimeters are equal.

The Experiment and Results

Participants (58 fifth- and sixth-graders) were divided into two groups. One group of students was tested first on the shapes as you see them above. The second group, however, was tested first on what researchers called the discrete mode of presentation. In this mode, the perimeters of the shapes are made more salient by drawing them using equal-sized matchsticks instead of as continuous lines:

Each group was tested on the other mode of presentation 10 days later. So, students who began with the discrete mode were tested in the continuous mode later, and students tested first in the continuous mode were tested 10 days later in the discrete mode.

The results are pretty staggering. And keep in mind that students in both groups took both tests. The order of the tests is what is primarily responsible for the dramatically different results.

Previous research in mathematics and science education has shown that specific modes of presentation may improve students’ performance (i.e., Clement 1993; Martin and Schwartz 2005; Stavy and Berkovitz 1980; Tirosh and Tsamir 1996). Our results corroborate these findings and indeed show that success rate in the discrete mode of presentation test is significantly higher than in the continuous mode of presentation test. This significant difference is evident in all conditions (congruent, incongruent inverse, and incongruent equal) and in all levels of complexity (simple and complex). The visual information depicted in the discrete mode of presentation strongly enhances the salience of the relevant variable, perimeter, and somewhat decreases the salience of the irrelevant variable, area. This visual information also emphasizes the perimeter’s segments that should be mentally manipulated when solving the task. The discrete mode of presentation, therefore, enhances the use of strategies that are regularly used when solving this task. In the continuous mode of presentation, however, no hint of such possibility of mentally breaking the solid line into relevant segments is given.

Be More Helpful

The researchers note that a similar study using discrete segments for perimeter found no effect for the order of presentation. The authors surmise that this was due to the fact that, in that study, researchers did not use equal-sized discrete units, so students could not manipulate these units to determine perimeter. That is, discreteness is not what mattered, but that the discrete units could be manipulated successfully to produce correct responses to perimeter questions. The key, still, is that insights gained from working first with these discrete, equal-sized units transferred to the continuous mode of presentation.

The positive effect of a previous presentation observed in the current study can be seen as “teaching by analogy.” In teaching by analogy students are first presented with an “anchoring task” that elicits a correct response due to the way it is presented and hence supports appropriate solution strategies. Later on, students are presented with a similar “target task” known to elicit incorrect responses. The anchoring task probably encourages appropriate solution strategies, and such a sequence of instruction was effective in helping students overcome difficulties (e.g., Clement 1993; Stavy 1991; Tsamir 2003). . . .

Performing the discrete mode of presentation test strongly enhances the salience of the relevant variable, perimeter, and somewhat decreases that of area. This enhancement supports appropriate solution strategies that lead to improved performance. This effect is robust and transfers to continuous mode of presentation for at least 10 days. In line with this conclusion, a student who performed the continuous test after the discrete one commented that, “It [continuous] was harder this time but I used the previous shapes, because I could do tricks with the matchsticks.”

Babai, R., Nattiv, L., & Stavy, R. (2016). Comparison of perimeters: improving students’ performance by increasing the salience of the relevant variable ZDM, 48 (3), 367-378 DOI: 10.1007/s11858-016-0766-z

The Perceptron

2016-05-26T20:20:00.000-04:00

I'd like to write about something called the perceptron in this post—partly because I'm just learning about it, and writing helps me wrap my head around new-to-me things, and partly because, well, it's interesting.

It works like this. Take two categories which are, in some quantifiable way, completely different from each other. In this example, let's talk about apples and oranges as the categories. And we'll give each of these categories just two dimensions—weight in grams and number of seeds. (I'll fudge these numbers a bit to help with the example.)

Studying this information alone, you're already prepared to know with 100% confidence whether an object is an apple or an orange given its weight and number of seeds. In fact, the number of seeds is unnecessary. If you have just the two objects to choose from, and you are given only an object's weight, you have all the information you need to assign it to the apple category or the orange category.

But, you have to play along here. We want to train the computer to make the inference we just made above, given only a set of data with the weights and number of seeds of apples and oranges. Crucially, what the perceptron does is find a line between the two categories. You can easily see how to draw a line between the categories at the left (where I have plotted 100 random apples and oranges in their given ranges of weights and seeds), but the perceptron program allows a computer to learn where to draw a line between the categories, given only a set of data about the categories.

Training the Perceptron

The way we teach the computer how to draw this line is that we ask it to essentially draw a prediction line. Then we make it look at each instance (apple or orange, one at a time) and see whether the line correctly predicts the category of the object. If it does, we make no change to the line; if it doesn't we adjust the line.

The prediction line takes the standard form \(\mathtt{Ax + By + C = 0}\). If \(\mathtt{Ax + By + C > 0}\), we predict orange. If \(\mathtt{Ax + By + C \leq 0}\), we predict apple. We can put the "or equal to" on either one of these inequalities, and we need the "or equal to" on one of them for this to be a truly binary decision.

Okay, so let's draw a prediction line across the data above at \(\mathtt{y = 119}\), say. In that case, \(\mathtt{A = 0, B = 1,}\) and \(\mathtt{C = -119}\). When we come across apple data points, they will all be categorized correctly (as below the line), but some of the orange data points will be categorized correctly and some incorrectly. On correct categorizations, we don't want the line to update at all, and on incorrect categorizations, we want to change the line. Here's how both of these things are accomplished:

Let's take the point at (10, 109). This should be classified as orange, which we'll quantify as \(\mathtt{1}\). The prediction line, however, gives us \(\mathtt{0(10) + 1(109) + -119}\), which is \(\mathtt{-10 \leq 0}\). So, the predictron mistakenly predicts this point to be an apple, which we'll quantify as \(\mathtt{-1}\). We want to adjust the line.

Subtract the prediction (\(\mathtt{-1}\)) from the actual (1) to get \(\mathtt{2}\). Multiply this by a small learning rate of 0.1, so \(\mathtt{0.1 \times 2 = 0.2}\). Finally, change \(\mathtt{A}\), \(\mathtt{B}\), and \(\mathtt{C}\) like this:

\(\mathtt{A = A + 0.2 \times 10}\)

\(\mathtt{B = B + 0.2 \times 109}\)

\(\mathtt{C = C + 0.2}\)

So, in our example, \(\mathtt{A}\) becomes \(\mathtt{0 + 0.2 \times 10}\), or 2, \(\mathtt{B}\) becomes \(\mathtt{1 + 0.2 \times 109}\), or 22.8, and \(\mathtt{C}\) becomes \(\mathtt{-119 + 0.2}\), or \(\mathtt{-118.8}\).

We have a new prediction line, which is \(\mathtt{2x + 22.8y + -118.8 = 0}\). This line now makes the correct prediction for the orange at (10, 109), as you can see at the right with the blue line.

But this point is only used to adjust A, B, and C (called the "weights"), and then it's on to the next point to see if the new prediction line succeeds in making a correct prediction or not. The weights change with each incorrect prediction, and these changing weights move the prediction line up (left) and down (right) and alter its slope as well.

Many Iterations

Suppose we next encounter a point at (10, 90), which should be an apple. Our predictron, however, will make the prediction \(\mathtt{2(10) + 22.8(90) - 118.8 = 2153.2 > 0}\), which is a prediction of orange, or 1. Subtract the prediction (1) from the actual (\(\mathtt{-1}\)) to get \(\mathtt{-2}\), and multiply by the learning rate to get \(\mathtt{-0.2}\). Our weights are adjusted as follows:

\(\mathtt{A = 2 + -0.2 \times 10 = 0}\)

\(\mathtt{B = 22.8 + -0.2 \times 90 = 4.8}\)

\(\mathtt{C = -118.8 + -0.2 = -119}\)

If you graph this new line, \(\mathtt{0x + 4.8y + -119 = 0}\), you'll notice that it's between the other two, but it would still make the incorrect categorization for the apple at (10, 90). This is why the predictron must cycle through the data several times in order to "train" itself into determining the correct line. If you can make sense of it, there is a proof that this algorithm always converges in finite time for data that can be separated by a line (with a sufficiently small learning rate).

