Retired statistics professor Paul Alper writes:
Here is a flap you might want to blog about. I just found out about in today’s Washington Post, January 21, 2022, although it is dated April 26, 2021, before GOP Youngkin was elected—indeed, before he was the nominee. The news article is called Virginia Not Moving to Eliminate Advanced Math Classes, and it says:
[Virginia] 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.
“The main thing I think the mathematics team is talking about at this time is, ‘How can we make sure that students have more skills in those mathematical areas that will help them after graduation?’ ” [Superintendent James] Lane said. “Every job in the future is going to need more focus on data.”
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.
Of course, there is opposition:
[Loudoun County School Board member Ian Serotkin] 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 always, more heat than light:
Republican gubernatorial candidate Glenn Youngkin said that, if elected, he would fire everyone involved with the proposal. Another Republican gubernatorial candidate, Del. Kirk Cox (Colonial Heights), tweeted that stopping the Pathways Initiative would form part of his “7-part plan [to] fight the radical left head on.”
“It’s time to put a stop to the left-wing takeover of public education in Virginia,” he tweeted.
So, is the contemplated change of the math curriculum worthwhile in this supposedly modern era and how deeply is it a sinister [redundant because “sinister” etymologically relates to left] left-wing takeover? Put another way, how extensive is the right-wing paranoia justified?
My reply:
The first step is to fix whatever aspect of math teaching led to that Williams College math professor endorsing the bogus election fraud claim using stupid null hypothesis significance testing.
Oops . . . I forgot! Statistics is trivial so any math professor who’s been assigned to teach a probability or statistics course is automatically a statistics expert!
More seriously, I don’t know anything about what’s going on in Virginia, but in general I’m supportive of more parent involvement in curricula. Parents are not the experts, but it seems like a good idea for schools to be able to justify their curricula to parents.
As a teacher, textbook writer, parent, and former student, I have the impression that much of education at all levels is about doing what was done before or doing what is convenient to teachers and school administrators. I’ve seen lots of incompetent teachers who keep doing the same empty thing. There’s no easy solution here—it’s not like there’s some untapped reservoir of great teachers who will do the job for the pay offered—but I like the idea that accountability to parents is an expectation, just to break the closed loop.
Regarding the Virginia thing, I like the plan to incorporate stronger foundations in data analytics: at least, it sounds good in general terms. I wouldn’t think that lots of parents would object to that, would they? Regarding the plan to eliminate all math acceleration prior to 11th grade: maybe they could have a test each year and allow students to jump ahead where they can? Otherwise, yeah, lots of boredom going on, in the same way that Spanish-fluent kids in elementary schools have to spend an excruciating hour or more a week, learning the numbers from 1 to 10, the colors, etc., year after year after year.
Finally, I’m unhappy about education reform getting politicized, but I guess it’s unavoidable; it’s been happening in one form or another my entire lifetime.
Great comments Andrew. I think schools have to offer good salaries to get top-notch instruction. In addition, some academic has to develop good math curricula.
I think a problem is that “accountability to parents” can mean a lot of different things, and some of them are deeply corrosive. For example, grade inflation in k12 is to no small extent a parental activism phenomenon. There’s also the “how much parental accountability” question. School boards exist and are often elected. Is that enough? I don’t know. I could write thousands of words about what is wrong with k12 ed, suffice it to say that partisans of most political persuasions would find things to like and things to hate in that list. “Dumbing down curriculum in the name of equity is bad” is one of those that tends to upset leftier types. “Parental involvement in many places is already excessive” is similar for the right and some centrists. “Most academic administration is a disaster” unites us all though. ;)
Republican gubernatorial candidate Glenn Youngkin became the Republican gubernatorial nominee Glenn Youngkin who became Republican Governor Glenn Youngkin. Both he and his Republican opponent, Kirk Cox, made headway with the electorate, not because of altering or opposing the math curriculum changes.
Their bête noire was in a different part of the academic sphere, “critical race theory,” a subject with a powerful resonance. So powerful that upon taking office, Youngkin issued
Executive Order #1 calls to restore excellence in education by ending the use of divisive concepts, including Critical Race Theory, in public education.
Executive Order #2 empower Virginia parents in their children’s education and upbringing by allowing parents to make decisions on whether their child wears a mask in school.
Executive Directive #2 to restore individual freedoms and personal privacy by rescinding the vaccine mandate for all state employees.
There were other orders and directives but none about math or statistics.
> There were other orders and directives but none about math or statistics.
I’m probably biased by my current job to think back to the math/science aspects, but a lot has gone on in my elementary school system since I’ve left.
* There’s now a large sports complex beside my old high school. It’s like a tourist thing where they have tournaments and whatnot — I think other schools/cities are doing it as well.
* We’re many generations into the laptops-for-kids/teachers thing. There’ve been laptops on loan, laptops for students, Google Drives and whatnot, yada yada
* It’s my understanding that all the anti-critical-race-theory legislation passed and is in action in my home state at least (all sorts of weird rules and red tape on what can be taught and processes for exceptions).
