Math, She Rote

My friends often have different educational backgrounds than mine. Some of them are younger, but even if they aren’t, they’re often from urban areas that had moved to more modern educational curricula before my school system had. The way I learned basic arithmetic remained unchanged from how it was taught from the early 1980s by the time I learned it in the late 1980s and early 1990s because that’s when our books dated from.

I learned during an interesting period in mathematical education history. It represented a kind of educational interbellum—a bit after the “New Math” of the 1960s and 1970s but before the “math wars,” instigated by the 1989 Curriculum and Evaluation Standards for School Mathematics. The latter 1989 publication has been called “reform mathematics,” which emphasizes processes and concepts over correctness and manual thinking. In other words, the educators promoting reform mathematics began to believe that the path students took toward the answer mattered more than whether they got the answer right. Many states’ standards and federally funded textbooks followed reform mathematics in the 1990s and beyond.

Reform mathematics emphasized constructivist teaching methods. Under this approach, instead of prescribing to students the best way how to solve a problem, teachers pose a problem and allow the student to surmount it by building on their own knowledge, experiences, perspective, and agency. The teacher provides tools and guidance to help the student along the way. Constructivist approaches involve experiments, discussions, trips, films, and hands-on experiences.

One example of a constructivist-influenced math curriculum, used in elementary school to teach basic arithmetic, was known as Investigations in Numbers, Data, and Space. It came with a heavy emphasis on learning devices called manipulatives, which are tactile objects which the student can physically see, touch, and move, to solve problems. These are items like cubes, spinners, tapes, rulers, weights, and so on.

As another example, someone I know recently described a system they learned in elementary school called TouchMath for adding one-digit numbers, which makes the experience more visual or tactile (analogous to manipulatives). They explained that for each computation, they counted the “TouchPoints” in the operands to arrive at the result.

I had never heard of TouchMath. In fact, I never solved problems using manipulatives, nor any analogue of them. I had little experience with this form of math education. We were given explicit instructions on traditional ways to solve problems (carrying, long division, and so on). Accompanying drawings or diagrams rarely became more elaborate than number lines, grids, or arrangements of abstract groupings of shapes which could be counted. They served only as tools to allow students to internalize the lesson, not to draw their own independent methods or conclusions.

I contrasted my friend’s experience with TouchMath to my experience. To add or subtract one-digit numbers, we merely counted. We were given worksheets full of these to do, and since counting for each problem would have been tedious and impractical, memorization for each combination of numbers would become inevitable. Given the expectations and time constraints, I’m certain rote memorization was the goal.

In a couple of years, we were multiplying and dividing, and we were adding and subtracting two- or three-digit numbers using carrying—processing the numbers digit-wise. At the same time, we were asked to commit the multiplication tables to memory. These expectations came in third grade, and it would be nearly impossible to make it out of fourth grade without committing the multiplication table and all single-digit addition and subtraction to memory (the age of ten for me).


Our teachers did not bother to force us to memorize any two-digit arithmetic operations. But I have some recollection a lot of years ago of my grandma telling me she had most two-digit additions and subtractions still memorized. It was just an offhand remark—maybe something she said as I was reaching for a calculator for something she had already figured out. Maybe we were playing Scrabble.

For context, she would have gone to school in rural Georgia in the 1940s and 1950s, and she graduated high school. (In that time and place, it was commonplace for many who intended to do manual, trade, or agricultural work not to continue through secondary school.)

I remember feeling incredulous at the time about the number of possible two-digit arithmetic operations that would imply memorizing. Of course, many would be trivial (anything plus or minus ten or one, or anything minus itself); others would be commonplace enough to easily memorize, while still others would be rare enough to ignore. But that still leaves several thousand figures to remember.