Other Notes

It's worth mentioning that the perceptron finds a line between the categories, but there are an infinite number of lines available. Also, in this example, we have two dimensions, or features, but the perceptron works for as many dimensions as you please.

So, to get schmancy, for each object in the data, \(\mathtt{z}\), our prediction function takes a linear combination of weights (coefficients + that C intercept weight) and coordinates (features, or dimensions) \(\mathtt{z = w_{0}x_{0} + w_{1}x_{1} + \ldots + w_{m}x_{m} = w^{T}x}\) and outputs \[\theta(z) = \left\{\begin{array}{11} \color{white}{-}1, & \quad z > 0 \\ -1, & \quad z \leq 0 \end{array} \right.\]

Even with multi-dimensional vectors, the weights are updated just as above, perhaps with a different learning rate, chosen at the beginning.

Below is a Python implementation of the Perceptron from Sebastian Raschka, using a classic data set about irises (the flowers).

Image credits: Joe King "Apple", Thomas Weldenhaupt "orange"

Is Area All About Rectangles?

2016-05-24T23:20:00.000-04:00

A very common way of teaching students the reasoning behind the formula for the area of a non-rectangular parallelogram is to demonstrate how to take apart such a parallelogram and turn it into a rectangle. Some U.S. states' elementary and middle-school mathematics standards used to be fairly explicit about this "way" of teaching. Here are two examples—the first from California, the second from Florida:

Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram).

Derive and apply formulas for areas of parallelograms, triangles, and trapezoids from the area of a rectangle.

The NCTM's 2000 publication Principles and Standards for School Mathematics (PSSM) also endorsed this method as consistent with stimulating students' understanding of, and investigation into, area:

One particularly accessible and rich domain for such investigation is areas of parallelograms, triangles, and trapezoids. Students can develop formulas for these shapes using what they have previously learned about how to find the area of a rectangle, along with an understanding that decomposing a shape and rearranging its component parts without overlapping does not affect the area of the shape.

And, from the third edition of this well-known book by Van de Walle, we have the following:

Parallelograms that are not rectangles can be transformed into rectangles having the same area. The new rectangle has the same height and two sides the same as the original parallelogram. Students should explore these relationships on grid paper, on geoboards, or by cutting paper models, and should be quite convinced that the areas are the same and that such reassembly can always be done with any parallelogram. As a result, the area of a parallelogram is base times height, just as for rectangles.

I should note that, of the four snippets I present above, only Van de Walle's even makes an attempt at a consistent distinction between non-rectangular parallelograms ("parallelograms that are not rectangles") and rectangles. But then even in the Van de Walle snippet, this distinction breaks down in the last sentence.

Consistent with the standards and the specific observations and suggestions in the widely referenced publications listed above, textbook lessons typically introduce students to the idea that the area of a non-rectangular parallelogram can be described by the same formula as that used to describe the area of a rectangle with the same base and height, using one or more illustrations like the one at right, accompanied by written instruction to help students understand the illustration.

Generally, the purpose of this written instruction is to clarify for students that (a) a part of the non-rectangular parallelogram was simply removed and then re-attached somewhere else on the figure--a transposition that does not change the area of the figure, and (b) the resulting figure is a rectangle whose height and base are the same as that of the non-rectangular parallelogram.

Maybe More About Parallel Lines

But, this progression has always struck me as a little backward, because rectangles and squares are special kinds of parallelograms. Deriving or developing a formula to describe the area of certain non-rectangular parallelograms based on what is taught about the area of rectangles--as is often done in this instruction--is a bit like arguing that the numbers 3, 1, 11, and 45 are all integers because you can count up from them or count down from them and reach the integer 9.

It turns out in both cases that what is stated is true—non-rectangular parallelograms and rectangles share the same area formula, and the numbers above, along with the number 9, are all integers. And it so happens that in both of these cases, the reasons are difficult to argue with as well—transforming a non-rectangular parallelogram into a rectangle without gaining or losing area is certainly a convincing demonstration, and so long as one construes "counting" as being restricted to integers, the reason given above for the "integer-ness" of 3, 1, 11, and 45 is satisfactory, albeit sloppy.

Still, the "integer-ness" of integers has nothing to do with the number 9, and the reason the area of a parallelogram can be described by the formula \(\mathtt{A = bh}\) or \(\mathtt{A = lw}\) may have nothing in general to do with rectangles and everything to do with the fact that parallel lines are at all corresponding points the same distance apart.

Is it not the case that the shaded figures shown have the same area (if their opposite sides are parallel)? Cavalieri's Principle, no? But this goes back to Euclid, too. Also, there's a connection to bivectors in higher-level math.

This idea--perhaps a slightly more general idea about area--awaits actual fleshing out in lessons, where the other principles (clarity, order, and cohesion), along with precision again, must be considered. If, though, we are to keep this topic (parallelogram area) where it typically falls in a mathematics curriculum, then we would need to move up discussions about parallel lines from where they typically fall, and such discussions could no longer be considered in isolation.

Making "Connections"

2016-05-07T10:58:00.000-04:00

I'm nearing the end of my read of James Lang's terrific book Small Teaching, and I've wanted for the last 100 pages or so to recommend it highly here.

While I do that, however, I'd also like to mention a confusion that piqued my interest near the middle of the book—a very common false distinction, I think, between 'making connections' and knowing things. Lang sets it up this way:

When we are tackling a new author in my British literature survey course, I might begin class by pointing out some salient feature of the author's life or work and asking students to tell me the name of a previous author (whose work we have read) who shares that same feature. "This is a Scottish author," I will say. "And who was the last Scottish author we read?" Blank stares. Perhaps just a bit of gaping bewilderment. Instead of seeing the broad sweep of British literary history, with its many plots, subplots, and characters, my students see Author A and then Author B and then Author C and so on. They can analyze and remember the main works and features of each author, but they run into trouble when asked to forge connections among writers.

What immediately follows this paragraph is what one would expect from a writer who has done his homework on the research: Lang reminds himself that his students are novices and he an expert; his students' knowledge of British literature and history is "sparse and superficial."

But then, suddenly, the false distinction, where 'knowledge' takes on a different meaning, becoming synonymous with "sparse and superficial," and his students have it again:

In short, they have knowledge, in the sense that they can produce individual pieces of information in specific contexts; what they lack is understanding or comprehension.

And they lack comprehension, even more shortly, because they lack connections.

Nope, Still Knowledge

As we saw here, with the Wason Selection Task, reasoning ability itself is dependent on knowledge. Participants who were given abstract rules had tremendous difficulties with modus tollens reasoning in particular, yet when these rules were set in concrete contexts, the difficulties all but vanished.

One might say, indeed, that in concrete contexts, the connections are known, not inferred. Thus, if you want students to make connections among various authors, it might help to tell them that they are connected, and how.

The Wason Selection Task, Part II

2016-05-02T20:25:00.000-04:00

Before we sink our teeth even deeper into the Wason Selection Task, we should look briefly at conditional reasoning arguments.

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

There are four types of conditional reasoning arguments that apply to the Wason Selection Task—two of them valid and two of them logically invalid. Each of these introduces a different second statement, after the "if P, then Q" formulation. That is, each of the following four types of conditional reasoning arguments can be identified according to the statement that comes 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 (P).
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. 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," as far as that goes. Formal logic does not concern itself necessarily with the contents of arguments, only their form.

Applying the Rules to the Selection Task

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" 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, 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 [∼P]) or a statement about Q (i.e., Q is true [Q] or Q is not true [∼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:

soda

vodka

Not Alcoholic
(∼P)

Not Legal Age
(∼Q)

Alcoholic
(P)

Legal Age
(Q)

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

D
(P)

Not D
(∼P)

3
(Q)

Not 3
(∼Q)

In each case, the statements which form a valid argument (i.e., P and ∼Q) indicate those cards (or people) that must be checked to determine the validity of the if-then statement.

The Wason Selection Task, Part I

2016-05-02T14:02:00.000-04:00

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.

soda

vodka

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." 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:

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.

