Independent Events

Honestly, sometimes my first off-the-cuff measure of how well we do collectively with a topic in mathematics is how well I do with it. This is terrible reasoning, of course, but it has some uses as a kind of first measurement that needs to be independently verified. It helps me notice potential weak spots in instruction, anyway, like with independent events.

The concept of independent events is certainly a candidate for being a weak spot. Most of what I’ve seen online doesn’t really crack the surface. Students are allowed to explore the concept of independent events, but all that seems to mean is to take a situation and tell me whether the events are independent. Big whoop.

And the definition often used—that two events are independent if the occurrence of one does not affect the probability of the other—has the potential of really confusing students (and me). Consider this situation.

independent events

Are the events “selecting a letter card” and “selecting a circle card” independent events? The way a student (and I) might reason, given just the definition above, would be to think, “Well, if I pick B, that would definitely change the probability of picking a circle, because the B-card is also a circle card. And, if I pick A, that would change the probability of picking a circle card, because then I would only have three cards to choose from. So, the events are not independent.”

Try This Instead

The above was perfectly sane reasoning; it’s just wrong because of a terrible explanation (or lack of an explanation in most cases). I thought of something maybe a little better. Here it is in sentence form, similar to how we worked on the impenetrability of trig ratios:

You have the same probability of choosing a letter card from all the cards as you have of choosing a letter card from just the circle cards.

This is what makes choosing a letter card and choosing a circle card “independent.” If I know that I’ve drawn a circle, the probability that I’ve also drawn a letter, \(\mathtt{0.5}\), is the same as if I didn’t know I’d drawn a circle. And of course this works automatically the other way around too: if I know that I’ve drawn a letter card, the probability of drawing a circle is the same as if I didn’t know. If I reduce the sample space from all cards to circle cards, the probability of “letter” is the same.

At the heart of independent events (besides the conditional probability flavoring above) are equivalent ratios, or a proportion . . . which gives me an ultra-short way of saying it a little more mathematically:

\[\mathtt{\frac{2}{4} = \frac{1}{2}}\]

Or, “2 letter cards out of 4 cards in all is the same as 1 letter card out of 2 circle cards.” In the symbolism of probability, we actually write this with complex fractions (by giving each numerator and denominator above a denominator of 4), and then disguise the complex fractions with \(\mathtt{P()}\) statements, which is all equivalent to the above proportion:

\[\mathtt{\frac{\color{purple}{\frac{2}{4}}}{\color{red}{\frac{4}{4}}} = \frac{\color{red}{\frac{1}{4}}}{\color{purple}{\frac{2}{4}}} \longrightarrow \frac{\color{purple}{P(\textrm{letter})}}{\color{red}{1}} = \frac{\color{red}{P(\textrm{letter and circle})}}{\color{purple}{P(\textrm{circle})}}}\]

And that gives us the definition of independence using conditional probability, from S-CP.A.3. If we remember our proportion work from way back when, then the “other” test for the independence of two events pops out of the equivalence of the products of means and extremes: \[\mathtt{P(\textrm{letter}) \cdot P(\textrm{circle}) = P(\textrm{letter and circle})}\]

The complex fraction part of this explanation seems to be the most important, actually. And we don’t really do a good job of letting kids in on that disguise either. But still, stapling the idea of independent events to a pair of equivalent ratios (a proportion) helps the whole idea make a lot more sense to me. And, truthfully, it makes the notion of “independence” as “not having an effect on another probability” seem almost wrong.

Update: This kind of reasoning works for the typical example of independent events. The situation involving separate spinners is fairly easy for kids to identify as being about independent events, but like a lot of other topics in mathematics education, we start off with examples that are easy and also completely misleading. Then we all opine that kids have difficulties because the material gets “harder.” Anyway, spinning a C on the first spinner and a 2 on the second spinner are independent events, but not because there are “independent” spinners.

