## Expert Knowledge: Birds and Worms

research

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.

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

## Modulus and Hidden Symmetries

research

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).

mod

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.

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.)

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.

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.

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.

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.

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.

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.

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.

## From Translations to Slope

If not before, students in 8th grade learn that a translation is a rigid motion that “slides” a point or set of points a certain distance. An important idea here that could stand to be emphasized a lot more is that the translations students study are linear translations—the translations move the set of points along a line. When this is understood prior to looking at slope, it can help with a deeper understanding of slope.

We can see the start of this in action when we play with the simulation below. Type positive numbers less than ten and greater than zero (3 characters max) into the blank boxes and then click on the arrow boxes to set the directions. This will create a translation sequence starting at (0, 0). For example, 9 ↑ 3 ← will continuously translate a point up 9 and left 3 (until it goes out of view). Click on the coordinate plane to run the sequence.

When the sequence is finished, a button should appear that allows you to click to show the line along which the point was translated using a repetition of the translation sequence. Click Clear to draw a new translation sequence (or repeat the one you just did). You can watch a (near) infinite loop if you’d like to put in things like 8 ↑ 8 ↓.

What Is Slope?

The example at right shows a finished sequence of repeated $$\mathtt{(x – 4, y + 6)}$$. There’s a whole lot to unpack here, which I won’t do. But, playing around with linear translations in this way can eventually reveal that the vertical and horizontal displacements form a ratio. For example, one can say that for every vertical move up 6 $$\mathtt{(+6)}$$, there is a horizontal move left 4 $$\mathtt{(-4)}$$. This simplifies to 3 : –2, and you can extend the sequence into the 4th quadrant to show that this is the same line as –3 : 2.

Referring to lines in terms of their slope ratios is pretty close to the finish line as far as slope understanding.

Y = Mx + B

We can ask about the corresponding y-value for an x-value of 5. The answer to this becomes the solution to a proportion, which we can generalize: $\mathtt{\frac{\color{white}{-}3}{-2} = \frac{y}{5} \quad \rightarrow \quad \frac{\color{white}{-}3}{-2} = \frac{y}{x}}$

So, we can arrive at $$\mathtt{y = -\frac{3}{2}x}$$. By this point, the slope ratio is ready for a special letter, and we can move up to the slope-intercept form. There are all kinds of catches and surprises in this development: zeros, the final b translation of the entire line, etc. But it is certainly an interesting connection between geometry and algebra for middle school, the key idea being that translations always move points along a straight line.

These ideas can essentially run alongside ratio development too, regardless whether the notion of translations is developed formally (there’s not much formality to it, even in 8th grade) or informally. See the Guzinta Math: Comparing Ratios lesson app for some more ideas about connections.

## Retrieval Practice with Kindle: Feel the Learn

I use Amazon’s free Kindle Reader for all of my (online and offline) book reading, except for any book that I really want that just can’t be had digitally. Besides notes and highlights, the Reader has a nifty little Flashcards feature that works really well for retrieval practice. Here’s how I do retrieval practice with Kindle.

Step 1: Construct the Empty Flashcard Decks

Currently I’m working through Sarah Guido and Andreas Müller’s book Introduction to Machine Learning with Python. I skimmed the chapters before starting and decided that the authors’ breakdown by chapter was pretty good—not too long and not too short. So, I made a flashcard deck for each chapter in the book, as shown at the right. On your Kindle Reader, click on the stacked cards icon. Then click on the large + sign next to “Flashcards” to create and name each new deck.

Depending on your situation, you may not have a choice in how you break things down. But I think it’s good advice to set up the decks—however far in advance you want—before you start reading.

So, if I were assigned to read the first half of Chapter 2 for a class, I would create a flashcard deck for the first half of Chapter 2 before I started reading. And, although I didn’t set titles in this example, it’s probably a good idea to give the flashcard deck a title related to what it’s about (e.g., Supervised Learning).

You still need to read and comprehend the content. Retrieval practice adds, it doesn’t replace. So, I read and highlight and write notes like I normally would. I don’t worry at this point about the flashcards, about what is important or not. I just read for the pleasure of finding things out. I highlight things that strike me as especially interesting and write notes with questions, or comments I want to make on the text.

Read a section of the content represented by one flashcard deck. Since I divided my decks by chapter, I read the first chapter straight through, highlighting and making notes as I went.

The reading doesn’t have to be done in one sitting. The important thing is to just focus on reading one section before moving on to the next step.

Step 3: Create the Fronts for the Flashcards

Now, go through the content of your first section of reading and identify important concepts, items worth remembering, things you want to be able to produce. You’ll want to add these as prompts on your flashcards. You don’t necessarily have to write these all down in a list. You can enter a prompt on a flashcard, return to the text for another prompt, enter a prompt on another flashcard, and on and on.

