Trig Ratios as Percents

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

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

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

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

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

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

  θ = °


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

An Example Problem: Testing Out the Percent Connection

41° 96 x

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

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

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

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

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


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).

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

The Problem of Stipulation

I think it would surprise people to read Engelmann and Carnine’s Theory of Instruction. (The previous link is to the 2016 edition of the book on Amazon, but you can find the same book for free here.) Tangled noticeably within its characteristic atomism and obsession with the naïve learner are a number of ideas that seem downright progressive—a label never ever attached to Engelmann or his work. One of these ideas in particular is worth mentioning—what the authors call the problem of stipulation.

problem of stipulation

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

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

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

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

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

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

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

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

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

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

Audio Postscript

Enhance the Salience of Relevant Variables

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

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

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


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

The Experiment and Results

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

salience of relevant variables

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

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

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

Be More Helpful

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

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

Performing the discrete mode of presentation test strongly enhances the salience of the relevant variable, perimeter, and somewhat decreases that of area. This enhancement supports appropriate solution strategies that lead to improved performance. This effect is robust and transfers to continuous mode of presentation for at least 10 days. In line with this conclusion, a student who performed the continuous test after the discrete one commented that, “It [continuous] was harder this time but I used the previous shapes, because I could do tricks with the matchsticks.”
Babai, R., Nattiv, L., & Stavy, R. (2016). Comparison of perimeters: improving students’ performance by increasing the salience of the relevant variable ZDM, 48 (3), 367-378 DOI: 10.1007/s11858-016-0766-z

Being Explicit About Symmetry


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

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

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

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

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

Another Observation

Working to orient oneself to the symmetries available in mathematical situations seems like one appropriate remedy to what I’ve called “left-to-rightism,” or “cinemathematics”—a syndrome that makes us teach concepts like the equals sign (unwittingly) in a left-to-right way, such that students take away (unwittingly) the misconception that the equals sign indicates that some answer is to follow, rather than that two expressions are equal. Some recent research points to the benefits of thinking about symmetry when teaching negative numbers as well.
Tsang, J., Blair, K., Bofferding, L., & Schwartz, D. (2015). Learning to “See” Less Than Nothing: Putting Perceptual Skills to Work for Learning Numerical Structure Cognition and Instruction, 33 (2), 154-197 DOI: 10.1080/07370008.2015.1038539

‘No Logic in the Knowledge’

Marilyn Burns left this really nice comment over at Dan’s a while ago. It’s one of the few times in my recent memory where I’ve seen an attempt to produce an understandable and functional rationale for a “no-telling” approach that doesn’t mention agency or character. Instead, it references the amount of ‘logic in the knowledge’ being taught. And it’s balanced and sensitive to context to boot. Here’s part of it:

Explicit instruction (teaching by telling?) is appropriate, even necessary, when the knowledge is based in a social convention. Then I feel that I need to “cover” the curriculum. We celebrate Thanksgiving on a Thursday, and that knowledge isn’t something a person would have access to through reasoning without external input―from another person or a media source. There’s no logic in the knowledge. But when we want students to develop understanding of mathematical relationships, then I feel I need to “uncover” the curriculum.

In a nutshell, some concepts are connected in such a way as to make it possible for students to derive one (or many) given another. There is ‘logic in the knowledge,’ and so explicit instruction is not strictly necessary to make connections between nodes in those situations; students can possibly do that themselves.

logic in the knowledge

A good example of a convention that math teachers might think of is the order of operations. One might argue that there’s no logic there; you just have to know the agreed-upon order, and so we have to teach it directly. By contrast, manipulating numerators and denominators when adding fractions has a logic behind it—if you’re adding win-loss records represented as fractions to determine total wins to total losses, then adding across is just fine. But it’s usually not fine, because the denominator often represents a whole rather than another part. Students do not necessarily have to be told this logic to get it. They can be led to discover that one third of a pizza plus two thirds of the pizza can’t possibly mean three sixths, or one half, of the pizza. Then they can build models to pin down exactly what fraction addition does represent, along with the connections to the symbolic representations of those meanings.