What’s the proportion (if the events are independent) that matches the situation?

independent events

Assuming we spin the first spinner and don’t know what we get, there are \(\mathtt{3 \times 1}\), or 3, outcomes that have 2 as the second spin, out of 12 possible outcomes. The outcomes are {(A, 2), (B, 2), (C, 2)}. But, if we know that we have spun a C first, then there is 1 outcome showing 2 on the second spinner, out of 4 possible outcomes. So, our proportion is \[\mathtt{\frac{3}{12} = \frac{1}{4}}\]

This is all we need to show that the two events are independent, actually. If that proportion is true, then the events are independent. But we can cue the complex fraction magic again for reinforcement: \[\mathtt{\frac{\color{purple}{\frac{3}{12}}}{\color{red}{\frac{12}{12}}} = \frac{\color{red}{\frac{1}{12}}}{\color{purple}{\frac{4}{12}}} \longrightarrow \frac{\color{purple}{P(2)}}{\color{red}{1}} = \frac{\color{red}{P(\textrm{C and 2})}}{\color{purple}{P(\textrm{C})}}}\]

Living in a World Full of Answers


I‘m really enjoying Ulrich Boser’s new book Learn Better. It is nicely balanced, humble, serious and informative, with a good tempo and a somewhat answers uncanny ability to make me spin to the side in my office chair every once in a while to rest my chin in a finger tent.

In particular, a section on the theme of embracing difficulties in learning got my chair spinning and fingers tenting. Here’s just one snippet:

The practical takeaway here is pretty simple. We need to believe in struggle. We need to know that learning is difficult. What’s more, we need the people around us to believe, too. . . .

This idea is at the heart of Lisa Son’s approach. She’s building norms around the nature of effort, the essence of struggle, the path to expertise. As Son told me, laughing, “I think I overdid it, but if someone gives my kid the answer, she’ll kill you.”

What I would add to this, though, is that, what typifies “struggle” for students in adulthood is not figuring things out for themselves when no one will give them the answers; it’s figuring things out for themselves in a world awash with answers. Students need to be able to deal with answers—from experts and from their peers—while keeping the lights of critical thinking on upstairs. That’s the struggle. And you don’t get practice with that struggle when you spend a lot of your schooling time stuck in a goobery game of hint-hint hide-and-seek with answers.

Students need practice dealing with “answers” from other people—from people of different races and religions, or no religion; from people whom you don’t like or who don’t like you; and from people who are more expert or less expert than you. And they need to be able to figure out that some of those answers are correct, and some of them are not, and other times there is no solid answer even when everyone else is convinced there is, or there is a solid answer when everyone else is convinced there isn’t. Students need practice listening to other people and understanding what they are saying, without feeling that their identity or cognitive liberty has been threatened.

I have to think that, while withholding answers is a good technique to use occasionally (and deliberately and skillfully), as a strategy it can run the risk of producing a generation of narcissistic idiots who close their ears to “answers” they themselves didn’t come up with.

Postscript: The subject of embracing difficulties comes up a little later in the book as well:

As a learning tool, DragonBox does not seem to teach students all that much, though, and people who play the game don’t do any better at solving algebraic equations, according to one recent study. Researcher Robert Goldstone recently examined the software, and he told me that the app didn’t appear to provide any more grounding in algebra than “tuning guitars.”

In the bluntest of terms, there’s simply no such thing as effortless learning. To develop a skill, we’re going to be uncomfortable, strained, often feeling a little embattled. Just about every major expert in the field of the learning sciences agrees on this point. Psychologist Daniel Willingham writes that students often struggle because thinking is difficult.

It’s true that thinking is hard work, actually, but it’s also true that learning is difficult in part because what you are learning, whatever it is, was created by people who think differently from you (at the moment, if you’re a novice). And school—when it is not mostly a game of “guess what’s in my head”—is one of the first places young people are exposed to this kind of thinking.

Trig Ratios as Percents

My audience is mostly folks interested in math education in one way or another, so it’s no use starting this post off with “All you may know about trigonometry ratios is likely captured in the gibberish mnemonic SOHCAHTOA.” Your understanding of trigonometry ratios is no doubt more sophisticated than that.