In a chapter from The Cambridge Handbook of Expertise and Expert Performance, Paul Feltovich, Michael Prietula, and K. Anders Ericsson indicate the one factor that explains these differential results: knowledge.

Some studies showed reasoning itself to be dependent on knowledge. Wason and Johnson-Laird (1972) presented evidence that individuals perform poorly in testing the implications of logical inference rules (e.g., if p then q) when the rules are stated abstractly. Performance greatly improves for concrete instances of the same rules (e.g., 'every time I go to Manchester, I go by train'). Rumelhart (1979), in an extension of this work, found that nearly five times as many participants were able to test correctly the implications of a simple, single-conditional logical expression when it was stated in terms of a realistic setting (e.g., a work setting: 'every purchase over thirty dollars must be approved by the regional manager') versus when the expression was stated in an understandable but less meaningful form (e.g., 'every card with a vowel on the front must have an integer on the back').

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.

Being Explicit About Symmetry

2016-05-01T11:08:00.000-04:00

After reading this piece about mathematician Terence Tao, I was turned on to a book about mathematical problem-solving Tao wrote as a 15-year-old, so I decided to check it out. The book contains a nice little nugget about symmetry in the first chapter, the importance of which sails by the author (and thus the reader) a bit, I think. Here's the problem that occupies all of the discussion in Chapter 1:

A triangle has its lengths in an arithmetic progression, with difference \(\mathtt{d}\). The area of the triangle is \(\mathtt{t}\). Find the lengths and angles of the triangle.

And here's the nice move with regard to symmetry that Tao makes, almost in passing. It doesn't seem to be a "code-cracking" idea in the context of the problem, but I think it highlights a key process in mathematical thinking that we can work to make more explicit for students. It might be helpful to tinker with the problem a little to put this move in some context:

We can use the data to simplify the notation: we know that the sides are in arithmetic progression, so instead of \(\mathtt{a}\), \(\mathtt{b}\), and \(\mathtt{c}\), we can have \(\mathtt{a}\), \(\mathtt{a + d}\), and \(\mathtt{a + 2d}\) instead. But the notation can be even better if we make it more symmetrical, by making the side lengths \(\mathtt{b - d}\), \(\mathtt{b}\), and \(\mathtt{b + d}\).

I can easily imagine, given what I know about most current curricula, a student writing down the side lengths of the triangle as \(\mathtt{a}\), \(\mathtt{a + d}\), and \(\mathtt{a + 2d}\), because this is how we teach arithmetic progressions. But we don't often explicitly make the fairly simple observation that any term, with the exception of the first, in an arithmetic progression can be written as \(\mathtt{a_n}\), with the previous term written as \(\mathtt{a_n - d}\) and the next term as \(\mathtt{a_n + d}\). It seems that we can find this observation by thinking a bit more about symmetry when we craft our explanations and instruction.

Another Observation

Working to orient oneself to the symmetries available in mathematical situations seems like one appropriate remedy to what I've called "left-to-rightism," or "cinemathematics"—a syndrome that makes us teach concepts like the equals sign (unwittingly) in a left-to-right way, such that students take away (unwittingly) the misconception that the equals sign indicates that some answer is to follow, rather than that two expressions are equal. Some recent research points to the benefits of thinking about symmetry when teaching negative numbers as well.

Tsang, J., Blair, K., Bofferding, L., & Schwartz, D. (2015). Learning to “See” Less Than Nothing: Putting Perceptual Skills to Work for Learning Numerical Structure Cognition and Instruction, 33 (2), 154-197 DOI: 10.1080/07370008.2015.1038539

Evident Even to an Ass

2016-04-23T10:45:00.000-04:00

Suppose there is a straight line segment \(\overline{\small\mathtt{AB}}\) between you, at point \(\small\mathtt{A}\), and some point \(\small\mathtt{B}\) you want to reach. Is there a shorter way to point \(\small\mathtt{B}\) that uses 2 line segment paths?

If you think the answer is yes, or if you think that the answer is no but it's not obviously no, the Epicureans have some pretty mean things to say about you:

It was the habit of the Epicureans, says Proclus, to ridicule this theorem as being evident even to an ass and requiring no proof, and their allegation that the theorem was "known" even to an ass was based on the fact that, if fodder is placed at one angular point and the ass at another, he does not, in order to get to his food, traverse the two sides of the triangle but only the one side separating them (an argument which makes Savile exclaim that its authors were "digni ipsi, qui cum Asino foenum essent"). Proclus replies truly that a mere perception of the truth of the theorem is a different thing from a scientific proof of it and a knowledge of the reason why it is true.

The theorem mentioned is the Triangle Inequality Theorem: the sum of the lengths of any two sides of a triangle is always greater than the length of the remaining side. And it is fascinating to me how seldom I have seen this theorem in textbooks phrased as a "shortest straight-line distance" statement. Most often, I see investigations about what side lengths can make up a triangle, which turns the ridiculously obvious into a complicated issue. But let's discuss that below. For now, a proof!

It's Okay To Be Both Obvious and Require Proof

We want to show that \(\small\mathtt{BA + AC}\) is greater than \(\small\mathtt{BC}\), that \(\small\mathtt{AB + BC}\) is greater than \(\small\mathtt{AC}\), and that \(\small\mathtt{BC + CA}\) is greater than \(\small\mathtt{AB}\).

Euclid starts by extending \(\overline{\small\mathtt{BA}}\) to a point \(\small\mathtt{D}\) such that \(\small\mathtt{DA = AC}\) as I've shown in the diagram. This means that \(\small\Delta\mathtt{ADC}\) is an isosceles triangle, and \(\small\measuredangle\mathtt{ACD}\) and \(\small\measuredangle\mathtt{ADC}\) are congruent. And since "the whole is greater than the part," m\(\small\measuredangle\mathtt{BCD}\) is greater than m\(\small\measuredangle\mathtt{ACD}\), which also makes m\(\small\measuredangle\mathtt{BCD}\) greater than m\(\small\measuredangle\mathtt{ADC}\).

Dizzy yet? Just remember the last rung of the ladder we got to: m\(\small\measuredangle\mathtt{BCD}\) is greater than m\(\small\measuredangle\mathtt{ADC}\). Therefore, we can say something about the line segments that those two angles "catch." Specifically, we can say that \(\overline{\small\mathtt{DB}}\) is longer than \(\overline{\small\mathtt{BC}}\), because Proposition 19. This is the same as saying that \(\small\mathtt{DA + AB > BC}\). And since \(\small\mathtt{DA = AC}\), we can make a substitution to get \(\small\mathtt{AC + AB > BC}\), which is the first of the three statements above that we wanted to prove: \(\small\mathtt{BA + AC}\) is greater than \(\small\mathtt{BC}\). Euclid tells us that we can prove the other two statements with a similar method, and I believe him.

Internalizing the Idea That Mathematics Is Complex and Intuition-Free

In the link is an example of what we often put students through to investigate the Triangle Inequality Theorem. This is consistent with what I have seen in a lot of lesson plans.

Were the authors of this document aware that there is a very intuitive way of looking at this theorem? It might be enough to suppose that they weren't and that they, like all of us, sometimes just repeat what we see or hear elsewhere.

But it's also worth entertaining the possibility that they did know and pressed on anyway. What good reasons might they have for doing so? I think the best answer is simply Proclus's reply above: "a mere perception of the truth of the theorem is a different thing from a scientific proof of it and a knowledge of the reason why it is true." To which I would respond, Indeed, a different thing. Not necessarily a better thing.

If we can give students the "mere perception of the truth" of a theorem (or of any mathematical idea), we should do so, even if it doesn't make us feel smart, leaves a lot of class time to fill, or runs counter to a set of standards. I would argue that students still have to prove those theorems and justify those ideas. But they can then do so correctly oriented to the reality of what they are doing: drawing on their own perceptions and knowledge to make their ideas plain to others.

Spaced Practice

2016-04-10T18:44:00.000-04:00

Spaced practice, also called distributed practice, refers to the practice—after initial learning—of leaving gaps in time between sessions devoted to reviewing previously learned material. This stands in contrast to its much less effective counterpart, massed practice, which "crams" all of the review together in time.

