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.

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

retrieval practice with Kindle

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).

Step 2: Read, Just Read

retrieval practice with Kindle, retrieval practice with Kindle, retrieval practice with Kindle, retrieval practice with Kindle, retrieval practice with Kindle,

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.

retrieval practice with kindle

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.

retrieval practice with kindle

Screenshot 1

retrieval practice with kindle

Screenshot 2

retrieval practice with kindle

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(,

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.

For more information on retrieval practice, go talk to The Learning Scientists (on Twitter as @AceThatTest).

ResearchEd: Getting Beyond Appearances

Not from ResearchEd.

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.

Knowing stuff is pretty exciting!

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.

Instructional Effects: Action at a Distance

I really like this recent post, called Tell Me More, Tell Me More, by math teacher Dani Quinn. The content is an excellent analysis of expert blindness in math teaching. The form, though, is worth seeing as well—it is a traditional educational syllogism, which Quinn helpfully commandeers to arrive at a non-traditional conclusion, that instructional effects have instructional causes, on the right:

The Traditional Argument An Alternative Argument
There is a problem in how we teach: We typically spoon-feed students procedures for answering questions that will be on some kind of test.

“There is a problem in how we teach: We typically show pupils only the classic forms of a problem or a procedure.”

This is why students can’t generalize to non-routine problems: we got in the way of their thinking and didn’t allow them to take ownership and creatively explore material on their own. “This is why they then can’t generalise: we didn’t show them anything non-standard or, if we did, it was in an exercise when they were floundering on their own with the least support.”

Problematically for education debates, each of these premises and conclusions taken individually are true. That is, they exist. At our (collective) weakest, we do sometimes spoon-feed kids procedures to get them through tests. We do cover only a narrow range of situations—what Engelmann refers to as the problem of stipulation. And we can be, regrettably in either case, systematically unassertive or overbearing.

Solving equations provides a nice example of the instructional effects of both spoon-feeding and stipulation. Remember how to solve equations? Inverse operations. That was the way to do equations. If you have something like \(\mathtt{2x + 5 = 15}\), the table shows how it goes.

Equation Step
\(\mathtt{2x + 5 \color{red}{- 5} = 15 \color{red}{- 5}}\) Subtract \(\mathtt{5}\) from both sides of the equation to get \(\mathtt{2x = 10}\).
\(\mathtt{\color{white}{+ 5 \,\,} 2x \color{red}{\div 2} = 10 \color{red}{\div 2}}\) Divide both sides of the equation by 2.
\(\mathtt{\color{white}{+ 5 \,\,}x = 5}\) You have solved the equation.

Do that a couple dozen times and maybe around 50% of the class freezes when they encounter \(\mathtt{22 = 4x + 6}\), with the variable on the right side, or, even worse, \(\mathtt{22 = 6 + 4x}\).

That’s spoon-feeding and stipulation: do it this one way and do it over and over—and, crucially, doing that summarizes most of the instruction around solving equations.

Of course, the lack of prior knowledge exacerbates the negative instructional effects of stipulation and spoon-feeding. But we’ll set that aside for the moment.

The Connection Between Premises and Conclusion

The traditional and alternative arguments above are easily (and often) confused, though, until you include the premise that I have omitted in the middle for each. These help make sense of the conclusions derived in each argument.

The Traditional Argument An Alternative Argument
There is a problem in how we teach: We typically spoon-feed students procedures for answering questions that will be on some kind of test.

“There is a problem in how we teach: We typically show pupils only the classic forms of a problem or a procedure.”

Students’ success in schooling is determined mostly by internal factors, like creativity, motivation, and self-awareness.

Students’ success in schooling is determined mostly by external factors, like amount of instruction, socioeconomic status, and curricula.

This is why students can’t generalize to non-routine problems: we got in the way of their thinking and didn’t allow them to take ownership and creatively explore material on their own. “This is why they then can’t generalise: we didn’t show them anything non-standard or, if we did, it was in an exercise when they were floundering on their own with the least support.”

In short, the argument on the left tends to diagnose pedagogical illnesses and their concomitant instructional effects as people problems; the alternative sees them as situation problems. The solutions generated by each argument are divergent in just this way: the traditional one looks to pull the levers that mostly benefit personal, internal attributes that contribute to learning; the alternative messes mostly with external inputs.