But have you thought about trig ratios as percents? This will be enough for most of you:

sin θ = \(\mathtt{\frac{opposite}{hypotenuse} = \frac{?}{100}}\) = percent of hypotenuse length

It makes sense when you dredge up the 6th-grade math you remember and start making connections between it and the trigonometric ratios sine, cosine, and tangent (for example). After all, opposite : hypotenuse is the sine ratio, but it’s also just a ratio. If we think of it as a percent, we could say that if the sine of a reference angle is equal to 0.75, that means that the side opposite the angle in a right triangle is 75% the length of the hypotenuse. If the cosine were 0.75, that would mean that the side adjacent to the reference angle is 75% the length of the hypotenuse, since cosine is the ratio adjacent : hypotenuse. And a tangent of 0.75 means that the opposite side is 75% the length of the adjacent side, because tangent is simply the ratio opposite : adjacent.

The percent connection (or fraction; doesn’t have to be percent) strikes me as being immediately more useful for seeing meaning in values for trigonometric ratios. They usually go by students as just values which can’t be put into a sentence—a long list of changing decimals in a lookup table. Yet, the percent connection is right there, waiting for us to combine our middle school math knowledge with new material. We could model what this process of meaning-making actually looks like, rather than just ask them to go make meaning and hope for the best.

Of course, it also helps to be able to visualize what a sine of 0.75 looks like. Try, say, \(\mathtt{49^\circ}\) below on the unit circle and press Enter. That gives me something that looks pretty close to a sine of 0.75 (an opposite side that is \(\mathtt{\frac{3}{4}}\) the length of the hypotenuse, right?).

  θ = °


But the interactive tool, while helpful maybe, isn’t necessary, I don’t think. One can think about drawing a right triangle, say, with an adjacent side length about 80% of the hypotenuse length (a cosine of about 0.8). It will have to be longer than it is tall, relative to the reference angle, to make that work. The percent connection thus links a trigonometry ratio value to a simple and accessible visual.

An Example Problem: Testing Out the Percent Connection

41° 96 x

The basic mathematical (as opposed to contextual) trigonometry practice problem looks like this: Determine the length of \(\mathtt{x}\).

I can’t say the percent connection makes this a faster or more efficient process. What I would say is that knowing that the sine of 41° means the percent of the hypotenuse length represented by the opposite side length makes me feel like I know what I’m doing, other than moving numbers and symbols around. (Thinking about percents also gives us a way to estimate what my \(\mathtt{x}\) will be, if I know that the figure is drawn to scale.)

The sine of 41° is approximately 0.65605902899. With the percent connection, I know that this means that the opposite length is about 65.61% the length of the hypotenuse. It’s hard to overstate, I think, how useful it is to be able to wrap all of this number-and-variable work into one sentence like this: 96 is about 65.61% of x. I can climb the last few steps from there, by either dividing or setting up an equation—however the work happens, I at least have some background meaning to the numbers I’m playing with.

We can continue from there, of course (as we can without the percent connection, but so rarely do because the tedium of setting up and solving for the variable has overloaded us). The tangent of 41°, approximately 0.86928673781, tells us that the opposite side is about 86.93% the length of the adjacent side.

This guy gets it, and he seems to be the only one. It shouldn’t come as any surprise that he’s an experienced mathematics teacher a computer scientist who’s never taught. But, you know, it really should surprise us. Someday.


Motivation Is Caused by Achievement


It doesn’t seem right that doing well in math should cause students to have more intrinsic motivation and not the other way around. But this is just what child development researchers found recently, published here at the beginning of last year. In a large sample of students in Grades 1 to 4, the paper’s authors discovered that

achievement predicted intrinsic motivation from Grades 1 to 2, and from Grades 2 to 4. However, intrinsic motivation did not predict achievement at any time.