If, for example, you're learning about deriving the formula for the area of a circle today, and you know that in 100 days you will be tested on this learning, then spacing out your review of the material up to test day will be much better for your final score than cramming the night before.

Indeed, this recent article by Kang, along with providing a review of the evidence for the effectiveness of spaced practice, goes into some detail about optimal timings:

Retention of . . . the study material . . . was highest when the lag was about 10% to 20% of the tested retention interval (see also Cepeda et al., 2009). In other words, there is no fixed optimal lag—It depends on the targeted retention interval. If you want to maximize performance on a test about 1 week away, then a lag of about 1 day [14% of a week] would be optimal; but if you want to retain information for 1 year, then a lag of about 2 months [16% of a year] would be ideal.

So, in my area-of-a-circle example, the optimal lag time between initial study and test is between 10 and 20 days. And this is if there is only one review session. (And, needless to say, that review session has to be of high quality; I should likely work to retrieve what I can and then comprehensively review the material.)

If there are multiple opportunities for reviewing material (as is almost always the case), it is not quite clear whether it is better for the review sessions to be spaced out equally or for the interval to expand over time.

Yes, Virginia, There Is a 'Higher Order'

There are a few ancient dualist sects still in existence, I hear, who believe that higher-order thinking is in competition with memory and who would dismiss evidence about spaced practice on the grounds that it ignores the former 'in favor of' the latter. While this is barely worth a response except to point to a modern library or university, Kang does take the time to type the obvious: "acquiring foundational knowledge and being able to quickly access relevant information from memory are often prerequisites for higher order learning."

In addition, the author cites some of the evidence that spaced practice has benefits not only for memory but also for generalization and transfer:

In one study, college students attended a 45-min lecture on meteorology and then reviewed the information (in a quiz with corrective feedback) either 1 or 8 days later (Kapler, Weston, and Wiseheart, 2015). On a final test 35 days after the review session, students in the 8-day condition performed better than those in the 1-day condition not just on the factual recall questions but also on the questions that required application of knowledge. Other studies support spaced practice of mathematics problems (Rohrer and Taylor, 2006) and ecology lessons (Gluckman, Vlach, and Sandhofer, 2014; see also Vlach and Sandhofer, 2012). In addition to improving mathematics problem solving and science concept learning, spaced practice benefits the long-term learning of English grammar in adult English-language learners (Bird, 2010). In all cases, the students were not just memorizing solutions but were instead applying their learning to solve new problems.

Some Classroom Evidence Too

Last but not least, we need to place spaced practice in an environment that is entirely unique and as chaotic as the inside of a quasar, with no control over any variables. That way we'll really know if it works:

Although most studies on spaced or interleaved practice have been conducted in laboratory settings (for better control over extraneous variables), students in actual classrooms can benefit from instructors using these learning strategies (e.g., Carpenter et al., 2009; Sobel, Cepeda, & Kapler, 2011). A few studies were conducted not only in real-world educational settings but also in the context of a regular curriculum (i.e., instructional manipulation on course content).

In one classroom-based study, the mathematics homework assignments for seventh-grade students were manipulated across 9 weeks (Rohrer, Dedrick, & Burgess, 2014). . . . On a surprise test containing novel problems (on the same topics), given 2 weeks after the final homework assignment, the students were substantially better at solving the types of problems that had been practiced in an interleaved manner than those under blocked practice.

Kang, S. (2016). Spaced Repetition Promotes Efficient and Effective Learning: Policy Implications for Instruction Policy Insights from the Behavioral and Brain Sciences, 3 (1), 12-19 DOI: 10.1177/2372732215624708

Interleaving

2016-04-09T21:47:00.002-04:00

Inductive teaching or learning, although it has a special name, happens all the time without our having to pay any attention to technique. It is basically learning through examples. As the authors of the paper we're discussing here indicate, through inductive learning:

Children . . . learn concepts such as 'boat' or 'fruit' by being exposed to exemplars of those categories and inducing the commonalities that define the concepts. . . . Such inductive learning is critical in making sense of events, objects, and actions—and, more generally, in structuring and understanding our world.

The paper describes three experiments conducted to further test the benefit of interleaving on inductive learning ("further" because an interleaving effect has been demonstrated in previous studies). Interleaving is one of a handful of powerful learning and practicing strategies mentioned throughout the book Make It Stick: The Science of Successful Learning. In the book, the power of interleaving is highlighted by the following summary of another experiment involving determining volumes:

Two groups of college students were taught how to find the volumes of four obscure geometric solids (wedge, spheroid, spherical cone, and half cone). One group then worked a set of practice problems that were clustered by problem type . . . The other group worked the same practice problems, but the sequence was mixed (interleaved) rather than clustered by type of problem . . . During practice, the students who worked the problems in clusters (that is, massed) averaged 89 percent correct, compared to only 60 percent for those who worked the problems in a mixed sequence. But in the final test a week later, the students who had practiced solving problems clustered by type averaged only 20 percent correct, while the students whose practice was interleaved averaged 63 percent.

The research we look at in this post does not produce such stupendous results, but it is nevertheless an interesting validation of the interleaving effect. Although there are three experiments described, I'll summarize just the first one.

Discriminative-Contrast Hypothesis

But first, you can try out an experiment like the one reported in the paper. Click start to study pictures of different bird species below. There are 32 pictures, and each one is shown for 4 seconds. After this study period, you will be asked to try to identify 8 birds from pictures that were not shown during the study period, but which belong to one of the species you studied.

Once the study phase is over, click test to start the test and match each picture to a species name. There is no time limit on the test. Simply click next once you have selected each of your answers.

Based on previous research, one would predict that, in general, you would do better in the interleaved condition, where the species are mixed together in the study phase, than you would in the 'massed,' or grouped condition, where the pictures are presented in species groups. (See this post, however, for some serious qualifications to this statement—interleaving is demonstrably effective for practice, but less so for initial learning, as is tested here.) The question the researchers wanted to home in on in their first experiment was about the mechanism that made interleaved study more effective.

So, their experiment was conducted much like the one above, except with three groups, which all received the interleaved presentation. However, two of the groups were interrupted in their study by trivia questions in different ways. One group—the alternating trivia group—received a trivia question after every picture; the other group—the grouped trivia group—received 8 trivia questions after every group of 8 interleaved pictures. The third group—the contiguous group—received no interruption in their study.

What the researchers discovered is that while the contiguous group performed the best (of course), the grouped trivia group did not perform significantly worse, while the alternating trivia group did perform significantly worse than both the contiguous and grouped trivia groups. This was seen as providing some confirmation for the discriminative-contrast hypothesis:

Interleaved studying might facilitate noticing the differences that separate one category from another. In other words, perhaps interleaving is beneficial because it juxtaposes different categories, which then highlights differences across the categories and supports discrimination learning.

In the grouped trivia condition, participants were still able to take advantage of the interleaving effect because the disruptions (the trivia questions) had less of an effect when grouped in packs of 8. In the alternating trivia condition, however, a trivia question appeared after every picture, frustrating the discrimination mechanism that seems to help make the interleaving effect tick.

Takeaway Goodies (and Questions) for Instruction

The paper makes it clear that interleaving is not a slam dunk for instruction. Massed studying or practice might be more beneficial, for example, when the goal is to understand the similarities among the objects of study rather than the differences. Massed studying may also be preferred when the objects are 'highly discriminable' (easy to tell apart).

Yet, many of the misconceptions we deal with in mathematics education in particular can be seen as the result of dealing with objects of 'low discriminability' (objects that are hard to tell apart). In many cases, these objects really are hard to tell apart, and in others we simply make them hard through our sequencing. Consider some of the items listed in the NCTM's wonderful 13 Rules That Expire, which students often misapply:

When multiplying by ten, just add a zero to the end of the number.
You cannot take a bigger number from a smaller number.
Addition and multiplication make numbers bigger.
You always divide the larger number by the smaller number.