It’s Not the Spoon-Feeding, It’s What’s on the Spoon

I am and have always been more attracted to the alternative argument than the traditional one. Probably for a very simple reason: my role in education doesn’t involve pulling personal levers. Being close to the problem almost certainly changes your view of it—not necessarily for the better. But, roles aside, it’s also the case that the traditional view is simply more widespread, and informed by the positive version of what is called the Fundamental Attribution Error:

We are frequently blind to the power of situations. In a famous article, Stanford psychologist Lee Ross surveyed dozens of studies in psychology and noted that people have a systematic tendency to ignore the situational forces that shape other people’s behavior. He called this deep-rooted tendency the “Fundamental Attribution Error.” The error lies in our inclination to attribute people’s behavior to the way they are rather than to the situation they are in.

What you get with the traditional view is, to me, a kind of spooky action at a distance—a phrase attributed to Einstein, in remarks about the counterintuitive consequences of quantum physics. Adopting this view forces one to connect positive instructional effects (e.g., thinking flexibly when solving equations) with something internal, ethereal and often poorly defined, like creativity. We might as well attribute success to rabbit’s feet or lucky underwear or horoscopes!

instructional effects

Contiguity Effective for Deductive Inference

research post

The discourse that surrounds the technicalities in this paper contains an agenda: to convince readers that the benefits of retrieval practice extend beyond the boring old “helps you remember stuff” caricature to something more “higher order” like deductive inference. But I’m not convinced that the experiments show this. Rather, what they demonstrate fairly convincingly is that informational contiguity, not retrieval practice, benefits inference-making. A related result from the research, on the benefits of text coherence, is explained here.

The Setup

The arch-nemesis of this research is a paper by Tran, et al. last year, which appeared to show some domain limitations on retrieval practice:

They found that retrieval practice failed to benefit participants’ later ability to make accurate deductions from previously retrieved information. In their study, participants were presented sentences one at a time to learn . . . The sentences could be related to one another to derive inferences about particular scenarios. Although retrieval practice was shown to improve memory of the sentences relative to a restudy control condition, there was no benefit on a final inference test that required integration of information from across multiple sentences.


So, in this study, researchers replicated Tran et al.’s methods, except in one important way: they did not present the sentences to be learned one at a time but together instead.

Participants were each presented with four scenarios (two of which are outlined at right) consisting of seven to nine premises in the form of sentences. In each scenario, deductions to specific conclusions were possible. For two of the four scenarios, subjects used retrieval practice. They were given a chance to read the sentences in a scenario at their own pace and then were shown the sentences again for five minutes—in cycles where the order of the sentences was randomized. During this five-minute session, subjects were asked to type in the missing words in each premise (between one and three missing words). The complete sentences were then shown as feedback. Each participant used restudy for the other two scenarios. During the restudy five-minute session, subjects simply reread the premises again, in cycles again, with the order of the premises randomized for each cycle.

The Results and Discussion

Two days later, participants were given a 32-item multiple choice test which “assessed participants’ ability to draw logical conclusions derived from at least two premises within each scenario.” And consistent with the researchers’ hypothesis, the retrieval practice conditions yielded significantly better results on a test of deductive inference than did the restudy conditions.

Yet, it’s not at all clear that retrieval practice was the cause of the better performance with respect to inference-making. There was another cause preceding it: the improved contiguity of the presented information, as compared with Tran et al.’s one-at-a-time procedure. It’s possible that the effectiveness of retrieval practice is limited to recall of already-integrated information, and the contiguity of the premises in this study allowed for such integration, which, in turn, allowed retrieval practice to outperform restudy. It is a possibility the researchers raise in the paper and one that, in my view, the current research has not effectively answered:

However, other recent studies have failed to find a benefit of retrieval practice for learning educational materials (Leahy et al. 2015; Tran et al. 2015; Van Gog and Sweller 2015). These studies all used learning materials that required learners to simultaneously relate multiple elements of the materials during study and/or test. Such materials that are high in element interactivity need constituent elements to be related to one another in order for successful learning or task completion (element interactivity may also be considered as a measure of the complexity of materials, see (Sweller 2010)).