One reason this may seem incorrect even though it may be correct is that the way we talk about—and thus think about—causality in human affairs evolved long before we were a species that conducted experiments on people. Each of us inherits a language developed by a predominantly dualist, animist, and creationist culture, which spoke about minds, separate from the natural world, that effect change on that world, not the other way around:

It is like quantum physics; we may intellectually grasp it, but it will never feel right to us. When we see a complex structure [like mathematics achievement—JF], we see it as the product of beliefs and goals and desires. Our social mode of understanding leaves it difficult for us to make sense of it any other way. Our gut feeling is that design requires a designer.

intrinsic motivation

Thus it seems backwards to us to suggest that the internal state of a designer (e.g., his ‘motivation’) should have no significant effect on his design (e.g., his mathematical performance). And it seems truly bizarre that the opposite, in reality, is the case. But, again, that is what the results reported here suggest.

The diagram shows the significant cross-grade correlations unearthed in the study. There was a significant correlation between achievement and motivation from Grade 1 to Grade 2 and from Grade 2 to Grade 4. There was no similar correlation from motivation to achievement across grades.

Don’t Stop. Believin’.

There are many caveats, as there are with any study. You can take a look yourself at the final manuscript available online. Motivation was self-reported. Achievement was measured using two standardized assessments. The whole study was an exercise in data mining. Etc. It is worth taking a look, too, at the authors’ discussion of previous research addressing similar questions and the weaknesses of those studies.

One thing I find interesting is this part of the authors’ conclusion, under the heading of “Implications for educational practice”:

Interventions in education try to increase intrinsic motivation, and hopefully achievement through promoting students [sic] autonomy in instructional setting [sic] (e.g., opportunity to select work partners and assignment tasks; Koller et al., 2001). The present findings could mean that these practices may not be the best approach in the early school years (Cordova & Lepper, 1996; Wigfield & Wentzel, 2007).

That’s it as far as implications, which seems a bit thin. Not even the vanilla suggestion that interventions designed to increase achievement may be better uses of time than those designed to increase motivation? Because that’s the real implication here.

Garon-Carrier, G., Boivin, M., Guay, F., Kovas, Y., Dionne, G., Lemelin, J., Séguin, J., Vitaro, F., & Tremblay, R. (2016). Intrinsic Motivation and Achievement in Mathematics in Elementary School: A Longitudinal Investigation of Their Association Child Development, 87 (1), 165-175 DOI: 10.1111/cdev.12458

Educational Achievement and Religiosity


educational achievement

I outlined a somewhat speculative argument that would support a prediction that increased religiosity at the social level should have a negative effect on educational achievement here, where I suggested that

Educators surrounded by cultures with higher religiosity—and regardless of their own personal religious orientations—will simply have greater exposure to concerns about moral and spiritual harm that can be wrought by science, in addition to the benefits it can bring.

Such weakened confidence in science may not only directly water down the content of instruction in both science and mathematics—by, for example, diluting science content antagonistic to religious beliefs in published standards and curriculum guides—but could also represent an environment in which it is seen as inartful or even taboo for educators of any stripe to lean on scientific findings and perspectives in order to improve educational outcomes (because nurturing children may be seen to be the provenance of more spiritual and less scientific approaches). Both of these effects, one social, one policy-level, could have a negative effect on achievement.

A new paper, coauthored by renowned evolutionary psychologist David Geary, shows that religiosity at a national level does indeed have a strong negative effect on achievement (r = –0.72, p < 0.001). Yet, Stoet and Geary’s research suggests a different, simpler mechanism at work than the mechanisms I suggested above to explain the connection between religiosity and math and science educational achievement. This mechanism is displacement.

The Displacement Hypothesis

It’s a bit much to give this hypothesis its own section heading—not that it isn’t important, necessarily. It’s just self-explanatory. Religiosity may be correlated with lower educational achievement because people have a finite amount of time and attention, and spending time learning about religion or engaging in religious activities necessarily takes time away from learning math and science.

It is not necessarily the content of the religious beliefs that might influence educational growth (or lack thereof), but that investment of intellectual abilities that support educational development are displaced by other (religious) activities (displacement hypothesis). This follows from Cattell’s (1987) investment theory, with investment shifting from secular education to religious materials rather than shifts from one secular domain (e.g., mathematics) to another (e.g., literature). This hypothesis might help to explain part of the variation in educational performance broadly (i.e., across academic domains), not just in science literacy.

