The Politicization of Math Class
The least controversial subject in school is suddenly controversial.
For as long as I can remember, there has been a fight between liberals and conservatives over school curricula. The former tend to want more focus on diversity and identity issues, while the latter want schools to stick to readin’, ‘ritin’, and ‘rithmetic. The biggest struggle has been over history and social science textbooks, with Texas and California often dictating the curriculum of other states because they’re such massive purchasers. We’ve seen this most recently with the controversies over the 1619 and 1776 projects.
In recent weeks, though, we’re seeing a kerfuffle over mathematics curricula. One would think that it would be hard to politicize something as concrete as numbers but it turns out to be more complicated than that.
It first came to my attention a couple weeks ago when my home state of Virginia seemed to be doing away with tiered math classes, keeping all kids on the same track until 11th grade. Except it turned out not to be quite true. As WaPo reported,
The kerfuffle began when Loudoun County School Board member Ian Serotkin (Blue Ridge) posted on his public Facebook page about the initiative. He wrote on April 22 that he had received a briefing on the initiative, which he wrote promised to “revamp the K-12 math curriculum statewide.”
He praised the initiative for “some noble goals,” including the fact that it “provides a pathway for every student to be able to take calculus or higher math by the end of high school if they so choose.” But he criticized it for what he said it would do to advanced math classes, claiming the proposal would force all seventh-graders to take the exact same math class, all eighth-graders to take the exact same math class, and so on through 11th grade.
“As currently planned, this initiative will eliminate ALL math acceleration prior to 11th grade,” Serotkin wrote. “That is not an exaggeration, nor does there appear to be any discretion in how local districts implement this.”
As best I can tell, Serotkin was acting in good faith and the problem was some incredibly awful workshopping of proposals that were being considered as part of the regular review process.
Superintendent James Lane said the Virginia Department of Education is in the early stages of a regularly scheduled revision of its mathematics Standards of Learning, which guide school systems in their course offerings across all areas of instruction. As part of that revision process, which takes place every seven years, state officials recently began workshopping some ideas as to how Virginia could teach mathematics in a way that better prepared children for college and the workforce, Lane said.
The ideas — detailed online as part of a program called the Virginia Mathematics Pathways Initiative — include rejiggering eighth-, ninth- and 10th-grade math courses to place a greater emphasis on fields including data science and data analytics, Lane said. Schools would still offer traditional courses such as Algebra I, Geometry and Algebra II, the superintendent said, but these courses would now “incorporate stronger foundations in data analytics,” for example.
Long story short and probably too oversimplified: no final decisions have yet been made but there’s a strong likelihood that the eventual policy will be delay getting students ready for college mathematics, particularly calculus, and instead focus more on “practical” math like statistics. This is outside my expertise but, offhand, that’s probably a sensible policy.
Another report, in the Virginia Mercury, notes that this is part of a nationwide rethinking of how we teach our kids math. And, naturally, equity and inclusion are a big part of the puzzle.
In webinars, VDOE officials have been upfront that the initiative is based on data, especially standardized test scores that show Black, Hispanic and low-income students have lower pass rates on state math assessments than White and Asian students. Those critical of the initiative have argued that Virginia is lowering its standards in favor of certain students instead of improving instruction to help more children reach advanced courses.
The idea of reforming math education, though, isn’t new, and it wasn’t pioneered in Virginia. At least 22 states have explored the idea, which is based on decades of research on how traditional math coursework is failing students from many backgrounds.
In Virginia, like much of the country, most math curriculum follows one traditional route. The Charles A. Dana Center at the University of Texas — whose research has informed pathways initiatives in multiple states — describes it as “a course in geometry sandwiched between two courses in algebra.” The larger goal is to send as many students as possible down the path toward calculus.
That’s because upper-level algebra courses are viewed as more rigorous by many colleges and often required for admission or to transfer credits. But over the years, educators have found that they alienate students who don’t need those courses in their future careers. Some, for example, might be better served by statistics classes that provide a strong foundation for the classes they’ll take in college. Other students might make the decision to stop taking math classes as soon as they can because the information just doesn’t apply to them.
“For some students, it’s just not relevant,” said Patricia Parker, who serves as an adviser for the Virginia initiative on behalf of the State Council of Higher Education for Virginia and Virginia Community College System. But even for advanced-track students, the status quo doesn’t always lead to success. According to the Dana Center, roughly a third of students at all public four-year colleges are placed in a remedial math course before they go on to college-level classes. That includes students who have taken courses like Algebra II and calculus in high school.
“What parents don’t know, or maybe don’t realize, is that so much mathematics gets retaken when you get to college,” Parker said. In 2015, VCCS embarked on its own effort to reform math education at the community college level after learning that many students weren’t gaining the background they needed in high school.
“What we learned was that it wasn’t the students’ ability to do calculus or not that was preventing them from being stellar engineers or stellar scientists,” Parker said. “It was that they were coming with weak foundational skills.”
I came through elementary school in the heyday of the much-derided New Math movement and came away with what I think was a pretty strong foundation. I was good at it. Until the more advanced mathematics classes in high school and college. I did fine, especially in high school, in terms of grades. But I didn’t really learn it. I memorized formulas and when to apply them but retained almost none of it. I took two semesters of college calculus and literally couldn’t tell you what the courses were about. Conversely, I still utilize things like the commutative and distributive properties I learned in early elementary school on a daily basis.
I’m quite amenable, then, to the idea that most people don’t need trigonometry and calculus in high school. For that matter, those who are going on to STEM degrees can probably wait until college—or, certainly, the last couple of years of high school—to master those skills.
That debate is inherently political, simply because it’s happening in a political arena. But the real controversy isn’t about how best to teach math but in an underlying philosophical dispute. That became clear a couple of days ago when California’s curricular review came under scrutiny.
The treatment in the press has been all over the place, mostly seen through ideological lenses, but you can read the official site for yourself. I draw your attention in particular to “Chapter 2: Teaching for Equity and Engagement.”
In California, all teachers strive to ensure every child has an equal opportunity to succeed. Teachers of mathematics can provide equitable education by making sure all students receive the attention, respect, and resources they need to achieve their potential. California classrooms combine diverse communities and students who bring a rich variety of cultural and linguistic resources that teachers can draw on to create culturally-relevant lessons (Ladson-Billings, 2009; Hammond, 2020; Milner, 2011). Cultural relevance is important for learning and also for expanding a collective sense of what mathematical communities look and sound like to reflect California’s diverse history. A focus on equity recognizes that mathematics, over the years, has developed in a way that has excluded many students (see Chapter 1). Because of these inequities, teachers need to work consciously to counter racialized or gendered ideas about mathematics achievement (Larnell, Bullock, & Jett, 2016). It is common for people to claim that avoiding aspects of race, culture, gender, or other characteristics as they teach mathematics, means they are being equitable; but the evolution of mathematics in educational settings has resulted in dramatic inequities for students of color, girls, and students from low income homes (Joseph, Hailu & Boston, 2017; Milner & Laughter, 2015). These inequalities include not only access to high-quality curriculum and resources, but also to instruction that appropriately leverages students’ diverse knowledge bases, identities, and experiences for both learning and developing a sense of belonging to mathematics (Langer-Osuna & Esmonde, 2017). A “color-blind” approach allows such systemic inequities to continue (Battey, 2013; Martin, 2007). The examples that follow are provided to help educators utilize and value students’ identities, assets, and cultural resources to support learning and ensure access to high achievement for all students in California—particularly English learners, who are linguistically and culturally diverse, and those who have been disenfranchised by systemic inequities.
In many circles, this alone will set off alarm bells. Math is math, damn it, and can’t possibly be racialized or gendered! But the basic idea that we need to provide examples that students can relate to is pretty uncontroversial. Some of the ways in which they propose to do this, though, may well be.
Beyond that, there is a basic questioning of the larger focus of the math curriculum, as exemplified by the Chapter 1 subtopic, “Mathematics as a Gatekeeper or a Launchpad?” And it results in a proposal that gets at the controversy over both the Virginia and California proposals:
Mathematics continues to play a role in how we conceive of our careers, evidence-based civic discourse and policy-making, and the examination of assumptions and principles underlying action. All students are capable of making these contributions and achieving these abilities at the highest levels. As a guide to implementing the Standards, this framework outlines innovative mathematical learning experiences with the potential to help all California students.