* Security stuff continues to go up — stuff like tinted windows and security booths. I think we had a school cop when I was a kid.
* All sorts of immigration stuff — legislation, ESL classes — lots of political tie-ins on these of course
* There was a pedo teacher and the principal there got removed by the school board after local protests (apparently this teacher had a history before joining the school — anyway this seems like an example of the local school-board-electorate-feedback thing doing something)
* There are whole new schools in the system!
When I went to school cell phones were just kicking in. I suspect the dynamic there has been relatively stable (if kid uses phone in class -> phone goes in teacher’s desk until X sorta thing)
(1) “fields including data science and data analytics” in practice means mindless “learning” of statistics via thoughtless application of rules, or bizarre computer-science-without-high school computer science classes that I wouldn’t have believed could exist.
(2) I don’t know about Virginia, but there’s definitely a truly depressing increase in people who believe that offering advanced math classes is somehow unjust. See e.g. https://scottaaronson.blog/?p=6146
Raghu:
1. Yeah, it’s really hard to know what to think about this. It seems that there could be good well-designed courses for the average middle schooler or high schooler in all sorts of math-related topics, including probability, statistics, computer programming, algebra, and geometry. But in real life it seems that there are approximately zero such courses—which makes me suspect there’s something about all this that I don’t understand. Just for example, I don’t see much rationale for the topics taught in the geometry class: all sorts of things about angles and similar triangles that I don’t think ever get used again, kinda like learning a foreign language that you’ll never be called upon to speak. As for the statistics curriculum: all those lessons about the mean and the median seem really to be contrary to the spirit of statistics, which is to learn from data, not to throw away information. Maybe one reason I can’t get so worked up and angry about proposed curriculum changes is that I’m not convinced that the existing curriculum is so wonderful—despite many decades of careful thought by experts in math, science, and education to make it better. Of course it’s possible to take any system and make it worse, so I’m not saying I would endorse every proposed change. I’m just saying that I’m not sure how to think about, for example, the replacement of a bad geometry class by a bad statistics class.
2. I followed the link, and . . . I get their point, but I found it off-putting somehow. Maybe it was all the bold face, maybe it was the mention of “Fields Medalists, Turing Award winners, MacArthur Fellows, and Nobel laureates”—that kind of thing always makes me suspicious. The open letter itself seems reasonable; there’s just something about the framing at that link that makes me bristle.
One of my favorite things about high school math were the competitions. I guess it was the day off school aspect? I’m not sure. I remember one we went to and it was a big bowl shaped arena thing you walked down into (like a college basketball court) and everyone got their tests and filled them out. It was very intense.
We always got dusted at these things but it was good times (I don’t really remember ever worrying to look at the scores too much lol).
Somehow the ACT and SAT tests weren’t nearly as fun (and even if our school we did tons of practice tests for those — I think we were taking practice versions yearly since late elementary school). Standardized elementary school tests were pretty bleh too. I don’t remember getting excited for them.
Ben:
I enjoyed high school math competitions too. Beyond the social aspects (camaraderie with my friends on the math team, the fun of getting in the car with them and going to the competitions, etc.), math team was good because it pushed me, and it helped me learn my limitations. A bad thing was that it gave a distorted view of what math is (see discussions here, here, and here). On the other hand, had we not had math team in high school, it’s not like then I would’ve had a better idea of what math was all about. I’d say that math team was an unambiguously good thing, at least for me.
“I’m not sure how to think about, for example, the replacement of a bad geometry class by a bad statistics class.” — I agree that a lot of what’s taught as geometry is archaic and could be replaced by other things, but it has a few virtues: (i) it introduces ideas of proofs and logical thinking in a visual way, as Dave noted below, and (ii) unlike statistics, the teaching of which seems often to *negatively* impact students’ understanding of the subject, poor geometry teaching is at least not harmful! I’m being a bit flippant, of course, but not too much…
About (2): I don’t really like the framing, either, but I suppose they’re trying to impress people who are impressed by such things.
Raghu –
Thanks for the link – it led to a lot of interesting material.
>… offering advanced math classes is somehow unjust.
Although I have been a big fan of the Algebra Project for many years (b/c, imo, they implement a sound pedagogical), I do think it’s a legitimately complicated issue and maybe one you should take time to understand the issues from different side?
As someone who paid close attention to the Virginia Pathways stuff (I’m a parent in Virginia), the initial reaction was driven by draft language from the Department of Education that would have disallowed taking advanced courses until 11th grade. As in other states (particularly Oregon and California) this was driven by “equity” concerns. The response from the department education was to say that nothing had been finalized and that this was only a draft. Yet in this presentation from the department of ed (https://www.youtube.com/watch?app=desktop&v=siS8jlTcUzo&ab_channel=VirginiaDepartmentofEducation), they are proposing heterogeneous groupings of students (instead of ability based grouping) all the way through grade 10 with traditional advanced courses (pre-calc, calculus, etc.) being reserved for grade 11 and 12. With that understanding, it does seem that saying “Virginia is getting rid of advanced math until grade 11” is kind of spot on. There are definitely many things that can be improved about math education (and the framework presented seems like not a bad start), but it’s a bit disingenuous to say that they are not planning to do what they are in fact planning to do.