In some sense, these are problematic because they are like the sparrows and finches above when presented only in groups—they are harder to stop because we don't present them in situations that break the rules, or interleave them. Appending a zero to a number to multiply by 10 does work on counting numbers but not on decimals; addition and multiplication do make counting numbers bigger until they don't always make fractions bigger; and you cannot take a bigger counting number from a smaller one and get a counting number. For that, you need integers.

Notice any similarities above? Can we please talk about how we keep kids trapped for too long in counting number land? I've got this marvelous study to show you which might provide some good reasons to interleave different number systems throughout students' educations. It's linked above, and below.

Images credits: All About Birds, Anne Davis 773

Birnbaum, M., Kornell, N., Bjork, E., & Bjork, R. (2012). Why interleaving enhances inductive learning: The roles of discrimination and retrieval Memory & Cognition, 41 (3), 392-402 DOI: 10.3758/s13421-012-0272-7

A Thought About the Chinese Room

2016-03-13T19:16:00.000-04:00

In 1980, philosopher John Searle proposed a thought experiment called The Chinese Room. Here's a brief—and a bit fast-moving—summary of it:

Although the thought experiment was apparently meant to illustrate the impossibility of "strong AI," or artificial intelligence capable of consciousness and human-like "mind," it has clear relevance for thinking about education as well. What differentiates "meaning-based" understanding and mimicry? for example.

Three Little Words: "I Don't Know"

You are invited to notice that the room's occupant doesn't actually know Chinese, but simply matches characters from an input message to a page in a rule book in order to generate an output message. Presumably, the particular method for processing inputs is not important, so long as the person inside the room does not understand the characters he is receiving or sending out.

The problem here is that this observation is not interesting unless you smuggle in some assumption about how the human mind works. It is not interesting that the person in the room doesn't understand Chinese unless you make the assumption that something in a Chinese speaker's brain does understand Chinese. Needless to say, this is not the case. Neurons don't understand Chinese. They are precisely as clueless as the man in the room.

The correct comparisons—at the comparable scales—should be Chinese room : Chinese speaker and person in room : inner workings of the mind. The thought experiment misleads us into comparing the person in the room with the Chinese speaker. This would work if the speaker were identical with the inner workings of her mind. Which is exactly the assumption we make about people, even though we know it has to be wrong.

Exactly how the human mind differs from elaborate mechanical rule-following is beside the point. The point is that for the Chinese Room thought experiment to have its intended effect, it seems you must assume that the human mind has some kind of "meaning" mechanism which is not built out of dumber parts. But John Searle didn't know this in 1980. And you don't know it now. It is an unwarranted assumption.

And almost certainly false.

Retrieval Practice Effective for Young Students

2016-03-05T17:30:00.002-05:00

These results are about as straightforward as they come in the social sciences. In an article published in Frontiers in Psychology, researchers report the results of three experiments which show that the benefits of retrieval practice (practice with retrieving items from memory) extends to children as much as to adults.

To get some sense of what retrieval practice is like, go read the abstract of the paper linked above. Then, crucially, without re-accessing the article, write down everything you can remember about the abstract on a blank piece of paper. So long as you can remember something from the abstract, this act of retrieving from memory what you read (i.e., retrieval) is much more beneficial to your remembering the information in the long term than just re-reading the passage.

Research tells us that there is surprisingly little else to say to qualify this statement about the benefits of retrieval. It works even in the absence of feedback, and, as is reported in the current study, it works (better than repeated study) perhaps regardless of students' processing speed and reading comprehension ability.

Still in Phase 2

The idea that retrieval must be at least in part successful for it to work seems to suggest a bit of a chicken-and-egg issue:

Successful retrieval is essential for retrieval practice; if students cannot recall much, there will likely be little or no benefit of retrieval practice (Karpicke, Blunt, Smith, & Karpicke, 2014). For example, Karpicke and colleagues (2014, Experiment 1) examined free recall—a retrieval-based learning strategy that has been shown time and again to be extremely effective for adult learners (e.g., Roediger & Karpicke, 2006)—in 4th graders and found no benefit on learning measured 5 days later. Initial retrieval success in this experiment was very low (around 8%).

This may read as though successful retrieval is essential for retrieval to be successful. But this is mostly because of the two different measurements of success being used—successful retrieval refers to recalling some part (greater than 8%, presumably) of already learned material at any given moment or short window of time; retrieval practice is considered successful when you remember the material over longer stretches of time than with other strategies.

Still, this potential confusion may bring into relief the importance of recognizing the place for retrieval practice in learning: it strengthens memory; it doesn't create memory. As such, it is situated firmly in Phase 2—it seems to work best once students have progressed at least part way, if not all the way, through the acquisition phase. We have to be careful about this. "Input less, output more" is a great message for Phase 2 learning. In a 'culture' (I guess) that has a hard time dealing seriously with acquisition issues of learning, it's a potentially dangerous one. We still have to provide students with rich inputs in order to make retrieval any good.

Not the Only Interpretation

To be honest, what prompted me to write up this study was my wondering how the success of retrieval might be interpreted in a regular classroom, by a teacher without access to the hypotheses and previous results that informed this study. What might a "layperson" conclude was the explanation for the superior performance of the retrieval group, given no information about the different groups in the study except for their final scores and their overt behavior during the study?

There are all kinds of different behavioral interpretations possible when hypotheses about cognition are out of reach. One might conclude, for example, that the retrieval group was allowed to "take more ownership of their learning"—you see, because they weren't helped as much; they had to fill in the missing letters of the words to be remembered rather than just reading them off of a list. One might say that students in the retrieval group were "learning by doing," especially if participants in that group were writing when other participants weren't. And this writing is very 'kinaesthetic', don't you know.

It's not even all that hard to see how parts of that Pyramid of Learning nonsense could be a folk-psychological, behavioral misinterpretation of benefits more appropriately accorded to retrieval practice. Engaging in retrieval "in the wild" may often look like those activities closer to the bottom of the pyramid than to the top. But of course those activities don't necessarily cause better performance; a positive confound, such as retrieval practice, may be responsible for both the activity and the better performance. When science is absent and we must rely on observations alone, misunderstanding the underlying cause of better retrieval may then naturally result in trying to replicate superficial features of situations correlated to improved performance. Misunderstanding then slowly becomes common wisdom and philosophy.

Jeffrey D. Karpicke, Janell R. Blunt, & Megan A. Smith (2016). Retrieval-Based Learning: Positive Effects of Retrieval Practice in Elementary School Children Frontiers in Psychology, 2-28 : 10.3389/fpsyg.2016.00350

Supporting Instructional Analogies

2016-02-29T23:18:00.000-05:00

An international comparison study published in 2007 found that, compared with Hong Kong and Japanese teachers, U.S. teachers offered students relatively little cognitive support for processing analogies presented during instruction, even though the use of analogies was essentially the same in all classrooms studied.

Teachers in both Asian regions used spatial supports for comparison more than twice as often as did their U.S. counterparts. These teachers also used far more gestures that emphasized comparison than did U.S. teachers, even though the latter used gestures of some kind almost equally often. Hong Kong teachers were almost twice as likely to prompt mental and visual imagery as were U.S. teachers, and Japanese teachers were even more likely . . . Hong Kong and Japanese teachers appear to be more attentive to the processing demands of relational comparisons than are U.S teachers. Their teaching reflects the use of strategies to reduce processing demands on their students. Such differences in adherence to sound cognitive principles may have a real impact on the likelihood that students benefit from analogies as instructional tools.

If you're interested, some of the 1999 TIMSS videos, on which this study was based, are here.

Idle Speculation: "Owning" Ideas

I've always found the notion of ownership compelling when thinking about East-West cultural differences with regard to instruction. Although it is a bit simplistic, there is still, I believe, a fairly robust difference between Eastern and Western cultures on the collectivist-individualist continuum, with Eastern cultures finding themselves, in general, closer to the collectivist side and Western cultures closer to the individualist side.

