Bloom’s Against Empathy

empathy

I‘m on my way out the door to be on vacation, but I wanted to mention (and recommend) Paul Bloom’s new book, Against Empathy: The Case for Rational Compassion, before I do—you know, to put you in the holiday spirit.

Bloom makes a strong case that empathic concern acts as a spotlight—inducing a kind of moral tunnel vision:

Empathy is a spotlight focusing on certain people in the here and now. This makes us care more about them, but it leaves us insensitive to the long-term consequences of our acts and blind as well to the suffering of those we do not or cannot empathize with. Empathy is biased, pushing us in the direction of parochialism and racism. It is shortsighted, motivating actions that might make things better in the short term but lead to tragic results in the future. It is innumerate, favoring the one over the many.

In line with Bloom’s narrative, I would say that the short-sightedness of empathy is what makes students’ boredom more salient than students’ lack of prior knowledge. The innumeracy of empathic concern leads to a valorization of personalization and individualism at the expense of shared knowledge of a shared reality. And its bias? I’m sure you can think of a few ways it blinkers us, makes us less fair, maybe leads us to believe that a white middle-class definition of “success” is one that everyone shares or that everyone should share.

Perhaps next year we can talk about how in-the-trenches empathy is not such a great thing, and that perhaps we need less of it in education—and more rational compassion.


The Repeated Addition Mindset

repeated addition

On the number line at the right, the tick marks are all evenly spaced and the values for the tick marks increase from left to right. One can perform repeated addition of 2 from the rectangle to arrive at the value for the circle or repeatedly subtract 2 from the circle to get to the rectangle. In other words, you can determine the additive relationship between the values for the circle and rectangle.

What you cannot do, however, is determine the multiplicative relationship between the values. If the rectangle is at 4, then the circle is at 18, and \(\mathtt{\frac{18}{4} = 4.5}\), which means nothing in the context of this number line.

repeated addition

We can also determine a multiplicative relationship without being able to identify an additive one. Given the same assumptions as above, we know that the value associated with the square is, without a doubt, \(\mathtt{\frac{1}{4}}\) the value for the rectangle. But there’s no way of telling what the distance between them is.

Multiplication Is Not Repeated Addition

Although getting into the math is not the reason for my writing this post, I want to stick with the above contrast briefly. The point is simple, but (believe me) hard to swallow: no matter how tightly connected the two operations are, no matter how many years you have taught it this way or how fine you turned out as an adult after learning it this way, no matter Peano or extensionality, multiplication is not repeated addition. It is not that it is “not just” repeated addition. And it’s not “just semantics.” Multiplication is not repeated addition. The two are not the same.

One important reason for the distinction, in my view, is that it forces us as educators to “level up” to the multiplicative and treat it as basic and fundamental. We are forced to connect multiplication to more intuitive operations, like scaling and stretching, which in turn means that students will have more direct psychological access to ratio, scale, unit rate, slope, and on and on, rather than having to build everything up from the numerical-additive every time a new concept is introduced.

In one of his famous classic lectures (39:42), Richard Feynman perfectly explains why multiplication is to be chosen over repeated addition, even if they lead to the same intuitive conclusions in every context in which they are applied. Except, of course, Feynman is talking about physics. He says, referring to three ‘different’ laws of motion:

These theories are exactly equivalent. The mathematical consequences in every one of the different formulations of the three formulations—Newton’s laws, the local field method, and this minimum principle—give exactly the same consequences. What do we do then? . . . They are equivalent. Scientifically, it is impossible to make a decision. . . . Psychologically, they are different because they are completely unequivalent when you go to guess at a new law . . . they become not equivalent in psychologically suggesting to us the guess as to what the laws might look like in a wider situation.

So it is with repeated addition and multiplication. Absorbing multiplicative reasoning into our bones, treating multiplication as fundamental rather than derived, allows us to “guess at” what’s going on in a lot of middle-school and later mathematics. For most of these topics, like slope and scale, a repeated-addition intuition will look like no intuition at all.

Insanity: Repeatedly Adding and Expecting Multiplication

The difficulties we have moving away from repeated addition and toward multiplication in mathematics are reflected in how we think about more everyday things too.