Screenshot 1

Screenshot 2

Screenshot 3

When you have at least one prompt, click on the flashcard deck and then click on Add a Card (Screenshot 1) and enter the prompt.

Enter the prompt at the top. (Screenshot 2) This will be the front of the flashcard you will see when testing yourself. Leave the back blank for the moment. Click Save and Add Another Card at the bottom right to repeat this with more prompts.

When you are finished entering one card or all the cards, click on Save at the top right. This will automatically take you to the testing mode (Screenshot 3), which you’ll want to ignore for a while. Click on the stacked cards icon to return to the text for more prompts. When you come back to the flashcards, your decks may have shifted, since the most recently edited deck will be at the top.

Importantly, though, Screenshot 3 is the screen you will see when you return and click on a deck. To add more cards from this screen, click on the + sign at the bottom right. When you are done entering the cards for a section, get ready for the retrieval practice challenge! This is where it gets good (for learning).

Step 4: Create the Backs for the Flashcards

Rather than simply enter the backs of the flashcards from the information in the book, I first fill out the backs by simply trying to retrieve what I can remember. For example, for the prompt, “Write the code for the Iris model, using K Nearest Neighbors,” I wrote something like this on the back of the card:

import numpy as np
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
iris_dataset = load_iris()
X_train, X_test, y_train, y_test = train_test_split(iris_dataset.data, iris_dataset.target)

There are a lot of omissions here and some errors, and I moved things around after I wrote them down, but I tried as hard as I could to remember the code. To make the back of the card right, I filled in the omissions and corrected the errors. As I went through this process with all the cards in a section, I edited the fronts and backs of the cards and even added new cards as the importance of some material presented itself more clearly.

Create the backs of the flashcards for a section by first trying as hard as you can to retrieve the information asked for in the prompt. Then, correct the information and fill in omissions. Repeat this for each card in the deck.

Step 5: Test Yourself and Feel the Learn

One thing you should notice when you do this is that it hurts. And it should. In my view, the prompts should not be easy to answer. Another prompt I have for a different chapter is “Explain how k-neighbors regression works for both 1 neighbor and multiple neighbors.” My expectations for my response are high—I want to answer completely with several details from the text, not just a mooshy general answer. I keep the number of cards per chapter fairly low (about 5 to 10 cards per 100 pages). But your goals for retaining information may be different.

But once you have a set of cards for a section, come back to them occasionally and complete a round of testing for the section. To test yourself, click on the deck and respond to the first prompt you see without looking at the answer. Try to be as complete (correct) as possible before looking at the correct response.

To view the correct response, click on the card. Then, click on the checkmark if you completely nailed the response. Anything short of that, I click on the red X.

For large decks, you may want to restudy those items you got incorrect. In that case, you can click on Study Incorrect to go back over just those cards you got wrong. There is also an option to shuffle the deck (at the bottom left), which you should make use of if the content of the cards build on each other, making them too predictable.

## ResearchEd: Getting Beyond Appearances

The Beuchet Chair shown at the right is a fun visual illusion—a trick involving distance and perspective—and illusions like it are solid, predictable go-tos for anyone trying to make the case for the importance of learning about science and research at events like ResearchEd.

The idea is to show you how appearances can be deceiving, how your own cognitive apparatus is not designed to present the world to you perfectly as it is, and that, most importantly, experiences alone, whether isolated or combined, do not reliably illuminate the hidden patterns and regularities which govern our lives and the natural world.

Once this doubt is sown, what we hope happens next is that you will re-evaluate your beliefs about the world as you continue to move through your life, strengthening some of them with better explanations and justifications, loosening the threads of others, and considering new beliefs and motives, too.

And central to this ongoing project for those of us both inside and outside of science are, I think, two tendencies, represented at some of the sessions I attended at the ResearchEd Washington event last week:

1. The tendency to distrust the superficial, shallow, easy, or popular—those things that are, like the illusion above, true only from a limited perspective. It is the tendency to be dissatisfied with everyday explanations, short-term thinking, folk wisdom, and faith-based certainty.
2. The tendency to seek out deep explanations rather than ephemeral ones—a preference for connected, theoretical (though still fallible), conceptual knowledge, which “constitutes the means society uses to transcend the limits of individual experience to see beyond appearances to the nature of relations in the natural and social world.”

Robert Pondiscio: Why Knowledge Matters

Robert Pondiscio’s session was as pure a distillation of this latter tendency as you’ll find. Robert memorably contrasted two reactions to President Obama’s inauguration: one which expressed an elation that the United States now had a president that looked like many underrepresented students, and one which expressed a deep connection to the nearly 50 years of American history that came full circle on January 20, 2009—a history that could not be accessed except by the knowledgeable.

He cautioned that the two reactions are not mutually exclusive, while still driving home the importance of conceptual knowledge and the school’s vital role in providing students access to it.