What we can say, with some confidence, is that even if the benefits of retrieval practice were limited to improvements in recall (as prior research has demonstrated), such improvements do not stand in the way of improvements to higher-order reasoning, such as inference-making. (And shaping the path for students, such as improving informational contiguity can have a positive effect too.)
Eglington, L., & Kang, S. (2016). Retrieval Practice Benefits Deductive Inference Educational Psychology Review DOI: 10.1007/s10648-016-9386-y

Entia Successiva: How to Like Science

entia successiva

The term ‘entia successiva’ means ‘successive entities.’ And, as you may guess, it is a term one might come across in a philosophy class, in particular when discussing metaphysical questions about personhood. For instance, is a person a single thing throughout its entire life or a succession of different things—an ‘ens successivum’? Though there is no right answer to this question, becoming familiar with the latter perspective can, I think, help people to be more skeptical and knowledgeable consumers of education research.

Richard Taylor provides an example of a symphony (in here) that is, depending on your perspective, both a successive and a permanent entity:

Let us imagine [a symphony orchestra] and give it a name—say, the Boston Symphony. One might write a history of this orchestra, beginning with its birth one hundred years ago, chronicling its many tours and triumphs and the fame of some of its musical directors, and so on. But are we talking about one orchestra?

In one sense we are, but in another sense we are not. The orchestra persists through time, is incorporated, receives gifts and funding, holds property, has a bank account, returns always to the same city, and rehearses, year in and year out, in the same hall. Yet its membership constantly changes, so that no member of fifty years ago is still a member today. So in that sense it is an entirely different orchestra. We are in this sense not talking about one orchestra, but many. There is a succession of orchestras going under the same name. Each, in [Roderick] Chisholm’s apt phrase, does duty for what we are calling the Boston Symphony.

The Boston Symphony is thus an ens successivum.

People are entia successiva, too. Or, at least their bodies are. Just about every cell in your body has been replaced from only 10 years ago. So, if you’re a 40-year-old Boston Symphony like me, almost all of your musicians and directors have been swapped out from when you were a 30-year-old symphony. People still call you the Boston Symphony of course (because you still are), but an almost entirely different set of parts is doing duty for “you” under the banner of this name. You are, in a sense, an almost completely different person—one who is, incidentally, made up of at least as many bacterial cells as human ones.

What’s worse (if you think of the above as bad news), the fact of evolution by natural selection tells us that humanity itself is an ens successivum. If you could line up your ancestors—your mother or father, his or her mother or father, and so on—it would be a very short trip down this line before you reached a person with whom you could not communicate at all, save through gestures. Between 30 and 40 people in would be a person who had almost no real knowledge about the physical universe. And there’s a good chance that perhaps the four thousandth person in your row of ancestors would not even be human.

The ‘Successive’ Perspective

Needless to say, seeing people as entia successiva does not come naturally to anyone. Nor should it, ever. We couldn’t go about out our daily lives seeing things this way. But the general invisibility of this ‘successiveness’ is not due to its only being operational at the very macro or very micro levels. It can be seen at the psychological level too. Trouble is, our brains are so good at constructing singular narratives out of even absolute gibberish, we sometimes have to place people in unnatural or extreme situations to get a good look at how much we can delude ourselves.

An Air Force doctor’s experiences investigating the blackouts of pilots in centrifuge training provides a nice example (from here). It’s definitely worth quoting at length:

Over time, he has found striking similarities to the same sorts of things reported by patients who lost consciousness on operating tables, in car crashes, and after returning from other nonbreathing states. The tunnel, the white light, friends and family coming to greet you, memories zooming around—the pilots experienced all of this. In addition, the centrifuge was pretty good at creating out-of-body experiences. Pilots would float over themselves, or hover nearby, looking on as their heads lurched and waggled about . . . the near-death and out-of-body phenomena are both actually the subjective experience of a brain owner watching as his brain tries desperately to figure out what is happening and to orient itself amid its systems going haywire due to oxygen deprivation. Without the ability to map out its borders, the brain often places consciousness outside the head, in a field, swimming in a lake, fighting a dragon—whatever it can connect together as the walls crumble. What the deoxygenated pilots don’t experience is a smeared mess of random images and thoughts. Even as the brain is dying, it refuses to stop generating a narrative . . . Narrative is so important to survival that it is literally the last thing you give up before becoming a sack of meat.

