# Writing good questions for fast finishers

Here’s the original activity, from Illustrative Math’s 8th Grade curriculum:

I love the main task. I don’t really know what to do with that “Are You Ready For More?” extension, though.

It’s definitely challenging and interesting, but it’s not really connected deeply with the math that comes before it. And really solving that HANGER + HANGER + HANGER = ALGEBRA problem likely would require a bunch of time — where is that time coming from? Usually I don’t need something that’ll take kids ~15 minutes to solve, I need something interested a fast-finisher can think about for ~3 minutes.

Put it like this: I’m not sure it would be worthwhile (or even possible) to talk about that HANGER/ALGEBRA puzzle with the whole class, if only a couple students even got to it. But if I’m not going to be willing to honor those sorts of problems with airtime, why would a kid ever dig into it?

I don’t mean to give the Illustrative Math curriculum a hard time, as I’m a huge fan. I’m really thinking more about how to improve the problems I’ve prepared for students who finish quickly. How can I make it clear that they really are part of our class?

I’ve really been playing with these ideas this year. Here was my replacement for the HANGER/ALGEBRA puzzle:

The math is close enough to the task itself that I really felt like I could talk about these problems with everyone — even briefly — and it was valuable. Kids who try it will get some airtime. I’m trying to bring the extensions closer to the main task.

I’ve also been trying this with geometry, where I built an activity on top of some Don Steward practice problems:

Understandably, Illustrative Math doesn’t have extensions like these ready for every problem. But something like this is an easy way to make an activity more useful for a broader range of students (which also gives kids who need it more time).

But Illustrative Math is a free and openly licensed curriculum. They’re actively seeking user contributions to their curriculum — all this seems like a nice match for the sorts of revisions I’m talking about here.

How long would it take to write these things? How many could 20 teachers write in an afternoon? One for each 8th Grade task?

A teacher can dream!

# Is there a good way that schools could better serve their most advanced students?

This essay is co-authored with TracingWoodgrains, and is the result of our adversarial collaboration project.

TW and I have deep disagreements about education. I’m a pessimist about personalization software and our ability to gamify learning; TW is an optimist. TW thinks there very well might be a deep reserve of untapped potential at the upper ends of student achievement; I don’t really think this is true. I don’t think a radical restructuring of schools would change much unless you could simultaneously change the thoughts and desires of parents and students; TW thinks we should seriously consider rebuilding schooling on the grounds of mastery learning without age-grading.

With all this disagreement, we set out to see if we could write something coherent about our areas of agreement. Not easy.

We also set out to win a cool \$1K from the blog SlateStarCodex, who hosted a contest for the best adversarial collaboration and has published our essay but has not yet told us if we won.

tl;dr THIS THING IS LONG. Here is the section breakdown:

Section One: Gifted underachievement is real, but there are no easy answers.

Section Two: Expanding or deepening tracking is unlikely to help even top students without deeper curricular modification.

Section Three: Acceleration could, though.

Section Four: The thing is that schools are not in the business of maximizing learning at all costs.

Section Five: Personalization software, as it exists today, is almost universally bad. The potential for software to help learning comes when it pays attention to the social, motivational aspects of learning.

Section Six: Our recommendations for parents, educators, kids.

Please skip around. Skip to Section 5 if you care about edtech and personalization software. If you only read one section, I’d personally prefer if you read Section 4, but hey, that’s just me. TW would (I’m guessing) want you to read Section 1.

Also this was written for a non-teacher audience, i.e. not my blog. So apologize if the tone is off. Also half of this is TW’s voice, so it probably reads different than what you’re used to hearing from me. Anyway, enjoy, and please comment with thoughts, disagreements, agreements, etc.

***

“What do America’s brightest students hear? Every year, across the nation, students who should be moved ahead at their natural pace of learning are told to stay put. Thousands of students are told to lower their expectations, and put their dreams on hold. Whatever they want to do, their teachers say, it can wait.”  – A Nation Deceived, p.3

“There is an apparent preference among donors for studying the needs and supporting the welfare of the weak, the vicious, and the incompetent, and a negative disregard of the highly intelligent, leaving them to “shift for themselves.” Hollingworth, 1926

1. Eager to Learn and Underachieving

Pretend you’re a teacher. With 25 students, who gets your attention during class?

There’s the kid who ask for it, whose hand is constantly up. There’s also the quiet kid in the corner who never says a word, but has been lost in math since October, who will fail if you don’t do something. There’s the student in the middle of the pack, flowing along. Finally, there’s the kid who finishes everything quickly. She’s looking around and wondering, what am I supposed to do now?

In a survey of teachers from 2008, just 23% reported that advanced students were a top priority for them, while 63% reported giving struggling students in their classes the most attention. A 2005 study found the same trend in middle schools, where struggling students receive the bulk of instructional modification and special arrangements. This was true even while 73% agreed that advanced students were too often bored and under-challenged in school. While teachers, it seems, are sympathetic to the smart bored kid, that’s just not a priority for them.

This isn’t to blame teachers who are under all sorts of pressure to carry low-performing students over the threshold and who, in any event, are only trying to do what’s best for their kids. Which is the most urgent concern? If you don’t equip a kid with the skills they need, next year’s class might be a disaster for them. Or maybe they’ll fail out of school. And behavior problems? Often those begin with academic struggles. Gifted children, on the other hand — they’re on the way to becoming gifted adults. They can take care of themselves, for a minute, the logic goes. More often than not, the teacher will encourage the early finisher to go read a book, or start homework, or do anything at all while the teacher works to help the quiet, lost kid in the corner.

If the kids are just a little bored, that’s nothing strange. It’s hard to find someone who wasn’t bored in school sometimes. For many top students, already poised for achievement, this turns out just fine. And yet, there are persistent stories of how the lack of challenge can turn into something more serious.

One version of the story goes like this: from a young age, a student finds the work in school easy. It doesn’t take long for them to expect school to be easy for them — it becomes a point of pride. Over years of floating through school, an identity takes hold. Then, one day, maybe after years of schooling, something finally becomes challenging for the student… but there’s nothing nice about this challenge. The challenge is now a threat. The student begins to find school challenging, and their world falls apart. They feel isolated and misunderstood at school. They lash out. They hate it, and they can’t wait to get out.

When we asked Reddit users and blog readers to describe their experience of school, we heard versions of this story:

• Miserable waste of time, was almost never offered opportunities to learn. Largely ignored teachers and read books during class. I felt like it was a profound injustice that I was punished for doing so. I now have kids of my own and will be home-schooling them.
• I was bored. The pace was too slow and work was not interesting. Being forced by law to get up early and go somewhere to learn things I already know means permanent and firm dislike.
• I went to local public schools for kindergarten through high school, and the experience wasn’t good. Academically, the classes were slow and poorly taught. Even the AP classes were taught at the speed of the slowest student, which made the experience excruciating. The honors and regular classes were even worse: I was consistently one or more grades ahead of the rest of the class in every non-AP class except honors math. I learned not to bother studying or doing homework even in the AP classes which probably wasn’t great for my work ethic.

The stories of student pain and underachievement in school get more intense as we consider cases of extremely precocious children. The pressures on the student increase, and without help a student often experiences isolation from their peers and a whole other host of difficult feelings. Miraca Gross studied students like these in Australia and found that precocious students were often suffering in silence. Speaking particularly about precocious students who underachieve, she writes:

The majority of the extremely gifted young people in my study state frankly that for substantial periods in their school careers they have deliberately concealed their abilities or significantly moderate their scholastic achievement in an attempt to reduce their classmates’ and teachers’ resentment of them. In almost every case, the parents of children retained in the regular classroom with age peers report that the drive to achieve, the delight in intellectual exploration, and the joyful seeking after new knowledge, which characterized their children in the early years, has seriously diminished or disappeared completely. These children display disturbingly low levels of motivation and social self-esteem. They are also more likely to report social rejection by their classmates and state that they frequently underachieve in attempts to gain acceptance by age peers and teachers. Unfortunately, rather than investigating the cause of this, the schools attended by these children have tended to view their decreased motivation, with the attendant drop in academic attainment, as indicators that the child has “leveled out” and is no longer gifted.

What do we make of these stories? How common are such experiences?

From the literature on “gifted underachievement” we get partial confirmation — underachievement is a real phenomenon, supported by numerous case studies. According to a survey of various school practitioners, underachievement is the top concern when it comes to gifted students. By definition, advanced students are only a small percent of each student body, so few are affected in any given place, but on a national scale it becomes a more serious problem.

This is not just a problem for the affluent. It has persistent impacts on Black students, poor students, and students who are learning English, who are less often recommended for gifted programs or special accommodations. Here’s one way this manifests itself: in one study, 44% of poor students identified as gifted in reading in 1st Grade were no longer academically exceptional by 5th Grade. For higher-income families, only 31% of 1st Graders experience this slide.

The lack of attention to this group extends to the research. It’s difficult to pin down the number of students impacted. While underachievement is a real phenomenon, current research doesn’t tell us very much about the factors contributing to gifted underachievement. What studies have been done tend to focus almost entirely on things like whether students with ADHD or unsupportive families underachieve, rather than looking at controllable factors like the sort of teaching students experience in school.

Schools are the institutions in charge of educating kids. Those who rush into school, eager to learn, should not walk out feeling rebuffed and ignored. This is doubly true for talented kids from at-risk populations, who may not have the support structure outside of school to ensure their success if school has no time for them. It’s clear, though, that we cannot degrade the experience of other students to help those who already have an academic leg up. Is there a feasible approach to address this problem without making things worse?

We have good reason to think that personalized attention makes a huge difference to a student’s learning. Research suggests that tutoring that supplements a student’s coursework is a very effective educational intervention. Benjamin Bloom caught people’s attention with the idea of a 2 standard deviation effect in the 1980s. More recent research has lowered that sky-high estimate to more realistic numbers, and a meta-analysis found an effect size of 0.36, still a powerful impact, enough to take a student from the 50th percentile of achievement to the 64th.

If supplemental tutoring works, the dream goes, what if we replaced classroom work entirely with tutoring? Can’t we just do that for gifted underachievers and precocious students? We have tantalizing success stories of this kind in the education for precocious children. In a famous case, John Stuart Mill’s father decided that the philosophy of utilitarianism needed an advocate, and planned a demanding course for him. Mill didn’t underachieve: he learned Greek at age 3, Latin at age 8, and flourished as a philosopher. László Polgár declared he had discovered the secret of raising “geniuses” and went about showing it by tutoring his daughters in chess from the age of 3. It’s hard to argue with his results: two grandmasters and an international master, one of whom became the 8th ranked chess player in the world and the only woman ever to take a game off the reigning world champion.

Though this sort of tutoring seems like a dream come true for underachieving gifted students, in practice it’s a non-starter in schools. (It lives on in homeschooling, to an extent). In a world where schools are struggling to help every kid learn to read, the ethics of only assigning tutors to gifted students is dubious and almost certainly a political impossibility. The cost of assigning a tutor to every child, meanwhile, would do something special to property taxes. This simple answer, then, can lead to a clearer understanding of the complexity of educational questions: It’s possible to focus on simple practices that work while disregarding nonacademic concerns and political feasibility.

To be useful, educational ideas should be effective, politically feasible, and economical. If tutoring for gifted underachievers isn’t workable, might there be some other way to approximate the benefits of personal, human attention? Here are three of the most common tools that advocates for gifted education propose:

What follows is an evaluation of how promising each of these tools is, both in theory and in practice.

Our favorite one-stop reading on tutoring: this.

1. Ability Grouping (a.k.a Tracking)

The case for placing students of similar abilities together in a classroom seems like it ought to be as simple as the case for tutoring. Teachers will be more effective if their students have similar pacing needs. So, group kids who need more time in one class and those who need less time in another. It’s not tutoring, but it should be the next best thing.

Things in education research are rarely that simple, though.

Bob Slavin, a psychologist who studies education, is one of the most-cited education researchers around. He seems like a compulsively busy fellow. He writes, he runs research centers, he designs programs for schools. (He blogs.) A journalist from The Guardian once asked Slavin for his likes and dislikes, and in case you were wondering he likes work and dislikes complacency.

In the late ‘80s and early ‘90s, Slavin performed a series of meta-analyses of the existing literature on tracking and between-class ability grouping. Overall, he found no significant benefits from ability grouping, even for “top track” students across elementary, middle, and high schools.

But the other surprising finding of Slavin’s was that nobody was academically hurt by ability grouping — not even the lowest track students. Slavin argued that when you consider all the non-academic concerns, the scales weigh in favor of detracking, i.e. avoiding ability grouping.

What are those non-academic concerns? In the conclusion of his review of the evidence from elementary schools, he writes:

“Ability grouping plans in all forms are repugnant to many educators, who feel uncomfortable making decisions about elementary-aged students that could have long-term effects on their self-esteem and life chances. In desegregated schools, the possibility that ability grouping may create racially identifiable groups or classes is of great concern.” (p.327)

That’s Slavin’s view. So, where is the debate?

One thing that is decidedly not up for debate in the literature is that Slavin’s non-academic concerns are real. Opponents and defenders of tracking alike agree that low-track classes are often chaotic, poorly taught environments where bad behavior is endemic, and that this is a major problem. Tom Loveless is a contemporary defender of tracking, and writes that “even under the best of conditions, low tracks are difficult classrooms. The low tracks that focus on academics often try to remediate through dull, repetitious seatwork.” Jeannie Oakes made a name for herself by carefully documenting the lousiness of a lot of low track classes.