I was reminded, again, of scenes we often see when something of astronomical importance has just happened—that roomful of jubilant scientists at, say, the Jet Propulsion Laboratory.

Sure, the images of, say, the Mars Rover’s safe landing come along eventually. But pictures are not what gets these folks excited. It’s data. Data that says the Rover has entered orbit, has deployed the parachute, has fired its rockets. What causes all the excitement is, quite literally, knowledge.

The Learning Scientists: Teaching the Science of Learning

The Learning Scientists continued to reinforce the power of investigating the deep and often hidden patterns and regularities involved in education as they presented evidence for the benefits of spaced practice and retrieval practice on student learning.

Many lifetimes lived out in close proximity to children and students have failed to systematically reveal these robust effects on learning. Yet, stand back, apply a little (1) and (2) from above, and you get results that help overturn the destructive notion that the brain is like a tape recorder. While it would be a mistake to assume that a result is true just because it’s counterintuitive, results around spacing and retrieval often are, even to the participants in the study.

Dylan Wiliam, Ben Riley

What I took away from Dylan’s keynote and Ben’s presentation (and from the Learning Scientists’ session)—other than what they were about (info on Ben’s session here)—is that while I am attracted to those ideas in education that feature a suspicion of everyday thinking and a search for deeper regularities, it is absolutely vital that we have people in our community who can bring this search for general meaning to our everyday thinking (and not the other way around! which is essentially searching for empirical justification for low-level theorizing; also called just-so stories)—people who understand the realities of the classroom, where much of what is discovered in education science must play out. People who are much more diplomatic than I, but with whom I could easily find common cause.

Because we all have a desire to see learning science and other education research have a tangible, practical, positive effect on students’ (and teachers’) lives. But we can’t pull it off alone. We have such a great start in connecting research with practice in groups like ResearchEd!

## Variable as a Batch of Numbers

There are a couple of interesting lines from the Common Core State Standards for Mathematics (CCSS-M), referencing the meaning of a variable in an equation, which have been on my mind lately. The first is from 6.EE.B.5 and the second from 6.EE.B.6. I have emphasized in red the bits that I think are significant to this post:

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?

Understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

One of the reasons these are interesting (to me) is that, almost universally as far as I can tell, curricula in 6th grade mathematics (CCSS-M-aligned) limit themselves to equations like $$\mathtt{5+p=21}$$ and $$\mathtt{2x=24}$$, which only have one solution. So, it’s not possible to talk about the values (plural) that make an equation true; nor is it possible to talk about a variable as representing any number in a specified set when all of our examples will essentially resolve to just one possibility.

How do curricula then cover the two standards above? Well, it’s possible to still hit these two standards when you interpret multiple solutions as something that belong with inequalities. Inequalities are part of the “or” statement at the end of 6.EE.B.5 and can be seen as part of “the purpose at hand” in 6.EE.B.6. This, it seems, is the interpretation that most curricula for 6th grade (again, as far as I can tell) have settled on.

Stipulation and Functions

A reason this may be problematic is that it introduces a stipulation (or continues one, rather)—one which, as far as I can tell, is not effectively stretched out in Grades 7 or 8. That stipulation is this: a variable in an equation represents a single number. We dig this one-solution trench deeper and deeper for two to three years until one day we show them this. In this object, a function, the $$\mathtt{x}$$ most certainly does not represent a single value.

But, crucially, $$\mathtt{x}$$ doesn’t have to represent a single value even back in 6th grade. That is, when solving an equation in middle school, the variable may wind up to be one number, but we don’t HAVE to make students think that it always will. An equation—even a simple 6th-grade equation—can have no solutions, one solution, or all kinds of different solutions. For example, $$\mathtt{x = x + 2}$$ has no real solutions; $$\mathtt{6x = 2(3x)}$$ has an infinite number of solutions; whereas the tricky $$\mathtt{x = 2x}$$ or $$\mathtt{x = 2x + 2}$$ each have one solution apiece. (The latter is a 7th-grade equation, though.)

Once Is Not Enough

The point that an unknown in an equation does not automatically represent one value could be made a little better if solving quadratics or absolute value equations typically preceded an introduction to functions. But even if the content were moved around to fit those topics before functions, the trench is dug mighty deep in middle school. Further, the 8th-grade standard that references different numbers of solutions as we did above, 8.EE.C.7a, is too late, and is often interpreted by curricula as comparing two linear expressions (e.g., $$\mathtt{y = x}$$ vs. $$\mathtt{y = x + 2}$$; parallel so no solutions), thus keeping the one-solution stipulation ironically intact.

Frequent reminders starting when variables are introduced through the introduction of functions would serve students better, I think, especially when they tackle concepts such as domain and range. The notion that an unknown can represent 0, 1, or multiple values could also help to make linear algebra a bit more approachable when it is introduced.

Check out John Redden’s and Paul Gonzalez-Becerra’s Open Graphing Calculator, which I used in this post.