From the open letter:
Such frameworks aim to reduce achievement gaps by limiting the availability of advanced mathematical courses to middle schoolers and beginning high schoolers.
I’m not entirely unsympathetic to good faith arguments in favor of tracking, but that kind of bad faith framing does no one any good, imo – and anyone who doesn’t see how/why that’s a bad faith framing should study the issued in more depth.
How is that bad faith arguing. That’s the explicit reasoning behind the move. Leveling down is how most people who are working for “equity” justify their arguments. Same thing for getting rid of entrance exams for magnet schools (like Thomas Jefferson) gifted programs. Instead of arguing for extra resources to bring struggling student groups up to par, they want to just get rid of things that “marginalized groups” don’t currently perform well at. As Ibram Kendi argues (regarding standardized testing) in his book: if there is a disparate outcome, it’s racist and has be gotten rid of.
Like I said, if you don’t k own why it isn’t bad faith you should sidu the issue more. It’s a implicated issue, and it would be hard to discuss reasonably comprehensively in blog comments. But simply on face value – portraying such a complicated issue so simplistic ally should be highly suspect.
But in short – the objective isn’t to equalize outcomes by holding down some students from learning more. That’s a ridiculous and obviously bad faith premise. Disagreeing, pedagogically, with the likely benefits of “de-tracking” doesn’t necessitate (nor justify) viewing it through such an an absurd framing.
And enlarging the discussion to Kendi only displays further that you would benefit from reading more. The rationale for “detracking” (or for discontinuing standardized testing) existed for decades prior to anyone ever hearing his name.
Again, disagreeing on the merits of the different views on these subjects doesn’t require such simplistic, bad-faith characterizations, and I’d suggest neither will dealing with these issues benefit from such an approach – although I can’t really rule out that a zero sum engagement might prove beneficial for those on one particular side as opposed to the other.
I’m always dubious of the rhetorical tactic of “If you think x, then you need to do more research because I’m not going to explain why you’re wrong.”
From my reading the main finding is that tracking benefits higher achieving students and potentially harms lower achieving students (with mixed results), with detracking benefitting low performing students at the expense of high performing students. Few studies (if any) of good quality have shown benefits to higher achieving students.
So sometimes we can read between the lines (much like we do with Republican rhetoric on “election security”). In the case of detracking, the equalizing of outcomes is achieved, in part, by holding higher achieving students back.
I’m honestly surprised that’s not obvious. An extremely talented basketball player benefits from playing with people of his/her own abilities or above. If that player is relegated to playing with people far below his/her skill level, that player’s development (as a b-ball player) will suffer.
I enlarged the discussion with Kendi because he represents the zeitgeist of current equity reasoning. I don’t think one can separate equity discussions regarding detracking from other equity discussions.
JFA –
> I’m always dubious of the rhetorical tactic of “If you think x, then you need to do more research because I’m not going to explain why you’re wrong.”
Sure, as a general rule that’s a bad form of argument. But again, having been intimately involved in these issues for a long time – I would say if you aren’t aware of a larger scope of the arguments made here in favor of “detracking” then a logical explanation is that you haven’t looked at the issues in much depth.
> From my reading the main finding is that..,
So then my question is how much reading have you done on this? It’s an incredibly complex topic, that ranges across pedagogy, epistemology, cognitive science, educational philosophy, educational psychology, politics…just to start with. I think that if you think that there is a “the” “main finding” you’re either dealing with the literature selectively or you haven’t dealt with it comprehensively. I don’t object to people having strong opinions one way or the other but anyone dealing with this issue in good faith has to recognize that the empirical findings are (1) pretty much all over the place and, (2) placed within a context where empirical study is incredibly complicated and of limited value – given the wide range of outcome variables for which control of confounding variables is immensely complex.
> Few studies (if any) of good quality have shown benefits to higher achieving students.
First of all, there are the caveats about any of the “good quality” studies. Second, your determination there is necessarily a function of your definition of “benefits.” You certainly have a right to define that as you wish, but you should be aware of how conclusions on this topic are, at least to some extent, a function of which definition you choose. Thus, to just embed one particular definition to generalize a global statement of outcome is analytically problematic.
> So sometimes we can read between the lines (much like we do with Republican rhetoric on “election security”). In the case of detracking, the equalizing of outcomes is achieved, in part, by holding higher achieving students back.
See above. But further, that statement makes all kinds of assumptions about methodology and pedagogy.
> I’m honestly surprised that’s not obvious.
Again, I refer back to my question of just how familiar you are with the range of this topic? I think if someone is familiar with this topic in-depth, it shouldn’t be surprising that simplistic, generic, and widely generalized conclusions aren’t “obvious.”
> An extremely talented basketball player benefits from playing with people of his/her own abilities or above. If that player is relegated to playing with people far below his/her skill level, that player’s development (as a b-ball player) will suffer.
I’d suggest that using that analogy isn’t particularly useful in context.