I know I’m not alone in having been exposed to the idea that simply adding on more opinions, more voices, more collaboration, is an absolute good—one that, if we just add enough, can accomplish anything. And I’m not alone in having watched that fail more often than not. Post-mortem analysis tends to reveal (surprise, surprise) that we didn’t add on enough, we didn’t work hard enough, we weren’t good enough.

This is the repeated addition mindset: add on to solve problems.

In education, it looks like this (to me): Whatever good thing we can do for students or whatever bad thing we can avoid doing, the message about this thing is delivered in the exact same way to the whole of education, no matter the grade level. If the message is that X is good, the message is to do it in first grade, then again in second grade, and third grade, add on, add on, repeatedly.

This way of delivering messages betrays an assumption about the purpose of schooling: that it is not designed to systematically scale up student knowledge and ability, but to provide practice for the same general “skills” over and over. Not only that, but it paints a picture of an education system that is not really a system at all—just a series of stations, manned by employees with no responsibility to each other.

The multiculturalists of the 1980s and 1990s accepted a too romantic, essentialist view of language that helped fragment the school curriculum. They seemed to believe that Americans could transcend particularity, that we did not need communal knowledge shared by all but could happily exist as a universe of separate cells: out of many, many. These cells could then all function together if students achieved critical-thinking skills. But neither the critical-thinking idea nor curricular fragmentation has worked out for the social groups that these ideas were supposed to help. Gap closing has stagnated; the achievement gap persists.

–E.D. Hirsch, Jr.


The Appeal to Common Practice

appeal to common practice

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. When we are tempted to justify a teaching practice based on these knowns, we can run the risk of making an appeal to common practice.

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.


Searching the Solution Space

creativity

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 topics that I am incredibly suspicious about, insight and 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 similar obstacle to learning, described by S. Engelmann in his work, is called the problem of stipulation.

A: 21
B: 127
C: 3
Target: 100
0
0
0







But Creativity 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 and creativity 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



Help Me Explain These Results

Two questions from a survey I had in my research stash from a while ago (it was real; it just wasn’t public): (1) What is your greatest challenge as a math educator? and (2) What do your students struggle with the most? The responses for motivation and prior knowledge are interesting. Here is a visual for only 4 of the responses to each question:


I was curious to see how folks in the K12 Math Ed Community on Google+ would respond to these questions (the wordings of the choices below are paraphrases of the wordings in the larger survey):

motivation

The results show essentially the same shape for both groups: in both (on a superficial reading), educators’ main challenges don’t seem to be a fit for students’ main struggles. And, importantly, they could have been. That is, if boredom and anxiety were higher, one could conclude that these problems were generating the motivation challenge for teachers, but in neither group of results do ‘boredom’ and ‘anxiety’ match ‘lack of foundational knowledge’ as a source of student struggle—even when you add them together.

So, what’s the right spin on these results? Do you see the same disconnect I’m seeing?

P.S.: I was reminded just before posting this about the study explained in the video at the right. This quotation from the video seems to be related to the results above:

As the percentage of students in the first grade classroom with math difficulties increased, teachers tended to increase their use of . . . movement or music to teach mathematics or increased use of manipulatives or calculators.

Reading Between the Lines

Frankly, these results make me worry. I worry because I think that while motivation is a potential problem for every student, motivation is a primary problem mostly for rich and otherwise well-resourced students. I’ve hinted at this before. And I’m not alone in this worry. Former teacher and administrator Eric Kalenze writes, in his book Education Is Upside Down:

The divide continues to grow. America’s “have-not” students spend much of their school career being coaxed into engaging with learning (and not necessarily learning what will help them academically or institutionally), while the “have” students, pre-engaged in school tasks by virtue of birth into an alignment with mainstream institutional expectations, receive the academic rigor that will propel them into rewarding post-secondary study and lucrative careers.