The more I thought about it, the more I saw that, in her world, it would make better sense to memorize literally thousands of things rather than work them out over and over. She had no way of knowing that affordable, handheld calculators would exist in a few decades after she graduated from school, after all. Each time she memorized a two-digit addition or subtraction, she saved herself from working out the problem from scratch over and over again for the rest of her life. This saved her effort and time every time she

  • balanced her checkbook,
  • filled out a deposit slip at the bank,
  • calculated the tax or tip on something,
  • tallied up the score for her card game,
  • totaled up a balance sheet for her business,
  • made change for a customer, or
  • checked that the change she was being given was correct,

to say nothing of all the hundred little things I can’t think of. She married young and has run small businesses for supplemental income all her life, so managing the purse strings fell squarely into her traditional gender role. Numbers were part of her daily life.

So for the first half of her life, none of this could be automated. There were no portable machines to do the job, and even the non-portable ones were expensive, loud, slow, and needed to be double-checked by hand.

I don’t believe she remembered these all at once for a test, the way I learned the multiplication tables in third grade. It seems likely she memorized them over time. It’s possible that expectations in her school forced a lot of memorization that I didn’t experience when I went many decades later, but maybe she was just extra studious.


I recall, as I went through school, having to rely more on a calculator as I approached advanced subjects. Before calculators became available to students, appendices of lookup tables contained pre-calculated values for many logarithms, trigonometric functions, radicals, and so on. Students relied on these to solve many problems. Anything else—even if it were just the square root of a number—came from a pen-and-paper calculation. (Many of my early math books did not acknowledge calculators yet, but this changed by the 1990s.)

Charles Babbage reported that he was inspired to attempt to mechanize computation when he observed the fallibility of making tables of logarithms by hand. He began in the 1820s. After a hundred and fifty years, arithmetic computation would become handheld and affordable, fomenting new tension around what rote memorization plays in both learning and in daily life.

Today, we’re still trying to resolve that tension. Memorization may feel like it has a diminished role in a post-reform education environment, but it’s by no means dead. Current U.S. Common Core State Standards include expectations that students “[b]y end of Grade 2, know from memory all sums of two one-digit numbers,” and, “[b]y the end of Grade 3, know from memory all products of two one-digit numbers.” That sounds exactly like the pre-reform expectations I had to meet.

All this means is that there has been neither a steady march away from rote memorization nor a retreat back to it. Research is still unclear about what facts are best memorized, when, or how, and so there’s no obvious curriculum that fits all students at all ages. For example, the Common Core Standards cite contributing research from a paper which reports on findings from California, concluding that students are counting more than memorizing when pushed to memorize arithmetic facts earlier. The paper reasons this is probably due to deficiencies in the particulars of the curriculum at the time of the research (2008).


I’m not an expert, and I don’t have easy answers, but my instinct is that rote memorization will always play an inextricable role in math education.

Having learned about the different directions in which the traditional and reform movements of math education have tugged the standards over the years, I tend to lean more traditional, but I attribute this to two things. One is that I was educated with what I remember to be a more traditional-math background, and though I didn’t like it, it seems serviceable to me in retrospect.

The other reason is that, for me, memorization has always come easily. I don’t really know why this is. It’s just some automatic way I experience the world. Having this point of view, though, I can easily see how beneficial it is to have answers to a set of frequent problems ready at hand. It’s efficient, and its benefits never cease giving over time. The earlier you remember something, the more it helps you, and the better you internalize it. Even for those who can’t remember things as easily, the returns on doing so are just as useful.

I do completely agree with the underlying rationale of the constructivist approach. Its underpinnings are based on Piaget’s model of cognitive development, which is incredibly insightful. It seems useful to learn early to accommodate the discomfort of adapting your internal mental model to new information by taking an active role in learning new ideas in order to surmount new problems.

I don’t necessarily believe that a constructivist learning approach is intrinsically at odds with rote memorization—that is to say, that memorization necessarily requires passive acquisition. In fact, the experience of active experimentation and active role may help form stronger memories. It’s more likely they compete in curricula for time. It takes longer to mathematically derive a formula for area or volume by independent invention, for example, than to have it given to you.

In fact, constructivist learning works better when the student has a broader reservoir of knowledge in the first place from which to draw to begin with when trying to find novel solutions to problems. In other words, rote memorization aids constructivist learning, which then in turn aids remembering new information.