> I enlarged the discussion with Kendi because he represents the zeitgeist of current equity reasoning.
That issue aside, again, your impression of the zeitfeist of “current reasoning” is based on a superficial view of the issues involved.
> I don’t think one can separate equity discussions regarding detracking from other equity discussions.
In that I agree. I do think they’re necessarily linked. Theres a lot of overlap. But they aren’t close to being fully congruent sets.
The proposed change had two parts: increased emphasis on data, and removal of acceleration. My reading of the educational literature is that, while many purported aptitude-treatment interactions (such as “visual vs. auditory learning”) are minor or nonexistent, there is one big one that is real, the fact that higher-IQ students do better when they go faster and lower-IQ students do worse. This is especially true in math. (And it is probably even more true if we substitute “math ability” for “IQ”.) Thus, allowing students to progress through the math curriculum at different rates is a way to optimize learning. My experience as a student and parent suggests that the status-quo is to allow some acceleration but not enough. Yet the amount is limited by “educational infrastructure”. It helps an advanced middle-schooler if there is a high-school across the street, but the last years of high school are usually difficult.
Perhaps I was being excessively subtle when I threw in the business about the views regarding how or how not to alter a mathematics curriculum in Virginia. Obvious to me that the real hot-button issue for galvanizing an electorate was to offer up red meat in the form of the so-called “critical race theory.”
But, what initially piqued my interest was that the Washington Post article of January 21, 2022 was a reposting of its article of April 26, 2021. What was the purpose of doing that? How often do such things happen? What am I missing? Was there a subtle ulterior motive? Was there an unsubtle ulterior motive? A Bezos directive? A secret Bezos directive?
It looks like there are two, almost completely separate issues at stake here
1. Adding data analysis and computer based courses
2. Restricting access to accelerated curriculum
Looking at this from a frictionless vacuum, and ignoring intent, I don’t see a reason to do number 2. More options is usually better.
Adding in some pragmatics, if data courses have to come at the cost of early acceleration for want of resources or teachers or whatever, I’d say that adding data analysis courses is obviously more important than shitty AP calculus, and so I’d say both of these changes are a net good on the utilitarian merits of it.
Considering intent, fixing educational inequities by restricting advanced students seems pretty stupid to me. Unless convinced that the content of education is irrelevant to outcomes and the whole thing is really just a set of arbitrary hurdles to sort students and gatekeep high society. Honestly, that’s probably true about K-12 education, but I’d rather we all try to make that not the case than accept it.
Getting snarky with it, it is kind of amusing that people are now willing to defend to the death shitty AP calculus courses because other people mentioned correcting inequities by removing them. Removing them for that reason is stupid, but that doesn’t mean removing them is necessarily a bad idea. I wonder if Democrats wanted to remove lessons on the dewey decimal system and snail mail writing, but cited inequities as a reason, then Republicans would then reflexively defend to the death writing snail mail as vital curricula.
My genuine opinion as a successful victim of these courses, is that pre-Calculus and AP calculus is a waste of your time. You should have the option to take them if you want, but you’re just gonna have to take them again, but for real this time, in college.
I was so bored in Algebra 2 that I taught myself Calculus using the book Quick Calculus… I then had to take Calc AB because it was all they had in my high school… I breezed it and then got a 5 on the AP test. This let me skip to Calc 3 (multivariable) my first semester in college.
So I think it is possible to skip those classes in college but I don’t think it’s the ideal situation.
my peers in high school were actually taking college math classes at the Junior college nearby. I didn’t have access to that for various reasons, transportation being one of them. But what I realized by the time I left high school was that the truly ideal situation is to get the hell out of high school by end of 10th grade and do 4 yes at a junior college, which is in fact often a great resource for the motivated student. Once you’ve had 4 years of junior college you can have tried out lots of classes and will be in a great position to transfer to a 4 year college with a major chosen and mainly higher level classes to take… And much more breadth than the usual path. You’ll come out of the 4yr college with dramatically less debt as well.
My impression is that the perceived value in the standard math curriculum comes from the idea that it enables people to “solve problems” in some way that they couldn’t without that education. I guess I agree with this in a broad sense, but “solving problems” is a weighty phrase. To solve a problem first requires recognizing and framing a situation as something that can be understood systematically. In other words, you have to build a model of a situation. Then a “solution” comes from finding a way to intervene on that model in order to yield a desired outcome and ALSO being able to realize—and critically evaluate—that intervention in real life, as opposed to just in the model.
You can use calculus and differential equations to build models of certain kinds of systems and then figure out, analytically, how the model will behave under different interventions. Historically, without computers, we were stuck with that basic type of “problem solving”, though even then the math curriculum didn’t really approach it this way. Little effort was/is spent in how you go from reality to model and back again. Indeed, I suspect that even high-performing students in either grade school or college don’t even view the math they learned as a way to model any aspect of reality, no matter how simple. And even for those who do appreciate that, the analytical tools provided by calculus are very limited in the complexity of the systems they can represent, making them much less useful when dealing with either complex systems (living things, societies, etc.) or even just dealing with “simple” systems in more detail (building a bridge, hitting a baseball, etc.). There’s a reason that engineers don’t just model things with math, but with physical models too.
