## 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 = °

cos-sin-1
1-tan-sec
cot-1-csc

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

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

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

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

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

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

## Motivation Is Caused by Achievement

research

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

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

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

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

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

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

Don’t Stop. Believin’.

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

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

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

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

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

## Educational Achievement and Religiosity

research

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

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

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

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

The Displacement Hypothesis

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

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

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

The Scattering Hypothesis

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

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

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

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

Religiosity or supernaturalism does not make you dumb. But perhaps it has the broader effect of making it more difficult to fasten minds onto reality, as it fills the solution space with only those possibilities that have little bearing on the real world we live in. This would certainly show up in measures of educational achievement.

Stoet, G., & Geary, D. (2017). Students in countries with higher levels of religiosity perform lower in science and mathematics Intelligence DOI: 10.1016/j.intell.2017.03.001

## Expert Knowledge: Birds and Worms

research

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

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

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

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

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

Interpret the Results

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

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

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

Experiment 3

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

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

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

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

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

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

## Modulus and Hidden Symmetries

research

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

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

mod

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

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

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

## Religiosity and Confidence in Science

research post

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

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

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

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

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

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

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

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

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

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

It may be, then, that educators surrounded by cultures with higher religiosity—and regardless of their own personal religious orientations—will simply have greater exposure to concerns about moral and spiritual harm that can be wrought by science, in addition to the benefits it can bring. Consistent with my own thinking about the subject, these concerns would be amplified in situations, like education, where science looks to produce effects on human behavior and cognition, especially children’s behavior and cognition.

Johnson, D., Scheitle, C., & Ecklund, E. (2015). Individual Religiosity and Orientation towards Science: Reformulating Relationships Sociological Science, 2, 106-124 DOI: 10.15195/v2.a7

## Provided Examples vs. Generated Examples

research post

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

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

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

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

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

Achievement Results

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

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

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

Also, Students’ Time Is Valuable

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

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

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

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

Some Commentary

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

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

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

## Bloom’s Against Empathy

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

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

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

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

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

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

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

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

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

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

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

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

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

Insanity: Repeatedly Adding and Expecting Multiplication

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

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

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

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

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

–E.D. Hirsch, Jr.

## From Translations to Slope

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

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

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

What Is Slope?

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

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

Y = Mx + B

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

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

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