One reason the displacement hypothesis makes sense is that religiosity is as powerfully negatively correlated with achievement in mathematics as it is with science achievement.

The Scattering Hypothesis

But certainly a drawback of the displacement hypothesis is that there are activities we engage in—as unrelated to mathematics and science as religion is—which don’t, as far as we know, correlate strongly negatively with achievement. Physical exercise, for goodness’ sake, is one example of such an activity. Perhaps there is something especially toxic about religiosity as the displacer which deserves our attention.

Maybe religiosity (or, better, a perspective which allows for supernatural explanations or, indeed, unexplainable phenomena) has a diluent or scattering effect on learning. If so, here are two analogies for how that might work:

  • Consider object permanence. Prior to developing the understanding that objects continue to exist once they are out of view, children will almost immediately lose interest in an object that is deliberately hidden from them, even if they were attending to it just moments earlier. Why? Because it is possible (to them) that the object has vanished from existence when you move it out of their view. If it were possible for a 4-month-old to crawl up and look behind the sofa to see that grandma had actually disappeared during a game of peek-a-boo, they would have nothing to wonder about. The disappearance was possible, so why shouldn’t it happen? This possibility is gone once you develop object permanence.
  • Perhaps more relevant, not to mention ominous: climate change. It is well known that religiosity and acceptance of the theory of evolution are negatively correlated. And it turns out there is a strong positive link between evolution denialism and climate-change denialism. How might religiosity support both of these denialisms? Here we can benefit from substituting for ‘religiosity’ some degree of subscription to supernatural explanations: If the universe was made by a deity for us, then how can we be intruders in it, and how could we—by means that do not transgress the laws of this deity—defile it? This seems a perfectly reasonable use of logic, once you have allowed for the possibility of an omniscient benevolence who gifted your species the entire planet you live on.

The two of these together seem pretty bizarre. But I’m sure you catch the drift. In each case, I would argue that the constriction of possibilities—to those supported by naturalistic explanations rather than supernatural ones—is actually a good thing. You are less likely to be prodded to explain how the natural world works when supernatural reasons are perfectly acceptable. And supernaturalism can prevent you from fully appreciating your own existence and the effects it has on the natural world. Under supernaturalism, you can still engage in logical arguments and intellectual activity. You can write books and go to seminars. Your neurons could be firing. But if you’re not thinking about reality, it doesn’t do you any good.

Religiosity or supernaturalism does not make you dumb. But perhaps it has the broader effect of making it more difficult to fasten minds onto reality, as it fills the solution space with only those possibilities that have little bearing on the real world we live in. This would certainly show up in measures of educational achievement.
Stoet, G., & Geary, D. (2017). Students in countries with higher levels of religiosity perform lower in science and mathematics Intelligence DOI: 10.1016/j.intell.2017.03.001

Expert Knowledge: Birds and Worms


Pay attention to your thought process and how you use expert knowledge as you answer the question below. How do you think very young students would think about it?

Here are some birds and here are some worms. How many more birds than worms are there?

Hudson (1983) found that, among a small group of first-grade children (mean age of 7.0), just 64% completed this type of task correctly. However, when the task was rephrased as follows, all of the students answered correctly.

Here are some birds and here are some worms. Suppose the birds all race over, and each one tries to get a worm. Will every bird get a worm? How many birds won’t get a worm?

This is consistent with adults’ intuitions about the two tasks as well. Members of the G+ mathematics education community were polled on the two birds-and-worms tasks recently, and, as of today, 69% predicted that more students would answer the second one correctly.

Interpret the Results

Still, what can we say about these results? Is it the case that 100% of the students used “their knowledge of correspondence to determine exact numerical differences between disjoint sets”? That is how Hudson describes students’ unanimous success in the second task. The idea seems to be that the knowledge exists; it’s just that a certain magical turn of phrase unlocks and releases this otherwise submerged expertise.

But that expert knowledge is given in the second task: “each one tries to get a worm.” The question paints the picture of one-to-one correspondence, and gives away the procedure to use to determine the difference. So, “their knowledge” is a bit of a stretch, and “used their knowledge” is even more of a stretch, since the task not only sets up a structure but animates its moving parts as well (“suppose the birds all race over”).