Some tracked schools seem to have done better with their low tracks. Gamoran, an opponent of tracking, speaks highly of how some Catholic schools handle lower tracks. Gutierrez identifies several tracked schools with strong commitments to helping students across the school advance in mathematics, and concludes that “tracking is not the pivotal policy on which student advancement in mathematics depends.” Making these experiences better is an important goal. These difficult dynamics are a genuine and widespread issue, though, and educators are rightly concerned about them.

Slavin’s concerns about exacerbating racism in schools are relatively uncontroversial as well. It’s not so much that race is a factor in track placement. Using a large nationally representative sample and controlling for prior achievement, Lucas and Gamoran found that race wasn’t a factor in track placement. (Though Dauber et al, found that race was a factor in track placement in Baltimore schools, so maybe sometimes racism is a factor in placement.)

But because of existing achievement gaps between e.g. Black and white students, there’s the potential in a racially mixed school that ability groups will effectively sort Black students into the lowest track and expose them to a lot of dynamics that are difficult to quantitatively measure but frequently discussed in education. A school where being Black is associated with poor performance and misbehavior will, according to many educators and researchers, lead to lower expectations and academic self-esteem for all Black students.

(Good news for people who like bad news: school segregation is getting worse, so the interaction between tracking and race is getting better.)

The main controversy surrounds Slavin’s claims about the academic impact of ability grouping. His meta-analyses were part of an extended back-and-forth with Chen-Lin & James Kulik, who wrote several competing analyses on the ability grouping literature. Slavin and the Kuliks each criticized the other’s methodology, but the core point the Kuliks made was that ability grouping did have positive effects on gifted students as long as curriculum was enhanced or accelerated to match, and that this typically did happen in dedicated gifted and talented programs. The Kuliks pointed out that both they and Slavin largely agreed on the data both analyzed, but that Slavin excluded studies of gifted programs from his research while the Kuliks made those studies a focus.

Tom Loveless, senior fellow at the Brookings Institution, summarized one important aspect of their dispute, which is that their debate centers more on values than their read of the extant evidence:

Slavin and Kulik are more sharply opposed on the tracking issue than their other points of agreement would imply. Slavin states that he is philosophically opposed to tracking, regarding it as inegalitarian and anti-democratic. Unless schools can demonstrate that tracking helps someone, Slavin reasons, they should quit using it. Kulik’s position is that since tracking benefits high achieving students and harms no one, its abolition would be a mistake (p.17)

Betts notes the studies the Kuliks and Slavin reviewed in their meta-analyses had some flaws, with relatively small N and non–nationally representative data. Using more nationally representative samples, a number of researchers (Hoffer, Gamoran and Mare, Argys, Rees and Brewer) came to the conclusion that conventional tracking benefits students in the high tracks and hurts those in the low tracks. But it’s really hard to control for the right factors in these definitely non-experimental studies, and Betts and Shkolnik raise questions about the results of these papers. And there was also a recent big meta-meta-analysis that found no benefits for between-class grouping, echoing Slavin, but that did find benefits for special grouping for gifted students, echoing the Kuliks.

Just to mess with everybody, Figlio and Page argue that by attracting stronger students to the school (because parents seek tracking) students in low-tracks benefit, secondarily.

So, in summary, what should we make of all this? Betts, an economist, says in a review of the literature that when it comes to the average impact of tracking or the distribution of achievement “this literature does not provide compelling evidence.” Loveless doesn’t disagree, but notes that for high achievers, the situation is clearer:

“The evidence does not support the charge that tracking is inherently harmful, and there is no clear evidence that abandoning tracking for heterogeneously grouped classes would provide a better education for any student. This being said, tracking’s ardent defenders cannot call on a wealth of research to support their position either. The evidence does not support the claim that tracking benefits most students or that heterogeneous grouping depresses achievement. High achieving students are the exception. For them, tracked classes with an accelerated or enriched curriculum are superior to heterogeneously grouped classes.” (p.22)

At the end of the day, all academic impacts of tracking are mediated by teaching and the curriculum. If a teacher doesn’t change what they teach or how they teach it, no grouping decision will help or hurt a student academically in a significant way. Tracking only could benefit gifted students if it came with some sort of curricular modification.

This is a conclusion with wide-reaching support. Even Slavin, who so staunchly opposed conventional ability grouping, was extremely impressed by something called the Joplin Plan, which involves three core features:

• Regular testing and regrouping of students on the basis of the tests
• A different curriculum for each group of students

Slavin, the Kuliks, and everyone else seemed to agree that students in the plan — at all ability levels — tended to get 2-3 months ahead of students in typical programs over a year of instruction. The Joplin plan involves ability grouping — the good kind of ability grouping.

So in 1986, when the Baltimore School Superintendent turned to Bob Slavin to design a program that would improve the city’s most dysfunctional schools, guess how Slavin grouped students?

Slavin worked with research scientist Nancy Madden (they’re married) to design Success for All for Baltimore, and it’s a prominent program in the school improvement world, implemented in thousands of schools and spreading. Those three features of the Joplin plan — assessment, regrouping along the lines of ability and targeted teaching — are core features of their program.

Success for All isn’t the only example of a successful curriculum implementing these ideas. Direct Instruction was created by Siegfried Engelmann and Wesley Becker in the 1960s, and it also groups students according to their current levels in reading and math while frequently reassessing and regrouping. DI has a strong body of research supporting its efficacy (for one, it was the winner of the famous-in-education Follow Through experiment), but fell largely out of favor outside of remedial classrooms. In early 2018, a new meta-analysis spanning 50 years of research reinvigorated conversation around Direct Instruction. It found an average effect size of 0.51 to 0.66 in English and math over 328 studies (p<0.001), — strong evidence that the program works.

While its effect on student performance is rarely disputed, the program remains controversial. Historian of education Jack Schneider writes: “Direct Instruction works, and I’d never send my kids to a school that uses it. The program narrows the aims of education and leaves little room for creativity, spontaneity and play in the classroom. Although test scores may go up, the improvement is not without a cost.” Ed Realist worries that its pedagogy is unsavory, has not been shown to work for older students, that wealthier parents are voting with their feet against the curriculum, and that DI could exacerbate gaps between students. Supporters, by contrast, paint the picture of a robust, effective system that has been ignored and disregarded.

Success for All and Direct Instruction are not simple programs for schools to adopt. Implementing them amounts to a major organizational change, and pushes at the extremely resilient notion that children in school should be grouped by their ages. Comprehensive ability grouping programs such as these seem to work, but in practice they are rarely used.

Our favorite one-stop source for reading on ability grouping: here, or maybe here to get a broader picture of the controversy.

1. Acceleration

Forget the comprehensive approach, then. Does it work to simply move an individual student (e.g. an underchallenged and frustrated student) through the curriculum at whatever pace seems to make sense?

There are a few different ways schools can help some students access the curriculum more quickly. A kid can skip a full grade, or several grades in extreme cases. They can stay in their grade for some classes, but join higher grade levels for some parts of the day. They might be assigned to two classes in one year (e.g. Algebra 1 and Geometry). Or, in some cases, a young student might start school at an even younger age than is typical.

If a child is ready for a higher level within a subject and studies it instead of the lower level, it’s almost a given that they’ll learn more. The real research questions are (a) from an academic standpoint whether accelerated children do tend to be ready, or if they do poorly in classes post-acceleration) and (b) whether acceleration exposes students to non-academic harm (e.g. stress, demotivation, loss of love for subject, poor self-esteem).

The Study of Mathematically Precocious Youth (SMPY) is an ongoing longitudinal study examining thousands of mathematically gifted students. In one SMPY study, researchers compared the professional STEM accomplishments of mathematically gifted students who skipped a grade to those who remained at grade level. They found that, controlling for a student’s academic profile in a pretty sophisticated way, students who skipped a grade tended to be ahead of the non-skippers in terms of degrees earned, publications, citations accrued, and patents received. From this work it seems skipping a grade in the SMPY cohort did nothing to hurt a kid’s learning or enthusiasm for their passions.

Acceleration has been one of the focuses of SMPY studies. A 1993 piece about SMPY findings reported “there is no evidence that acceleration harms willing students either academically or psychosocially.” This is supported by various meta-analyses, going back to the 1984 Kulik & Kulik paper and confirmed by more recent work such as a 2011 analysis of existing studies. Beyond the “does no harm” findings, these meta-analyses also report academic benefits to students.

It can be confusing, when reading these studies, to keep track of just how gifted the students happen to be. For example, SMPY has studied five cohorts so far, ranging from students who assessed in the top 3% to those who assessed in the top 0.01%. As we consider students farther away from the mean of achievement, the need for acceleration becomes more acute.

Lots of teachers encounter “1 in 100” students every year, but the education of “off the charts” students is necessarily more a matter of feel than policy. Still, there are success stories to learn from, and they show a remarkable sensitivity to both the academic and social well-being of the student.

Terence Tao is a famous success story of this kind. He surprised his parents by discovering how to read before turning two, and as a child he started climbing through math at a blistering rate. He was identified as profoundly gifted from a young age, and his education was carefully tracked by Miraca Gross as part of her longitudinal study of profoundly gifted children:

His parents investigated a number of local schools, seeking one with a principal who would have the necessary flexibility and open-mindedness to accept Terry within the program structure they had in mind.

This set the pattern for the ‘integrated,’ multi-grade acceleration program which his parents had envisaged and which was adopted, after much thought and discussion, by the school. By early 1982, when Terry was 6 years 6 months old, he was attending grades 3, 4, 6 and 7 for different subjects. On his way through school, he was able to work and socialize with children at each grade level and, because he was progressing at his own pace in each subject, without formal “grade-skipping,” gaps in his subject knowledge were avoided.

His education continued in much the same fashion, culminating in a Ph.D. by the age of 21 and a remarkable and balanced life since. He has since given his own advice on gifted education.

Given the success of acceleration, are we accelerating enough? On the one hand, it appears that acceleration is a widely used tool for giving gifted students what they need. When looking at the top 1 in 10000 students in terms of mathematical ability as identified by the SMPY, nearly half of the group skipped grades, and almost all of them had some form of acceleration, whether that meant advanced classes, early college placement, or other tools. About two-thirds reported being satisfied with their acceleration, rating it favorably across many categories:

[Source: SMPY]

The dissatisfied third of those 1 in 10000 students, for the most part, reported wishing they had been offered more acceleration. And advocates for gifted education strongly endorse the notion that acceleration is under-used. A Nation Deceived is premised on this idea — though besides for “more” the report doesn’t get specific concerning how many students ought to be accelerated, and the report mostly makes a cultural argument in favor of acceleration, citing stories like Martin Luther King Jr. graduating high school at 15.

We wanted to know more about how educators think about acceleration, so we surveyed (via twitter) twenty-one teachers, academic coaches, tutors and administrators. The survey prompted educators to respond to the following scenario:

In your school there is currently a 1st Grader who does math above grade level, e.g. he performs long division in his head. His parents initiated contact with the teacher after hearing their child complain that math at school was boring. They’re concerned that he isn’t being challenged. The classroom teacher knows that he is above grade-level in math, and is trying to meet his needs in class. The parents, however, do not think the current situation is working. The teacher reports that the student is difficult to engage during math class, and that sometimes he misbehaves during math.

From their responses, it certainly seems that acceleration was on the table, but almost always the last option after a number of in-class or non-classroom options (e.g. after school clubs) were explored. That acceleration in math should be a “break in case of emergency” response is also the line offered by the National Council of Teachers in Math: tracking is morally indefensible, acceleration should be viewed with suspicion but can sometimes be appropriate.

In many ways, mainstream education is living in Bob Slavin’s world. He was a leading opponent of tracking, but was impressed by certain forms of ability grouping. He took the research on ability grouping that actually works (through assessment, frequent regrouping, and curricular modification) and used it to create a program for failing schools. He expresses suspicion about acceleration of gifted students in general, but agrees that at times it is a useful and necessary tool. If you broach the conversation about acceleration with your child’s teachers, you might hear some version of Bob Slavin’s take.

There is more to say about where this skepticism comes from. But it’s important to note that just because a student could be accelerated doesn’t always mean that they should. While some gifted students fit the profile we sketched above — frustrated with school, bored and underchallenged, and finding it hard to connect to peers — many equally capable students are happy in their school lives. (We heard some, but not many, happy stories from online commenters.) If a child is happy and successful without acceleration, they are likely to remain happy and successful regardless of whether they are accelerated, and if they don’t want to accelerate, it should not be forced on them. At least some of the suspicion towards acceleration comes from parents who inappropriately push schools to accelerate their happy, satisfied children.

Acceleration is also not the only option. There is much more to learn than is taught in regular courses. Even in a normal class, a well-designed curriculum or an experienced teacher can create “extensions” to the main activity, so that students who are ready for more have something valuable to engage with. Enhancement or exposure to new, similar topics can serve students as well. A student who has jumped ahead in arithmetic may be entranced by a glance at Pascal’s triangle and number theory. One who is fascinated by English might find similar joy in learning Spanish or Chinese. Both of these, alongside acceleration, follow a simple principle: if a child wants to learn more and is able to do so, let them learn more. Overall, the balance of evidence suggests that acceleration is a practical and resource-effective way to help gifted, underchallenged students flourish in schools.