To develop learning that can lead to mathematical power for all California students, the framework has much to correct; the subject and community of mathematics has a history of exclusion and filtering, rather than inclusion and welcoming. There persists a mentality that some people are “bad in math” (or otherwise do not belong), and this mentality pervades many sources and at many levels. Girls and Black and Brown children, notably, represent groups that more often receive messages that they are not capable of high-level mathematics, compared to their White and male counterparts (Shah & Leonardo, 2017). As early as preschool and kindergarten, research and policy documents use deficit-oriented labels to describe Black and Latinx and low-income children’s mathematical learning and position them as already behind their white and middle-class peers (NCSM & TODOS, 2016). These signifiers exacerbate and are exacerbated by acceleration programs that stratify mathematics pathways for students as early as sixth grade.
Students internalize these messages to such a degree that undoing a self-identity that is “bad at math” to one that “loves math” is rare. Before students have opportunities to excel in mathematics, many often self-select out of mathematics because they see no relevance for their learning, and no longer recognize the inherent value or purpose in learning mathematics. The fixed mindset about mathematics ability reflected in these beliefs helps to explain the exclusionary role that mathematics plays in students’ opportunities, and leads to widespread inequities in the discipline of mathematics. Some of these include:
*Students who are perceived as “weak” in mathematics are often informally tracked before grade seven in ways that severely limit their experiences with and approaches to mathematics (Butler, 2008) and their future options (Parker et al, 2014). See also Chapter 8.
*Students who do not quickly and accurately perform rote procedures get discouraged and decide not to persist in mathematically-oriented studies.*Students who are learning the English language are deemed incapable of handling, and denied access to, grade-level authentic mathematics.
*Students with learning differences that affect performance on computational tasks are denied access to richer mathematics, even when the learning differences might not affect other mathematical domains (Lambert, 2018).
*Students who are tracked into lower mathematics courses in middle and high school can be denied entry into prestigious colleges.
Many factors contribute to mathematics exclusion. As one example, consider a system described in more detail in Chapters 7 (Grades 6-8) and 8 (Grades 9-12): Though many high schools offer integrated mathematics, high school mathematics courses are often structured in such a way (e.g., algebra-geometry-algebra 2- precalculus) calculus is considered the main course for Science, Technology, Engineering, Arts, and Mathematics (STEAM)-oriented students, and is only available to students who are considered “advanced” in middle school—that is, taking algebra in eighth grade. In order to reach algebra in grade eight, students must cover all of middle grades math in just two years (or else skip some foundational material). This means that many school systems are organized in ways that ultimately decide which students are likely to go into STEAM pathways when they begin sixth grade. This reality leads to considerable racial- and gender-based inequities and filters out the majority of students out of a STEAM pathway (Joseph, Hailu, Boston, 2017). Moreover, English learners have disproportionately less access, are placed more often in remedial classes and are steered away from STEAM courses and pathways (National Academies of Sciences, Engineering, and Medicine, 2018). High school mathematics courses such as data science should exist as a viable option whether students consider STEAM or non-STEAM career options.
Considering that many competitive colleges and universities (those that accept less than 25 percent of applicants) hold calculus as an unstated requirement, the inequitable pathway becomes even more problematic. Many students remain unaware that their status at the end of fifth grade can determine their ability to attend a top university; if they are not in the advanced mathematics track and on a pathway to calculus in each of the subsequent six years of school, they will not meet this unstated admission requirement. This mathematics pathway system, typical of many school districts, counters the evidence that shows all fifth graders are capable of eventually learning calculus, or other high-level courses, when provided appropriate messaging, teaching, and support. The system of providing only some students pathways to calculus, or statistics, data science or other high-level courses has resulted in the denial of opportunities too many potential STEAM students—especially Latinx and African American students. At the same time, arbitrary or irrelevant mathematics hurdles block too many students from pursuing non-STEAM careers. Mathematics education must support students whether they wish to pursue STEAM disciplines or any other promising major that prepares them for careers in other fields, like law, politics, design, and the media. Mathematics also needs to be relevant for students who pursue careers directly after high school, without attending college (Daro & Asturias, 2019). Schooling practices that lead to such race- and gender-based disparities can lead to legal liabilities for districts and schools (Lawyers’ Committee for Civil Rights of the San Francisco Bay Area, 2013). A fuller discussion of one example is included in Chapter 8. The middle- and high-school chapters (Chapters 7 and 8), and the data science chapter (Chapter 5) outline an approach that enables all students to move to calculus, data science, statistics, or other high level courses, with grade level courses, 6, 7, and 8 in middle school. The new provision of a data science high school course, open to all students (not only those considered “advanced” in middle school), that can serve as a replacement for algebra 2, has the potential to open STEAM pathways to diverse groups of students, both through its engaging content and its openness to all students—as described further in Chapter 5, and Chapters 7 and 8. [emphases all mine-jhj]
So, again, I don’t claim to be an expert in primary and secondary education, much less mathematics education. That we stigmatize students as “bad at math” if they struggle early and that this becomes self-reinforcing and thus self-fulfilling strikes me as more than plausible but likely.
But, like Robbie Soave, I’m exceedingly skeptical at the foundational claim that every single student is capable of achieving at the highest levels.
The framework’s overriding perspective is that teaching the tough stuff is college’s problem: The K-12 system should concern itself with making every kid fall in love with math.
Broadly speaking, this entails making math as easy and un-math-like as possible. Math is really about language and culture and social justice, and no one is naturally better at it than anyone else, according to the framework.
Contrary to what this guidance seems to suggest, math is not the end-all and be-all—and it’s certainly not something that all kids are equally capable of learning and enjoying. Some young people clearly excel at math, even at very early ages. Many schools offer advanced mathematics to a select group of students well before the high school level so that they can take calculus by their junior or senior year. It’s done this way for a reason: The students who like math (usually a minority) should have the opportunity to move on as rapidly as possible.
For everyone else… well, advanced math just isn’t that important. It would be preferable for schools to offer students more choices, and offer them as early as possible. Teens who are eager readers should be able to study literature instead of math; young people who aren’t particularly adept at any academic discipline might pick up art, music, computers, or even trade skills. (Coding doesn’t need to be mandatory, but it could be an option.)
The essence of good schooling is choice. Individual kids benefit from a wide range of possible educational options. Permitting them to diversify, specialize, and chart their own paths—with helpful input from the adults in their lives—is the course of action that recognizes vast differences in interest and ability. Holding back kids who are gifted at math isn’t equitable: On the contrary, it’s extremely unfair to everyone.
Yet the framework seems to reject the notion that some kids are more gifted than others. “An important goal of this framework is to replace ideas of innate mathematics ‘talent’ and ‘giftedness’ with the recognition that every student is on a growth pathway,” it states. “There is no cutoff determining when one child is ‘gifted’ and another is not.” But cutoffs are exactly what testing and grading systems produce, and it’s absurdly naive to think there’s nothing innate about such outcomes, given that intelligence is at least partly an inherited trait.
If California adopts this framework, which is currently under public review, the state will end up sabotaging its brightest students. The government should let kids opt out of math if it’s not for them. Don’t let the false idea that there’s no such thing as a gifted student herald the end of advanced math entirely.
Now, I think this overstates things. While I agree that we ought to let the gifted kids move faster lest they grow bored and that we shouldn’t torture those who are incapable of learning calculus by pretending otherwise, there’s obviously a foundational level of math that every high school graduate needs to possess. How to best achieve that is debatable but that should be the goal.
The notion that every kid is a potential Einstein is just absurd, though. We recognize that not everyone can be LeBron James. Or even make the high school basketball team. Why do we need to pretend otherwise with academic subjects?
Anyone who thinks every student is capable of the same level of math is seriously deluded. And I say that as an engineer who would love to believe that he has the same level of math ability as say Euler or Gauss, but who somehow hasn’t changed the world of mathematics the way they (and numerous other people had). Seriously, I can’t believe the people suggesting that everyone is equally gifted in math have ever done any math at all — as you say, its as ridiculous as suggesting everyone can potentially play basketball at Lebron’s level. In fact, I’d say they’re openly lying, as I suspect they ran into their own limits of math ability at a point far below the limits met by say the Field’s prize winners.
Teaching statistics without calculus and algebra is basically just teaching people to plug into formulas, and typically leads to a great lack of understanding of what the statistics mean and when to apply them. Stats and data analysis in general are based on calculus and algebra, in the same way that reading and writing is based on understanding a language. You could probably learn how to grammatically structure a sentence in a language you don’t understand by plugging words into some formula, but it’d be pointless because it would have no meaning for you.