You’ll note, I hope, that not only does the report above disclose how our very mental lives are entia successiva—thoughts and emotions that arise and pass away—but the report assumes this perspective in its own narrative. That’s because the report is written from a scientific point of view. And from that vantage point, people are assumed (correctly) to have parts that “do duty” for them and may even be at odds with each other, as they were with the pilots (a perception part fighting against a powerful narrative-generating part). The unit of analysis in the report is not an entire pilot, but the various mechanisms of her mind. Allowing for these parts allows for functional explanations like the one we see.

An un-scientific analysis, on the other hand, is entirely possible. But it would stop at the pilot. He or she is, after all, an indivisible, permanent entity. There is nothing else “doing duty” for him, so there are really only two choices: his experience was an illusion or it was real. End of analysis. Interpret it as an illusion and you don’t really have much to say; interpret it as real, and you can make a lot of money.

Entia Permanentia

Good scientific research in education will adopt an entia successiva perspective about the people it studies. This does not guarantee that its conclusions are correct. But it makes it more likely that, over time, it will get to the bottom of things.

This is not to say that an alternative perspective is without scientific merit. If we want to know how to improve the performance of the Boston Symphony, we can make some headway with ‘entia permanentia’—seeing the symphony as a whole stable unit rather than a collection of successive parts. We could increase its funding, perhaps try to make sure “it” is treated as well as other symphonies around the world. We could try to change the music, maybe include some movie scores instead of that stuffy old classical music. That would make it more exciting for audiences (and more inclusive), which is certainly one interpretation of “improvement.” But to whatever extent improvement means improving the functioning of the parts of the symphony—the musicians, the director, etc.—we can do nothing, because with entia permanentia these tiny creatures do not exist. Even raising the question about improving the parts would be beyond the scope of our imagination.

Further, seeing students as entia permanentia rather than entia successiva stops us from being appropriately skeptical about both ‘scientific’ and ‘un-scientific’ ideas. Do students learn best when matched to their learning style? What parts of their neurophysiology and psychology could possibly make something like that true? Why would it have evolved, if it did? In what other aspects of our lives might this present itself? Adopting the entia successiva perspective would have slowed the adoption of this myth (even if were not a myth) to a crawl and would have eventually killed it. Instead, entia permanentia, a person-level analysis, holds sway: students benefit from learning-style matching because we see them respond differently to different representations. End of analysis.

Finally, it should be noted, though it goes without saying, that simply putting one’s ideas into a journal article does not guarantee that one is looking for functional explanations. Even nearly a decade later, Deborah Ball is still good on this point, though the situation since then has improved, I think:

Research that is ostensibly “in education” frequently focuses not inside the dynamics of education but on phenomena related to education—racial identity, for example, young children’s conceptions of fairness, or the history of the rise of secondary schools. These topics and others like them are important. Research that focuses on them, however, often does not probe inside the educational process. . . . Until education researchers turn their attention to problems that exist primarily inside education and until they develop systematically a body of specialized knowledge, other scholars who study questions that bear on educational problems will propose solutions. Because such solutions typically are not based on explanatory analyses of the dynamics of education, the education problems that confront society are likely to remain unsolved.

Update: Okay, maybe one last word from a recurring theme in the book Switch:

In a pioneering study of organizational change, described in the book The Critical Path to Corporate Renewal, researchers divided the change efforts they’d studied into three groups: the most successful (the top third), the average (the middle third), and the least successful (the bottom third). They found that, across the spectrum, almost everyone set goals: 89 percent of the top third and 86 percent of the bottom third . . . But the more successful change transformations were more likely to set behavioral goals: 89 percent of the top third versus only 33 percent of the bottom third.

Why do “behavioral” goals work when just “goals” don’t? Behavioral goals are, after all, telling you what to do, forcing you to behave in a certain way. Do you like to be told what to do? Probably not.

But the “you” that responds to behavioral goals isn’t the same “you” whose in-the-moment “likes” are important. You are more than just one solid indivisible self. You are many selves, and the self that can start checking stuff off the to-do list is often pulling the other selves behind it. And when it does, you get to think that “you” are determined, “you” take initiative, “you” have willpower. But in truth, your environment—both immediate and distant, both internal and external—has simply made it possible for that determined self to take the lead. Behavioral goals often create this exact environment.

Inference Calls in Text

research post

Britton and Gülgöz (1991) conducted a study to test whether removing “inference calls” from text would improve retention of the material. Inference calls are locations in text that demand inference from the reader. One simple example from the text used in the study is below:

Air War in the North, 1965

By the Fall of 1964, Americans in both Saigon and Washington had begun to focus on Hanoi as the source of the continuing problem in the South.