Our favorite one-stop source for reading on acceleration: here.

1. Educational Goals in Conflict

Through acceleration, tutoring, or ability grouping, some kids could learn more. Why aren’t schools aggressively pursuing that? Shouldn’t they be working to teach kids as much as possible? Isn’t that what a school supposed to do? That educators are skeptical of ability grouping or acceleration can be maddening from the perspective of learning maximization: Why are schools leaving learning on the table?

Here’s something we don’t talk about nearly enough: schools are simply not in the learning-maximization business. It turns out that parents, taxpayers and politicians call on schools to perform many jobs. At times, there are trade-offs between the educational goals schools are asked to pursue, and educators are forced to make tough choices.

Historian David Labaree has one way of thinking about these conflicting educational goals, which he expands on at length in Someone Has to Fail. For Labaree, there are three competing educational goals that are responsible for creating system-wide tensions:

• democratic equality (“education as a mechanism for producing capable citizens”)
• social efficiency (“education as a mechanism for developing productive workers”)
• social mobility (“education as a way for individuals to reinforce or improve their social position”)

As Labaree tells it, these goals end up in tension all the time. A lot of things that seem like gross ineptitude or organizational dysfunction are really the result of the mutual exclusivity of these goals:

These educational goals represent the contradictions embedded in any liberal democracy, contradictions that cannot be resolved without removing either the society’s liberalism or its democracy … We ask it to promote social equality, but we want it to do so in a way that doesn’t threaten individual liberty or private interests. We ask it to promote individual opportunity, but we want it to do so in a way that doesn’t threaten the integrity of the nation or the inefficiency of the economy. As a result, the educational system is an abject failure in achieving any one of its primary social goals … The apparent dysfunctional outcomes of the school system, therefore, are not necessarily the result of bad planning, bad administration, or bad teaching; they are an expression of the contradictions in the liberal democratic mind.

Ability grouping and acceleration fit nicely within the tensions Labaree exposes. These learning-maximizing approaches could find support from those who see education as a national investment in our defense or economy. Of course, the strongest demand for acceleration in schools can come from parents, who want schools to give their children every possible opportunity to be upwardly mobile. (“We want to make sure they can go to a good college.”)

Those act as forces in favor of ability grouping and acceleration. But schools also know that they are held responsible for producing equitable outcomes for a citizenry that sees each other as equals. A program that raises achievement for top students without harming others has an appeal an economist could love, but within schools this can count as a problem.

The way this plays out in practice is that many schools are inundated with requests to accelerate a kid. Parents — especially financially well-off, well-connected parents — can typically find ways to apply pressure to schools in hopes of helping their children reach some level of distinction. They’ll sometimes do this even when it wouldn’t benefit a child’s education (it would be educationally inefficient), or when it would exacerbate inequality (by e.g. letting anyone with a rich, pushy parent take Algebra 1 early).

In short, from a school’s standpoint those are two problems with acceleration. First, parents will push for it even when it’s not academically or socially appropriate. Second, it can exacerbate inequalities. That could explain where the culture of skepticism within education comes from.

This is meant entirely in terms of explaining the dynamic. The way this plays out can be incredibly painful. Systems designed to moderate parental demand can keep a kid in a depressing and frustrating situation:

My older son wanted to move up to a more advanced math course for next year. He took two final exams for next year’s course in February and answered all but 1/2 of one question on each. So roughly 90% on both and his request to skip the course was denied. (source)

Districts sometimes have extensive policies that can be incredibly painful to navigate when trying to get a student who truly needs acceleration out of a bad classroom situation. We heard from one educator who had a very young student expressing suicidal ideations. It was all getting exacerbated by the classroom situation — the kid said he felt his teachers and peers hated him because he loved math. The parents and the educator tried to find a better classroom for the child, and were met with all the Labaree-ian layers of resistance. Off the record, the educator advised the parents to get out of dodge and into a local private school that would be more responsive to his needs.

A happy ending: the 4th Grader moved to a private school where he was placed in an 8th Grade Honors class. He likes math class now. He seems happier, he’s growing interested in street art and social justice work.

But without a doubt, there are some unhappy endings out there.

1. Personalization Software

[source: Larry Cuban]

“Ours is an age of science fiction,” Bryan Caplan writes in The Case Against Education.  “Almost everyone in rich countries — and about half of the earth’s population — can access machines that answer virtually any question and teach virtually any subject … The Internet provides not just stream-of-consciousness enlightenment, but outstanding formal coursework.”

The dream of using the Internet to replace brick-and-mortar classrooms is a dream that is entirely in sync with the times. This is reflected in the enormous enthusiasm directed towards online learning and personalization software. Bill Gates, Elon Musk, and Mark Zuckerberg have all invested heavily in personalization and teaching software. And the industry as a whole is flush with funding, raising some 8 billion dollars of venture capital in 2017, while reaching 17.7 billion in revenue.

Finally — a way out of the school system and its knot of compromises! If schools are institutions whose goals are in tension with learning-maximization… then let’s stay away from schools and their tensions and give the children the unfettered learning they want. Let’s create the ideal tutor as a piece of software.

This dream isn’t just in sync with our times — it has a long history. This history is particularly well-documented by historian Larry Cuban (author of Teachers and Machines and Tinkering Toward Utopia) and by Audrey Watters (she’s writing a book about it). Watters’ talk “The History of The Future of Education” is as good a representative as any of the major thesis: that the dream is larger than any particular piece of technology. Motion pictures, radio, television, each of these was at times promoted as an educational innovation, able one day to free students from lockstep movement through school and into a personalized education. From Thomas Edison to B.F. Skinner, tech advocates have long envisioned the future that (at least according to Caplan) we’re living in now.

Then again, tech advocates in the past also thought they were living in the age of personalized learning. In 1965, a classroom that used a program called Individually Prescribed Instruction was described this way:

Each pupil sets his own pace. He is listening to records and completing workbooks. When he has completed a unit of work, he is tested, the test is corrected immediately, and if he gets a grade of 85% or better he moves on. If not, the teacher offers a series of alternative activities to correct the weakness, including individual tutoring.

For comparison, here is the NYTimes in 2017, and the headline is A New Kind of Classroom:

Students work at their own pace through worksheets, online lessons and in small group discussions with teachers. They get frequent updates on skills they have learned and those they need to acquire.

The similarity between modern day and historical personalization rhetoric doesn’t settle the matter — in a lot of ways, clearly the Internet is different — but personalization software seems to have arrived at a lot of familiar, very human frustrations.

Anyone who has gone online to learn has, at some point, come face to face with this dilemma: On the internet, you can study almost all human knowledge, but usually you don’t. In a world with virtually every MIT course fully online for free, a world with Khan Academy and Coursera and countless other tools to aid learning, why has the heralded learning revolution not yet arrived?

In a way, the revolution has arrived — it just hasn’t improved things much. Rocketship Schools, a California charter using online learning for about half of its instruction, has had solid results. Lately, though, they’ve moved away from some of their bigger bets on personalization and rediscovered teachers, saying “We’ve seen success with models that get online learning into classrooms where the best teachers are.” School of One was a widely hyped high school model in NYC that was preparing to scale up its offerings… until a fuller picture of the results came in and it was pilloried. Online charter schools, meanwhile, seem to actively depress learning.

Part of the problem is that it’s hard to get solid research on the efficacy of various ed tech products. Many tools, particularly those sold directly to schools or used by online charters, are proprietary and stuck behind paywalls, selectively presenting their best data and limited demos. The ed tech sector in general seems to deliver mixed results to students.

Why is it so hard to make effective teaching software?

For one, teaching is complex. A good human teacher does a lot of complicated things — gets to know their students, responds to the class’ moods and needs, asks “just right” questions, monitors progress, clarifies in real time as a look of confusion dawns on the class, etc., etc. — and it’s simply hard to get a computer to do that.

Maybe, theoretically, a piece of software could be designed that does these things. But in practice, many software designers don’t even try. It’s easier and cheaper to make pedagogical compromises, such as providing instruction entirely through videos. Yes, there are some thoughtful tools made by groups like those at Explorable Explanations, such as this lesson on the Prisoner’s Dilemma. But building high-quality tools well-adapted for a digital environment is difficult and time-consuming, and for prospective designers, destinations like Google or Blizzard tend to be more glamorous than working with schools. In practice, humans currently have a lot of advantages over computers in teaching.

Even if we overcame all the design issues, though, would students be motivated to stick with the program? Studies of online charters point to student engagement as the core challenge. When you put a kid in front of a computer screen, they jump to game websites, YouTube, SlateStarCodex, Google Images — anything other than their assigned learning. Many educational games that try to fix this resort to the “chocolate covered broccoli” tactic, trying to put gamelike mechanics that have nothing to do with learning around increasingly elaborate worksheets.

To be fair, student engagement is also the core challenge of conventional schools. But that’s precisely what the much-maligned structures of school are attempting to confront. The intensely social environment helps children identify as students and internalize a set of social expectations that are supportive of learning. The law compels school attendance, and schools compel class attendance. .And, once a child is in the classroom, their interactions with actual, live human instructors can set high academic expectations that a child will genuinely strive to meet.

The conventional story is that school is incredibly demotivating, but compared to their online counterparts schools are shockingly good at motivation. MOOCs like those on Coursera have an average completion rate of 15 percent — public schools do much better than this. Popular language app Duolingo’s self-reported numbers from 2013 would put their language completion rate at somewhere around 1%. If all a user has to rely on is their daily whim to continue a course, the most focused and conscientious may succeed, but those are the ones who already do well in schools. That’s a big part of why people lock themselves into multi-year commitments full of careful carrots and sticks to get through the learning process. Writers such as Caplan think that people are revealing their true interests when they skip learning to fart around on the web, but we might as well see a commitment to attend school as equally revealing. People need social institutions to help do things we’d truly like to do. As such, even as computers become better teachers, the motivational advantage of schools seems likely to persist.

How might tech-based learning tools address these factors, so they might stand a chance at holding students’ attention long enough to teach them? Art of Problem Solving, an organization promoting advanced math opportunities to children, makes a good case study. It’s found a balance worth examining. First, it provides accessible gamelike online tools that center on a careful sequence of thought-provoking problems. Second, it offers scheduled online classes with the promise of a fast pace, challenging content, and a peer group of similarly passionate students taught by subject matter experts. The online classes are more expensive offerings, but they preserve the human touch.

What does that balance mean for students? If they’re in the conscientious, self-motivated crowd that wants to learn everything yesterday, they can gorge themselves on software designed to be compelling. No barriers keep them from progressing. Software can always point to a next step, a harder problem. On the other hand, if they want to lock a motivational structure around themselves and keep the social benefits of school in a more challenging setting, they can.

Not every successful tool need look identical, but that core idea is worth repeating: software should enable the passion and self-pacing of eager kids, but should not rely on that to replace the power of social, human motivational structures. Yes, sometimes even the same structures used in “regular” schools.

Online learning, then, fits squarely within the history of attempts to automate teaching. Over and again we make the same mistakes and forget the lessons of history: that teaching is more complex than our machines have ever been, that motivation is largely social, and that schools will have a hard time distinguishing between altrustic designers and opportunistic profit-seekers.

For those in the market for online learning there are a lot of mediocre tools available, and many truly bad ones. Right now, there’s nothing that seems ready to serve as a full-on replacement for school without consistent, careful human guidance.

That said, depending on your passions, there are some excellent resources for learning out there. Especially if a student has a caring mentor or a passionate peer group, they can learn a lot online. As educators and designers create more tools that respect both the power and limitations of machines, that potential can grow. But it’s not quite science fiction.

Our algorithm has determined that you should watch the following two videos: here and here to balance realism and idealism

Education is complex and resists easy generalizations. That said, here are some generalizations.

On navigating school for your child:

• The brightest students do not thrive equally in every setting. Even the best students achieve more with teachers than on their own. Unless tutoring or some other private arrangement is possible, this means that a school is the best place to be for learning.
• But school right now doesn’t work for all kids. One fix: if a child wants to be accelerated and seems academically prepared for it, acceleration will usually help them.
• Most schools aren’t in the business of maximizing learning for every student, and in particular they tend to be skeptical of acceleration.
• Therefore: If your kid needs more than what school is offering, be prepared to be a nudge.
• But if you think your kid needs to be challenged more and your kid is perfectly happy in school, try really hard not to be a nudge.
• Don’t fight to move your child to a class that covers the exact same material at the exact same pace but has the word “Honors” next to it. That sort of ability grouping makes no educational difference.
• Prioritize free, open online tools. Don’t expect online tools to do the work for you or your child. Expect more distraction and less progress if online learning time is unstructured or unsupervised.

For educators:

• If you are an elementary teacher or administrator and your school is looking to try new things, consider cross-grade ability grouping by subject, especially in math and reading.
• Gifted kids are usually not equally talented in all fields. Consider options to accelerate to different levels in each subject based on demonstrated skill in that subject.
• A lot comes easily to smart kids, and sometimes they never get the chance to learn to struggle. Find something they think is hard, academic or not, so they are able to handle more important challenges later.
• If a child is bored in your class and knows the material, they probably shouldn’t be in your class.