However, it occurs to me that if everyone does have the same ability to do math, then we don’t need to teach it in school at all. We can just give every student a math book, and they’ll learn it for themselves like Ramanujan did (one of the greatest mathematicians of all time). Think of all the money we’d save on schools and teachers.
I don’t know how much this factors into all of this, but Educational research is some of the sh*tiest research I’ve ever seen. I had a close up look at the research of one particular educator when I briefly worked for a company that was trying to convert a program she had developed into a commercial product. The methodology of her research was… laughable. She had a bunch of pre-schoolers in a class as part of the university research arm. At specific times of the day they would break into two groups, half the students going with the tenured professor to work with her program and half going with the grad student to use a “traditional” early development computer program. At the end of a couple of months the students were evaluated for performance gains by (drum roll)… she and her grad students. When I fully understand the lack of rigor I was horrified, only to find out that she had been asked to present this peer reviewed paper at a major educational conference as one of the keynote speakers.
And I looked at other “prestigious” research, which turned out to be more of the same. I’m pretty sure the reason we are improving so glacially in educational attainment has a lot to do with the fact that all new teaching methods are essentially based on someone’s theory, with virtually no meaningful research to back it up.
That is simply scary. Education is vital, especially of the very young.
Sometimes I think that good performance in an academic subject has as much to do with temperament as ability. I did relatively poorly in math because it bored me senseless–it was just memorizing things, whereas the study of history and literature, say, provided room for speculation. If this weren’t true, why wouldn’t all people with I.Q.s of 125 or 135 do the same things? Instead, you have in the range people who become medical doctors, language professors, writers, engineers, actors, police detectives…a whole range. I know this is simplistic, but it seems to me true to a great extent.
And, of course, individual people have individual talents. But the point is, I was certainly smart enough to do well in math. I just didn’t like it.*
*I should add here that when I decided to do well in math, I overcame my aversion and did.
I’m terrible at math when it comes to solve actual math problems.
That said, in high school I manged to get through algebra, trigonometry, analytic geometry*, and calculus (derivatives and integrals if I’m translating the terms correctly), through persistence and summer school to make up for failing the subject twice.
Since then, as I predicted, I’ve had little need to use much math in daily life, beyond arithmetic and percentages (elementary school stuff), and some probability and statistics.
Would my life had turned out different had I skipped analytic geometry? I can’t think of a reason why.
*That’s the reason I cannot hear the name Descartes without a rise in my blood pressure. the subject is like drug addiction, in a bad way. The equations for lines are so easy, you feel good about yourself for getting it. Next thing you know, you’re miserable while trying to comprehend how to rotate or move the coordinates of a parabola.
I’m bad at math. Not because anyone told me I was bad at math but because I actively dislike math. I knew perfectly well, all through my ten years of schooling, that I hated math. It has all the attraction for me of doing the dishes or taking out the trash. I’m very happy that other people feel differently, I like buildings to not fall down, and I love all my tech toys, but there is no power on earth which now, or all those years ago, could have caused me to GAF about numbers.
In fact the tedious insistence of educators that I spend time trying to bully my own brain into sharing their obsession with math was a major contributor to my generalized dislike of formal education. I’m not a computer, I will not be programmed – I decide what goes in my brain, and how it is stored, accessed and used.
(As an aside, if you need proof that standardized tests are bullshit, I pulled a 790 out of 800 on verbal SAT (ahem, one of the test questions was incorrectly formulated) and a 610 in math which is fair to middlin’, despite not knowing any of the answers beyond long division. Literally no clue to any of it, but I can beat a test.)
As for acceleration – every sports team is accelerated gym. We can recognize physical excellence but treating intellectual excellence the same way is fraught? It’s bizarre. The kids who swim easily in math should be allowed to race ahead of people like me who hold them back. In part it’s a small d democratic instinct to insist that we are all created equal and all we need to do is work hard and blah blah blah. No. Talent is a thing. It exists and it is unequally endowed.
If someone argued that argued that students didn’t need to read and analyze literature because only a tiny percentage of them were going to be English majors in college and the ones that were could wait until then for “advanced reading” courses, that would never fly.
A big underlying problem here is that while STEM majors are rightly expected to still study arts and humanities as part of a well-rounded education, liberal arts majors are routinely permitted to go through life as innumerate boobs because “they don’t like math”. A lot of this is really about politicians and bureaucrats trying to normalize their own lack of education in the subject so they don’t have to feel insecure.
The curriculum, even math, has always been politicized. This is inevitable for at least two reasons.
First, since (to deploy a cliche) knowledge is power, any determination of what gets taught to whom is a political choice.
Second, since K-12 is largely a publically-funded concern (and private schools are regulated by states), choices are directly political.
The New Math issue noted in the OP fits that bill, as does the incessant griping about Common Core over the last decade or so.
And disagreements over what to do with “gifted and talented” programs and whatnot have been another since at least when I was in middle school (and I am not a young man).
I am not saying all of this is unimportant, but I feel like this kind of story is a perennial maelstrom of less than meets the eye (often because more-conservative leaning media outlets love to get upset about them).
I will say, from a university administrator’s POV, it is empirically true that students in my state (and, I believe, nationally) are increasingly less prepared for college math (and English!) than they were in the past (a long-term trend).
Well, I could argue that what you learn from literature and history, as well as art and music, is more universally applicable to one’s daily life than, say, a quadratic equation. As Kathy noted, most of us need only the arithmetic we learned in grade school to get by.
Oh yeah, my boss is asking me all the time, “so what was the symbolism in the monthly status report that just came out?”
I was a math guy in school and for a time I was a math-ish (very mathy CS) educator. I had no accelerated courses in high school. This made me feel behind in college – for about a year. Then no problem at all.
I feel that if one focuses a lot of effort on kids like me, then you create two tiers – tracks, they are called in ed – of student, and this means you end up not scooping up certain people. And it’s easy to believe that certain identities are left behind more often. It’s not that math as a subject is biased, but the institutions and methods we use to teach it certainly can be.
If you tell someone, or they tell themselves they are no good at math, they will do less well. I can’t tell you how many times I spent long periods of time trying to crack a tough problem. I persisted and eventually succeeded in almost all cases. If you believe you aren’t good at math, would you do that, or give much sooner?
@Steven L. Taylor:
One big thing I’d like to note here, because it’s one of my peeves: unless you’re in your 70s or 80s, all the math you know is New Math. They just stopped calling New Math after a while because after being used for a few decades it was now just math.
Which leads to bizarre stuff like people complaining about “New Math” while demanding schools go back to teaching the things which are New Math.
It’s been an interesting journey, over the last year, watching the public education system make a better argument for home-schooling than every libertarian over the last 30 years combined.
Likewise, Common Core is the English and Math curriculum in 40 of the 50 states this very moment. It just stopped being a Thing and people generally aren’t calling it Common Core any more. It’s just the state standard curriculum now.
(The realize bizarre thing at the time it was coming out was how much of the complaints were about history education when Common Core doesn’t even HAVE a history curriculum)
Come on, you know that’s not what I’m saying. You can’t tell me people aren’t influenced by what they read, see, and hear.
But I’ll bet you recognize archetypes in TV shows and movies, and debate what the helmet styles in the Mandalorian represent in relation to the culture they come from. 🙂
And people are influenced by the science and technology around them too. But the point is that just as most people don’t need calculus to get by in their day to day lives (which is always the excuse for not teaching them math), they don’t actually need literature analysis skill to get by in their day to day lives.
We teach everyone art and humanities because it makes them more well rounded people, even if they don’t strictly “need” to know it. Likewise, we should teach everyone math and science because it makes them more well rounded people even if they don’t strictly “need” to know it.
Maybe we wouldn’t be a country of anti-vax, climate denying, pseudoscience fans if primary and secondary math and science education got the same public love from politicians, journalists, artists, etc. that arts and humanities does.
Ya think? Most of the research I’ve read–which was pretty considerable because I was a teacher certificate in grad school and finished my program with a course of study involving teachers doing research–because 1) a fair amount of it is first time work by grad students (so, basically, practice dissertations) and 2) another fair portion of it is “ethnographic” research where, as far as I could see, the rule IS that anecdote=data (provided that you can find another “researcher” with similar anecdotes and conclusions). My favorite was a meta research where the author concluded that researchers in Education prove there thesis ~85% of the time–except when studying theses that are mutually exclusive, where the validation rate increases to ~95%. (WA!)