My feeling is that math will always require a traditional approach at its very heart to set in place a broad foundation of facts, at least at first, before other learning approaches can have success. Though the idea of critical periods in language acquisition has detractors and heavy criticism, there is a kernel of truth to the idea that younger minds undergo a period of innate and intense linguistic fecundity. Maybe as time goes by, we can learn more about math acquisition and find out which kinds of math learning children are more receptive to at which ages. Until then, I feel like we’re figuring out the best way to teach ourselves a second language.

I am grateful to Rachel Kelly for her feedback on a draft of this post.

Privacy Policy Updates: Data Storage

I updated WordPress today to version 4.9.6. I noticed this version comes with support for implementing privacy policies throughout the site. I seem to have been ahead of the curve in implementing my own, but when the GDPR in the EU comes into effect this month, it will clarify and simplify data privacy for much of Europe. This implies enforcement will become a more direct matter as well. Any web service accessible to Europe and which does business in Europe now has updated their privacy policies to ensure it complies with the GDPR—which is why everyone has gotten a raft of privacy policy updates.

Most of these privacy policy updates pertain to what rights customers or users have to their own data. Often, they grant new rights or clarify existing rights. This week’s new version of WordPress is yet another GDPR accommodation.

Today, I have to announce my own GDPR update. Yes, I’m just a tiny website no one reads, and I provide no actual services. But having already committed to a privacy policy, which I promised to keep up to date (and announce those changes), I’m here to make another update.

One nice thing that came with the the WordPress update is a raft of suggestions on a good privacy policy (and in what ways WordPress and its plugins may cause privacy concerns). I found that I had covered most of them, but one thing I needed to revisit was a piece of functionality in Wordfence.

I use Wordfence for security: It monitors malware probes and uses some static blacklists of known bad actors. It also, by default, sends cookies to browsers in order to track which users are recurring ones or which users are automated clients. The tracking consisted only of an anonymous, unique token which distinguished visitors from one another. Unfortunately, this functionality had no opt-out and did not respect Do Not Track.

Although my tracking was only for security purposes—not for advertising—and although did not store any personal information, nor did I share with anyone else, I realized I would have to disable it.

I had made explicit mention of this tracking in my previous revision of my privacy policy:

I run an extra plugin for security which tracks visits in the database for the website, but these are, again, stored locally, and no one has access to these.

This is unfortunately more vague than it should have been, since it doesn’t mention cookies. It also provides no provision for consent. It merely states the consequences of visiting my site.

The GDPR makes it clear that that all tracking techniques (and specifically cookies) require prior consent. Again, I’m not a company, and I don’t provide any service. I’m not even hosted in the EU’s jurisdiction. My goal, though, is to exist as harmoniously with my visitors as possible, whomever they may be, and have the lightest possible touch.

So I’ve disabled Wordfence’s cookie tracking. I’ve added a couple of points to my privacy policy which clarify more precisely which data is logged and under which circumstances cookies may be sent to the browser.

This interferes my analytics, unfortunately—it’s no longer possible to be sure which visitors are humans anymore. I think it’s worth it, regardless.

I also made a couple of other changes based on WordPress’s suggestions. I moved a few bullet points around to put some points closer together which feel more logically grouped. I also added a point which specifies which URL my site uses (meaning the policy would be void if viewed in an archived format, within a frame, or copied elsewhere).

How Transgender Children Are Made and Unmade

Note that the following post discusses the sensitive topic of conversion therapy for transgender children, along with mentions of outmoded terminology and psychodynamic models, ethically questionable studies and treatment practices, and links to some sources which may misgender or mislabel transgender people.

I have also added some clarifications to my final points on 12 May 2018.

Today, a friend pointed me to a news article out of the UK covering a new study by Newhook et al. released in the International Journal of Transgenderism. The study was published a couple of weeks ago and criticizes a handful of other studies made in the last decade which bolster a myth that the vast majority (more than 80%) of children who have presented as transgender have since “desisted” (reverted to being cisgender) as adolescents or adults. Those studies, all released in the years since 2008, analyze children who were researched in the years since 1970 up until the 2000s.