This is all to say that if a greater focus on “data analytics” meant a greater focus on how you go from a collection of observations (“data”) to a formal model to a realized intervention, then I think that would be a big advance. It would capture the aspects of “problem solving” that the current curriculum doesn’t approach well. But as Raghu says above, this is often not the case when people talk about “data science” education—it amounts to replacing one mindless system with another.
“Indeed, I suspect that even high-performing students in either grade school or college don’t even view the math they learned as a way to model any aspect of reality, no matter how simple”
Agree that it requires maturity to apply knowledge to everyday problems.
So simple rules: What is the problem (frame), observe the entire field (count), sum like with like and then scale the observations, only then analyse and conclude. That comes from diff equations although some may posses that view innately. Lack of that ability in daily life comes across strongly in obvious biases. Calculus informs how to make good-enough approximations. High school math deals with learning some techniques needed for above but that is not explained at all. Except for sorting it is as useful as Greek.
This is a fascinating discussion. I find it difficult to come down on any particular side, because my experience tells me that HS math overall (with some exceptions obviously) is terrible, even counterproductive for many students.
I came to this through teaching for two decades at an open admissions college, and before that a range of institutions from CC to elite. Whatever my ostensible topic, I always taught math and stats, initially forced into it but eventually by design. What I observed was (a) widespread, and I mean really widespread, math anxiety, even among students who excelled in other areas of the curriculum, (b) an inability to apply ordinary reasoning to math/quantitative problems, and (c) a handful of students who came adequately prepared and had insufficient opportunity to apply their skills because of their classmates. This is on HS.
I know there is a vast scholarly literature on teaching K-12 math, and I have read almost none of it. Bearing that in mind, here are my impressions:
1. There is a lack of connection between the ordinary reasoning kids apply to problems in their own world or even academic problems in fields like history and lab or field sciences and the reasoning they ought to apply to math problems. When most of them bump up against something they don’t already know or can immediately see a path to, they shut down. This must be a product of faulty pedagogy along the way.
2. The kinds of students who grow up to be math teachers are different from most of their peers. They excel at deductive reasoning, and when they get their jobs they assume everyone else should be like them. But most students are primarily inductive.
3. There seems to be a culture in K-12 (and perhaps later) math instruction premised on the belief that there are two types of students, those who have what it takes to do math well and those who don’t. Despite protestations to the contrary, much of the assessment process is about identifying who’s in which group. This could be the source of a lot of the math anxiety I’ve seen, where students are either ashamed that they lack this “math gene” or are defiant that, yes, they lack it but it’s unfair to demand that they should have it.
4. Math instruction is way too isolated from substantive topics that ought to be in dialogue with it. How many HS’s are using the opportunity of the pandemic, for instance, to do a dive into exponential functions, the use of logarithms, etc.? You could have a whole generation that really got this stuff.
So this informs my reaction to the Virginia business. I’m all for customization of learning to benefit differences in interest and native skill, and that could include advanced tracks in various fields, but not if it exacerbates the pathology of sorting. And the first call on resources has to go to fixing what doesn’t work for the vast majority. As for the shift to data, well yes and at last. But as most of us agree, it can’t just replicate the errors of the past. There has to be a basis in critical reasoning, engagement with uncertainty and difficulties in measurement and a two-way engagement with a range of subject knowledge.
And when so few students are served well, “equity” tends to merge with overall reform.
Peter –
There is a LOT in your comment that I agree with strongly, and which aligns with my own experiences, but I do want to comment on this:
> …(c) a handful of students who came adequately prepared and had insufficient opportunity to apply their skills because of their classmates.
I would suggest there is an assumption there which is not well-grounded. Consider that the problem was pedagogical and methodological, and related to resources, training, and the like. Necessarily finding casualty in “classmates” might be conflating the symptoms with the disease, so to speak.
I would also quibble with your point #2…but I think it’s more that I have my own version of describing a similar structural issue (i.e, teachers having a limited ability to understand, experientially, the difficulties many of their students encounter).
Peter –
> (a) widespread, and I mean really widespread, math anxiety, even among students who excelled in other areas of the curriculum, (b) an inability to apply ordinary reasoning to math/quantitative problems, and
Some thoughts on this…with my theorizing to help make it a little less black box-like, and related to developmental psychology.
I think a lot of this starts for many students around 12ish or so. If you think Piaget was on to something, that would be around the time when a lot of kids begin formal operational thinking. Of course, the age of transition into particular developmental stages varies across the individual for a variety of reasons.
Anyway, if you accept a basic premise that intellectual development often follows along a basic developmental path of a necessary foundational concrete understanding preceding abstract or theoretical understanding….
So at around 12 or so, kids begin to be expected to process a lot of mathematical understanding at a more abstract or conceptual level. But if particular kids haven’t reached a developmental stage of formal operations, or don’t have a sufficient understanding and experience at a more concrete level – and then are expected to do more abstracted reasoning and to produce correct answers, and further are sorted into categories on that basis, then problems crop up.