Okay, So That’s a Good Start Anyway

Yet, while it seems reasonable to me to suggest that one must teach explicitly when there is ‘no logic in the knowledge,’ it’s too strong, I think, to suggest that when the logic is there one must not teach that way. (And I note that Ms Burns does not go this far in her comment.) Explanations do not make it impossible nor even difficult for students to traverse those conceptual nodes in ‘logic-filled’ knowledge—unless they are bad explanations or they are foisted on students who have a lot of background knowledge or both.

Regardless, it seems like a pretty good start to say that when deciding where on the “telling” spectrum we can best situate ourselves, we can think about, among other things, how well connected a concept is to other concepts (for the students), the conceptual distance between two or more nodes (for the students)—there are a lot of good testable hypotheses we might generate by staying in the content weeds while boxing out distracting woo.

Ultimately, what I think is pleasing about Burns’s comment is that it is the beginning of a good explanation. It provides a functional rationale—tied directly to content, but not ignoring students—for choosing one or another general teaching method. And it can be connected to other things we know about, such as the expertise-reversal effect and the generation effect. These are the kinds of explanations we should be producing and looking for in education in my humble opinion. It is not necessary to be a researcher or academic to generate or appreciate them.

Image credit: Mauro Entrialgo.

Gricean Maxims and Explanation


When we converse with one another, we implicitly obey a principle of cooperation, according to language philosopher Paul Grice’s theory of conversational implicature and the Gricean maxims.

This ‘cooperative principle’ has four maxims, which although stated as commands are intended to be descriptions of specific rules that we follow—and expect others will follow—in conversation:

  • quality: Be truthful.
  • quantity: Don’t say more or less than is required.
  • relation: Be relevant.
  • manner: Be clear and orderly.

I was drawn recently to these maxims (and to Grice’s theory) because they rather closely resemble four principles of instructional explanation that I have been toying with off and on for over a decade now: precision, clarity, order, and cohesion.

In fact, there is a fairly snug one-to-one correspondence among our respective principles, a relationship which is encouraging to me precisely because it is coincidental. Here they are in an order corresponding to the above:

  • precision: Instruction should be accurate.
  • cohesion: Group related ideas.
  • clarity: Instruction should be understandable and present to its audience.
  • order: Instruction should be sequenced appropriately.

Both sets of principles likely seem dumbfoundingly obvious, but that’s the point. As principles (or maxims), they are footholds on the perimeters of complex ideas—in Grice’s case, the implicit contexts that make up the study of pragmatics; in my case (insert obligatory note that I am not comparing myself with Paul Grice), the explicit “texts” that comprise the content of our teaching and learning.

The All-Consuming Clarity Principle

Frameworks like these can be more than just armchair abstractions; they are helpful scaffolds for thinking about the work we do. Understanding a topic up and down the curriculum, for example, can help us represent it more accurately in instruction. We can think about work in this area as related specifically to the precision principle and, in some sense, as separate from (though connected to) work in other areas, such as topic sequencing (order), explicitly building connections (cohesion), and motivation (clarity).

But principle frameworks can also lift us to some height above this work, where we can find new and useful perspectives. For instance, simply having these principles, plural, in front of us can help us see—I would like to persuade you to see—that “clarity,” or in Grice’s terminology, “relevance,” is the only one we really talk about anymore, and that this is bizarre given that it’s just one aspect of education.


The work of negotiating the accuracy, sequencing, and connectedness of instruction drawn from our shared knowledge has been largely outsourced to publishers and technology startups and Federal agencies, and goes mostly unquestioned by the “delivery agents” in the system, whose role is one of a go-between, tasked with trying to sell a “product” in the classroom to student “customers.”