Those recent desistance studies might hint at a couple of interpretations of transgender children who desist. The most neutral one is that such children were “going through a phase,” playing out the vagaries of youthful whims and later changing their minds. However, these studies also permit a more sinister interpretation—one in which children were subject to external influences that “confused” them about their gender, a confusion which time and therapy later allowed them to outgrow and reject.

It stands to reason that, because each child included in the original studies had contact with researchers, it was likely they were seeking treatment which included therapy, which might seem to support the latter interpretation. The standard of care for whichever diagnosis they received, which would have varied by location and time—more on this below—would possibly have focused, in fact, on influencing the child away from transgender or homosexual behaviors. Many research studies and forms of treatment, especially in earlier years, would have taken the form of conversion therapy. That also creates interpretative concerns from the original studies—they affect their own outcome. (This is referenced below as well.)


First, I want to briefly discuss the flaws from the desistance studies so that we can begin to erode the desistance myth. The news article above sums up the critique introduced by the new study quite well.

The ‘desistance’ figure come from studies conducted between the 1970s and the 2000s in the Netherlands and Canada, which assessed whether the kids that sought services at the gender clinic turned out to be trans as adults. The new publication concludes that the figure included all kids that were brought to the clinic, many of who never experienced gender dysphoria in the first place nor saw themselves as trans. Kids that shouldn’t have been a part of the figure were therefore being used to ramp up the numbers.

The news article elaborates that, not only is there uncertainty in how many children should have been counted as transgender in the first place, the earlier studies make blanket assumptions as to what happened to those children afterward.

Another flaw is that in the follow up, all participants that weren’t included for whatever reason were simply brushed off as ‘desisters’. This was done without having any factual evidence or knowledge about the children involved.

In what should have been simple division, the numbers on both sides of the division sign have become suspect. Now the question becomes, do we have the actual figures? Here’s where the real problems start. We need to delve into the primary source, the Newhook et al. study, itself.


The study is called “A critical commentary on follow-up studies and ‘desistance’ theories about transgender and gender-nonconforming children,” authored by Newhook et al. It contains a methodological meta-analysis of four previous studies. As it states in its introduction,

In the media, among the lay public, and in medical and scientific journals, it has been widely suggested that over 80% of transgender children will come to identify as cisgender once they reach adolescence or early adulthood. This statement largely draws on estimates from four follow-up studies conducted with samples of gender-nonconforming children in one of two clinics in Canada or the Netherlands (Drummond, Bradley, Peterson-Badali, & Zucker, 2008; Steensma, Biemond, de Boer, & Cohen-Kettenis, 2011; Steensma, McGuire, Kreukels, Beekman, & Cohen-Kettenis, 2013; Wallien & Cohen-Kettenis, 2008).

The critiques in the Newhook et al. study aren’t new, and the authors take pains to mention some of their forebears in their introduction as well. They contextualize their new study by explaining that they hope to guide the eighth upcoming version of the WPATH standards of care, which will determine how transgender children for years to come are treated.

Newhook et al. mention older follow-up studies from before the year 2008 of gender-non-conforming children, but the authors explain those studies are tainted by methodological and sampling problems. They are also likely irrelevant since they were not cited in the meta-analysis’s contribution to the 80% figure. So they skip these earlier studies in their meta-analysis.

We recognize that numerous follow-up studies of gender-nonconforming children have been reported since the mid-20th century (e.g., Green, 1987; Money & Russo, 1979; Zucker & Bradley, 1995; Zuger, 1984). In that era, most research in the domain focused on feminine expression among children assigned male at birth, with the implicit or explicit objective of preventing homosexuality or transsexualism.

(I’d like to draw your attention for a moment to the fact that Kenneth Zucker was an author both in the 1995 study above and in the later 2008 study mentioned earlier. We’ll return to him later.)

