Boosters are given to improve immune response so the idea is that you improve the immune response so vaccinated people generate more antibodies against an active infection and you get less breakthrough cases.
My recollection is that the showdown aspect is something new in the sense that it has happened only under a Democrat President and started in Obama's presidency.
Do you mean "shutdown?" The first shutdown occurred in 1980 under the Carter administration. There's been 20-ish shutdowns of varying degrees since then.
Shutdowns have occurred due to no appropriation bill being passed before the previous one expires. Raising the debt ceiling is a different thing. It used to be a routine thing to raise the ceiling because the monies have been appropriated. It was only under Republicans in the Senate trying to sabotage Obama that it became an issue.
This is one reason why it will be hard to get rid of the silly requirement to take shoes off. DHS has a financial interest in keeping the theater going.
Not just financial interest, but also getting a free pass to make people do what authorities want without having to convince them, and to crack down on anyone they don't like for any reason.
Rules like this that don't really have a point and are applied unevenly (like drug laws, many traffic laws, public intoxication...) are just a lazy way to make sure authorities can make anyone or any group they want "fall in line" while not actually preventing favored individuals or groups from doing what they want. It's the most unfair and unequal kind of regulation, and in general people cant6get enough of it, precisely because they know it will be used mostly against groups they don't like, but they dont have to explicitly identify those groups or otherwise acknowledge their prejudice.
In Minneapolis they want to dismantle the existing police force and replace it with another. A big reason for wanting to do this is to get rid of the problem cops. Police in the U.S. do a horrible job in solving even major crimes. They currently commit a large number of acts of violence without accountability. They system definitely needs to be dismantled and rebuilt.
> In Minneapolis they want to dismantle the existing police force and replace it with another. A big reason for wanting to do this is to get rid of the problem cops.
So throw out the whole bunch because of a few bad apples? And what happens when the replacements have problem cops? Do it over again?
Five police officers in Miami recently attacked two men while thirteen of their fellow officers watched. [1] The idiom "bad apples" comes from the saying, "A few bad apples spoil the whole bunch," so it's an appropriate term here.
And only charged with misdemeanor "battery" without any enhancements. Had the perps not been police it would be multiple felonies + enhancements for things like gang activity and firearm possession. Looks to me like the DA does not want to upset the police and their unions and is only doing the very minimum to make an impression that he is working.
"Bad apples" should lose their jobs before more shit happens under their watch. Instead, they become a symptom of something bigger: the police is unable to police itself.
Presumably the so called good apples will get hired by the newly formed police force. Though I would argue there are very few good apples. You aren’t a good cop when you allow and fail to report on your fellow officers meting out non judicial punishments.
There are lots of things stopping the DFL from doing what it wants with the police force. I suggest you look into how police forces hold cities hostage when city government acts in a way they don’t approve of.
He didn't. By stating that he has now way of verifying he most likely meant that he exhaust all the ways, within his ability, to verify it. While the other poster made no effort to verify if Jacques is not a bot (it's easy). There's a difference in actively verifying given statements and passively accepting all that is written or all doubts that a busy mind can produce.
I think you may have missed it a bit too. jacquesm wrote:
...I have no way of verifying it then I don't believe them. Worked pretty good so far.
This can't possibly be true since one can't verify everything. If one truly lived this then they'd be in an endless quest of verifying things. For instance, verifying verifiers.
We are about to enter into an age of hard to detect AI generated articles and research so this isn't philosophical silliness.
Thanks for your rant! It was wells stated. I did not know anything of these things in the sense of having any academic knowledge about this stuff.
I didn't know about the counting one and am not surprised by this. In a thread on another post I wrote that there is a lot of brain washing that occurs in the teaching of low level mathematics. It's easier to brain wash children in this sense than it is for adults.
I don’t give a definition. The students know real numbers are the “decimal numbers”. They don’t need to know a definition at this point and giving one more precise than “decimal numbers” will just confuse them. They gain familiarity by practice in the same way a Fahrenheit person like myself can only get intuition with Celsius temperatures is by using them.