Further, questions about whether or not students are using knowledge they possess raise questions about whether or not students are, in fact, determining “exact numerical differences between disjoint sets.” On the contrary, it can be argued that students are simply watching almost all of a movie in their heads (a mental simulation)—a movie for which we have provided the screenplay—and then telling us how it ends (spoiler: 2 birds don’t get a worm). The deeper equivalence between the solution “2” and the response “2” to the question “How many birds won’t get a worm?” is evident only to a knowledgeable onlooker.

Experiment 3

Hudson anticipates some of the skepticism on display above when he introduces the third and last experiment in the series.

It might be argued, success in the Won’t Get task does not require a deep level of mathematical understanding; the children could have obtained the exact numerical differences by mimicking by rote the actions described by the problem context . . . In order to determine more fully the level of children’s understanding of correspondences and numerical differences, a third experiment was carried out that permitted a detailed analysis of children’s strategies for establishing correspondences between disjoint sets.

The wording in the Numerical Differences task of this third experiment, however, did not change. The “won’t get” locutions were still used. Yet, in this experiment, when paying attention to students’ strategies, Hudson observed that most children did not mentally simulate in the way directly suggested by the wording (pairing up the items in a one-to-one correspondence).

This does not defeat the complaint above, though. The fact that a text does not effectively compel the use of a procedure does not mean that it is not the primary influence on correct answers. It still seems more likely than not that participants who failed the “how many more” task simply didn’t have stable, abstract, transferable notions about mathematical difference. And the reformulation represented by the “won’t get” task influenced students to provide a response that was correct.

But this was a correct response to a different question. As adults with expert knowledge, we see the logical and mathematical similarities between the “how many more” and “won’t get” situations, and, thus we are easily fooled into believing that applying skills and knowledge in one task is equivalent to doing so in the other.

expert knowledge

Hudson, T. (1983). Correspondences and Numerical Differences between Disjoint Sets Child Development, 54 (1) DOI: 10.2307/1129864

Modulus and Hidden Symmetries


A really nice research paper, titled The Hidden Symmetries of the Multiplication Table was posted over in the Math Ed Community yesterday. The key ideas in the article center around (a) the standard multiplication table—with a row of numbers at the top, a column of numbers down the left, and the products of those numbers in the body of the table, and (b) modulus. In particular, what patterns emerge in the standard multiplication table when products are colored by equivalence to \(\mathtt{n \bmod k}\) as \(\mathtt{k}\) is varied?

The little interactive tool below shows a large multiplication table (you can figure out the dimensions), which starts by coloring those products which are equivalent to \(\mathtt{0 \bmod 12}\), meaning those products which, when divided by 12 give a remainder of zero (in other words, multiples of 12).


When you vary \(\mathtt{k}\), you can see some other pretty cool patterns (broken up occasionally by the boring patterns produced by primes). Observing the patterns produced by varying the remainder, \(\mathtt{n}\), is left as an exercise for the reader (and me).

Incidentally, I’ve wired up the “u” and “d” keys, for “up” and “down.” Just click in one of the boxes and press the “u” or “d” key to vary \(\mathtt{k}\) or \(\mathtt{n}\) without having to retype and press Return every time. And definitely go look at the paper linked above. They’ve got some other beautiful images and interesting questions.


Barka, Z. (2017). The Hidden Symmetries of the Multiplication Table Journal of Humanistic Mathematics, 7 (1), 189-203 DOI: 10.5642/jhummath.201701.15

Religiosity and Confidence in Science

research post


In response to a question posed on Twitter recently asking why people from the U.K. seemed to show a great deal more interest in applying cognitive science to education than their U.S. counterparts, I suggested, linking to this article, that the differences in the religiosity of the two countries might play a role.