For tech designers and users:

• If you’re making online tools, make the learning the most interesting part of them. Don’t rely on chocolate-covered broccoli or assume that just presenting the material is enough. Take the problem of motivation seriously.
• Look for passionate groups with robust communities, whether online or offline. Don’t overlook the social aspect of learning.

And for advocates of educational reform, in general:

• People almost only talk about educational efficacy. But don’t be fooled — educational debates are only sometimes about what works, and frequently about what we value.

One last thing: if you’re an educator or a parent or just somebody who spends time around children, take their feelings seriously, OK? If a kid is miserable, that’s absolutely a problem that has to be solved, no matter what district policy happens to be.

Acknowledgments: Thanks to /u/Reddit4Play from reddit, JohnBuridan from the SSC community, blogger Education Realist, and many others who read drafts and offered ideas along the way.

# Noticing Humans & Noticing Wonders – Guest Post by Benjamin Dickman

This is a guest post by friend of the blog Benjamin Dickman

• Noticing Humans refers to the importance of students being aware that they are seen by others – including, but not limited to, their math teachers – and that we, as teachers, would do well to interrogate how our external perspectives match or don’t match students’ internal perspectives.
• Noticing Wonders refers to a specific in-class activity related to Paul Lockhart’s essay, “A Mathematician’s Lament” (pdf), that I designed – based on suggestions from my Middle School English teaching colleagues – to gain insight into how my students were thinking about mathematics.

In noticing and wondering:

How can we follow up on our wonders about the ways in which students see mathematics?

Being Noticed

During a summer visit to my childhood home in Boston, Massachusetts, to see my family, I made a point of asking my mom[1] for a very specific suggestion: What can I say to students who voice anxiety or discomfort around an upcoming (math) assignment, especially when my prior knowledge of their quality of work and preparation suggests that they will do well? (For example, I would consider “Don’t worry about it!” or “I’m sure you’ll do fine” to be suboptimal responses.) Paraphrasing, her idea was that sharing evidence of strong past performance is fine, but that a helpful additional sentence would be one to the effect of: “As your teacher I have confidence in you, and while I know that you may not share in that confidence right now, I wish that you could see yourself through my eyes.” Less than a month later, I was reading a piece in Quanta Magazine about 2018 Fields Medal (an award sometimes called the Math Nobel) recipient Akshay Venkatesh, from which I excerpt (emphasis added):

Clearly, [Venkatesh’s] adviser must have written a glowing letter of recommendation for him — but why? Venkatesh took this question to Jordan Ellenberg, a friend and fellow mathematician. Ellenberg’s reply has stayed with Venkatesh over the years: “Sometimes, people see things in you that you don’t see.”

About a week after, I noticed on Twitter an open-request from Professor of Mathematics Education Ilana Horn about her son:

I happen to be a math teacher who had 5 minutes, so I watched the video and left a couple of comments. I thought that my first comment got to the heart of the matter, and left a second comment even though I deemed it less relevant. Here are the comments with their respective responses:

In retrospect, I conjecture that my first comment resonated primarily with other math educators, who can see what I see about habits of professional mathematicians, but that it would require more time and evidence to be fully believed by a child. My second comment had a different outcome – to “jump up and down with happiness” – because it comes from a different perspective: how one is seen (or even not seen) by their peers, and the ways in which one’s presence and absence are noticed, even from afar. The through line that I perceive in these three items – my mom’s suggestion to me, Ellenberg’s advice to Venkatesh, and the excerpted tweets – is that, although there is much talk in math education about noticing and wondering, I am concerned that we are (or: I am) not doing enough to ensure that I let my students know that I am noticing them for who they really are: as mathematical thinkers, as students, as humans.

I need to attend better to voicing what I notice in them, and thinking intentionally about how they share what they notice in one another. This is true for students who retain an anxiety around (math) assignments; it is true for future Fields Medalists; it is true for students who experience tracking systems in ways that are too often unjust; it is true for so many other students who cannot – or do not yet – see themselves through a caring adult’s eyes; and, I strongly suspect, it is true for so many teachers who long to be noticed and seen – knowingly or not – for who they really are: as mathematical thinkers, as teachers, as humans.

In noticing and wondering:

How can we follow up on our wonders about the ways in which students see mathematics?

Noticing Wonders

About once a month for each of the past two years, I have met with a group of teachers at my school – some Middle School, some Upper School, some both – to talk about “Writing to Learn” strategies. One of these is called a Focused Free Write, and although it was introduced as a way for Middle School students to get a foothold on Ovid, I decided to adapt it for use with an essay that is somewhat well-known in certain math education communities: Paul Lockhart’s “A Mathematician’s Lament,” which is often referred to simply as “Lockhart’s Lament” (PDF re-link). If you’ve never read it, or even if you have, you might try to complete the activity below; the students in our Problem Solving & Problem Posing course had no prior familiarity with any of the author’s writings. To summarize (over)simplistically, I feel that this essay, or Lament, speaks in a way that resonates with many mathematics teachers. I wanted to use it to gain insight into the specific mathematics learners with whom I was working. Here is the full prompt, for which I wrote the directions in red. No additional information was provided about the source text or its author:

We spent 10 minutes on the quiet writing part of this task, and, once we wrapped up and it came time to share out, the students organically decided that they would each read aloud one of their classmates’ focused free writes. I share this task here because I think it is a meaningful example of how strategies from another subject in another grade can be successfully transferred to the context of a math class. Moreover, I am compelled to point out explicitly that other teachers could pick a different text and/or a different excerpt and/or a different set of words to replace with ellipses. I picked this text, excerpt, and its removals as a function of what I hoped to accomplish with and learn from my students. Below are six student examples, followed by Lockhart’s original words. In each case, the portions that replaced the ellipses are in blue. Rather than analyzing the student work, I prefer to let it speak for itself; so, I will conclude with these images. There are many ways in which additional context can frame how each student-generated example reads; here, I note only that I work at an all-girls day school. Part of what made this so meaningful to me was knowing the individual students. For the reader without this shared connection, I simply suggest reading as much or as little as you’d like, and in whatever order you wish – perhaps comparing corresponding parts across the provided examples. Finally, I hope that, if you try this activity with your students – or with other teachers, or with other humans – then you will consider sharing the wonders that you notice.

[1] I happen to think “being my mom” is a sufficient criterion for giving good advice, but my mom also happens to be a child psychiatrist whose work includes advising for the PBS show Arthur, which created a therapist cartoon character named after her!

# I don’t focus my classroom on solving problems

This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theoristsmusicians and artistsexplorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.

I’ll update this post with links to all the pieces that are part of this group of posts. So far that includes: Anna, Dylan, CarmelJoshuaLara and Nadav, Dan, and David.

Here’s my entry.

I.

I’m not exactly sure how to say the thing that I want to say, because the thing I want to say isn’t quite true, and can easily be misunderstood. But the thing I want to say is that problem solving is overrated in math teaching.

A story may help explain what I mean. I’ve been planning a conference for math educators, and I work on recruiting people to lead sessions. In the middle of each day, we have a block of time for doing some math together. But the thing is, it can be a tricky session to lead. Some of our participants have advanced degrees in mathematics; others teach younger students and haven’t used algebra in years. How do you make sure you’re doing math that’s meaningful for everybody?

This gets especially hard if the session is framed as the attempt to solve a math problem. The problem is one of aim — will the problem be productive for everyone in the room? What if someone immediately sees the problem for what it is, and makes quick work of it? As Benjamin Dickman put it, “if the plan can be brought down with Oh I’ve Seen This Already…then to me it’s a risky move.”

(Sure, you can tell people what to do if they’ve seen it already, but doesn’t that feel like a consolation prize?)

It’s the same problem that I had to grapple with at math camp during the summer. One of my jobs at camp was to teach the counselor class. Counselors ranged from 16 to 20 years old. The class involved giving counselors time to grapple with some of the challenging math that the camp asked kids to do battle with.

Here is a typical problem from camp:

What is the sum of all the digits of the numbers from 1 to 1000, i.e. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 + 0 + 1 + 1 + 1 + 2 + … + 9 + 9 + 9 + 1 + 0 + 0 + 0?

That’s tricky. But it was significantly trickier for some of the youngest counselors I taught, who were themselves high school students, and significantly less so for the seasoned math major who is headed straight toward a PhD program in math.

What to do?

In this case, I started class by emphasizing that the goal was not to solve the problem, but to find math to understand.

I put a representation — 10 through 15, with the digits all splayed out like 1 + 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1 + 5 — up on the board, and asked the class what they saw and noticed.

I presented the original problem. Then I asked each counselor to come up with a question that was related, but not the same, as the original problem.

I asked everyone to pursue one of these questions, or maybe the original problem.

Explicitly framing class as a search for understanding, calling on students to notice whatever catches their attention, focusing attention on new representations, asking students to think about a problem as living in the context of a network of similar problems…these are all aspects of math that I emphasize, at the expense of solving a specific problem.

Getting a bunch of counselors who signed up to work at a math camp to work on math is not exactly teaching on the hardest difficulty setting. I mean, damn, if every group of kids that I worked with could spend a half hour essentially poking around mathematically after the set up I left these counselors…my job would be so much easier.

I’m not saying that my counselor class should be anyone’s model.

But I am trying to say that I felt like I was subverting problem solving. Because when you make problem solving the goal, you create a lot of challenges for yourself. What happens if someone finishes first? If the goal is solving the problem, why should the teacher ever help? or why should they refuse to help? How do you make sure everyone feels valued if they don’t solve the problem? How do you keep people from feeling that your extension is a consolation prize?

I’m still not quite sure what to say about problem solving, but maybe it’s like this: a lot of things get much harder once you tell your class that the goal is to solve a problem. So most of the time, I don’t.

II.

I think some people think of math as essentially focused on problems, but this is not how mathematicians necessarily describe their work.

For example, while Timothy Gowers (in Two Cultures of Mathematics) does describe himself as oriented towards problem solving, he sees this as the minority position within mathematics, where the dominant orientation is towards theoretical understanding. Michael Atiyah is his chosen representative of this dominant orientation. Atiyah describes himself as unmotivated by problems:

I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.

Talking, learning, discussing math: these are distinctly mathematical activities. It’s not all about problem solving — it’s not even primarily about problems.

There is a line of thought that argues that math education must be focused on problems and problem solving, because that’s what mathematicians do. But mathematicians do so many things beyond solving a problem.

More than anything else — Atiyah says this — mathematicians are interested in learning about math. They learn from colleagues, who explain things to them. They learn from classes and seminars. They ask questions. They teach, and learn from teaching. They learn new ways to represent ideas. They notice new things, learn to see new things as important. And, yes, they work on problems.

There’s another thing too. If you’re oriented towards problem solving, you’re oriented towards the achievement of the few. If mathematical success is about solving valuable problems, that’s going to be inherently inequitable. Your history of math becomes the history of individual achievement, mostly represented by men from privileged cultures. It’s the same inequities that show up in the classroom around problem solving — some have a head start, some finish first — but played out over an entire field.

And it’s the sort of thing that can get inside your head. Am I worthy of working on a problem? Will I ever contribute by solving something of value? (Or even: Has my culture ever contributed anything of value? Have people like me ever made important contributions?)

These worries make perfect sense in a world full of unsolved puzzles waiting to be solved. But, if we’re oriented towards understanding instead of problem solving, the whole situation gets turned on its head. Achievement is understanding, and there is more than one way to spread understanding. It’s not just about being first — in fact, it’s not about that at all.

These two ways of seeing achievement in math can clash. For example, when a graduate student worried online if he could meaningfully contribute to math, Bill Thurston answered that he was thinking of “contribution” too narrowly:

Mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining — they depend very heavily on the community of mathematicians.

If we focus too much on solving a problem as a goal, then we end up focusing on the “solving.” And then we can end up paying too much attention on who did the solving, instead of looking at the way clarity was spread through some mathematical community.

(There are ways of combating this slide, but it’s hard work!)

If our culture and classrooms are focused on understanding, equity has a better shot.

III.

So much for teacher conferences, summer camp, and mathematical culture. But the last piece of this sort-of-critique of problem solving comes from my regular classroom work during the school year.

I have definitely found that, when students are starting off learning something new, solving a specific problem has limited use. The drive to answer the question can narrow attention in unhelpful ways, focusing students on whatever it is that will help them most solve the problem — even if that’s not a new and interesting feature.

So I try to hold off on problem solving, at first, and I very rarely frame that as the goal of a class session.

I remind students: the goal is spreading understanding.

I ask students to notice new things.

I ask students to study examples of student work, both correct and incorrect. Because learning from existing mathematical work is mathematical. It’s only second-rate if your goal is problem solving.

I ask students to write their own problems, and to see problems as existing within genres of problems. (If Monday was about finding the volume of prisms, very likely my first ask on Tuesday will be for students to come up with a prism to sketch and to find its volume. These sorts of tasks are surprisingly challenging! Remembering stuff is hard, but the work is deeply mathematical.)

I ask students to compare representations.

And if all these things aren’t exactly in conflict with problem solving, that’s also not our goal. We’re aiming for understanding, and problem solving happens along the way.