Back to math and my own “research” (i.e. anecdote collection) over 25 years of occasionally teaching arithmetic/mathematics in many classrooms and at many levels, from remedial to precalculus. Just like any other method of symbolic expression of abstract concepts (that is language [and thank Singapore for the idea]), math may not be so much hard as it is labor intensive. In addition, some number of students may well be more interested in the subject theoretically more than they are in a sense of mastery of the skill set. (In much the same way as Archimedes theorized the water lifting screw, but, according to the sources in the book I was teaching from at least, the Greeks didn’t build pumps.) Either way, and taking both factors into account, it might be wisest to move our theories for how to teach math to a more exploratory track and also allow students to progress on the mastery-based skill development at their own pace. This suggestion has the unfortunate side effect of shifting some of the labor intensity to the teacher and letting computer-based efforts such as Khan Academy take over on the labor-intensive work at the student level (thus bollixing up the current wailing over “accountability on the part of the district and the teachers”), but no one can come up with systems that will make everyone happy.
@Michael Reynolds: “Literally no clue to any of it, but I can beat a test.” To paraphrase the words of the TESOL/TESFL highest score scholarship winner the next to last year I was in Korea:
That and teaching kids that a multiple choice test doesn’t measure your knowledge as much as it measures your ability to recognize responses that won’t answer the question…
*–I would note that this idea will work for arithmetic/mathematics, too, but that your experience reinforces the other side of the proposition. You can’t win if you don’t/won’t play. That’s one of the biggest obstacles that plagues education from about grade 6 or 7 on–students who are tired of/not interested in playing school.
@Stormy Dragon: I will go a step further and note that most people (I am not addressing to this to JJ) who use the terms “New Math” and “Common Core” have no idea what they mean.
TBH, I don’t know what the New Math is, save that it was new at some point and that some people used to make fun of it on TV.
@Stormy Dragon: No, he isn’t. But I’m willing to bet that at least occasionally he asks “what do we know/are we learning from the monthly status reports.” N’est pas?
Exactly. It drove me a bit nuts, TBH. “Common Core” became a term to mean “something at my kid’s school I don’t like.”
@just nutha: And to ask them how to take a big pile of information and compile it into readable form. Something that one learns in comp and lit (and in social science classes as well).
@Stormy Dragon: History is lucky then. English as a curricular phenomenon has something on the order of two or three hundred separate learning benchmarks that have to be measured and validated in Common Core systems. It’s a mess–at best.
F**K!!!! Doesn’t anyone generalize anything anymore???
Speaking of tests, let me offer this anecdotal experience from my own high school years:
I noticed that the less intellectually students always begged for what they called “objective” tests, meaning multiple choice questions or fill-in-the-blanks. I, on the other hand, always wanted essay tests, because I loved to write, and, frankly, I probably figured I could bullshit my way through them.
I never, though, met a bright student who balked at an essay test.
Should be “intellectually endowed” in the second line.
As a dyslexic who didn’t know they were til college, I *HATED* math. I understood it and if you didn’t expect me to show my work, I was right. The problem was proving I was right -I simply couldn’t without making a mistake somewhere along the way that gave me a diff answer from the one I knew to be correct and nobody would accept “I just know” even if I answered correctly each time. As I got older, I realized for most it was the opposite – they could show the work for what it should be but never got it right because they didn’t grasp what they’d memorized. The more abstract the math, the harder it got for people because they learned concrete basics without the knowledge of how and why they fit together. Once things start get flexible, confidence goes off the rails.
1+1=2. Why? I dunno, just is. Cuz you can count on fingers??
-1* -1 = 1. Why? I dunno, they made us memorize it.
ax^2 + bx + c = 0. Why? WTF is that?! Why are there letters? Why do I need to know that??
We teach math very poorly. We make people feel like they can’t understand fundamentals of the universe because it’s hard and not because we teach it in an ineffective manner that turns people off the more complex it gets, especially when you start getting to “well, you know that thing we told you never changes? Turns out we lied to young you because it’s complicated”. Not everyone is going to be great at it but virtually everyone is capable of more math then they suspect because they internalized at a young age “this is hard and I’m not good at it”
@CSK: I always thought that essay tests allowed me to capitalize on my wide ranging associative and interdisciplinary inclinations to present my sophisticated understanding of the topic.
But considering that in lit classes I probably never actually read all the way through a text until I reached graduate school (and immediately stopped reading Portrait of the Artist as a Young Man once I decided that I wasn’t going to write on it), “bullshit my way through” is probably accurate enough. Potayto/potahto.
“I always thought that essay tests allowed me to capitalize on my wide ranging associative and interdisciplinary inclinations…”
“Oh. A bullshit artist.” — Bea Arthur, History of the World, Part One.
That’s it! I have never quite figured out how to explain it, but that’s a very big part of it. You start with four possible answers. You eliminate two because they’re obviously not right, which makes your odds 50/50 on any given question. Of those two possible answers, both will be plausible, but one will feel like a riff on the other, and can be eliminated. Finally, look at patterns – is the answer likely to be ‘B’ on every question? If nothing else tips the balance pick the letter that is under-represented.
And bingo, suddenly Princeton is offering admission to a high school drop-out.
@just nutha: @CSK:
The power of bullshit. As many here will attest, I can bullshit convincingly on a wide range of topics I know nothing about. OTOH, many of the people who do theoretically know what they’re talking about are also bullshitting. And my bullshit is going to be more interesting.
There are also transferable skills. For example, I beat the shit out of a biology SAT (we used to have to take one extra SAT) despite having never taken a bio class and not being there for 11th and 12th grade. How? The above explained methods plus a grasp of Greek and Latin root words. Ta da! I coulda been a doctor.
@CSK: “As Kathy noted, most of us need only the arithmetic we learned in grade school to get by.”
But maybe… that’s because the only math we know is the arithmetic we learned in grade school, and if we understood higher math we would be getting by at a higher level.
There are things in this world I can’t begin to understand — literally, can’t even read introductory explanations — because I don’t have a good grounding in math. It’s not going to kill me that I’ll never understand how black holes work or what string theory is or, you know, why buildings don’t fall down, but I am aware it’s a deficiency, not something to brag about.
I’d like to know those things, too, but my point about mathematical expertise is that most people just need basic arithmetic to get through life. And we’re more likely to pursue advanced knowledge in the areas that interest us.
There’s also the fact that most people lack any real intellectual curiosity.
* I work with computers, and most Computer Science degrees require Calculus III these days, with many schools having a “Calculus I for non-STEM people”. As a result, people without Calculus exposure during high school sometimes find themselves too scared to even try a CS degree program, and end up doing something else. This despite the fact that very few CS fields actually need any calculus (signal processing and machine learning benefit more strongly from linear algebra, but you can make do without it for most other fields). So having calculus classes in high school helps people be less scared of being in college, but it might be better to ask if you really need either one.
* The “your course is set by 5th grade” hit us personally. My identical twins are both ADD, and their attention span wanes throughout the day. Their different schedules in 5th grade here in Virginia, with which classes were at which point in the day, meant they had different levels of success, which meant they were pigeonholed into different classes from there on. They’re now Juniors, with one ready for AP Calculus and the other not eligible for it. They’ll be evaluated completely differently for school. Colleges here look at what high schools offer and judge you by that, not by what you were offered a chance to take.
American kids rank 31st among 79 other countries in math skills and have a smaller percentage of high achievers in math.
The self-appointed mission of our public schools is to level out scholastic accomplishments to avoid esteem issues and exacerbating inequality. As always with leftists, the issue is with society. Actually the real problem is with the political goals of the educational establishment and the poor way math is taught.
Other countries teach three straight years of integrated math. Concepts of algebra, geometry, probability, statistics and data science are taught together. Math education is taught with the goal of developing complex reasoning not memorizing formulas and procedures.
The public educational system was already failing. More and more parents are dropping out in favor of home schooling. Watch out for the left to try to outlaw homeschooling.
The power of bullshit.
I prefer the phrase my Theatre History prof used: Witty Repartee 🙂
During Theatre Criticism class* I quickly learned to read Act I Scenes 1-2, skim the middle, then read the final scene of Act III.
When we started discussing the play, I’d jump right into the conversation at the start. When discussing the motivations of Hedda Gabler, I’d bring up “Ah, but that changes by the third act.” Then go quiet during Act II (just asking questions of the other students: “Why do you attribute it to that?”), and then get back into it in Act III.
I don’t think I ever read an entire play in that class.
I did, however, learn how to “compare and contrast [name two strong female characters]”.