Princeton economist Roland Bénabou led a study, for instance, which found that religiosity and scientific innovation were negatively correlated. Across the world, regions with higher levels of religiosity also had lower levels of scientific and technical innovation—a finding which held even when controlling for income, population, and education. Bénabou commented in this article:

Much comes down to the political power of the religious population in a given location. If it is large enough, it can wield its strength to block new insights. “Disruptive new ideas and practices emanating from science, technical progress or social change are then met with greater resistance and diffuse more slowly,” comments Bénabou, citing everything from attempts to control science textbook content to efforts to cut public funding of certain kinds of research (for instance involving embryonic stem cells or cloned human embryos). In secular places, by contrast, “discoveries and innovations occur faster, and some of this new knowledge inevitably erodes beliefs in any fixed dogma.”


The study’s analysis also includes a comparison of U.S. States, which showed a similar negative correlation, as shown at the left.

Importantly, this kind of analysis has nothing to say about the effects of one’s personal religious beliefs on one’s innovativeness or acceptance of science. This song is not about you. It is a sociological analysis which suggests that the religiosity of the culture one finds oneself in (regardless of income and education levels) can have an effect on one’s exposure to scientific innovation.

Religiosity can have this effect at the political and cultural levels while simultaneously having a quite different effect (or no similar effect) at the personal level.

But About That Personal Level

Perhaps more apropos of the original question, researchers have found that individual religiosity is not significantly correlated with interest in science, nor with knowledge of science—but it is significantly negatively correlated with one’s confidence in scientific findings.

More religious individuals report the same interest levels and knowledge of science as less religious people, but they report significantly lower levels of confidence in science. This means that their lack of confidence is not a product of interest or ignorance but represents some unique uneasiness with science. . . .

Going a little further, the researchers provide this quote in the conclusion, which is as perfect an echo of educators’ qualms with education research (that I’ve heard) as can likely be found in literature discussing a completely different topic (emphases mine):

Religious individuals may be fully aware of the potential for material and physical gains through biotechnology, neuroscience, and other scientific advancements. Despite their knowledge of and interest in this potential, they may also hold deep reservations about the moral and spiritual costs involved . . . Religious individuals may interpret [questions about future harms and benefits from science] as involving spiritual and moral harms and benefits. Concerns about these harms and gains are probably moderated by a perception, not entirely unfounded given the relatively secular nature of many in the academic scientific community (Ecklund and Scheitle 2007; Ecklund 2010), that the scientific community does not share the same religious values and therefore may not approach issues such as biotechnology in the same manner as a religious respondent.

It may be, then, that educators surrounded by cultures with higher religiosity—and regardless of their own personal religious orientations—will simply have greater exposure to concerns about moral and spiritual harm that can be wrought by science, in addition to the benefits it can bring. Consistent with my own thinking about the subject, these concerns would be amplified in situations, like education, where science looks to produce effects on human behavior and cognition, especially children’s behavior and cognition.
Johnson, D., Scheitle, C., & Ecklund, E. (2015). Individual Religiosity and Orientation towards Science: Reformulating Relationships Sociological Science, 2, 106-124 DOI: 10.15195/v2.a7

Provided Examples vs. Generated Examples

research post

The results reported in this research (below) about the value of provided examples versus generated examples are a bit surprising. To get a sense of why that’s the case, start with this definition of the concept availability heuristic used in the study—a term from the social psychology literature:

Availability heuristic: the tendency to estimate the likelihood that an event will occur by how easily instances of it come to mind.

All participants first read this definition, along with the definitions of nine other social psychology concepts, in a textbook passage. Participants then completed two blocks of practice trials in one of three groups: (1) subjects in the provided examples group read two different examples, drawn from an undergraduate psychology textbook, of each of the 10 concepts (two practice blocks, so four examples total for each concept), (2) subjects in the generated examples group created their own examples for each concept (four generated examples total for each concept), and (3) subjects in the combination group were provided with an example and then created their own example of each concept (two provided and two generated examples total for each concept).

The researchers—Amanda Zamary and Katharine Rawson at Kent State University in Ohio—made the following predictions, with regard to both student performance and the efficiency of the instructional treatments:

We predicted that long-term learning would be greater following generated examples compared to provided examples. Concerning efficiency, we predicted that less time would be spent studying provided examples compared to generating examples . . . [and] long-term learning would be greater after a combination of provided and generated examples compared to either technique alone. Concerning efficiency, our prediction was that less time would be spent when students study provided examples and generate examples compared to just generating examples.