# Worked Examples and Loop-De-Loops

This is a Loop-de-Loop:

At camp, I taught a session on Loop-de-Loops, a mathematical object from Anna Weltman’s book. I had about 30 people in the room (15 counselors, 15 kids) looking for patterns and trying to figure out a bit about what Loop-de-Loops are all about.

I introduced Loop-de-Loops using Chris Lusto’s fantastic site, asking everyone what they noticed and were wondering after I hit “Show Me.” (Go try it; it’s a blast.)

This was perfect for us, as I wanted the focus of the session to be about asking questions. (Lord knows that the kids spend enough time solving problems in this camp. Asking questions is at least half of a full mathematical picture.) Lusto’s site makes it easy to quickly explore many different lists of numbers, generate theories and see if those theories hold up.

In fact, it’s too easy to generate these Loops with the software. For the sake of understanding, it’s always good to construct stuff by hand a couple times. Often we notice different things when we construct objects by hand than we do on the computer.

My first time running this session, though, I realized something: it’s hard to make a Loop-de-Loop!

Students and counselors struggled to draw these things — though, when they saw these objects on the computer, they were extremely confident that they understood how to make them. Surprise!

Today, I ran the session for another group of kids and counselors. This time, I came prepared. I wrote a worked example activity that aimed to help everyone better learn how to make these cool Loopy things:

I quickly made a handout right before class today. (Hence the marker.) I knew that I needed to include the three things I always include on one of these things:

1. The example, clearly distinguishing between the “task” and the “student work,” and trying to make sure to only include marks that contribute to understanding. (See those little arrow heads? I noticed that people had a hard time keeping track of directions while drawing these.) I tried to remove any distracting text or clutter — reading about the split attention effect helped me learn how to do this.
2. Prompts for noticing the most important stuff. Research suggests that students often don’t explain things to themselves, or do so superficially. (And experience totally confirms this.) Prompts, along with a clear call for students to spend a minute responding to the prompts, helps a lot with this I find.
3. A chance to try it out on your own, with the model nearby to help, if you get stuck.

I first read about this structure while reading about Cognitive Load Theory, but things didn’t click for me until I also saw the Algebra by Example project. Two other pieces have helped me better understand this bit of my teaching:

Here’s how I do these things. First, I put the “student work” up on the board and ask everybody to silently study it. (People need time to think before talking!) Then, I ask everyone to check in with a neighbor and to take turns making sure each can explain what’s happening. (This is usually where there are “ah!”s and “oh wait!”s.) Then, we talk about the prompts. After that, I tell everyone to try it on their own.

This part of the session went so much better today than my first pass. Everyone was still challenged by drawing these, and there were still a lot of mistakes. But the difference between these two sessions was precisely the difference between productive and unproductive struggle. Instead of flailing around when they got lost in the construction, everyone had something to go back to. Ah, OK, so this is how it’s done.

Part of my job is also to help the counselors support the students in their math work. A lot of the counselors tell me they struggle with knowing about how much to give away to a student when they are stuck. And while I totally know what they mean, I always tell them that their job isn’t to give away stuff or to avoid giving stuff away. Their job is to get the ball rolling for the student, get thinking happening, as quickly as possible — and then to step back and let that thinking happen.

Today I needed to get the ball rolling. It was a session about posing problems, and I started with asking everyone to notice and ask questions. But an important part of getting the ball rolling was a worked example.

# ‘Power Works By Isolating’ [#tmc18]

Tina wants to start talking about diversity at TMC (the conference), so here’s a quick thought in that direction:

Would you rather travel to a foreign country alone, or with a group of friends?

One way of thinking about diversity — not a great way, I think — is about being welcoming to minorities. The problem with this is that it’s never comfortable to be the minority in that position. It doesn’t matter how warm or welcoming the majority is, especially when we’re talking in particular about race. There is loneliness and discomfort that’s inherent to that kind of “inclusion.”

I am not a person of color, so I can’t speak directly to that. But the loneliness is a feeling that’s somewhat familiar to me from walking around new cities with my yarmulke on, trying to keep Shabbat in the middle of a big edu gathering, and (especially) from trying to find kosher food at these conferences.

From my conversations with Marian and others, this loneliness is a common experience for educators of color. It’s a powerful force that functions to exclude entire groups from spaces, even if a community is really, really trying to be welcoming to them. This is one reason why, so far, people of color are so scarce in many math edu gatherings.

There is another way to think of diversity, though, and it has to do with creating spaces that aren’t dominated by the majority. In particular, it’s about creating opportunities for people of color to gather together and travel through white educational spaces.

One group that I think has learned this lesson — about the difference between the first and second kinds of diversity — is Heinemann, the education publisher. They have a two-year fellowship for educators to write and think about their work. I was a member of their first cohort, from 2014 to 2016.

The first cohort looked like a lot of diversity-minded groups in education. Which is to say, though we were “diverse” in many ways, there was not a cadre of people of color amidst the whiteness.

I’m not privy to the decision-making of the Heinemann folks, but they’ve clearly made a conscious choice to go in a different direction in the fellowship. Here is a picture of the 2nd cohort (see Kent, top right):

Here is a picture of their third cohort (see Marian, center right):

Heinemann now sees it as a priority to invite a contingent of educators of color into their fellowship.

This is not to suggest, by the way, that any of the members of color in these groups are otherwise under-qualified for the fellowship. Heinemann gets hundreds of applicants for the handful of spots they have to offer. At a certain point of quality, how do you even choose between two excellent applications? The folks running things get to choose which over-qualified applicants make the cut and which don’t — they just do it in a way that battles isolation for people of color.

Spaces like this simply don’t exist in math education. There is no mainstream math edu group that intentionally cultivates “diversity” in this way, TMC included.

It seems, from what I’ve heard, that Heinemann’s approach worked. The Need For and Needs of Teachers of Color” is a podcast episode where the second cohort’s “fellows of color” discussed what the fellowship has meant for them.

The loneliness of being the only educator of color comes up. Here it is, as described by Tricia:

I’m only one of handful of teachers of color in my entire district. A student once asked me, “how many teachers of color are there?” I’m like, “that’s a good question.” And we never see each other. We never speak to one another. Issues that teachers of color face in particular are never addressed, so power works by isolating.

The flipside of this, though, is that bringing together a group that breaks down that isolation can be extremely empowering to its members. It can help arm educators of color to go further into elite educational spaces, places that otherwise could be dominated by that loneliness.

One of my favorite parts of the podcast is when Anna describes how she considered dropping out of a doctoral program. It was her friends of color from the fellowship who gave her the encouragement to continue traveling in a space that might otherwise be uncomfortable to pass through:

So I started a doc program…and I sat down and I thought today’s the day I’m going to quit. And Kim texted me and she said, “Anna, don’t quit. We need you.” And not quit teaching, but quit the program and so that really made a difference, and being able to know that there are people who are going to tell us, “don’t quit” is just so important.

I’m sure the doctoral program wanted to make their space comfortable for all. But there are limits to that approach. The second approach to diversity is different. It’s less like offering someone your couch for a weekend, and more like offering your home for a friend and their family.

I don’t know who exactly I’m calling to action in this post, because I don’t know where there are opportunities to do things like this in math education. (Concretely, I’m not sure who would fund something precisely like the Heinemann fellowship, but for math educators.) I’m not sure if this could be the initiative of a yearly conference, of a rambling online community, or maybe it could be done by a company or one of the professional organizations. I don’t know.

But I do think that if we think about the loneliness that’s inherent in crossing into majority spaces, we’ll do a better job thinking of ways to do better. And one tool could be the intentional creation of spaces for math educators of color, within these larger gatherings.

# Book Review: Masters of Theory

Masters of Theory: Cambridge and the Rise of Mathematical Physics by Andrew Warwick

***

What does it mean to excel at mathematics? And how should you teach towards this kind of excellence?

A common way of telling it is that for as long as people have been learning math, there has been a “traditional” way of teaching it. The tradition of traditional math teaching involves, but is not limited to: textbooks, nightly homework, in class problem solving performances, corrections and grades.

When did it start? Where? One of the things that really stunned me about Masters of Theory is that there is an entirely concise and known answer to this question: it started in Cambridge University, and coalesced in the 1840s and 1850s. And as to the question of “why?,” the answer is largely: because of competition and testing.

More on that development in a moment. First, some trivia.

• Did you know that the word “coach,” as in “mathematics coach” or “basketball coach,” came into practice in the 1830s in England to describe private mathematics tutors? The word was borrowed from “coach” as a mode of transport — the most efficient, fastest, classiest way to travel. When Cambridge undergrads were reaching for a word to describe the work of someone you could hire (if you had money) to help you smoothly move through the mathematical terrain of the mathematics curriculum, they turned to analogy: coaches.
• For a top student, James Clerk Maxwell had a reputation as a sloppy student of math at Cambridge, and his attempt at capturing his thoughts on electromagnetism in a text was basically incomprehensible.
• Before the 19th century, paper had very little to do with learning mathematics.

These pieces of trivia fit together nicely in Warwick’s detailed narrative. There are clear stages marking the shift. Here is a summary.

Step 0: What was learning university math like before the shift?

The goal in the medieval university wasn’t technical facility with a problem solving apparatus. You weren’t supposed to be able to solve problems. The goal was knowledge of works of mathematics.

A central text was Euclid:

“The main form of undergraduate teaching was the thrice weekly college lectures in which students were taught the basics required for a pass or mediocre honors degree…The main job of the mathematics lecturer was to go through required sections of such important texts as Euclid’s Elements, ensuring that the majority of students had at least a minimal grasp of geometry, arithmetic, algebra, mechanics, hydrostatics, optics and astronomy. Lectures were run at the pace of the average student and appear…to have consisted mainly in the lecturer asking students in turn to state proofs and theorems or to solve simple problems orally.”

If you were really great at mathematics, that meant that you were the absolute best at oral discourse in mathematics. You could recite proofs, split hairs, debate and give convincing rhetorical presentations of various proofs or arguments, etc.

Step 1: Newton, Leibniz and the emergence of an extremely powerful technical apparatus for solving problems in math and physics

Then, Calculus happened:

“In the early seventeenth century, most of what was known at the time as ‘mixed mathematics’ was not overly demanding from a technical perspective. However, it was considered by the majority of scholars to be of only secondary importance to the study of ‘physics,’ the search for the true causes of natural phenomena…It was with the gradual translation of Newton’s mechanics into this new [algebraic] language in the early years of the eighteenth century that the fundamental techniques of mathematical physics, indeed the discipline itself, began slowly to come into existence.”

And this was completely different than the mathematics that was previously seen as valuable for undergraduates to learn:

“The aspect of these developments of most immediate relevance to our present concerns is that the increasingly technical nature of physico-mathematics from the mid-seventeenth century made it ever more alien to undergraduates students.”

The new technical apparatus was useful in solving a variety of problems. Yet university mathematics education at Cambridge didn’t significantly change. It continued teaching more or less as it always had:

“Even at Trinity the primary function of a college lecture was still to inform students what they should read and to test their recall and understanding of that reading by catechetical inquisition…In order to find out whether students had learned the definitions, proofs, and theorems they were required to know, the lecturer would go round the class asking them in turn to enunciate propositions and even to solve simple problem orally. the paper and pens provided in the lecture room were not therefore central to the teaching process, but enabled students to take notes as they saw fit during the oral exchanges between the lecturer and individual members of the class…The only visual aid employed by the mathematical lecturer was ‘a cardboard, on which diagrams were drawn relating to the mathematical subject before us’ (Prichard, 1897, 36). This cardboard was handed from student to student as the lecturer went round the class and seems to have functioned as a kind of primitive blackboard.”

This is 100 years after Euler, 150 after calculus, and university math class still didn’t involve learning how to solve problems.

(Going off script here for a moment to note that you were still supposed to be able to solve arithmetic problems. The point is that if you wanted to assess this, you’d do so in an oral recitation. Written mathematics wasn’t expected, and so anything beyond basic mental math wasn’t oriented towards solving problems.)

Step 2: The new mathematics is perfect for ranking students and assigning scholarships

As this is going on, Cambridge University began using the Senate House examination (a.k.a. the Mathematical Tripos) to award valuable college fellowships. If you were one of the top 2 or 3 “wranglers” (as they are apparently called in Cambridge) you were guaranteed “one of the few recognized routes by which a young man from a relatively poor background, but with academic ability, could make his way up the social scale in Georgian Britain.”

At first, it was an oral test and not focused on mathematics. But the new technical mathematics was perfect for clearly ranking students (you either solve the problem accurately, or you don’t) and the exams changed in three ways that Warwick identifies:

1. the exam became focused more on math and natural science
2. if you wanted to rank in the top few, you had to do a written exam, not just an oral one
3. the exam became increasingly competitive

So, the exam, its content, and the incentives students encountered were changing rapidly. But the teaching in the university classes didn’t change at all. Which meant that before long…

Step 3: An extensive system of private tutors (“coaches”) emerged to prepare students for the examination

“The main point to take from this brief survey of Cambridge pedagogy is that professorial and college lectures, tutorial sessions, public disputations and private study were all forms of learning based in the first instance on reading or oral debate. With the gradual introduction of written examinations the preferred form of teaching began to change to suit the new form of assessment. Success in the Senate House examination depended increasingly on the ability to write out proofs and theorems and to solve difficult problems on paper. Ambitious students accordingly turned to private tutors.”