I think it’s clear you’ve never taught low level mathematics courses. There is a lot of hand waving and brain washing that happens. The vast majority of people don’t know what a number is in a precise, mathematical sense. At the level of the intended audience it would be wholly inappropriate talk about the definition of a number.
My background on this topic is that I’ve taught intermediate algebra for over 20 years.
Assuming you already know what a rational number is, the next step is to tell you what a real number is. A real number is defined as the equivalence class of all sequences of rational numbers that converge to the same value. For example, every sequence of rational numbers that gets arbitrarily close to the square root of two as you go to higher terms is considered "the square root of two."
If you don't know what a rational number is, it's the equivalence class of every pair of integers that can be simplified to the same fraction. For example, (4,6) and (2,3) are both rational numbers, and in fact are the same rational number: two thirds.
If you don't know what an integer is, it's the natural numbers, but with negative numbers.
If you don't know what a natural number is, it's either zero, or a number that follows a natural number. For example one is the number that follows the natural number zero, and two is the number that follows the natural number that is the natural number that follows zero.
Instead of assuming an understanding of what natural numbers are, you could have continued to define all of them as equivalence classes, as that is what they are.
The integers are the equivalence classes of differences of natural numbers, while the natural numbers are the equivalence classes of finite sets having the same number of elements (i.e. which may have a bijection between themselves), including the empty set.
If you look closely, I didn't assume natural numbers, I stopped at Peano axioms. Stopping at set theory would have been reasonable but it would have added a line at the end about sets that wasn't describing numbers per se.
In order to decide that two sets have the same number of elements (for this relationship various names have been used, e.g. equipotence, equipollence, equinumerosity), you do not need numbers or being able to count.
You just need to be able to show an one-to-one correspondence between the elements of the two sets. If an one-to-one correspondence cannot exist, then the sets have different numbers of elements.
This relationship divides then the sets in equivalence classes. If you choose a representative of each equivalence class that you use to compare to other sets to see if they have the same number of elements and you give a name to each of those representatives, you have defined the so-called natural numbers.
This is actually how the numbers originated, for humans and for many other animals.
Nobody conceived a system of axioms and then thought about what could satisfy them. That came much later and is useful only for establishing which are the essential properties of some mathematical objects. Most of the definitions of various mathematical objects as equivalence classes correspond to their real historical origin, because recognizing that some things are equivalent according to some criterion is how abstract concepts are created based on concrete things.
When you see a red apple and a red rose, you understand that they have a common property, being red, and then you name this property "red" and you can recognize the same property in other objects.
When you see 5 sheep and 5 crows, you understand that these groups have a common property, having 5 members, and the same property characterizes the set of fingers of your hand. You name this property "five" and when you see another group of things you can compare it with the set of fingers of your hand to see if it also has 5 members.
you can show two sets have the same number of elements without having an intrinsic notion of "number" - find a bijection between them, mapping every member of set A to set B and vice versa, and you know you have two identically-sized sets without doing any counting.
First define natural numbers as sysops did (or you can use peanos axioms).
Then add the negative numbers (I actuallly don't remember how this is done, ig it's usually hand waved as trivial). The negative and natural numbers together make up the integers.
The rationals are introduced as a pair of numbers (a, b) where a is an integer and b is a positive integer. (a, b) is considered the same rational as (c, d) if a * d = b * c.
The reals are finally introduced as sets of rationals with a certain property, namely that if p is in S, then all smaller rationals must also be in S. Edit: there are a few more properties, see https://en.wikipedia.org/wiki/Dedekind_cut
The book Numbers [1] by Ebbinghaus et. al. is a comprehensive treatment, starting from natural numbers up to complex numbers (and quaternions iirc). It also has lots of historic context, how our number systems evolved.
I'll preface this by saying that I got bored and didn't finish it (Axioms? Rubbish, where's my field theory etc.) but Terence Tao's book on Algebra seemed like a somewhat gentle and very thoughtful introduction to the subject. Not necessarily easy by any means but it looks like he has put a lot of work into the pedagogy (whereas some mathematicians just shit out theorem and proof onto the page with no regard whatsoever for the prose, justification or flow - but I (am forced to) digest)
Numbers start with counting. You can build up more kinds of numbers, but tearing down counting into smaller pieces is surprisingly hard. Personally, I think it's good enough to start with any two distinguishable states. You can then keep combining those states in various ways to form the counting numbers. Note that more complex symbols like "1 2 3" are themselves effectively a form of tally, where you are counting the number of angles. You can combine your method with geometry or a clock and call it "measurement" of space and time, respectively.