* With a pleasant, but hard-core prof who made every student change the en-dash on page ### to an em-dash, because, dammit, it was used incorrectly!
If COVID has taught us anything, it’s that just about nobody understands exponential growth. Perhaps if more people had learned calculus…
Kevin Drum had a good, common sense, post on this subject a couple days ago. He’s talking about a CA report, but I expect it’s not much different than VA.
As with the Emma Green Atlantic piece in the Following the Science Isn’t Our Nature post, this seems significant ado about not much.
But we’re also influenced by what we believe is true of the physical world (and you need math to understand most of our theories of that), as well as by numbers (everything from finance to the statistics we constantly see thrown around, and without algebra and actually calculus you can no more understand what those numbers mean than you can understand Shakespeare without understanding English or whatever language you’ve seen it translated into).
Math is like learning how to speak and understand a language — its a pre-requisite for understanding science, medicine and the physical world (and increasingly finance and economics) in the same way that understanding basic grammar and the meanings of words is a pre-requisite for understanding the literature and history you rightly say we all should learn (and I agree that literature and history are vital studies — however so are math and science).
The point of algebra is no more being able to solve the quadratic equation than the point of literature is being able to recite one of Shakespeare’s soliloquy’s or the point of history is to remember the year Julius Caesar died. The point of all of these is that they’re keys to understanding the physical, cultural and social universes. Its like being healthy and fit even though you’re never going to be a professional athlete.
The number of humans with the intellectual candlepower to really understand black holes is tiny. I doubt you could assemble five thousand people out of the 7 billion or so population of earth. Lots of people can get it at a superficial level, and lots of people think they understand, but very, very few do, and those people would be the first to tell you that what they understand is not necessarily the final truth, but rather an understanding of the current understanding.
In any case, I don’t think cramming physics down the throats of the 99% who are never going to come to a serious understanding even of the current understanding, is useful. The average American is probably more likely to need to understand how to shoe a horse or cure a barrel stave.
<a href="@Kurtz:”>I commented on this the other day when @Doug Mataconis posted it in an open thread.
Something I would like to call attention to is the Scientific American article that Robbie Soave referenced. The link is also in OP’s quoted passage.
Years ago as a waiter, I had a family of regulars. As I got to know them, one of the kids expressed his frustration with division. I said, “It’s the opposite of multiplication, you are just undoing it.” He said, “I understand that, but I don’t get how to do it.”
@Northerner is engaging in a bit of a straw here. Even the most gifted math student is unlikely to grow into the next Einstein. It seems pretty clear that Albert wasn’t a generational talent, he was likely well beyond a 5-sigma event. But it seems to me the most plausible explanation is to say that the kid at my table is unlikely to love math, but can probably reach an adequate level of competency. If the research referenced in Scientific American is close, he is still in the stage where the approach to math instruction matters most. Indeed, even in adulthood, when genetics play a much larger role in determining reception of instruction, students can learn math.
To @Michael Reynolds‘s point, it seems to me that frustration in early math classes is likely a factor in how receptive one becomes to math as age progresses. An adult engaging in a retrospective exercise is unlikely to remember or know the source of dislike. “I’m not a computer” is simply not true. Computers don’t function at the level of a conscious entity–in my view, an emergent property of the structure of a whole brain–but they do function similarly to or the same as certain independent processes of regions of a brain.
Some young students may be able to grasp math procedures intuitively, but that hardly proves that they can’t grasp them with a different type of instruction. One thing that occurs to me is that an advanced student may be able to communicate to a less intuitively inclined student more effectively than an adult. That is an argument for keeping all students at the same level for much longer than is currently practiced. This provides a benefit to the advanced student as well–it sharpens communication skills that many scientifically inclined adults often lack.
Soave’s piece is utter trash. It doesn’t even reach the level of hot rubbish, as there isn’t enough substance in it to retain thermal energy. He mischaracterizes the content of the California proposals and actively distorts it by not including the citations found in the original when he quotes them. It’s an example of an ideologue engaging in reflexive response in a publication with an ironic name given that its purpose is to evangelize for a particular viewpoint rather than engage in systematic evaluation. It’s propaganda, pure and simple.
James, the NBA comparison is not very good here. The differential between wingspan and height and height itself are both key to playing basketball at the highest level. Genetic factors influence height (70-90%) much more than they do intelligence (60% in adults, 20% at infancy.) Also, there is some suggestion that environmental influences on height are not restricted to negative effects. Childhood diseases and nutrition are well studied, height is not. But animal studies show that exercise at an early age can increase the length of bones via an increase in bone growth hormones and testosterone. So even in a trait that is mostly determined by genetics, activity matters in outcome.
We need to stop with this silly notion that people are born with x intellectual ability.
To continue doing it is to make the same mistake Postmodernists made insisting that the mind is blank slate and that the only thing that determines outcomes is environment. If you want to be biological determinism’s version of Derrida, fine by me, but don’t be surprised if I engage in a bit of trolling at the suggestion.
But the issue is that it isn’t restricted to physics. This is a point that some of us keep trying to make to you on the topic of partisanship. Both social science and natural science are dependent on math. Black holes is a poor example, because political decisions aren’t made on them.
If we increased math competency as a whole, it would be much harder for The Heartland Institute to push false narratives about AGW. People would understand the ‘tricks’ used to produce the hockey stick graph weren’t a form of manipulation. And we would reduce the influence of AFP and Heritage to cook the books on economic policy. The point being, improving education is the only way to reduce the influence of the bullshitters in the long term.
Our minds are formed by a complex interaction of DNA, environment, free will and random chance. But DNA is a thing, and yes it gives you some abilities and deprives you of others. I’d acknowledge for example that having spent a couple years living in France as a kid and attending French schools contributed to my much later writing career. But the same exact environment might have had a negative effect for someone else. In fact it did in the case of my younger sister.
Almost everyone can keep time to music. I can’t. I mean that if I’m listening to a song, even if that song is very familiar, I cannot keep track of the beat for more than a few seconds, not even with intense focus. Effortless for most people, nearly impossible for me. That’s not because I was raised in a cave and never exposed to music. For whatever reason brain no like rhythm.
OTOH I’ll hear a false note in a story from miles away, it jumps out at me as clearly as the beat jumps out to most people. Did GOT go bad in season 8? No, it started going off the rails in season 5. I hoped they’d rescue it in 6, (nope) and by season 7 the damn thing was unsalvageable. Season 8 was doomed years before.
American hate the very idea of natural abilities, or natural limits, but they’re quite real.
@Michael Reynolds: No. You explained it just fine. What you said is pretty much what I’ve said in countless mini presentations about how to take MC tests. The only thing I leave out is the “look at patterns – is the answer likely to be ‘B’ on every question? If nothing else tips the balance pick the letter that is under-represented” part because I think that’s not reliable enough (but is as good a way to make a blind guess as any other, I’ll admit).
Likely true, at the margins, but not likely to have a major effect. It’s easy to talk about teaching math to people who don’t have an interest in math, but so far every effort has failed. Old math, new math, common core math, extra special secret math, and year after year the tests show the same levels of competence. Little Johnny still can’t do algebra.
Look at an educational initiative that should be much easier. JFK launched the whole school fitness craze with various tests and benchmarks and within mere years we had the fattest country on earth. Since then we’ve labeled foods and educated people and had millions of hours of TV devoted to haranguing people, and discovered an endless array of negatives around obesity, and the capitalists have made billions off fitness and. . . we are the fattest country on earth.
Education is not a panacea. As a nation we were much better-educated in 2016 than we were in 1860. Now compare and contrast our presidential elections in those two years. 156 years of ever-lengthening educations got us from Lincoln to Trump.
And this is why I think the CA plan is a bad one. In the name of equity and inclusion they’re recommending forcing everyone into the same mold. Different students need different learning environments–and that includes letting those who excel in something advance faster than those around them–in that area.
2 examples from my personal life:
A guy I went to school with from k-12 was an excellent artist, but in all the “remedial” classes for reading, math, etc. His passion was drawing airplanes. In HS he took the basics in math and science and did okay, but not great. The last I heard he was designing aircraft for Boeing or Lockheed. He struggled with math, but he intuitively knew what shapes were appropriate. Let someone else do the math.
A good friend of mine was the same–in all the remedial classes. But he could pull apart an engine and rebuild it with his eyes closed. He joined the army straight out of school, got injured, got a job as a security guard at a factory. When they automated the security, they moved him to maintenance and found out what we’d all known for decades: He’s a mechanical genius. Now he’s part of the R&D department developing parts for industrial fire-suppression systems and nuclear subs.