Step 4: These tutors, competing with each other for top students, soon land on a paper-based pedagogy that is distinct from what the university offers, and closely resembles what we think of today as “traditional” math teaching

There was a whole industry of tutors surrounding Cambridge University, but some tutors stood out and gained reputations for reliably producing top scorers on the examinations. The first of these elite tutors was William Hopkins:

“Hopkins’ success derived from his teaching methods, his own ability and enthusiasm for mathematics, and his reputation as a tutor. Unlike the majority of private tutors, he taught students in small classes–between ten and fifteen pupils–composed of men of roughly equal ability. This meant that the class could move ahead at the fast possible pace, the students learning from and competing against each other. Hopkins considered it an ‘immense advantage in Class Teaching when there is a sufficient equality in the ability and acquirements of each member of the class’ that ever student would: ‘hear the explanations which the difficulties of others might require, and thus be led to view every part of the subjects of his studies, through the medium of other minds, and under a far greater variety of aspects than those under which they would probably present themselves to his own mind, or would be presented by any Tutor teaching a single pupil (1854, 19-20).”

“Hopkins’s teaching methods were thus designed to optimize the benefits of intensive, progressive, and competitive learning. He also developed and exploited an avuncular intimacy with his pupils which would have been quite alien to most college lecturers and university professors…In the friendly atmosphere of his teaching room, Hopkins combined the admiration of his students with his own infectious enthusiasm for mathematics to promote the competitive ethos and a dedication to hard work.”

An entire chapter is dedicated to a later tutor, Edward Routh, whose students dominated the exams over many years. Here’s what Warwick has to say about Routh’s teaching:

“The primary method of teaching, around which Routh’s whole system was built, was the one-hour lecture to a class of not more than ten pupils using blackboard and chalk. The blackboard was a fairly recent pedagogical innovation in Cambridge at this time, private tutors having previously worked on paper with their pupils sitting next to them…Routh would begin with a ‘swift examination of exercise work’ set for the class at the end of the previous meeting. These exercises generally required students to reproduce proofs and theorems, and to solve related problems, as they would have to in the examination, and Routh would quickly discuss any errors common to several members of the class and those of an individual from which he felt the class might learn.”

Going over homework on a blackboard — this was state of the art!

More:

“Despite the public nature of these corrections there was ‘no jesting, no frivolous word over a blunder,’ and Routh would neither give any ‘tips’ on which exercises he thought likely to appear in forthcoming examinations nor express an opinion on the relative abilities of members of the class. Having corrected the work of the previous lecture he would at once launch into a ‘continuous exposition’ of the material to be covered that day,each member of the class taking notes as fast as he was able. Routh generally led students through which he considered the best textbook accounts of each subject…”

“Revision sessions apart, Routh devoted little time to solving difficult examination problems in his lectures. At the end of each hour, rather, he would hand out about six problems ‘cognate to the subject’ of the lecture’ which were to be solved and brought to the next class.”

“Once a week Routh gave a common problem sheet to all his students, regardless of year or group. In one week the students were allowed as much time as they required to puzzle out the solutions, but, every other week, the problems had to be solved ina timed three hours under examination conditions. Each students was required to leave the problems in the pupil-room on Friday or Saturday in order that Routh could mark them over the weekend. The following Monday the marked scripts would reappear in the pupil room together with Routh’s model solutions (to save him having to waste precious minutes in the lecture) and a publicly displayed mark sheet ranking all students according to the marks they had scores. These biweekly ‘fights’ gradually accustomed students to working at the pace required in the Senate House, incited and preserved an atmosphere of fierce competition, and provided that objective measure of relative merit upon which Routh himself took care never to comment.”

He gave feedback to everyone, and everyone had room to improve:

“Routh was extraordinarily scrupulous in marking student scripts to the extent that ‘it was one of his peculiarities that he was never wholly satisfied with any work shown up to him’ (Moulton 1923). On one memorable occasion when a brilliant student, Fletcher Moulton, mischievously prepared a problem paper ‘on which no criticism could be offered,’ Routh still found room for improvement by urging Moulton to ‘Fold neatly.'”

In contrast, less successful tutors tended to group students in two and threes, didn’t recreate exam conditions, and individualized instruction more (rather than following a systematic and carefully constructed syllabus). Make of that what you will.

Step Five: All this time, university lecturers still weren’t teaching any of the new technical math that calculus and mathematical physics produced! The only way to learn any of this stuff was through the tutors. This went on for way longer than you might expect.

Step Six: Eventually, both the methods and curriculum of this system of private tutors were mainstreamed and brought into the colleges. From there, it spread to top prep schools and eventually to public schools.

***

Honestly, none of this is really the main point of the book. The book’s major argument is far more ambitious than just telling this story. It aims to detail how mathematical training at Cambridge involved an extensive local, distinct, technical manner of doing math. If you wanted to learn this way of doing math, you had to be socialized into it at Cambridge. And this manner of doing math, Warwick argues, can help explain the distinctive research that into mathematical physics that happened at Cambridge.

(Historians of science had already argued that the technical apparatus of science — the tools, machines, etc. — involved a local, distinctive area of knowledge that grounds whatever theorems or ideas they shared. Warwick is making a similar argument about paper, examinations, the classes that coaches offered. All of these things are like the microscopes of Cambridge research into mathematical physics. It sounds better when Warwick says it.)

All of this is pretty heady; extracting just the pedagogical story from it is sort of cheap.

(Full disclosure: I only read the first half of the book! That’s where the pedagogical history mostly is.)

At the same time, all of this info deserves to be more widely known in math education. I think it’s particularly relevant to a number of perennial discussions:

• Has there only ever been one mathematics, or has mathematics changed drastically depending on culture and time?
• What is “traditional” mathematics teaching? When did it arise, and why did it become widespread?
• What’s the relationship between the current way we teach mathematics and past ways of teaching it?

I’m eager to read more books like Masters of Theory. Reading recommendations are seriously appreciated.

I.

Here’s the situation:

You’re the principal of a large elementary school

Ooh, spooky. Read on — it gets worse!

You’re the principal of a large elementary school, and one of the decisions your school faces is what to do with young students who aren’t challenged by their class’ mathematical work.

A certain segment of the population of educators is rolling their eyes right now. Read on, read on.

For example, in your school there is currently a 1st Grader who does math above grade level, e.g. he performs long division in his head.

His parents initiated contact with the teacher after hearing their child complain that math at school was boring. They’re concerned that he isn’t being challenged.

The classroom teacher knows that he is above grade-level in math, and is trying to meet his needs in class. The parents, however, do not think the current situation is working. The teacher reports that the student is difficult to engage during math class, and that sometimes he misbehaves during math.

That’s the situation I posed in a little survey I recently shared on twitter. I then asked two questions:

1. As principal, what policy would you want your school to do for someone like this 1st Grader?
2. Do you encounter this situation, or something like this situation, in your current role in education? If so, how is it typically handled, in your experience?

Twenty-one people responded (responses here). This post is about what they said, and what I think about what they said.

II.

Here was my process for writing this scenario. First, I thought about what I’ve seen happen in the schools I’ve worked at. Then I emailed someone with more experience with this stuff than me and asked her how this usually goes down at her schools. Then I basically ripped off her email for this scenario.

Which is a long way of saying that I was pretty sure that this was a realistic situation. Still, it was good to hear people say that it sounded right to them:

“Yes, there is a large range of abilities even in kindergarten.”

“My son IS this kid (and also in 1st grade).”

“Yes!  Currently have similar situations in Grades 3 and 4; it’s a very rare year where this issue doesn’t come up.”

And let’s cut to the chase: according to people who are not me, how well do schools tend to handle this scenario?

A bunch of people thought their schools were handling this pretty well. A lot of these people, but definitely not all, were talking about high schools:

“I have high schoolers and would be willing to place a student in a higher level class if they could study and demonstrate mastery on a department final exam. I have tried assigning Khan Academy on an individualized basis.”

“Rarely [is this a problem] since math classes can easily be leveled, especially on a HS block schedule. And I think it’s easier for HS math teachers to connect to more advanced courses if they’ve taught them before. (In my experience HS math teachers change courses more often than elementary teachers change grade levels).”

“Yes, there is a large range of abilities even in kindergarten… It just takes patience.”

Then a bunch of people said that this was just not working at all in their schools:

“It’s pretty terrible at our school. We don’t have the resources to do anything when kids transfer having already knowing trig.”

“Tough one in High School, especially since we have no honors programs.”

“My son IS this kid (and also in 1st grade). I completely respect the teacher for trying to engage my kid, but he needs something else. Either allow him to work with some 2nd graders (this would be some massive coordination between grades), or give him new puzzles (logic stuff or solving mysteries or…), or help him with particular weaknesses during this time (perhaps social skills or OT), or become a librarian’s helper twice a week while others work with on-level stuff.”

“Typically slight pressure (or guidance) is applied to the teacher to differentiate and provide math learning for the student at the level and depth that they need. Of course, in reality this usually means that the teacher is given a few resources, tries hard, but is unable to challenge the student in the way that they need. Strategies such as open ended questions and 3-act problem solving help, but teaching is hard with so many diverse brains in a class and many teachers do not have experience or knowledge to extend math for such a student.”

“It varies wildly by building in our district but typically the response will be a lot of hand waving about how differentiation in the class is already occurring.”

“Student is given independent work while teacher works with others. Student is given consequences for behavior when not enriched.”

Some schools do seem to have figured out ways of making sure this problem isn’t just dumped on the classroom teachers:

“In grades K-2, these students are often pulled-out for additional math enrichment opportunities. Say the math block is 60 minutes. These students are pulled-out for around 15-20 minutes daily to work on enrichment tasks. This tends to happen at the discretion of the teacher and the groups change for every unit. There are 12 units.

In grades 3-5, there’s actually an advanced math class. The class is grade-level accelerated (3rd grade learns 4th grade concepts) and the criteria for that class depends on standardized test scores. These groups are not flexible and rarely change.”

But, overall, it seemed to me that this is a problem that people have experienced, and many (most?) find their school’s response frustrating.

III.

If the status quo is often bad, what would educators prefer? The responses hardly coalesced around one idea, but I was able to group them:

1. Find a pedagogical solution (differentiate, use groups, better tasks, etc.) [4]
2. Assign challenges for the kid to work independently on [4]
3. Create something outside of the school day (e.g. a club, a course) for the kid [2]
4. Connect the kid(s) with an enrichment specialist [3]
5. Let the kid go to a different math class during math [4]

And, finally, there were people [4] who recommended all of these things, essentially treating #1-5 as a ladder of intervention.

Which raises the possibility that everybody on my survey would agree with this, but they were all imagining slightly different situations.

Meaning, I didn’t read these as really disagreeing much. Of course, if you could provide a pedagogical solution — including extension work — that would typically be easier in any situation. And everyone would probably agree that it would be amazing to have another teacher who could come in and handle some of the kids. AMAZING.

And I bet that 95% of teachers would agree that moving kids to another class is the most complicated solution, the break-in-case of emergency solution.

But why? Why should moving to another grade’s math class be the last resort? After all, to people outside of schools it seems like it would be the simplest approach — it requires no extra work on the part of the teacher, no extensions, it’s compatible with whatever pedagogy, and you don’t need extra personnel to make it work. Why is this unpopular?

The easiest answer would be an ideological one — teachers don’t like inequitable solutions — but on the other hand do we really mind creating inequitable solutions?

Every single thing that educators recommend besides for this kind of acceleration also exacerbates inequities. If you have an interventionist providing special challenges to a few kids — that’s inequitable. If you give extension work to some kids — that’s inequitable. If only some kids go to a math club after school — that’s inequitable, even if it’s open to all.

Some may balk at my use of “inequitable,” and I get that, because we only tend to use “inequitable” to describe certain things in education. Like, we’re used to a world where some kids learn more and some kids learn less, and that’s not “inequitable” because everybody got a fair (“fair”) chance to learn the same material in class. But just looking at the situation — one person gets one outcome, one kids gets another — that is an inequity. And if you give only some kids a chance to learn e.g. cryptography in an extension worksheet that’s inequitable too.

That’s a long way of saying that I don’t think teachers are ideologically opposed to acceleration, in the sense that teachers don’t like it because it creates inequities.

IV.

So why is moving a 1st Grader to a 2nd Grade math class behind a “break in case of emergency” label? I don’t know. Here are two possibilities:

1. Schools think it’s educationally risky for students.
2. It creates a huge pain in the neck for administrators.

I think both are true, but if you know your way around a school you might also know that a lot of things happen because they solve administrative problems. Which is another way of saying that they help administrators deal with parents.

Especially for younger students, moving to another room might backfire. This is both experientially true (some of my 3rd Graders do much better with consistency and a familiar environment) and also something that seems like it might be starting to trickle through in research. See: “Two studies point to the power of teacher-student relationships to boost learning”.

But I think the more likely explanation is that some parents really, really, really want their kids to be accelerated, even if it’s not appropriate. This is especially true as parents become wealthier, and this is especially true of white parents.

So when a school starts opening up the option for moving up a grade for math, all of the sudden some other parents start calling up the school.

Seriously, so many people in teaching have experienced this situation. Parents talk to a young kid, then (though they’ve seemed totally happy in class up until this point) the kid says, “Hey Mister, I was just wondering if you had any, umm, like extra math? or more challenging stuff? Because sometimes class is too easy for me.”