The kids who aren’t ready to start with the more abstract reasoning processes, for whatever reason, start to get a message of deficiency. Learning becomes less about the topic at hand, and more about performing, getting evaluated by a teacher, and then getting sorted into a group.
This pattern is exacerbated by a couple of factors. One, is that when you get to things like Algebra, it becomes more difficult for teachers to help students to see the linkages between the concrete and the abstract. Teachers used to be able to talk with younger kids about cutting pizzas into slice or adding apples together, to help students link to abstracted algorithms for addition or division. But now, making those linkages explicit becomes inherently more difficult (this is where curricula like that used by the Algebra Project comes into play).
So in addition to running up against an obstacle purely on the students’ end, there’s also a roadblock on the teachers’ send. A tendency when confronted with this complicating factor is to construct curricula to just have students memorize algorithms and then be judged on their performance on tasks that effectively just measure their ability to memorize those algorithms. One unfortunate side-effect from this is that some students who really want to understand concepts deeply – to know “why” you cross out the seven and carry the one – or who aren’t inherently inclined to just follow instructions – are the ones who wind up feeling the most alienated.
So then kids begin to personalize a deficiency in the curriculum or a limitation in the teacher’s skills, and reach the conclusion that they aren’t “good at math” – merely because they’re particularly inquisitive, or because they are being asked to perform abstracted operations before they’ve mastered the concrete underpinnings or really become comfortable with formal operational thinking. Once they start to look at the learning task in that fashion, it’s hard for them to get past it. Math instruction tends to take on the message that they are deficient, and thus math (and failure to perform on math tasks) is something best avoided.
I had an interesting window into this process when I trained on using Montessori materials for math instruction. A particularly useful example were the materials used to teach kids how to square and cube binomials and trinomials. It was fascinating to watch my fellow adults, who went through the process of transitioning from using squares and rectangles and cubes and rectangular parallelepipeds when working at a concrete level, to learning how abstract and then use formulas to square and cube binomials and trinomials. It wasn’t at all uncommon to see adults, some of whom were teaching math, saying that now they actually understood the abstracted algorithms the had been expected to just memorize as kids.
Of course, the further up the chain you go in math instruction, the harder it is to help students follow the linkages from the concrete to the abstract. But hopefully, if you’ve used a richer curriculum at lower levels then students are less likely to have embedded a message that they “can’t do math.”
> who really want to understand concepts deeply – to know “why” you cross out the seven and carry the one
Well, I don’t think it’s necessarily bad to think about different types of math as little games where you memorize. I think that’s a more honest characterization (and I assume there’s reasons why we call one game algebra, and one game arithmetic, and one game calculus…).
I think it’s part of the Proofs and Refutations introduction that as much as mathematics is presented as some clean axioms -> theories -> more theories construction etc. etc. — that’s not really how it was constructed or understood during the construction and so probably isn’t such a great framework to understand it.
Maybe some here would find this interesting:
In this short note, I argue that focusing on the role of simulation to learn about possible worlds (represented/mediated by probability models) enables making better sense of what happens in the actual world (observations in hand). Statistics just being cast as the scientific process laid out in diagrammatic form and animated. In many ways, statistics is just a formalization of more common everyday methods of reflection. However many including scientists find it difficult to fully grasp what to make of statistics in their work. This in part may be due to the use of probability models that are abstract objects or mathematical entities that seem divorced from observations in hand. The mathematics can seem to take on a life of its own disconnected to observations. Nothing is further from the truth in that probability models implicitly represent (or misrepresent), in an idealized sense, how the observations came to be.
A common everyday reflection is to consider if some action is beneficial or not. This starts with an assessment of was it beneficial this time. Now, just because something seemed to work this time does not ensure it will work next time. In many contexts, what to expect to happen next time is more important that what did happen this time. Especially if next times seem inevitable (e.g. next time up to bat). Most important reflections are future oriented. The dead past is behind us.
Statistics tries to formalize this kind of reflection regarding future possible observations. For instance, in a clinical trail trial where there was an observed benefit for the treatment, should we expect observed benefits would happen in future trials? Scientific statistics uses probability models to do these reflections about the future. On the other hand, statistics focused on the recent past might be best termed accounting statistics – simply summarizing what happened.
Science in general is thinking, observing and then making sense of thinking and observing for some particular future oriented purpose. Making sense here can be formalized in assessments of what would repeatedly happen in some reality or possible world. If it would repeatedly happen, a habit of either an organism, community or physical object/process, it is real. “A habit extends beyond any single act, encompassing tendencies to act in distinctive ways across relevantly similar situations, unless those tendencies are actively inhibited.” [C Legg] Hence, they have some degree of uncertainty and are future oriented. “habits replicate the formal structure of knowledge” [C Legg] which is what scientific statistics tries to learn and quantify from observations.