Now, the Newhook et al. critical commentary study notes that the four desistance studies arrive at a figure of over 80% desistance. Then it begins to note what abilities and limitations these studies have. The methodological concerns center around what we can know and can’t know, given what information was collected at the time and afterward.

What I found was that because the studies used children from times spanning from 1970 onward, the basis for diagnosis itself seeded the flaws in mis-categorization, both in mis-categorizing children as transgender in the first place and then again on follow-up.

Back in 1970, no formal diagnosis for gender identity disorder or gender dysphoria existed. Doctors and researchers had only informal descriptions. As Newhook et al. explain,

However, the plain-language meaning of gender dysphoria, as distress regarding incongruent physical sex characteristics or ascribed social gender roles, has been established since the 1970s (Fisk, 1973). When these four studies refer to gender dysphoria, they are referring to this plain-language context of distress, and not the newer DSM-5 diagnostic category.

The DSM-III would not exist until 1980, so the meanings applied here may vary from person to person, as experience and prejudice allow.  I do not know all the criteria which were applied. (I have been unable to locate the Fisk source, but he appears to be the source of the term “gender dysphoria.”)

Then, in the 80s and 90s, the DSM-III, DSM-III-R, DSM-IV, and DSM-IV-TR each included a “gender identity disorder” diagnosis which came with a “GID/Children Transsexualism” or “gender identity disorder in children” category. The symptomatology of these were similar in general shape and included distress (a gender dysphoria component) but also certain behaviors (e.g., crossdressing), timeframes (e.g., six months), and so on. This is a very definite case of moving the goalposts, where the diagnostic criteria shifted. In some ways, they became more lax. Diagnostic criteria often state that only a certain number out of all of the above need be satisfied over a period of time, so if every component but gender dysphoria is present, the diagnosis of gender identity disorder can still apply.

At the same time, the standards of care also were shifting, evolving through time to match the competing typologies and psychosexual models of the providers. Adults learned to conform to expectations (such as crossdressing for a year before receiving treatment or professing attraction to men where no such attraction existed).

Children who may not have been aware of these standards and criteria, acting on their needs and wants, might have very well fallen in and out of the categorizations changing around them. Through no fault of their own, the category of transgender might one day have landed upon a child and then another day slipped away from them.

The Newhook et al. study describes the problem this way:

Due to such shifting diagnostic categories and inclusion criteria over time, these studies included children who, by current DSM-5 standards, would not likely have been categorized as transgender (i.e., they would not meet the criteria for gender dysphoria) and therefore, it is not surprising that they would not identify as transgender at follow-up. Current criteria require identification with a gender other than what was assigned at birth, which was not a necessity in prior versions of the diagnosis. For example, in Drummond et al. (2008) study […] the sample consisted of many children diagnosed with GIDC, as defined in the DSM editions III, III-R, and IV (American Psychiatric Association, 1980, 1987, 1994). Yet the early GIDC category included a broad range of gender-nonconforming behaviors that children might display for a variety of reasons, and not necessarily because they identified as another gender. Evidence of the actual distress of gender dysphoria, defined as distress with physical sex characteristics or associated social gender roles (Fisk, 1973), was dropped as a requirement for GIDC diagnosis in the DSM-IV (American Psychiatric Association, 1994; Bradley et al., 1991). Moreover, it is often overlooked that 40% of the child participants did not even meet the then-current DSM-IV diagnostic criteria. The authors conceded: “…it is conceivable that the childhood criteria for GID may ‘scoop in’ girls who are at relatively low risk for adolescent/adult gender-dysphoria” and that “40% of the girls were not judged to have met the complete DSM criteria for GID at the time of childhood assessment… it could be argued that if some of the girls were subthreshold for GID in childhood, then one might assume that they would not be at risk for GID in adolescence or adulthood” (p. 42). By not distinguishing between gender-non-conforming and transgender subjects, there emerges a significant risk of inflation when reporting that a large proportion of “transgender” children had desisted. As noted by Ehrensaft (2016) and Winters (2014), those young people who did not show indications of identifying as transgender as children would consequently not be expected to identify as transgender later, and hence in much public use of this data there has been a troubling overestimation of desistance.