And a lot of the time it’s…well, it’s a lovely kid, but a kid who wouldn’t be at the top of your “I’M WORRIED THAT THEY’RE NOT CHALLENGED” list.

So what do we do? Anything, as long as we keep it in the classroom. Because as long as it stays in the classroom, the teacher is in charge.

Like Cuban & Tyack say in Tinkering Toward Utopiaclassrooms are places that are more safe from outsider touch. At the end of the day, as long as it’s happening in a classroom, it’s sort of invisible from an outside perspective. And this can be bad (see: all the sad stories above) but in a way it can be good — it’s one of the only ways schools have to protect kids from parental demands.

I’m not saying that anyone is doing this on purpose, but there’s a structure to how schools respond to unchallenged kids, and it likely exists for a reason. That reason could be to protect schools from parents pushing their kids ahead against their kids’ needs.

V.

It seems to me that the status quo almost works. If more schools had interventionists who could come in and focus on the needs of the unchallenged, that would be amazing. (Those specialists along with teachers and parents could then decide if a kid would be better off in a different math situation.)

This raises an interesting philosophical question, which is whether schools should spend their money on the needs of kids who aren’t challenged by their grade-level material.

Of course, any “should money get spent” question in education is complicated, since money for something means money away from something else. And a lot of people think that any money for students who are doing well is money that is effectively being taken away from students who are waaaay over-challenged by the curriculum.

But I also think it’s fair to say that, as a matter of funding, as a matter of research, as a matter of journalism, which is to say “in general,” in education we do mostly focus on under-achieving students. I’m not saying that this is wrong — it makes sense for us to focus on kids that are losing the game we made for them — but I think it is true.

There is a group of kids, though, who are unchallenged and as a result school is not working for them. In a lot of situations, these kids don’t have much to do. And if they’re in your classroom, and they present classroom management issues when they’re bored? Good luck with that, teacher.

(The exception is probably high-SES districts where parental demand forces schools to come up with a policies, plus they’re more likely to have resources for handling unchallenged kids who are working above grade level in math or other subjects. The rich get richer, etc.)

So here’s my conclusion:

• If your school has a pile of money sitting around, it might be good to spend it on something like a coach or interventionist who can focus on enrichment.
• If your school is strapped for cash, you might still consider whether it would improve the overall situation by hiring someone who can focus on enrichment. It might improve the classroom situation enough that everyone benefits.
• If you’re a researcher, this might be something interesting to study.

# “Taking a Knee in Math Education”: Danny Martin’s NCTM talk, partially transcribed

Annual Meeting and Exposition – National Council of Teachers of Mathematics

At the most recent NCTM Annual convention, Prof. Danny Martin gave a talk titled “Taking A Knee in Math Education.” (See Annie and Wendy’s posts for a summary of the talk.) It’s pretty dense at times, and he hasn’t yet published (as far as I can tell) on the second half of the talk, which focuses on what he calls a “black liberatory fantasy” of math education. I wanted a transcript of the talk to refer back to and take a closer look at, and I figured that maybe others would find that useful too.

I started transcribing from around 35 minutes into the talk.

***

Equity for black learners in math education is a delusion — a compromise consistent with other historical compromises; undergirded by antiblackness; rooted in the fictions and fantasies of white imaginaries and white benevolence; held hostage by white sensibilities and sensitivities; and characterized, at best, by incremental changes that do little to threaten the maintenance of white supremacy and racial hierarchies inside or outside of mathematics education.

Given this position what do I propose as a different framing and vision of math education for black children — one committed to black self-determination, black liberation and black joy?

In the last part of this presentation I draw from recent work with two colleagues where we engage in what we are calling liberatory fantasy. Specifically, black liberatory fantasy, in order to imagine what we’re calling a black liberatory mathematics education.

We define black liberatory mathematics education as the framing and practice of math education that allows black learners to flourish in their humanity and brilliance, unfettered by whiteness, white supremacy and anti-blackness. We view liberation as a means to a radical end rather than an end in itself. We imagine a world in which our relationality is not to whiteness, anti-blackness, systemic violence, a world in which we are not defined by survival, resistance and a fight for freedom. We imagine a world in which we define ourselves, our joys and our desires in infinite multiplicities and in which we are committed to individual and collective black fulfillment.

In planting the seeds — the initial seeds — of BLM education, we draw inspiration from black liberatory struggles of the past such as the Black Panther 10 Point program as well as recent programs like Black Lives Matter where radical demands have been made within and against racial capitalism and state violence against black people.

We note that some contemporary efforts have been made to address the needs of black students in mathematics education including work that has framed math in terms of civil rights and citizenship and work that focused on culturally specific pedagogy. However in our view these approaches share a focus on liberal notions of reform and inclusion into the system of existing math education.

We are not suggesting here that there be a singular black liberatory mathematics education and we recognize that framings under this umbrella could be appropriated in many ways — even in ways that support the existing system of math education

Our goal here is to offer one perspective in the spirit of liberatory fantasy moving beyond efforts that focus on incremental change that have historically framed math education for black learners in ways that are deferential to white logics, white imaginaries, white sensitivities and white benevolence.

In terms of framing, a black liberatory math education prioritizes liberation over integration and freedom. This form of math education is skeptical of liberal notions of inclusion and equity, of appeals to democracy and citizenship, neoliberal multiculturalism and refuses all forms of systemic violence against black learners.

Moreover we recognize that freedom is never free. The freedom to be included in and to participate in anti-black spaces characterized by systemic violence is not freedom. We are cognizant that in calling for and framing a black liberatory math education we risk valorizing mathematics in a way that maintains its status and power. However our position is aligned with S.E. Anderson who in 1970 expressed that black people should learn mathematics not because of American capitalism’s advanced forms of technology require this background but because black liberation struggle against the capitalistic system requires this knowledge.

A second critical component of black liberatory education is the ability to exercise the right of refusal of the dominant education system of math education institutions and organizations that maintain their status as white institutional spaces and schooling practices and policies that instantiate anti-black violence and white supremacist orientations. Reformists often use this as a cover for the ongoing brutality of education against black learners. In this sense, reforms in math education can be viewed as sustaining the dehumanization process because these reforms are beholden to the overall anti-black system in which math education is embedded. The goal of reform is slight modification of teaching, curriculum and assessment processes, not radical dismantling. Recognizing that many children are seemingly trapped and imprisoned by the dominant system of math education we suggest resistance in the form of refusal in and refusal of.

Principle refusal in the existing system of math education will ultimately take black people to the precipice of refusal of the system. Examples of refusal in the existing system include — and its coincident that we see this nowadays — sit-ins, walkouts, and boycotts locally and nationally should be employed by black children and parents as a way to disrupt anti-black violence and dehumanization in math education. Inclusion into anti-black spaces should not be the goal of these walkouts and boycotts. Black parents and caregivers should also protest under-assessments and nullifications of their children’s abilities and refuse the tracking of black children to lower level and remedial tracks. Black parents  should refuse their children’s participation in remediation and research programs that are premised on deficit orientations and are designed to diagnose, repair and remediate — fix — black children.

These calls for refusal should not be construed with a call for segregation or racial isolation. Just as we are calling for humanizing treatment of black people we expect black people will recognize and value the humanity of others.

With respect to everyday practice: we suggest that a black liberatory math education is designed and directed first and foremost by liberation-seeking black people including parents, caregivers, community members, black teachers, and black students.

Within this new system we believe that every black child should learn mathematics for the purpose of understanding the history of black people of the United States. A k-12 curriculum — or several — devoted to the numbers of black life and history would at a minimum: help black children to understand black peoples’ incorporation into US society, quantify the ways in which the US racial state and economy were built on the labor of black people, and understand the disparate impact of systems and structures like education and prisons on the lives of black people.

We do not propose the implementation of curricula in top-down fashion nor suggest that it’s the only way to proceed. However we do believe that knowledge of self is key in meaningful math education, and the spirit of self-determination in a curriculum in support of a black liberatory math education should be co-developed first and foremost by liberation-seeking black people including parents, caregivers, community members and black students. We propose annual community reviews of all math curricula and assessments used with black children.

We propose that black children be taught by knowledgeable, liberation-seeking black teachers and independent community-controlled schools that stress collectivity and black humanity. All teachers, black and non-black, should be vetted by black parents, community members and children. Teachers should be required to live in or near the neighborhoods where they teach, and required to take training in restorative justice practices.

Drawing again on Anderson, we propose that free tutoring and math classes in community settings that are open to adults and children outside of school context, black college students and knowledgeable community members would teach these classes. Black college students with strong knowledge of math for example would serve weekend and summer internships in black communities, paid for with work study and summer research funds.

We propose the development of easy to understand and up-to-date resources such as a black parent’s guide to math education that allows black parents and caregivers to understand how school mathematics functions from many different perspectives including: curricular, assessment, teaching, and how practices like tracking and teacher recommendation for gifted programs are used against black children. Relatedly, we propose distributing comprehensive easy to read pamphlets which explain the pitfalls of financial shortcuts found within the world of the black consumer.

We propose that by the time they finish elementary, middle and high schools, all black children engage in at least one capstone project where they apply mathematics to propose solutions to challenges faced by black children — by black people. We propose that black parents, community members and children be consulted on community development projects so they can suggest ways to embed culturally relevant mathematics installations and activities in community spaces such as parks, playgrounds, barber shops, beauty salons, bus stops, community centers, community gathering spaces, health services, waiting rooms, neighborhood museums and other such contexts.

In stating these minimal components of a black liberatory math education we also assert that these are necessary but not sufficient. Revolutionary change and the building of a new and different system requires a commitment to such components.

Some people might say that my perspective is too pessimistic. [laughs] They will say that it ignores the good will of allies in the ongoing struggle for black liberation. They will point to the gains made in civil rights over the past few decades. In response, I offer this analysis of civil rights by Carol Anderson, professor of African American Studies at Emory:

She says in her book “Eyes Off the Prize”:

How could all of the blood, all of the courage, and all of the martyrs of the Civil Rights Movement still leave in its wake a nation whose schools are more segregated than ever, where more than half of all black children live in poverty, and where the life expectancy of African Americans has actually declined? And how could a movement with so much promise still leave more than six million African Americans trapped and dying in the “underclass”? The answer lies, I believe, not so much in the well-documented struggle for civil rights, but in the little known, but infinitely more important, struggle for human rights.”

For too long civil rights has been heralded as the prize for black equality. Yet those rights no matter how bitterly fought for could only speak to the overt political and legal discrimination that African Americans faced. Human rights, on the other hand, especially as articulated by the United Nations…had the language and philosophical power to address not only the legal inequity that African Americans endured, but also the education, health care, housing and employment need that haunted the Black community.”

It turns out, the civil rights, the prize, was historical compromise, a historical compromise from a demand for human rights. Equity is a similar compromise in math education.

In conclusion, I say that mainstream math education has traditionally invited black people to participate on its terms. Expecting this system to reform itself from its foundational purposes and fundamental character to a new state of validating and valuing the humanity of of black people is unrealistic in the face of evidence otherwise. Traditional discourses of equity and inclusion have been self-serving within liberal white imaginaries, white supremacy and anti-blackness. They have been inadequate for black liberation. The kinds of changes advocated for within mainstream math education discourses are welcomed and accommodated within the self-correcting systems of white supremacy and anti-blackness because they represent no real threat to these systems. Much in the same way that one can not fix capitalism, only replace it with a different system, because capitalism will seek to exploit, under any conditions or threats, liberation-seeking black people will recognize that mainstream math education cannot be fixed in service to black liberation.

Citing W.E.B. Du Bois, historian Ibrahim X. Kendi expressed the following: Instead of using our energy to break down the walls of white institutions, why not use our energy to refurbish our own? I will add that we must continue to find ways to take a knee on behalf of black children and in service to black liberations. Thank you.

Q&A:

Q: Thank you Dr Martin for your talk and for all of your wisdom all the time. Could you maybe…in your framework could you maybe say like a little bit about where you think people should start like this room is full of people who might be interested in doing this with different intersections like are there things that you think people could do to start doing this work?

A: I always have this response to what I’ve been called solution-on-demand — I don’t think you’re doing that right now.

Well, but I would say, Step Number One, Step Zero even, would be to hear me, first of all, to just hear me open ears, open heart, let it soak in, it may not be today, maybe tomorrow, maybe a week from now, maybe a month from then. But just hear me first of all. What am I saying? Why am I saying it? I think that’s Step One: what sense are you making of it?

Follow up. If you have questions about something I’ve said, ask for a clarification so we can begin a conversation.

In terms of the pragmatics of what do I do when I go back tomorrow: Obviously I can’t tell any particular person what they should do because I don’t know the context the children and I don’t want to essentialize whiteness, white people, black people, blackness…

But maybe Step One after Step Zero is sort of the internal work. The self-reflective work. One simple question is when you go back to a classroom with black children you have to ask yourself, why am I here? Why am i here? We have pre-service students at UIC Chicago who say that I’m here because I love all children, and it’s not true. It’s absolutely not true. And as hard as that may be to hear and to think about and to process, there are some of us in this room, despite all of our best intentions, who don’t love all children. And particularly we don’t love all black children for whatever reasons. And those reasons can be found internally. I think one has to come to grips with that. Why am I here? Because certainly the children are asking. Why are you here? What purpose are you going to serve in my life?