Achievement Results

All participants completed the same two self-paced tests two days later. The first assessment, an example classification test, asked subjects to classify each of 100 real-world examples into one of the 10 concept definition categories provided. Sixty of these 100 were new (Novel) to the provided-examples group, 80 of the 100 were new to the combination group, and of course all 100 were likely new to the generated-examples group. The second assessment, a definition-cued recall test, asked participants to type in the definition of each of the 10 concepts, given in random order. (The test order was varied among subjects.)

provided examples

Given that participants in the provided-examples and combination groups had an advantage over participants in the generated-examples group on the classification task (they had seen between 20 and 40 of the examples previously), the researchers helpfully drew out results on just the 60 novel examples.

Subjects who were given only textbook-provided examples of the concepts outperformed other subjects on applying these concepts to classifying real-world examples. This difference was significant. No significant differences were found on the cued-recall test between the provided-examples and generated-examples groups.

Also, Students’ Time Is Valuable

Another measure of interest to the researchers in this study, as mentioned above, was the time used by the participants to read through or create the examples. What the authors say about efficiency is worth quoting, since it does not often seem to be taken as seriously as measures of raw achievement (emphasis mine):

Howe and Singer (1975) note that in practice, the challenge for educators and researchers is not to identify effective learning techniques when time is unlimited. Rather, the problem arises when trying to identify what is most effective when time is fixed. Indeed, long-term learning could easily be achieved if students had an unlimited amount of time and only a limited amount of information to learn (with the caveat that students spend their time employing useful encoding strategies). However, achieving long-term learning is difficult because students have a lot to learn within a limited amount of time (Rawson and Dunlosky 2011). Thus, long-term learning and efficiency are both important to consider when competitively evaluating the effectiveness of learning techniques.

provided examples

With that in mind, and given the results above, it is noteworthy to learn that the provided-examples group outperformed the generated-examples group on real-world examples after engaging in practice that took less than half as much time. The researchers divided subjects’ novel classification score by the amount of time they spent practicing and determined that the provided-examples group had an average gain of 5.7 points per minute of study, compared to 2.2 points per minute for the generated-examples group and 1.7 points per minute for the combination group.

For learning declarative concepts in a domain and then identifying those concepts in novel real-world situations, provided examples proved to be better than student-generated examples for both long-term learning and for instructional efficiency. The second experiment in the study replicated these findings.

Some Commentary

First, some familiarity with the research literature makes the above results not so surprising. The provided-examples group likely outperformed the other groups because participants in that group practiced with examples generated by experts. Becoming more expert in a domain does not necessarily involve becoming more isolated from other people and their interests. Such expertise is likely positively correlated with better identifying and collating examples within a domain that are conceptually interesting to students and more widely generalizable. I reported on two studies, for example, which showed that greater expertise was associated with a significantly greater number of conceptual explanations, as opposed to “product oriented” (answer-getting) explanations—and these conceptual explanations resulted in the superior performance of students receiving them.

Second, I am sympathetic to the efficiency argument, as laid out here by the study’s authors—that is, I agree that we should focus in education on “trying to identify what is most effective when time is fixed.” Problematically, however, a wide variety of instructional actions can be informed by decisions about what is and isn’t “fixed.” Time is not the only thing that can be fixed in one’s experience. The intuition that students should “own their own learning,” for example, which undergirds the idea in the first place that students should generate their own examples, may rest on the more fundamental notion that students themselves are “fixed” identities that adults must work around rather than try to alter. This notion is itself circumscribed by the research summarized above. So, it is worth having a conversation about what should and should not be considered “fixed” when it comes to learning.

provided examples
Zamary, A., & Rawson, K. (2016). Which Technique is most Effective for Learning Declarative Concepts—Provided Examples, Generated Examples, or Both? Educational Psychology Review DOI: 10.1007/s10648-016-9396-9

Bloom’s Against 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.