Such differences are likely to extend to conceptual spaces as well, with ideas themselves attached to "owners" (even temporarily) in more individualistic communities while having no such attachment in more collectivist ones. This would make Western educators much more likely than their Eastern counterparts to evaluate instructional situations (one way or another) based on the contributions of the participants. Intellectual "ownership" as a salient characteristic can set up a kind of knowledge economy in Western education—one where concepts are given to students, who must "take ownership" of the ideas.

Perhaps this economy goes some way to explain why the students in every classroom in this study were exposed to comparable instructional opportunities, while those in Eastern classrooms experienced twice as much cognitive support (in some categories) during those opportunities: because actual instructional support (in this case, strategically using cognitive science principles to support analogies) blurs the lines of ownership and personality and zooms in on the ideas instead—a situation that is completely unremarkable in the East and deeply unsettling in the West.

I recently came across a very simple technique for animating smooth back-and-forth motions (and just back or just forth too) like you see above if you have the right browser—using the sine and cosine functions. And I have to admit a small feeling of embarrassment when I saw it. I'd never thought of it before, and it felt shockingly obvious after I'd read it. The 'ownership' construct is, I think, in part responsible for this feeling.

You see, because along with the new learning came the feeling that the idea had to be assigned a seller (the book) and a buyer (me)—or at least a supplier and a recipient. I am perfectly free to share this idea with others, who may repeat the seller-buyer assignations, creating a long (and evaporating) chain of ridiculous transactional interpretations about learning.

Richland, L., Zur, O., & Holyoak, K. (2007). MATHEMATICS: Cognitive Supports for Analogies in the Mathematics Classroom Science, 316 (5828), 1128-1129 DOI: 10.1126/science.1142103

Acquisition

2016-02-28T00:22:00.002-05:00

I've come around to thinking about acquisition again—that window of time in which learners make first contact with something to be learned. Ohlsson refers to this as the "Getting Started" phase, one of three phases in "skill acquisition":

The first stage begins when the learner encounters the practice task and ends when he completes the task correctly for the first time. . . .

There are five distinct types of information that might be available at the outset of practice: direct instructions; declarative knowledge about the task; strategies for analogous tasks; demonstrations, models and solved examples; and outcomes of unselective search. . . .

The second stage of practice begins when the learner completes the task for the first time and it lasts until he can reliably perform the task correctly. During this stage, the main theoretical problem is how the learner can improve his incomplete and possibly incorrect version of the strategy-to-be-learned. . . .

The optimization stage begins when the target skill has been mastered, that is, when it can be executed reliably, and lasts as long as the person keeps practicing.

At right is a graph from the same book which shows "how the relative importance of four learning mechanisms might shift across the three phases of skill acquisition." Instruction and solved examples are of utmost importance in the acquisition phase, with feedback about performance taking over in Phase 2, and self-directed learning from the statistical regularities of the domain in Phase 3.

You Are Here

At the moment, I would argue that modern educational thought does not pay sufficient attention to that first acquisition phase of learning. This does not seem to be a deliberate shifting of attentional resources away from Phase 1; rather, it is more a matter of conceptualizing "learning" as not having a Phase 1 at all—or a Phase 1 so straightforward and inevitable that it is of little interest to either practitioners or researchers.

We hear a lot about the testing effect, for example—as well we should, and more of it!—but we hear comparatively little about whether the representational content of what we are testing is any good. There is a lot of talk about methodologies such as using worked examples or uncovering the curriculum (differences that are important to talk about), but comparatively little talk about the quality of the content of those examples and whether what is being uncovered are just in-the-moment rules or in-the-moment 'sense-making' that won't make sense later on.

A colleague of mine recently reminded me that we still teach both rectangles and squares at the Kindergarten level, with no discussion about how squares are rectangles. It's right there in the CC Standards with no caveats.

Consider for a moment that teaching this false distinction requires extra energy on our part. We're not letting things slide here; we're reinforcing a misconception, purposely and actively. Consider also that there is no pedagogical justification for this, except within a system that always starts at Phase 2. We take 'em as they are after watching Sesame Street and playing with pre-school toys and try to roll back their misconceptions with feedback and exams. This process then repeats itself at every grade level. Finally, consider that this is Kindergarten math. If we can be moved, by some benevolent or malevolent philosophizing, to talk about rectangles and squares as two different things, there's no telling what else we can be talked into.

Recommendations

In my view, the study of acquiring representational content, which amounts to the study of representational content itself, is a promising avenue for AI research—which I think will gradually weigh in with increasing credibility on educational issues.

Let's "teach" one computer to form a solid schema in which rectangles and squares are distinct (but related) objects and then measure the cost (if any) of trying to undo this schema when we try to teach it about quadrilateral classification. I wouldn't be surprised if there was already work on this. I wouldn't be surprised if there wasn't either.

Update: The Weak Argument About the Weakness of Instruction

Education theorizing that features an emaciated acquisition phase is a good fit for the occupational constraints of classroom teaching (as I mention here); conversely, theories that promote a robust acquisition phase may likely be seen as threatening.

As such, there is a healthy amount of rationalizing discourse in education about "telling" that keeps this dissonance at bay. Ridiculous deepities like "the two lies of teaching" passed around in pep-rally seminars and online are of course far more common and easy to dismiss, but this, research reported in Ohlsson's book, demonstrates how virtually anyone can conclude that instruction is ineffective when they are resigned to terrible instruction:

What could the adult mean by saying that the Earth is round? The word "round" is ambiguous; it is used to refer to both circular (a two-dimensional property) and spherical (a three-dimensional property). Which of these meanings will be activated? Unless the child conceives the Earth as extending indefinitely in all directions, the flat Earth must have an edge somewhere, and an edge is a kind of thing that can be circular. The child is likely to conclude that the adult is saying that the flat Earth has a circular edge. . . . In short, if the listener believes that the Earth is flat, the apparently contradictory discourse, the Earth is round has no power to teach him or her otherwise, because the listener’s prior beliefs hold too much power over its interpretation.

Note the bold conclusion that teaching has no power over prior knowledge; and that teaching here is conceptualized as saying a sentence to students. The entire book, of course, rather than a blog post, provides the right context in which to interpret Ohlsson's thoughts. But I thought I'd point out that the above is repeated a little later, even more boldly:

Because new information is interpreted in terms of a person's current concepts and beliefs, such information has little power to change those concepts and beliefs. There is no obvious way to circumvent this assimilation paradox. As we cannot see without our eyeballs, so we cannot understand a discourse without our prior concepts.

It's bizarre that the obvious objection is not dealt with, if only to try to dismiss it. That is, it seems far more likely that children are told that the Earth is round, and this represents incomplete instruction, which students fill in with false notions.

The researchers even noted that demonstrating the shape of the Earth with a globe was sometimes not enough to dislodge misconceptions—children modified their ideas to make the Earth a hollow sphere partially filled with dirt, creating a flat plane on which people walked. Yet this did remove the misconception, because it succeeded in modifying children's ideas about the shape of the Earth, and the instruction was about the shape of the Earth, not about how people move around on it.

The idea that gravity is like a magnet holding people to the Earth, even "upside down" from a certain perspective, is an idea that you rarely see explicitly demonstrated. But this missing idea is clearly a possible reason for the misconceptions mentioned.

Vosniadou, S., & Brewer, W. (1992). Mental models of the earth: A study of conceptual change in childhood Cognitive Psychology, 24 (4), 535-585 DOI: 10.1016/0010-0285(92)90018-W

Are You Smarter Than a Belgian 8th Grader?

2016-02-21T23:47:00.000-05:00

A central point of this paper is something close to my heart—the notion that how one represents a certain piece of mathematics knowledge is often dramatically important. For this research in particular, the authors looked at fraction knowledge across three different countries: the U.S., Belgium, and China. (Emphasis below is mine.)

Despite country-specific differences in absolute level of fraction knowledge, 6th and 8th graders' fraction magnitude understanding was positively related to their general mathematical achievement in all countries.

What is meant by "fraction magnitude understanding"? In short, it means to understand fractions as values that "can be located and ordered on number lines." But I think this wording deserves to emphasized a little more strongly (even if it goes beyond what the researchers seem to say), because the mention of number lines might trigger what Carl Bereiter has referred to as the "We already do that" syndrome. So: understanding fractions as magnitudes does not mean just that a student can translate her part-to-whole mental representation of fractions onto a number line. It means that, whatever else they could also be, fractions are magnitudes to her—singular values which can be ordered along a number line.