So I think it’s hearing me, reflecting on the self, and working with those parents, valuing and respecting and seeing their humanity through all the things that I mentioned in the talk. If you can’t get over your ability to see black children as children, and allow them to be children, to make mistakes, to not have to rip up their paper in their presence, to not have to be upset — and I’m not romanticizing — to be upset because someone didn’t put away their cell phone that it escalates to the police and arrest. There’s a lot of work in between that thought and that action. So a lot of it comes back to this.

And we don’t have the utopia. We don’t have the system that I’m talking about. So it’s the work in the current thing. So you have to find ways in your own context to refuse inside the system. There are risks involved, clearly. There is risk involved. People have to decide what level of risk they want to do.

So I could keep going on and on and move out to the bigger and bigger and bigger, but a lot of folks want those take home answers and I can’t give take home answers, but I can say, think about these issues and how they might apply. Twist them and turn them in whatever way you think is oriented in the direction of black liberation, liberation of black children and people.

Black children should not have to go to school and experience the kinds of violence that we know that they experience. There is no road to justice. Anybody that talks about a road to justice is not really interested in justice. Justice is right here, right now, in this moment. I don’t have to travel a road to get to equity. It’s here! It’s here!

So I could go on, but you get the idea.

Q: Dr. Martin again, thank you for your presentation. Do you have any examples of programs either in or out of school that you could point to that might exemplify what you’ve been talking about?

A: There are some traditions, independent black schools obviously, and you (to the questioner) know a lot about those.

And there are some efforts that are sort of on the cusp because again if we had the answer we would be doing it.

The Algebra Project is an interesting — really interesting — context in the sense of its underpinnings, its commitment to black life, civil rights, black humanity. There are things that could be done differently, clearly.

Some of the work that’s been done with the production of scholars in HBCUs in mathematics for example. Not the fact that we’re trying to fit into the system but the work that those people have undertaken. Dr. Shabazz. We learned about some folks here at Howard. Those efforts that people create those counter-spaces and it might not be well-publicized in the news, EdWeek, etc. but there are probably many examples that we’re not aware of where people are working locally and doing these things, working with groups of black children. I’ve done Saturday programs, “math for moms,” working with black mothers and other mothers to help them help their children in ways that they think are effective.

I think there is scholarship that needs to take place. There are folks like myself, like my colleague here and other colleagues — we need to be doing work where we are. We’re in the academy. We need to be countering the knowledge production that says black children come to school with no pre-mathematical knowledge for example. The teacher in the classroom, the parents — they can’t counter that up close. We can.

So our writing, our voice, is another thing that can be doing some work. But I think it can be multiple venues, big and small, Algebra Project is the essence, the idea of that, liberatory Bob Moses own historical beginnings, Saturday community-based programs, taking over community spaces and trying to figure out how to make those culturally relevant as I said, being involved in the redevelopment of your community, gentrification not to be involved in that, but to say stop, we refuse this, we want other options and other kinds of things, and then it gets very personal too.

So this notion of refusal, I’ve said this many many times. I have a five year old on the verge of going to Kindergarten. Very nervous, very very nervous. Five year old male child. And it’s not giving up on public education because the public education that we have now is the public education that many children don’t deserve. We need new and different public education. We may have to at some point homeschool him. Because my level of trust in the system — not you as individuals — but in the system is getting lower and lower every day. I don’t want to come home one day and find that my precocious brilliant son is sitting in the principal’s office in handcuffs because of some silliness, some anti-black silliness.

So I think that individual black families to the degree that they can (if they don’t have the resources other people should be supporting them) have to make tough decisions to go against many of the things that we’ve been led to believe we should be following to be, you know, to be “American.” Those lures are very powerful. Democracy. Citizenship. You can problematize every one of those. There’s black citizenship, and there’s American citizenship, and those are not the same and those have never been the same.

Democracy. Democracy for whom? Look at the history of democracy for us. It’s not something we should be swallowing in a non-critical way. We have to make individual decisions, we have to look at efforts like the big things, the community things, it’s multi-faceted.

And if we had the answer we’d be doing it yesterday, but obviously we have one system and that might be part of the problem. I’m not suggesting chargers or anything of that nature, but if the response is — what are you going to do with all those black kids if you take them out of the system. Think about that response for a second. It says, black people have no option, no alternatives, but to go back to the system that is dehumanizing. That makes no sense to me! That makes absolutely no sense! When you have cornered a people where they have nothing else except to go back to the fire, that people might be in trouble. They might be in trouble.

Q: My name is Lindsey Black and I’m an elementary teacher in the DC area. I really appreciate your insights in the topic at hand. I specifically appreciated your identification of a liberation seeking black teacher. I was wondering if you could highlight what the vetting process would be like to find a liberation-seeking black teacher.

A: So one thing we know is that we lose a lot of potential black teachers. They come to the universities, struggle for various reasons, all sorts of institutional structural reasons, and they’re committed to black people, black children, black joy, but the things we’ve set up to vet them at that level, take them in a different direction.

We know that there are people there who are interested in black people’s humanity — humanity. We need to find ways to keep those folks engaged. Not rescue them, not save them, but keep them engaged. Somehow, some way. And it’s difficult work.

The vetting process beyond that? As I said, we need to find ways and avenues for parents to be at that front, first, second, third, level to say no way — or, yes way, we feel comfortable with this teacher teaching our children. I’m turning my child over to you 8, 9 hours a day, expecting them to be safe, expecting them to learn and grow, etc. I need to have a say in that.

Not necessarily just the selected few, but come out to a community meeting, fill up a room like this. Why do you want to teach in this school? I know it’s not in our existing system, but this is the liberatory fantasy beast. Be vetted by the black children! Why can’t black children be allowed to have conversations with the people who might be teaching them at some point — of all ages. Who are you? Where do you come from? Why do you want to teach me? What do you know about me? What do you know about my people? What do you know about my neighborhood? What do you think about me? Do you think I’m brilliant? Tell me! Do you think I’m brilliant?

“Umm well…” No, no no, no, it’s yes or no. It’s yes, or no.

We can fantasize in many different ways and there are some practicalities to it as well. I think parents should be right there. One, first line of defense. It’s not the parent, it’s the caregiver. Somebody’s gotta do the vetting. Alright. I know we have these systems and structures where other people do the vetting for us, but I have to raise the question: is this good enough? I don’t think it is. Well-intentioned, you do the hard work everyday, it’s not about individuals, it’s about structures and systems. And the system that we have, in my view, is not the one that is committed to black liberation, and to black people and to black communities. And that may be trouble for other children and groups as well, and people have to fight those fights.

Thank you.

# My Talk on Recent Growth Mindset Research

Last night I gave a talk about growth mindset research at the Global Math Department. They just posted the recording, here, and I previously shared links to some of the research we discussed last night here.

My main point from the talk was that the world of teaching for a growth mindset is talking about something quite different than what mindset researchers talk about. Dweck’s Mindset and Boaler’s Mathematical Mindsets are not talking about same thing.

For instance, YouCubed and Jo Boaler’s Mathematical Mindsets lays out a comprehensive vision of what “growth mindset” classrooms look like. This includes things like ending timed work, using more visual representations, ending tracking, plus many more things besides.

This has very little direct impact on mindset, though. This becomes especially clear after a careful look at what psych researchers themselves are doing with mindset research.

The original work comes from Dweck in a series of papers in the 70s and ‘80s. She noticed that, when presented with a challenge, some students would behave helplessly (e.g. giving up, “I’m bored,” getting upset, etc.) while others seem motivated by the challenge. What could explain these different patterns of behavior?

Dweck retreated to goals to explain behavior. Different people react to challenges differently because they have different goals. Some people have learning goals, while others have performance goals — basically a desire to look good.

OK, but then why do people have different goals?

That’s when Dweck lands on implicit theories of intelligence, i.e. mindset. People who have fixed mindsets believe that intelligence is essentially something you’re stuck with, hence they don’t have learning goals and are only left with looking good. (Presumably everyone likes looking good, it’s just that if you believe that intelligence is malleable you believe in learning.)

(As an aside, what does it mean to believe that intelligence is malleable other than believing in learning? In other words, is Dweck’s theory tautological?)

(Another question: why stop at mindset? Why doesn’t Dweck explain where mindsets come from? I guess, from the interventions they’ve designed, we can get a sense for one place that mindsets can come from.)

This leads mindset researchers to essentially make two predictions:

• Mindset matters. Having a growth mindset leads to good things (e.g. academic achievement, good relationships, professional success, etc.)
• We can do something about it — we can turn some of those bad outcomes into good outcomes with our interventions.

Those interventions can be very brief, as short as 45 minutes, and they can be done using completely standardized online materials. Pretty much every intervention has the same basic structure: science tells us that the brain can grow, that challenge is good, and hard work can lead to good things happening. Then there are some reflective prompts for discussion or writing.

(Some really good intervention materials from mindset researchers devoted to making this stuff free can be found on PERTS.)

One last question to round off the researcher version of growth mindset: how could such a brief intervention have any significant impact at all? It seems like magic. Not so, say Yeager and Walton in “Social-Psychological Interventions in Education: They’re Not Magic.” The key is recursive effects, positive feedback loops where the intervention marginally increases mindset, which then marginally promotes good learning, which then further effects mindset, which then &c., &c.

So the claim is that growth mindset interventions can have major effects. But just how major? That’s the question of the first of the recent papers we talked about, the 2018 meta-analysis of existing research.

How important is growth mindset for academic achievements? When the researchers pooled a comprehensive selection of results together, they found a very modest average effect. (Check out the paper and you’ll see some discussion of a fairly high gap between some studies that find fairly strong effects and others that don’t, more heterogeneity than you might expect. They aren’t able to easily find an explanation.)

And what about mindset interventions? The main point is that they find extraordinarily weak average effects for mindset interventions, as shown in this forest plot below. (The diamond represents the mean effect. Note just how many interventions are touching or less than 0 in their effect sizes.)

And these results seem roughly in line with what Dweck and her recent collaborators at PERTS are finding in their top-notch mindset experiments. They’ve been trying to address criticisms of mindset research while doing work that would prepare mindset research to scale and reach many students.

This move of the focus of mindset experiments to scale has quietly involved a number of other shifts as well.

• Away from testing in just one context
• Less researcher involvement in the mindset treatment
• Standardized materials
• Big studies

Correspondingly, there have been two changes in the sorts of results that the PERTS team have been publishing with Dweck.

• Moving away from claims that mindset interventions have an impact on everybody, now a focus on at-risk students.
• Much more modest effects being claimed.

You can see this in the figures from the 2015 pilot study. There was really not very much of an effect at all from the growth-mindset intervention for the general population. This is in line with the meta-analytic results.

There is an effect — again, a modest one — from the growth mindset intervention for at risk students on their GPA.

But these results came from what was supposed to be a trial run before a much larger experiment. And those results were briefly released earlier this year in draft version. The official version hasn’t been published, but the study was widely praised for its rigor and again found modest results, mostly for at-risk students.

Here’s where I want to start putting the pieces together. What I want to ask is what these results should entail for advocates of teaching for a growth mindset. My answer is: not much, because the relationship of all this research to what goes on in teaching was never clear.

Here’s what I think happened. Dweck’s research begins with descriptions of helpless behavior from students, and she points to motivational problems. All of this resonates deeply with teachers, so we associate motivational concerns with growth mindset. When Boaler talks about teaching for a growth mindset, usually she means motivational teaching, e.g. if you teach with visual representations more students will experience success and they’ll be more motivated to continue engaging.

But mindset researchers were only ever trying to impact the very bottom of this causal chain, the mindset stuff. And maybe, it turns out, that wasn’t as important as they thought it was, and all the stuff in the middle (i.e. teaching) is really important.

When we dialogue with research, there are always a couple different games we can play. One game we can play is “how can this be wrong?” There’s a time for that. Another is “how can this be right?” and there’s a place for that too, but my favorite is “how can both of these things be right?”

I think it’s entirely possible that some mindset interventions had huge successes, but I also think that when we try to get clear enough about why these worked, the generalizations don’t hold particularly well. And that’s because we’re working with motivation, fickle motivation, and it depends on so many different things — what the kids happen to need, who’s running the intervention, how the kids happen to hear the message, which of their other adults were involved, etc. It seems to be very hard to standardize motivation.

But teachers who advocate for growth mindset can be right — mostly right — also, because they were never talking about just changing a student’s theory of intelligence. They were talking about the entire causal chain of motivation, including social safety, helping kids see success, making the curriculum more accessible, &c., a million things that have to do with motivation but that are waaaay beyond the purview of mindset research.

I don’t mean to be too wishy washy. Mindset is not a revolution — that’s not a hypothesis that finds a ton of support in 2018. And teaching for a growth mindset doesn’t have all that much to do with mindset, and I’d prefer that we talk about motivation rather than mindset — I think that’s what we really mean to say.

I’ll end with two questions:

What other research would we be curious to see in 2019 or any other year, relating to mindset?

What are the narrower, more specific, less potentially revoluationary but more potentially connected to the classroom, questions that we can ask about motivation? Because revolutions like Dweck’s are consistently overrated at the outset, and in education we tend to place teaching on these boats that end up being much more rickety than they at first seem.