The major problem with K-12 education is the content. Until principles-based school mathematics is being taught, fiddling with the course structure won’t make much difference. For more information, see
“To Accelerate, or Not”
https://math.berkeley.edu/~wu/Huffington_Post_op-ed.pdf
“Learnable and unlearnable school mathematics”
https://math.berkeley.edu/~wu/AE2020A.pdf
David:
I was curious so I clicked on that second link of yours, and I couldn’t make sense of it. I guess it’s just too abstract, maybe too much of how a pure mathematician thinks for my taste. For example, it says:
I get the “arbitrary” thing—any base other than 2 could be considered arbitrary, I guess—but “unreasonable”? It seems very reasonable to me that in the number 432, the 4 stands for 400, the 3 stands for 30, and the 2 stands for 2. What’s wrong with that??
It’s just as arbitrary and unreasonable in base 2. For that particular example it’s arguably even more so: the digit 1 in the number 110110000 stands at the same time for 100000000, for 10000000, for 100000 and for 10000.
Base 2 is special because it’s the smallest positive integer for which place value is possible. I think there is something less arbitrary about choosing that compared to choosing say base 60 like the sumerians
> The first obstacle that children face in learning about whole numbers is the concept of place value. Children are taught, for example, that the digit 4 in the number 432 stands for 400, the digit 3 stands for 30, and the digit 2 stands for 2 itself. It is well-known that many children have difficultly learning this convention because they consider it arbitrary and unreasonable, and rightly so.
Actually, I would disagee. Children should be taught that the digit 4 stands for four hundreds, the 3 stands for three tens, and the 2 stands for two ones (or units). It might seem like nit-picking, but imo that’s an important element of teaching place value.
Andrew,
I’m afraid you need to try to think like a child in elementary school. I suspect most people think something like Roman numerals are more reasonable. Of course, there is a reason we use place value rather than Roman numerals. But, I doubt most people could tell you why.
I recall when I was in elementary school writing out the numbers in order when I was at home. I recall it because I did it wrong. At some point, I started adding another zero at the end every time I incremented a digit.
You could read Wu’s six books. They are textbooks for K-12 math teachers. Then compare them with what math teachers are currently taught, and with what they teach their students. They are remarkable books: It isn’t easy to take math and present it in a grade-appropriate way.
David:
I’m not denying that learning place value for numbers can be difficult. I just don’t see how Wu can go around calling place value unreasonable. Our number system is very reasonable, and it seems unreasonable for Wu to say otherwise. Again, this just gives me the impression that he’s coming from some super-abstract ivory tower.
But, sure, his books might be great; it could just be that in the linked article he’s engaging in hyperbole. Hyperbole can be fun sometimes; I just don’t really see the point in this particular case.
He wrote that students rightly consider it unreasonable because the reason is not explained to them. You are misquoting him.
> maybe too much of how a pure mathematician thinks for my taste
> he’s coming from some super-abstract ivory tower.
I wonder what characteristic of how a pure mathematician thinks you are referring to. I also wonder how you define “ivory tower”. Here is Professor’s Wu’s resume as of 2011:
http://ecolereferences.blogspot.com/2011/06/hung-hsi-wu-papers-about-mathematics.html
Is that what an “ivory tower” is?
>I’m afraid you need to try to think like a child in elementary school
>You could read Wu’s six books … They are remarkable books …
>
Oh boy, I just had a flashback to Evans Hall.
A view from the Student Section …
https://web.archive.org/web/20061025011720/http://www.jiggscasey.com/slappy/book_of_wu.html
You should talk to children and try to understand why some of them are not willing to go along with place value. You should also talk to elementary teachers and see how much trouble they have teaching this.
Look at history: Fibonacci introduced the Hindu-Arabic system to European math around 1200, but merchants knew about this system since around 1000. Yet this system did not replace Roman numerals until at least 1500. Before the Hindu-Arabic system, there were the Babylonian base-60 system, Egyptian hieroglyphic system, and the Greek Alphabet system, not to mention the Roman system. None use only 10 symbols or place value. The Chinese have used the Hindu-Arabic number system from day 1 (since ~1000 B.C.?) though not the 10 symbols of the West, of course.
The most natural system is essentially the Greek Alphabet system: If a new large number comes up, use a new symbol.
David:
Again, I can accept that place value is difficult and nonintuitive for many kids. But that’s not what Wu is saying. He’s saying that place value is “unreasonable.” I disagree. I think it’s very reasonable. I’ll buy your statement that the Greek system is “more natural,” but that just makes place value less natural; it’s more abstract. I don’t get why this would make it “unreasonable.”
Andrew –
> He’s saying that place value is “unreasonable.”
It may or may not be relevant, but that isn’t exacgly what he said in the quote you provided – where he said that kids consider it unreasonable. Maybe reading it in context clarifies, and it’s not clear to me exactly what that does mean – but in that quote he didn’t exactly say he considers place value unreasonable.
I left this out above…the “and rightly so” could mean that children consider it unreasonable because of aspects of the manner in which (or when) it’s taught.
Again, I don’t know exactly what it means to assign “unreasonable” to place value but I think that place value does in a sense seem “unreasonable” to many kids as it’s taught to them – as becomes evident given that many kids don’t gain a solid understanding of place value.