Those guys needed shop classes and practical engineering classes, not “one size fits all” math.
This video is about cancelling college debt, but it’s also relevant to the way that we’re teaching kids, and what we expect of them.
This is one of the greatest failings of the left – to think that one must merely be educated in order for them to understand an issue. After all, if you understand how bad something is, you must be willing to change it right? No sane person would let something stay in a terrible condition if they can fix it. Ignorance drives bad behavior and showing someone a better way gives them a logical choice and path to improvement.
We continually deny that we are a nation that prides ourselves on our rights to be asshats. That one can choose not to accept what they understand and be perfectly comfortable with uncomfortable facts. That “I don’t care, do U?” is normal because education do not grant one the ability to give a damn. You can’t teach people to care – you can show them why they should but you cannot get them to cross the gap if they don’t want to. Willful ignorance is a choice and increasingly it’s one that comes about from people rejecting bountiful educational instances that doesn’t fit their personal or preferred narrative.
Yes, and what we learned is usually as much about analysis of the data in the status report as it is about analysis of the verbiage in the status report.
And most people just need basic literacy to get through life.
Again, why is it that while you’re capable of understanding that arts and humanities beyond the bare minimum improves life because it’s a worth subject of study in and of itself, yet math and science is treated like chemotherapy where the goal is to be exposed to the bare minimum possible and then avoided unless absolutely necessary?
I knew I hated math by 6th grade. Hated it.
But I was also a straight A student. What to do?
Simple, when I got to high school, I took every math class as soon as I could. I finished all the math I needed to graduate high school before the start of 11th grade, with the help of two summer schools, and an after school class 1oth grade.
I was my high school valedictorian, but didn’t take any math the last two years of high school.
In college I took only was was required, and there wasn’t any advanced math for a Theatre Arts major.
That’s not what I’m arguing. I’m saying that most people get through life knowing less about math than they do about literature and history. That’s all. I didn’t say math was unimportant. I didn’t say it wasn’t worth studying.
Beyond counting and basic arithmetic operations–the point of math to the common person not destined to be a scientist is mental development. There is only so much real estate in the brain so it one engages in activities that develop that real estate–other activities using the same real estate are improved.
You want and improved citizenry able to reason and critically examine things? The classical trivium quadrivium education has always been the method to build the soft gray matter than enables that. However, the industrial mindset that has infected the culture measures worth in terms of “What does this mean for employment?” The answer can’t be nothing or its canned from the curriculum. Cursive? Well that’s not marketable to employers despite tangential benefits related to well-being.
Oh, and let’s dispel the notion that basic algebra, trig, and calculus are “advanced”. They are not. And many students hate math because many math teachers do not understand how to build repore with a students and adjust methods to their understanding.
@Michael Reynolds: Indeed it is an enduring idea on the earnest educted Left that somehow if the right information is presented, that by God, the Heathen will be converted.
Of course climate skepticism I have seen from perfectly math educated engineers and the like, so the idea having a command of maths would change political positions is a fuzzy fantasy.
I say this as a software engineer, but engineering is basically the extreme case of only caring about math and science for their utility, so if anything this demonstrates precisely why the “but do they really need it for day to day life?” approach to math and science education is a bad idea.
Well, we clearly don’t have an innate understanding of what the word “postmodernism” means.
I’m not sure why you think I’m engaging with straw — the quoted articles clearly state that they believe everyone has the same ability at math, and I’m responding to that with the observation that almost anyone who has done any mathematics (especially at undergraduate and graduate) will confirm — there are great differences in people’s ability to do math, arguably greater than the differences in ability to play basketball. For instance, twenty people of my ability could probably be competitive against Lebron in a game of basketball. But there are math problems that twenty people of my ability could not solve despite years of study and research, but which are solved by top rank mathematicians.
There is very likely a skewed Gaussian distribution in math abilities, with a very long tail towards the higher end. Most children will be close in abilities, and perhaps that’s a good target for teaching (gifted children tend to be self learners in any case, so all the school has to do in their case is not impede them). But that’s not what the article says — they say there is no difference in ability, and I find it hard to believe that they (or you) truly believe that. I learned high school and undergraduate mathematics very easily compared to most of my classmates (hence graduate degrees in physics and engineering). But at the graduate student level (actual research rather than classes) I ran into my limits and had to work hard on problems that a few gifted individuals picked up as easily as I had basic algebra and calculus. And as much as I’d like to flatter myself that these individuals were the Einstein level geniuses you refer to, it appears that none of them have won Field’s awards, let alone turned into once in a thousand year geniuses. They were simply higher on the math ability distribution than I was, and it would have been doing them (and society) a great disservice if they were held back to my level (ie if they weren’t allowed to carry on to do learning and research beyond my capabilities).
Gifted individuals should be encouraged in developing their gift. And most people are gifted in one way or another — some are gifted mathematically, some artistically, some at trades, some at sports. Pretending that everyone has the same natural talent and that anyone can become whatever they want to be with enough effort is dishonest and eventually hurts students when they find out it isn’t true. I wanted to play hockey like Bobby Orr. That didn’t work out, but luckily I was encouraged to work on what were my gifts, rather than told that I had the same talent as Orr and just had to work harder to make it.
8 + 21 + 13 + 2 + 12 + 5 + 2 + 18 + 1 + 7 + 7 + 9 + 14 + 7 + 7 + 9 + 14 + 7 = ?
Oh please, you are arguing that the only mold that matters is academic achievement measured by one metric that isn’t even all that good of one in the first place. The only thing the current and previous methods teach us is that learning math by rote and teaching it via operations on abstractions only works well for part of the mass of students.
And by the way, we already do “one size fits all” math. The only difference is that we put students in three different buckets but the methods of instruction are the same, it’s only a question of expectations and speed.
And beyond, not a single person here or ReasonBro has even tried to explain what’s so good about advanced math in high school. Does it actually help? Well, let’s see:
So your contention is that we shouldn’t hold back advanced students from on average getting a college grade five points higher? Or that we need to keep them advanced so they can take Calculus in high school just so they can . . . retake it the next year or get sent back to College Algebra?
And just in case we all get tripped up on the progressiveness of that article, here:
I guess even Forbes is joining the woke train of SJWs. Sarcasm switch off. Notice something here…Even the advanced students you’re so keen to Helen Lovejoy lose their confidence. Now apply that to elementary and early-middle school students who have struggled and their buddy who used to help them got shipped off to the advanced class.
On top of that, you ignore the argument that having more advanced students helping classmates having a harder time also improves skills for the ‘smarter’ ones. It’s not just communication skills. They’re likely to absorb a deeper understanding of the material if they have to figure out several ways to understand it.
I’m going to say the same thing I said the other day, it appears you didn’t read much of the actual documentation outlining the change. If you did, you would realize that it isn’t about connecting to minority students by changing the examples of word problems. Which, by the way illustrates the shortcomings of the current method of teaching.
Forever, it’s been: Here is a procedure. Here are a bunch of problems, do what I just did on the board. Now, here’s a situation that this procedure would be useful for…Why do you think students hate word problems? They’re contrived, sure. But it’s also that they just learned a procedure that is just a dangling abstract thing. They’re smart enough to figure out they have to use the process they were just shown and then practiced, but that does little to a.) show how and why that procedure works, and b.) prepare them for the word problems in later chapters that will require them to use a process they learned months ago.
I like you. To quote Michael Clayton, “you [have led] a rich and interesting life,” and that makes your posts a must read for me. I think you’re intelligent. But you’re stubborn and most often refuse to engage once discussions get more substantive. So either you tldr’ed the California documents or you read it with the expectation that it was bad and ignored what it presented.
None of this is to say we shouldn’t have shop class. But what you and Reynolds seem to willfully ignore is that part of the reason math skills are only superficially improving and/or declining is because we haven’t fundamentally changed the method of instruction. New Math, Integrated math, etc. never fundamentally changed the structure of courses–it’s still procedure–>practice procedure–>apply procedure to word problem. The goal of the program appears to be reversing that. And it offers an enormous amount of evidence from multiple disciplines as the basis for the change.
But sure, guys. Your personal experience and anecdotes outweigh all that research.
[I kept a quote from the same article at the bottom. It is the continuation of the above quote. I didn’t want the specter of woke language to cause further confusion for those who turn off their brains once something makes any hint of disproportionate impact on minority groups. Say it with me, disproportionate impact means it impacts everyone, it just hurts minorities more. So policies that are applied to everyone . . . keep saying it with me, now . . . improves the situation for those the previous policy has less of an impact, but an impact nonetheless.]