There are at least a few inferences that readers need to make here. Readers need to infer the causal link between “the fall of 1964” and “1965,” they are asked to infer that “North” in the title refers to North Vietnam, and they need to infer that “Hanoi” refers to the capital of North Vietnam.

The authors of the study identified 40 such inference calls (using the “Kintsch” computer program) throughout the text and “repaired” them to create a new version called a “principled revision.” Below is their rewrite of the text above, which appeared in the principled revision:

Air War in the North, 1965

By the beginning of 1965, Americans in both Saigon and Washington had begun to focus on Hanoi, capital of North Vietnam, as the source of the continuing problem in the South.

Two other versions (revisions), the details of which you can read about in the study, were also produced. These revisions acted as controls in one way or another for the original text and the principled revision.

Method and Predictions

One hundred seventy college students were randomly assigned one of the four texts–the original or one of the three revisions. The students were asked to read the texts carefully and were informed that they would be tested on the material. Eighty subjects took a free recall test, in which they were asked to write down everything they could remember from the text. The other ninety subjects took a ten-question multiple-choice test on the information explicitly stated in each text.

It’s not at all difficult, given this set up, to anticipate the researchers’ predictions:

We predicted that the principled revision would be retrieved better than the original version on a free-recall test. This was because the different parts of the principled revision were more likely to be linked to each other, so the learner was more likely to have a retrieval route available to use…. Readers of the original version would have to make the inferences themselves for the links to be present, and because some readers will fail to make some inferences, we predicted that there would be more missing links among readers of this version.

This is, indeed, what researchers found. Subjects who read the principled revision recalled significantly more propositions from the text (adjusted mean = 58.6) than did those who read the original version (adjusted mean = 35.5). Researchers’ predictions for the multiple-choice test were also accurate:

On the multiple-choice test of explicit factual information that was present in all versions, we predicted no advantage for the principled revision. Because we always provided the correct answer explicitly as one of the multiple choices, the learner did not have to retrieve this information by following along the links but only had to test for his or her recognition of the information by using the stem and the cue that was presented as one of the response alternatives. Therefore, the extra retrieval routes provided by the principled revision would not help, because according to our hypothesis, retrieval was not required.

Analysis and Principles

Neither of the two results mentioned above are surprising, but the latter is interesting. Although we might say that students “learned more” from the principled revision, subjects in the original and principled groups performed equally well on the multiple-choice test (which tests recognition, as opposed to free recall). As the researchers noted, this result was likely due to the fact that repairing the inference calls provided no advantage to the principled group in recognizing explicit facts, only in connecting ideas in the text.

But the result also suggests that students who were troubled by inference calls in the text just skipped over them. Indeed, subjects who read the original text did not read it at a significantly faster or slower rate than subjects who read the principled revision and both groups read the texts in about the same amount of time. Yet, students who read the original text recalled significantly less than those who read the principled revision.

In repairing the inference calls, the authors of the study identified three principles for better texts:

Principle 1: Make the learner’s job easier by rewriting the sentence so that it repeats, from the previous sentence, the linking word to which it should be linked. Corollary of Principle 1: Whenever the same concept appears in the text, the same term should be used for it.

Principle 2 is to make the learner’s job easier by arranging the parts of each sentence so that (a) the learner first encounters the old part of the sentence, which specifies where that sentence is to be connected to the rest of his or her mental representation; and (b) the learner next encounters the new part of the sentence, which indicates what new information to add to the previously specified location in his or her mental representation.

Principle 3 is to make the learner’s job easier by making explicit any important implicit references; that is, when a concept that is needed later is referred to implicitly, refer to it explicitly if the reader may otherwise miss it.

Britton, B., & Gülgöz, S. (1991). Using Kintsch’s computational model to improve instructional text: Effects of repairing inference calls on recall and cognitive structures. Journal of Educational Psychology, 83 (3), 329-345 DOI: 10.1037//0022-0663.83.3.329

Are Teaching and Learning Coevolved?

some hummingbirds coevolved with some flower species

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

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

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

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

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

The Experiment

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

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

The Results

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

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

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

Coevolution of Teaching and Learning

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

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

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

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