I happen to think that’ outcome is a function of how it’s often taught (as an abstract concept, almost like an algorithm).
Andrew: I’m sure the time that you’ve spent arguing about one word could have been better spent.
David:
Who are you kidding? All of us have better things to do than respond to blog comments! The fact that we’re here in this discussion at all is revealing our desperate efforts at procrastination.
Not really a contribution to this ongoing discussion, but a really interesting take on place values; https://www.explodingdots.org/
Thanks for this! I’m having my son watch some of these, he’s incredibly bored with the math he’s getting in school and finds this guy entertaining. About 1 minute into the first video he exclaimed “He’s doing base 2, that’s cool!” So it’s at least giving him some enjoyment.
Joshua:
Wu says, that children “consider it arbitrary and unreasonable, and rightly so.” The “rightly so” implies that Wu agrees that place value is unreasonable. I think that’s an unreasonable position on Wu’s part, or at the very least a position for which he offers zero justification!
I’m going to speak more to general, but related, trends I’ve seen in high school curriculum debates, and hopefully this doesn’t come off as a strawman:
People have discussed replacing traditional math classes like geometry and calculus with statistics and computer science in high schools for awhile now. I’ve long thought this would be a mistake, and that stats and CS would be better placed as high school science electives. For instance, an inclined student could do biology, chemistry, physics, and stats or CS for science courses, with algebra II + trig, geometry, pre-calc, and calculus for math. I think of high school as providing students with foundational knowledge they’ll need for college and teaching them how to learn, rather than direct training for knowledge economy jobs. And for those who are not college-bound (still a large portion of high schoolers), it’s unlikely they will be professional statisticians. But at least they’ll have an option to take a stats class elective if they’d like.
Students often learn how to do proofs in geometry, where they tend to be more intuitive because of the visual/spatial aspect of the subject, which is important for more advanced math topics in college and beyond (including stats and CS). Second, the foundations of statistics rely on calculus, so it’s hard to argue that statistics is more important than calculus. Calculus is the foundation of a lot of important topics in math and science.
If they want more data analysis in high school, they could work more of it into the various science classes. Provide more empirical case studies into the curriculum and teach the basic statistical concepts as needed. Things like mean, median, variance, and histograms don’t require a full class. Even linear regression could be taught as part of an empirical case study.
Regarding advanced classes, I can tell you as someone who entered a new school system for high school and was initially placed in the regular algebra II class as a freshman, it was painful. I was finishing the tests in 15 minutes, correcting the wording on exam questions while taking them, and was just generally not challenged until they moved me up sophomore year. This was at a relatively good school, and I still consider the honors geometry class I took as a sophomore to be one of the best classes I’ve taken at any level. Teachers need to teach to the middle of the class, so lumping in advanced students with ones that need more help doesn’t really benefit either group. That’s true in any subject. When I was in early elementary school, I was in the low reading group. I don’t think I would’ve benefited from being lumped in with the more advanced readers, because I needed patience and frequent assistance that the more advanced early readers didn’t need.
well said
Math != Stats. If they want to teach this, they should either make it its own standalone subject or thread it throughout the curriculum. The only sincere justification for cutting core subjects is limited resources. Algebra ii isn’t stopping kids from getting quant jobs.
Business leaders are fond of saying what they really need the schools to teach is critical thinking skills. I think one way of interpreting that is to make what goes on inside a student’s head look like what goes on inside an information scientist’s. But that’s really resource intensive, and hard to measure with a standardized test. I guess school leaders think the affordable version of that is to make what students do on their computer screens look like data analyst’s screens. Call it the cargo cult curriculum!
One thing I have not seen discussed relative to the California math proposal is the likely effect on schools. Parents who disagree with the program and have the means to do so, will increasingly send their children to private or charter schools. Parents without the means to do so will have no choice. I would suggest that those most harmed will be gifted math students from low to middle income families (and yes I do believe there is such as gifted math students, just as I believe, as a long-time coach, that there are gifts athletes)
> Finally, I’m unhappy about education reform getting politicized, but I guess it’s unavoidable; it’s been happening in one form or another my entire lifetime.
Yah. As I think is pretty obvious from this comment thread, it’s an inherently politicized topic.
Somewhat late to the party here, but as a 30-year high school math and statistics teacher I have some thoughts. One of the problems in revising secondary math curricula – with parent involvement – is that what parents primarily care about is what college their children will get into. So the pressure is always to prepare them for the SAT, ACT, or AP Classes. No one actually cares about whether or how they learn math, just what kind of test scores they’ll ultimately get. We’ve developed some interesting courses in our department, for example a Discrete Math course which covers things like graph theory, game theory, risk analysis (think that was useful these last 2 years?), social choice, emergent behavior, etc. Think we can get students to take it? Ha.
This makes me so sad, because I’m one of those tiny minority of parents who really does mainly care about what my kids learn. Its so hard to get schools to care because they’ve been taught by lots of experience that what you say is true. The pandemic has put this in sharp perspective. I have my kids in the online independent study program, we’ve been appalled at how it focuses on punching tickets and checking boxes rather than teaching what the child needs.