Because of the meaningful shifts in diagnostic criteria over the last fifty years, there’s little hope of reconstructing the true figures of desistance, such as they may be. We would need both detailed notes (interviews, etc.) from the original cohorts to attempt to assess the children’s self-reported identities and then those same cohorts’ adulthood identities, assessed the same way from follow-ups, to compare. I suspect the paucity of detailed qualitative data from the original studies would undermine such an effort, due to the primacy of researchers’ diagnoses over self-described experiences and identities.

In most studies, it appears we do not have such detailed notes and the like available. Newhook et al. do cite Steensma et al. (2011) as having some unique qualitative research, but quantitative data are very limited—there are only two interviews mentioned.


The Newhook et al. study also brings up many ethical concerns, and here I turn back to the problem of Zucker in particular. The authors identify three ethical concerns, of which the second is particularly insidious—the questionable goals of treatment itself.

In describing their second concern, the authors write,

A second ethical concern is that many of the children in the Toronto studies (Drummond et al., 2008; Zucker & Bradley, 1995) were enrolled in a treatment program that sought to “lower the odds” that they would grow up to be transgender (Drescher & Pula, 2014; Zucker, Wood, Singh, & Bradley, 2012; Paterson, 2015). Zucker et al. (2012) wrote: “…in our clinic, treatment is recommended to reduce the likelihood of GID persistence” (p. 393).

As I write, Zucker’s words are only six years old. To be clear: he is both espousing and practicing conversion therapy of children.

Zucker is not a marginalized figure in the world of psychiatry. He is not only respected and accepted; he was the head of the “Sexual and Gender Identity Disorders” group that revised the DSM-5, appointed by the American Psychiatric Association. A heartbreaking account of his attempt at conversion therapy may be found in this NPR story (with some misgendering).

He was not the only person in the group to favor controversial theories, either. Blanchard (who favors an outmoded typology of transgender people based on sexual attraction and also attempts conversion therapy) and Lawrence (who has expressed the belief that transgender people have a kind of body integrity identity disorder) also formed part of the group.

Why do I mention their role in shaping the DSM-5? Well, they believe children should be dissuaded from transgender identities, which they regard as pathological or maladaptive. Under their influence in shaping the diagnostic criteria for children and adults, they moved the goalposts for fitting the model. That then allowed studies to tally up how past children fit current, different diagnostic criteria to determine that they have “desisted.” In turn, these fudged figures can be used to justify further conversion therapy, resist affirmative care models of treatment, and influence the WPATH standards of care to inhibit access to treatment and personal safety.

I therefore question whether—after influencing or directly authoring new diagnostic standards for gender dysphoria—advocates for conversion therapy then revisited older studies to make follow-ups, aware of how the results would skew toward their desired outcome: an interpretation of a seeming tendency toward desistance, which marks transgender identities as “unnatural” aberrations which only emerge later in life and which can be headed off earlier in childhood. Buried underneath this interpretation is an implicit assumption about how children form transgender identities due to extrinsic influences. They conclude that they can prescribe a model of care which essentially counteracts those influences with their own.

Wiser people than I have already explained why better models of care, such as the affirmative care model, practiced in most North American clinics, provide better outcomes. The news article I began with also concludes with some great sources on treatment outcomes, which I cannot possibly outdo, so I’ll leave you to revisit Owl’s article.

Denying children bodily autonomy and agency over their identity is a form of abuse. The long-lasting confusion may result in self-denial, withdrawal, self-harm, or even suicide later in life. Unlike many forms of abuse, which happen privately, transgender conversion therapy coopts institutions toward its own ends by shaping the standards of care for treatment (via its influence on the WPATH with influential studies) and by writing the diagnostic manual itself. The prevalent myth of desistance of childhood gender dysphoria has been a powerful tool used to abuse children. It must be dismantled. To do so, we must expose pernicious and specious studies, using critical meta-analysis such as Newhook et al.’s.

I am grateful to Zuzu O. for feedback on this post.