It is well understood by most math grad students that some professors should never be allowed anywhere close to the beginners because those professors have been away from the early stuff for so long that they’ve forgotten it can be difficult, and it shows constantly in their attitude. TA’ing for one of those profs is horrible.
As a former “intuitively inclined math student”, dear please God no. It makes me the worst possible person to try and explain. Since it just made sense to me right away, I have no ability to understand why some other child doesn’t get it. All you’re going to do is end up with two extremely frustrated children.
To use a sports analogy, it’s the reason superstar players tend to fail as coaches, where is the best coaches are former players who were decent but not amazing: because they struggled in the sport, they tend to better understand what it takes to get other struggling players to get better, whereas the superstars don’t know how to translate their “I just always got it” to people who didn’t.
Denial of a human nature is central to postmodernism as a whole. See: Chomsky vs. Foucault. Chomsky expressed frustration because Foucault refused to consider universal grammar as a model for language acquisition on the grounds that it implied an innate human nature. (On the other hand, he also differentiated private Foucault from public Foucault. But that’s a different discussion.)
Either way, it’s safe to say that biological determinism and on the other extreme, mind as a blank slate imprinted by cultural processes are dogmatic assertions that should be discarded as too extreme. Genetics matter, environments matter.
Because the article itself is a straw. James even engages in it at the end by invoking “highest” rather than “high.” Rather than sift through the article, let me provide you with a blogpost from a Mathematical Education professor from Stanford. She is cited throughout the California proposal.
Let me be clear, there is a difference between these two statements:
Some people can learn math because they have a math brain, whereas some people can only learn up to 5th grade math, because their brain isn’t suited for it.
Everyone has the capacity to learn math at a high school level, even if some grasp it at a quicker pace naturally.
That is the argument. Immediately, everyone here recoiled because it seemed to deny that some people are more talented than others. But that isn’t the thread throughout the massive proposal. Rather, it is using a passage or two, divorced from context that implies something that isn’t required to accept the proposal. It isn’t a denial of any difference, but rather an argument that the bulk of students have the capacity to learn if instructed correctly.
I’m going to repeat something I said above as well. One of the issues I see is that everyone is taking it as a given that this would hold advanced students back. Those students can be engaged within this framework, and they don’t lose out on the opportunity to take calculus. Even then, I’ve provided links that show that calculus in high school does little to prepare most of the students who take it for college math. And those links, and the California proposal highlight this by drawing from publications by college math professors.
The fact is, this is a massive 200+ page document that cannot be summarized by a blog post or an op-ed. It also cites an enormous amount of research from neuroscience, psychology, math education faculty, and math professors. It isn’t some proposal based on leftistwokeberkeleyidealistsocialists. It’s a proposal based on data gathered by professionals. It’s worth looking into.
As @Jim Brown 32 puts it:
first pass without a calculator I got 173…
Fair enough, but being able to communicate idea helps clarify and enhance education. Nobody is advocating that students become the teachers. But discussion in groups amongst themselves isn’t inherently bad. And well, it occurs in every subject.
Plus, you are n=1. Also, innately getting it isn’t necessarily getting it in the sense of a deep understanding. Regardless, group discussion is and has been a part of education since Socrates. I think you’re overapplying your own experience.
Either way, it was just an off-hand comment by me, not meant to be a definitive assertion.
Hmm…is that in support of my suggestion? A lot of people in this thread are basing their opinions on short summaries of a long document and their own experience rather than figuring out exactly what is being said in the proposal and what it is based on.
James does a much better job in this post, even though it uses fewer words of his own than the other summaries I’ve read.
But, I do think “the highest level” in the original document needs to be either struck entirely or clarified to mean, highest level of secondary education. I see why people are getting tripped up on that particular phrasing.
Yeah, that woke earnest leftist James Madison was just engaging in a fuzzy fantasy:
Go back through my posts and find where I argued it’s a panacea. However, perhaps if some of the engineers and mathematicians you reference weren’t elevated by the educational system since they were six, some would have the humility to understand that other fields require research too.
But to some extent you and Michael are correct, partisanship can ensnare really smart people.
But I’m unlikely to convince the Brit who makes the rest of his fellow subjects appear humble in comparison.
Not quite what I was getting at. Think “translation” but not in a Euclidean geometry sense.
Debates over nature vs. nurture go back to ancient Greece, so its weird to suggest that the nurture view is the province of some recent intellectual movement.
In aggregate, superstar players may fail often as coaches. But most coaches fail. Not sure if there are studies that show they fail more often and that it’s statistically significant.
There are plenty of examples of solid players becoming great coaches, but there are so many moving parts in sports–team construction, competition level within division vs. outside of it, etc–that untangling all of it is kind of a mess.
The two sports I follow closely, NFL and NBA, are good examples of inflation and deflation of records and thus coaching reputation.
In the former, scheduling imbalance due to a relatively small number of games in a season plays a large role in determining who makes the playoffs. Also, because the playoffs are decided by single games, a coach can “earn” more seasons as coach based on a small sample size. See: Dan Quinn and Doug Pederson.
In the NBA, a coach in the Eastern Conference from 1997-2006ish could hang on by virtue of making deep playoff runs playing against teams with losing records in the early rounds. Many of those top few seeds would get crushed playing a regular season Western Conference schedule.
Sports, especially at the highest levels, resists simple explanations like great players make terrible coaches.
Similarly sound education policy doesn’t lend itself to a concise plan. So what we all end up doing is reading summaries that may or may not be communicating everything the plan rests upon. Especially from a publication like Reason by a dude who seemed to look at it, and say “tl;dr. I’ll just criticize this woke-seeming language because that gets clicks. I did my job.”
As I said above, James does a much better job than the other summaries I have seen. But I think some of the language in the document needs to be altered, because it gives a different impression of goals from what the research cited and the rest of plan implies.
The irony – if by irony you mean bitter disappointment in humanity – is that for the first time in human history virtually every single person has immediate access to essentially all of codified human knowledge. The result has not been gratifying.
Boom! I really, really, reeeeeeeaaaaaallllly wish that more people (including many teachers that I have known 🙁 ) understood how extinction/unlearning curves work.
@Mister Bluster: I used a calculator, and got 163. I suspect that your answer is the correct one because my answer spells “E[rror]93” when turned upside down, whereas yours spells “ELI,” and Eli is the guy who committed the murder in a mystery I saw on TV years ago. I think it was an episode of Columbo, but I don’t remember anymore. (Maybe wr knows. 😉 )
This is my fault, I used misleading language.
Postmodernists took it much further than say, John Locke. The overall Nature vs. Nurture debate was more partiuclarized than the pomo movement in the 20th century.
Locke argued that the mind was a blank slate and that the senses provided knowledge and the rules for processing them. But Postmodernists went further than that and argued that the institutions that produce and distribute knowledge shape those processes to the point that they pre-empt empiricism.
In a broad sense, they are similar. But Locke was arguing that the mind is a blank slate, and that human nature included faculties, senses and reason, that provided the potential for human freedom. But pomo rejected the notion of knowledge as neutral and that there were any faculties at all that were not constructed by notions of difference.
So while both could be described as “blank slate,” pomo argued that empiricism itself carries with it embedded truths that are confining rather than emancipatory. Or that rather than access to the natural world, our senses and our processing of them only convey socially constructed information–no claim on objective truth can be derived outside of social forces.
@Just nutha ignint cracker: [No edit for me. 🙁 ] And my answer should be “E[rror]91” not 93.
This made me smile. For all our disagreements on pragmatic politics and differences in details–we can always share in the taste of bitterness from human potential squandered. Next time you smoke, take a toke for me, please.
Oh yes math class is political. I’ve been a math tutor for 20+ years. When people complain about Common Core math, what they’re really complaining about is their kids learning algorithms for solving math problems that they don’t understand. But if you take the time to actually learn the algorithms, like I did, you’ll see, oh this is a really intuitive way of understanding how the distributive property works. The newer algorithms are better. But many people don’t take the time to learn it, they just babble, “this teacher ain’t teachin it right durr durr!”
@Mimai: 163, says Wolfram Alpha.
@Teve: A+ for the cheating method! But not the answer.
@Mimai: it’s not cheating, it’s the real world. In the real world you can use tools.
@Teve: Point taken. Though I was hoping for a spicy Australian nickname for this particular tool.
If I may humbly ask, what are the last 4 digits for?