Test Your Magnitudinal Understandings

I'll just briefly describe the methodology and results of this study here. I recommend the original paper (linked at the top) highly for its background discussion and detailed development.

The critical assessment used in the study was similar to the one I've created below. Researchers presented participants with 29 target fractions (centered above a number line) and asked them to "click the mouse on the correct location of the value of the fraction on the number line." Go ahead and try it! You don't need to be concerned with clicking right on the number line; only the horizontal position of the mouse in relation to the number line is recorded. Your score is recorded as a percentage of absolute error (PAE), so lower scores are better.

Your PAE average is 0, for 0 trial(s). By way of comparison, the best averages for students in each of the three countries in the study were as follows: China (8th grade): 3.88, Belgium (8th grade): 5.05, U.S. (8th grade): 14.93.

A second task, which I've also simulated below, involved 12 fraction comparisons (with \(\mathtt{\frac{3}{5}}\) being the right-side number for every comparison). Students were asked to click on the lesser fraction for each trial. Scores were recorded as correct response percents. Give it a shot!

\(\mathtt{\frac{3}{5}}\)

Your percent correct is 0%. The best scores for the three countries were China (8th grade): 88.46, Belgium (8th grade): 86.52, U.S. (8th grade): 76.81.

Finally, researchers also gave students two fraction arithmetic tasks, one addition and one division, using the exact same (4) pairs of fractions in each task. Students were allowed to use scratch paper and were not required to write their answers in simplest form. (Please, oh please, researchers, get with the times and stop saying "reduce" to "lowest form".) I did not simulate these tasks here, but the results (percent correct, 8th grade) for the three countries followed the same pattern as in the other tasks. Addition: 91.03, 86.76, 51.09; Division: 92.92, 84.56, 31.52^*.

Magnitude Understanding Correlates Strongly

Across the three countries, student performance on the number line task was the most strongly predictive of overall mathematics achievement (as measured by country-specific standardized tests). Indeed, results on this task alone delivered a statistically significant correlation with general mathematics achievement in all three countries (emphasis mine again):

In all three countries, fraction magnitude understanding was the best predictor of mathematics achievement scores, even after controlling for variation in fraction arithmetic skill . . . Taken together, these results indicate that the relation between fraction magnitude understanding and mathematics achievement is not attributable to specifics of the U.S. educational system. Instead, the strong relation between fraction magnitude understanding and mathematics achievement can be interpreted as a general cognitive characteristic of all students, regardless of their specific cultural and educational background.

The researchers also discuss some instructional implications, noting that improving understanding of whole numbers as magnitudes has been shown to have positive effects on mathematics learning. They also note studies by Gabriel, et al. and Fuchs, et al. which demonstrated similarly positive effects with fractions—the latter of which producing stronger results than an alternative approach emphasizing the part-to-whole perspective.

* OMFG

Torbeyns, J., Schneider, M., Xin, Z., & Siegler, R. (2015). Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents Learning and Instruction, 37, 5-13 DOI: 10.1016/j.learninstruc.2014.03.002

Searching the Solution Space

2016-02-13T21:59:00.001-05:00

My reading in education has been a bit disappointing lately. This has everything to do with the relationship between what I'm currently thinking about and the specific material I'm looking into, rather than the books and articles by themselves.

But Ohlsson's 2011 book Deep Learning is so far a wonderful exception to the six or seven books collecting digital dust inside my Kindle, waiting for me to be interested in them again. The reason, I think, is that Ohlsson is looking to tackle a topic that I am incredibly suspicious about—insight, creativity—in a smart and systematically theoretical way. The desire to provide technical, functional, connected explanations of concepts is evident on every page.

Prior Knowledge Constrains the Solution Space

Of particular interest to me is the idea that prior knowledge constrains a 'problem space,' or what Ohlsson wants to re-classify as a 'solution space':

A problem solution consists of a path through the solution space, a sequence of cognitive operations that transforms the initial situation into a situation in which the goal is satisfied. In a familiar task environment, the person already knows which step is the right one at each successive choice point. However, in unfamiliar environments, the person has to act tentatively and explore alternatives. Analytical problem solving is difficult because the size of a solution space is a function of the number of actions that are applicable in each situation—the branching factor—and the number of actions along the solution path—the path length. The number of problem states, \(\mathtt{S}\), is proportional to \(\mathtt{b^N}\), where \(\mathtt{b}\) is the branching factor and \(\mathtt{N}\) the path length. \(\mathtt{S}\) is astronomical for even modest values of \(\mathtt{b}\) and \(\mathtt{N}\), so solution spaces can only be traversed selectively. By projecting prior experience onto the current situation, both problem perception and memory retrieval help constrain the options to be considered to the most promising ones.

So, prior knowledge casts a finite amount of light on a select portion of the solution space, illuminating those elements which are consistent with representations in long-term memory and with a person's current perception of the problem. It may even be the case that the length of the beam from the prior-knowledge flashlight corresponds to the limitations of working memory.

Crucially, this selectivity creates a dilemma. It is necessary to limit the solution space—otherwise, a person would be quickly overwhelmed by multiple, interacting elements of a problem situation—but, as is shown, prior knowledge (among other things) may restrict activation to those elements in the solution space which are unhelpful in reaching the goal.

It may be a goal, for example, for students to have a flexible sense of number, such that they can estimate with sums, products, differences, and quotients over a variety of numbers. Yet, students' prior knowledge of working with mathematics can lead them to activate (and thus 'see') only 'narrow' procedural elements of solution spaces. The result can be that procedural mathematics is activated even when it serves no useful purpose at all.

This 'tyranny' of prior knowledge effects can be seen in the classic Einstellung experiments, a version of which is below—originally included in Dr Hausmann's write-up on the topic, which I recommend highly. The goal below is to simply fill up one of the "jars" to the target level (the first target is 100). When you're done, head over to Dr Bob's site for the explanation.

A: 21

B: 127

C: 3

Target: 100

But Insight Theories Are Not Learning Theories

If one wanted to provide a slightly more serious intellectual justification for much of the popular folk-theorizing in education over the last decade—and then essentially replay its development, idea by idea—misinterpreting insight and creativity theories like Ohlsson's would be an excellent strategy for doing so. (He never says, for example, that simply prior knowledge constrains the solution space, but that unhelpful prior knowledge does.)

It all seems to be there for the taking in these kinds of theories: the notion that solving problems is education's raison d'etre, the idea that an unknowable future—rather than being just a fact that we must accept—can play a part as a premise in some chain of reasoning, the bizarre thought that removing instructional support can represent a game-changing way of restructuring the majority of learning time, a fluttering emphasis on collaboration and distributed cognition. All of this that has been humming in the background (and foreground) for a while in education fits comfortably and rationally inside creativity theories rather than learning theories.

From a learning-theory point of view, the problem of, say, thinking flexibly about number is primarily a problem of constructing better solution spaces—bring the goal within the flashlight's view by instructing students (and thus making activation of 'number sense' more likely). Unfortunately, this requires a longer-term view, greater political will, and a bit of distance from everyday reality. Insight theories, on the other hand, assume that this number sense is already there but remains inert (students are only ever experiencing 'unwarranted impasses'). The problem for insight theory becomes simply how to redirect the flashlight's beam so that it uncovers the right knowledge. Along with the background assumptions listed above, these further assumptions of insight theories make them remarkably well tuned to the constraints of institutional teaching, both self-imposed and externally imposed. The practical work of teaching is still mostly a one-year-at-a-time affair, and sixth grade teachers, for example, do not have the time to remake solution spaces anew over the course of one year. What is within reach are redirection techniques suggested by insight theories. In this context, misinterpreting theories about insight for learning theories is practically inevitable.

Perhaps I'll find out where Ohlsson makes learning theory and insight/creativity theory connect as I read further. But it's worth noting that research Ohlsson himself conducted after the publication of this book has produced conclusions that run counter to certain predictions within it.