Fifteen year veteran of the classroom and current dean of the School of Education at the University of Michigan has noticed too. Her presentation at the 2015 meeting of the National Council of Supervisors of Mathematics implored educators to be explicit in their instruction for the sake of equity:

Requesting is not the same as teaching; when rich mathematical tasks and situations are used and students are left to puzzle about them on their own, likely will privilege those who have had opportunities with “experimenting with possibilities” and overcoming “broader societal and cultural views of what mathematics is and who is good at it.” [Slide 17]

And this post, “Knowledge Equality”, by Lisa Hansel from E.D. Hirsch’s Core Knowledge Foundation makes a passionate call for an education system that prioritizes knowledge for all:

What I mean by knowledge equality is all children having equal opportunities to learn the academic knowledge that opens doors. The knowledge that really is power. The knowledge that represents the history of human accomplishment. The knowledge that stands the test of time because it is beautiful.

The knowledge that privileged children acquire at home, in libraries and museums, and in school.

While it certainly makes some sense for motivational techniques to be used to address deficits in foundational knowledge, it doesn’t make that much sense. And I see in these results yet another indication of what my own observations and conversations with teachers and other education stakeholders both on and offline point to: that schooling as an institution is being pulled increasingly toward serving the needs of the few and well connected and away from serving the educational needs of the many. It’s a view of education’s priorities that is being sold by business and technology “gatekeepers” and their accomplices rather than demonstrated and proven by careful public scientific work.

For the next school year, and the one after that, administrators and policymakers should summon the courage to refocus the energy and resources of schools toward the “boring” technical work of building all students’ foundational knowledge—and help their people develop an immunity to untested, unrealistic motivational jibber-jabber.


Image mask: Marc Smith

motivation

Whited Sepulchres

A central—and remarkable—argument in Steven Pinker’s recent work, The Better Angels of Our Nature, is that a decline in collective moralization may be a significant cause of the decline over time in human violence. In other words, less morality (or rather, “morality”), less violence:

The world has far too much morality, at least in the sense of activity of people’s moral instincts . . . . the biggest categories of motives for homicide are moralistic. In the eyes of the perpetrator, of the murderer, it’s capital punishment—killing someone who deserves to die, whether it’s a spouse who’s unfaithful or someone who dissed him in an argument over a parking space or cheated him in a deal. That’s why people kill each other. . . .

The human moral sense does not consist of a desire to maximize well-being, to prevent people from harm. But it is a hodgepodge of motives that include deference to a legitimate authority, conformity to social and community norms, the safeguarding of a pure divine essence against contamination and defilement.

This idea helps me put some language around my discomfort with a lot of education discourse outside the policy and research levels. We moralize far too much about teaching and learning there. Or, rather, we moralize badly too often. Our “ought”s are not centered in the empirical, but in the ideal. Consider:

“Children, go get dressed for dinner. A family should look their best at mealtimes together” is moralizing. “Children, go get dressed for dinner. I have an important client coming over, and I want to impress her” is not. The reasoning attached to the second request is embedded in a real-world reality. He wants to impress a client, so he asks the children to get dressed for dinner. By contrast, the reasoning used to support the first request is rooted in “conformity to social norms.” The speaker wants the children to get dressed for dinner because doing so will bring them (and him) closer to an ideal he has in his head. Similarly, “Doctors are gentlemen, and gentlemen’s hands are clean” is moralizing—an idealistic “ought” (in this case, an “ought not”) untethered to reality.

In education, we have that students shouldn’t just sit in rows and listen to a teacher; that they should persevere and fail; that we should be less helpful; that students ought to create on their own, collaborate, and behave like real scientists and mathematicians do. To the extent that these are simply ideals for what students “ought” to be like, disconnected from evidence, they are moralizings: visions of a “pure and divine essence”; pictures in our heads of self-reliant, creative, free, and mature students; pictures that are, however well-intentioned, divorced from reality. It seems to me that in many ways the reforms inspired by these moralizings simply succeed in making children pretend they are accomplished, so that the adults can feel good about themselves.

If, as the research Pinker references suggests, our moral instincts are not as well calibrated as we think they are for modern life, and the population of “ought”s in our community is not controlled by predatory “is”s delivered by scientific thinking, we should be, at the least, increasingly wary of educational moralizing rather than increasingly comfortable with it.

Image mask: Ivar Gullord


Audio Postscript

morality