@restless: Transcription fail – this comments engine is a capricious bitch. Are you the ghost of Mr. Wittels?
Nope, had to look him up 🙂 Merely randomly insightful.
To the OP – long ago, in high school, I found calculus quite straightforward because I had been exposed to it early. As my father was a rocket engineer (lunar lander attitude control) he tried to teach me calculus in eighth grade when I was doing rate times time equals distance in pre-algebra.
Calculus made no sense at the time, but it must have sunk in a bit because 12th grade AP Calculus was a breeze. So, I was lucky in my parentage, and I’m all for finding ways to get more kids exposed to concepts in multiple ways. “What you tell me three times is true…”
I remember Sal Khan having some insightful ideas regarding math. He is great.
Math is math. You start with basic building blocks and add on and add on through the years. Problem is if you get say 70% on all your math tests, eventually all those 30% of math concepts you did not get over the years catch up to you and you disengage from the subject as you will hit a wall. I tend to agree with the statement that any kid could do calculus, if they are taught to master each math concept throughout school. But school is not designed to do that so you get what you get. Math is math, you cannot BS or half ass your way through it like many other subjects. I would have loved to have all the resources kids have today at their fingertips for learning and even mastering just about anything. But with everything kids can do and access for entertainment today, I would still likely be doing my math homework on the bus or before class in this generation. You could argue there are more distractions today for youth than any prior generation
I “understand” exponents, which I was exposed to in something like 8th grade math, just fine. I don’t think they’re intuitive for me, though, in the way linear progression is.
I completely agree that almost everyone can learn math at a high school (I suspect children with Down’s syndrome will be an exception — we had a class of them in my school, brought in from around the city, and they were great and very popular kids, but I suspect high school math would have been beyond them. I’m not just nitpicking here, I think children like them should be in regular schools for a variety of reasons I won’t go into here).
I also agree that gifted children won’t be held back by not having classes geared to them — they tend to be self-motivated. What I’d suggest for them (based solely on my experience) is that teachers simply leave them alone — hand them a text book for increasing levels and let them learn at their own pace. As Stormy says above, using them as tutors tends not to work, few children are gifted at that age at both some subject and in teaching.
Perhaps it is as you suggest, everyone is concentrating on an unfortunate turn of words by one of the presenters– ie saying that everyone has the ability to do math at the highest levels in the context of talking about society. I’d argue that only a few thousand people in the world have the ability to do math at the highest levels, for the same reason that only a few hundred people in the world have the ability to play basketball at the highest levels. For instance, as I said I’ve graduate degrees in math and physics, but I’m far from capable of doing math at the highest levels, and according to his own words, Einstein wasn’t either (though of course he was capable of doing physics at the highest level). That odd statement is certainly what caught my attention, as it was so obviously wrong. It’s strange that they would even say it, rather than saying the much clearer and truer statement that everyone (with a few exceptions as noted) has the ability to do high school math.
In terms of gifted children, I’ve heard it argued (by Japanese physicists) that one advantage the much easier American high schools have over the much more rigorous Japanese high schools is that the courses are in fact so easy that gifted children can explore ideas to a much greater extent, developing a creativity in the field that is damped out by the constant assignments and homework of the Japanese schools.
And interestingly enough, for the first time a vast majority of the population of the developed countries have access to adequate nutrition, and yet are less physically fit than they were in the past.
@Mimai: 8 + 21 + 13 + 2 + 12 + 5 + 2 + 18 + 1 + 7 + 7 + 9 + 14 + 7 + 7 + 9 + 14 + 7 = ?
@Zachriel: Yep, minus the last 4 digits (which were copied by mistake…I blame the comments engine). It was my mathematical characterization of this comments section.
@Pat S.: @Mu Yixiao: “I don’t think I ever read an entire play in that class.”
And somehow you think that’s something to be proud of.
Congratulations, you really showed those people who were determined to teach you something. Fooled ’em good!
@Michael Reynolds: “The average American is probably more likely to need to understand how to shoe a horse or cure a barrel stave.”
Well, then, let’s all save some money and stop teaching anything more than that to average students. Of course then we have to decide who has the opportunity for higher learning and who learns how to use the vacuums at the carwash.
Hey, here’s an idea! Let’s just assume that poor folks and minorities are simply too thick to benefit from a strong education, so they can have all the shop classes they want. Then we can save the math and science and other smarty stuff for those who really can appreciate it, the offspring of rich whites.
That kind of what your system looks like?
@Michael Reynolds: “American hate the very idea of natural abilities, or natural limits, but they’re quite real.”
Sure. But we can accept them as real without using them to define people at an early age. I don’t have a natural facility for math, but that doesn’t mean that I couldn’t have learned a lot more of it if the system weren’t designed specifically for those who do have the natural facility.
@Just nutha ignint cracker: “Eli is the guy who committed the murder in a mystery I saw on TV years ago. I think it was an episode of Columbo, but I don’t remember anymore. (Maybe wr knows. )”
Dude, SPOILER ALERT!!!
Yes, Americans hate the idea of natural limits–well, unless your Black, Brown, or a woman. /s
DNA is a real thing, yes. This is all answered above by me, by the proposal, and ironically at the link provided in Soave’s piece. The current state of research on DNA’s influence on intelligence shows that it’s about 50%.
Most importantly, it shows that the genetic influence on intelligence is low early in life. It tops out at 40% before adulthood.
Again, nobody is claiming that we can produce generations of mathematicians that push the profession forward. The claim is that all students can learn foundational math concepts that we teach in public schools.
It’s not politicians making these claims. It’s geneticists, neuroscientists, mathematicians, and professors of education. You’re not arguing against a concept of idealism you see in my arguments–you’re arguing against them.
Yeah, I’m fighting against people who don’t appear to be willing to look at what’s actually being proposed and the basis for it. I don’t fault them. I’ve read through way more of those documents than I wanted to. More importantly, I explored the citations quite a bit too. And it makes sense based on current research.
But this is part of the problem for me in a lot of these discussions, because I find that no matter how much nuance I include and how many times I make very clear distinctions about what I am and what I am not saying, I get responses that ignore all of that in favor of pre-existing beliefs and personal experience.
I thought of Steven’s recent posts about comparative social policy. I pointed this out earlier in the thread without citing his post. One of the problems with the op-ed he discussed was it included no weighing of the risks of change vs. the problems of the status quo. No one currently involved in this discussion pushed back against that, yet in this thread they engage in the same class of poor argumentation.
Indeed, the rush to calculus has been detrimental to the majority of students people fear a policy change would hurt. The most predictive stat for success in college calculus classes is good grades in high school algebra.
I was caught off guard by the quote highlighted by James. Because I had read enough of the rest of it to have formed a solid grasp of the basic argument being made. And none of it implies what can be read by looking at the phrase “highest level.” That’s one of the downsides to political documents. It’s easy to get caught up in language designed for political persuasion and focus on that rather than the substantive arguments being made.
That’s a very good point, and makes me wonder why they even added that phrase. I can’t think of a single advantage to it (as opposed to just saying “high school math”), whereas the disadvantage is as shown in this thread — its such a ridiculous phrase that, unless someone takes the time as you did to read the whole document, it leaves the impression that the writer had no idea what they were talking about and they and their suggestions should be ignored. Sometimes I can understand why someone might overstate their case (that is, I can see whose support they’re trying to gain), but in this case I can’t imagine that anyone supports the idea that everyone can do math at the highest level (anymore than everyone could play basketball at the highest level), so why say it?
Agreed. It’s not like many people are going to read it all. So any benefit they gain from the warm, open political consultant language is more than outweighed by the way the proposal will be consumed–via media who are likely to summarize based on the intro or filter it through their ideogical lensing.
Thanks for your insight.
Mimai, shouldn’t there be a space between the e and the b?
Teve, so much that!
Also thanks to Kurtz for actually reading through the proposal and cites that most of us don’t have time to read through.
What you have said comports with other studies I have been reading, with my classroom experience, and with the direction that education professors are talking about moving now.
Making math, science, etc relevant to the students, at least anecdotally, increases engagement and so increases learning.
I have certainly noticed a difference in learning outcomes when I’m able to make an abstract concept like wave propagation more relevant to kids here by tying it back to surf and where the storm waves are generated.
@Grewgills: Hey, don’t go shoulding all over my math (psychology joke). I think a space is acceptable but not essential. In this case, it makes for an uglier equation, so I left it out.
@Grewgills: shouldn’t there be a space between the e and the b?