We'll see!

Audio Postscript

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Holding Back: Inhibition

2016-01-27T17:42:00.000-05:00

The difference between successful people and really successful people is that really successful people say no to almost everything.

I doubt I'm much different from other people in that I tend to think about intelligence, academic success, smarts, whatever you want to call it, as something achieved by adding good things, whereas the opposite indicates a lack of good things. Or, in other words, positive academic outcomes are—it occurs to me, before I self-censor—caused by turning on some internal and external power sources, whereas with poorer outcomes there are stuck valves, leaky pipes, broken or frayed wires, somewhere.

These quasi-unconscious metaphors about people's mental states show up in idioms like bright (positive luminescence) and dim, not playing with a full deck (lacking cards), on the ball (positive causal force), beyond him/her (not the causal force), made sense of (positive), lost the plot/lost the thread/miss the point (negative), and sharp or dull.

It is natural, then, that I often forget that education doesn't work in just that one way. Positive outcomes are not always (or solely) achieved by making things happen, but often by stopping things from happening.

Huge Mice and Tiny Elephants

This study, which you can read online for free (thank you!) demonstrates one way in which "stopping things from happening" is an important—in fact, possibly critical—factor in learning and excelling at mathematics.

Researchers gave 58 children between the ages of 3 and 6 a Stroop task using pictures of animals (such as mouse and elephant) presented together either with their correct relative sizes (mouse much smaller than elephant) or their incorrect relative sizes (both pictures the same size). The children were asked to choose the animal that was largest in real life.

In order to perform on Stroop tasks, participants must inhibit responses to more salient characteristics of stimuli. And this inhibition can be measured. In the classic Stroop task, participants are asked to name the color of text. Color names are printed so that the color of the text matches the word (e.g., red, green, blue) or doesn't match (e.g., red, green, blue). It has been found that the words themselves interfere significantly with performance; i.e., it takes less time to respond that the color of blue is blue than it takes to say that red is green. In the animal size Stroop task, children must inhibit responses to the pictured sizes of the animals in order to produce a correct response regarding the animals' relative sizes in real life.

What researchers discovered in this study is that performance on the animal size Stroop task was a significant predictor of mathematics achievement, as measured by the Test of Early Mathematics Achievement—a finding that simply adds to a large and growing evidence base:

Cognitive development in the domain of number representation may therefore mean learning to ignore competing task-irrelevant dimensions of the stimulus. Indeed, a study in 3- to 5-year-old children from low-income families found that performance on a magnitude comparison task related to math achievement, but that this relationship was driven by trials that required inhibiting an irrelevant stimulus dimension (surface area) to select the larger numerosity (Fuhs and McNeil, 2013). This suggests that the ability to ignore irrelevant perceptual information and focus on number may explain why inhibitory control relates to early numeracy.

Discussion

It is not clear to me—certainly not from this study—that inhibition can be taught. I remember years ago being required to write math problems containing unnecessary information so that students would have to choose the information that they needed. But just making kids do something is not the same thing as teaching them something. It is, rather, a total cave to assessment obsession—we just found a way to call assessment "instruction".

If inhibition can be taught, then it seems to me that we should try to be more knowledgeable about its effects and importance and then explicit in our instruction about how to apply this skill—particularly in mathematics. (See this post for a related discussion.)

Our intuitions—and even our language, as I mentioned above—prime us to see the clip below as an example of brilliant connection-making (positive). But try the problem if you haven't seen it already, and watch the solution. How much of the difficulty of this problem is captured in the unnecessary salience of the concrete scenario?

How much closer to hand might mathematics be to the unbrilliant if we could teach them how to strategically ignore the buzzing confusion of the real world?

Merkley, R., Thompson, J., & Scerif, G. (2016). Of Huge Mice and Tiny Elephants: Exploring the Relationship Between Inhibitory Processes and Preschool Math Skills Frontiers in Psychology, 6 DOI: 10.3389/fpsyg.2015.01903

Perplexity Is Not Required for Learning

2016-01-16T23:35:00.000-05:00

Let me see if I can accurately describe the results of the first experiment from this study. But before I do, watch this video:

The video features a student, Justine, faced with a 'cognitive conflict'—one between her notion that objects fall at different rates depending on their weight and the scientific reality that objects fall at the same rate under the force of gravity.

Importantly, according to the video this conflict seems to be a necessary part of teaching and learning, as Miss Reyes suggests at about 0:54:

Look, I've been teaching for 12 years, and trust me, these students are anything but blank slates. They may be here for 8 hours a day, but for the other 16, they're in all kinds of classrooms: the dinner table, karate lessons, their sneeper-peeper feed, trashy vampire novels . . . My point is that these kids are out there, living life, constantly learning things.

Thus, for learning to occur, student thinking must be probed for misconceptions, and these misconceptions must be patiently challenged—i.e., cognitive conflict (or in some versions, perplexity or incongruity) must be induced. This I believe is the most widely subscribed version of the conceptual change model of learning. It is the prescriptive cousin of empirical and philosophical work that has assigned the cause of conceptual change to "the drive to make sense of anomalous observations that are inconsistent with existing concepts."

Change Without Conflict

Yet, in recent years—more recent than the papers cited in the link at the end of the video—the construct of conceptual change has met with a great deal of challenge and has not found much robust empirical support, leading the author of the study I examine here to write this fairly bold statement (emphasis mine):

Since Limon’s (2001) review, cognitive change researchers have investigated the influence on conceptual change of a broader range of cognitive and non-cognitive factors, including affect and motivation, individual differences, metacognition, epistemological beliefs, and intention to change. . . .

Taken at face value, the relative lack of effect of such conflicts across a broad range of studies falsifies the cognitive conflict hypothesis: The difficulty of conceptual change must reside elsewhere than in conflict, or rather the lack thereof, between misconceptions and normatively correct subject matter.

In fact, in their first experiment, Ramsburg and Ohlsson found that even inducing conflict via the age-old method of telling students they were wrong did not have a relatively significant impact.

Over several trials, the researchers first taught 120 undergraduates a misconception regarding visual characteristics of bacteria that make them oxygen-resistant (e.g., 'contain black nuclei'). Seventy-four of these students learned the misconception 'to mastery.' Then, for one group of students, called the complete condition, a further set of trials switched up the critical characteristic in the images, allowing students to see immediately the visual evidence disconfirming their misconception. For example, images of bacteria with black nuclei were shown with accompanying feedback informing students that these were not oxygen-resistant.

A second group of students, called the confirmatory-only condition, by contrast, were never presented with disconfirming evidence of their prior misconception in subsequent trials. All images that did not show the new oxygen-resistant characteristic also did not show the black nuclei. Thus, this group saw only positive examples of the new characteristic; their prior misconception was not explicitly challenged in the image trials.

As you might expect at this point, though it is still somewhat suprising in the context of common wisdom, no significant differences were found between the groups' abilities to learn the new characteristic after the misconception had been taught. No significant differences were identified in the rate of learning either. The second and third experiments in this study replicated these results, even after attempting to control for some weaknesses in the first experiment and strengthening the complete 'conflict' condition.

What Does This Mean for Justine?

Given the results of this study and similar results over recent years, we can at least conclude that it is wrong to suggest that provoking perplexity or inducing dissonance or cognitive conflict is necessary for learning. And even 'softer' claims about conceptual change probably deserve a great deal of suspicion and scrutiny. However, as the authors describe in some detail in the study, there are a number of methodological and conceptual challenges one faces in experimenting with this construct. So, it is by no means time to cut the funding for conceptual change research.

An intriguing question, I think, is why perplexity and incongruity don't work (to the extent that they don't). Why might it not be necessary—or even important—to induce a conflict between what students know and what they don't? I suspect the beneficial effects of cognitive conflict are mediated by prior knowledge (well, what isn't?), in favor of those with more of it.

Ramsburg, J., & Ohlsson, S. (2016). Category change in the absence of cognitive conflict. Journal of Educational Psychology, 108 (1), 98-113 DOI: 10.1037/edu0000050