(Real numbers are my favorite construction because I was enamored with Cantor's diagonal argument when I first learned it. It's quite clever and hints at the magic mathematicians are capable of. Although most mathematicians (algebra peeps) seem to like the classic "root 2 is irrational" proof more.)
I think that set theory and other analytic can be instructive, but can also obscure what's happening. The exact encoding of numbers using sets is just an "implementation detail", in the sense that there's many ways you can build natural numbers (for example) using set theory and they are all equivalent.
So it's like learning data structures by coding in assembly, which is what Donald Knuth thinks is the right thing to do anyway, but some other teachers would disagree. But if you want to see some high level construction, you could look to eg. Tarski's synthetic construction of reals
Which uses integers rather than sets as the building block, and is simpler than many constructions. And, of course integers themselves can be constructed out of sets, but they can be constructed out of lambda calculus terms as well https://en.wikipedia.org/wiki/Lambda_calculus#Encoding_datat... among many other constructions - but when we finally define integers, we can abstract away the implementation details (and that's really the crux of the question!)
It depends on your mathematical background. I don’t know of any books about this at the level below senior undergraduate mathematics. If you are familiar with sets I can outline the idea behind how to define non negative integers.
We assume the empty set exists and call this 0. We define 1 to be the set containing 0. So 1 = {0}. We define 2 to be the set containing 0 and 1. So 2 = {0, 1}.
Let’s look at this set: {a, b}. I know this set has size 2 and not 1 because I can map {a, b} to {0, 1} in a one-to-fashion. I can’t map {a, b} to {0} in a one-to-one fashion. We say any set has size 2 if it can be mapped to {0, 1} in a one-to-one fashion.
Sounds like a philosophical question. Mathematicians define axiomatisations and definitions that attempt to characterise in a rigorous way our intuition of numbers. But they don't tell us what numbers are.
I don't have any specific resources to recommend, however I'll give my take on the foundations of 'numbers'.
We use the label 'number' to refer to a broad swathe of mathematical objects, objects that are different but also so similar they often appear interchangeable (for example counting numbers and fractions).
In the formal mathematical sense, a specific type of number is a group of objects which have been defined to have specific properties. I'm going to leave object and property as defined in the usual sense, I think most people have a good idea about what those are and not sure I can add anything to them.
There are no rules as to what properties you are allowed to give to objects, nor what group of objects you want to include, but generally if you are learning about some specific thing it's because people find them useful or interesting; the definitions we have for different types of numbers are the ones we have found useful or interesting.
Remember that the different types of number appear very similar. It is common to build a 'hierarchy' of differnt types of numbers, where we start with a simple type of number and then add new properties and objects when we find limitations we don't want.
The first type of number in this hierarchy are the natural numbers (sometimes called counting numbers). The most common properties defined for these today are called the Peano axioms [0]. There are quite a few of them, and the history of how we came to the formalisation is very interesting (a lot of it is about avoiding inconsistencies/contradictions) but the key ideas are:
- there is a natural number called 0
- every natural number has a successor, which is also a natural number - we can write S(n) is the successor of n
Most of the other axioms define what it means for two natural numbers to be equal (=).
Just having these objects isn't particularly useful, we typically want to do things like add, multiply, and compare numbers. To do that we include some operations: addition (+), multiplication (*), and total ordering(<=).
These are defined as, taking a, b, c as natural numbers:
- a + 0 = a
- a + S(b) = S(a + b) (this is recursive, so if we define 1 as 1:=S(0) then we have 1+1 = 1+S(0) = S(1+0) = S(1))
- a * 0 = 0
- a * S(b) = a + (a * b)
- a <= b if (and only if) there exists some c such that a + c = b
Importantly, using these definitions, we can say that the natural numbers are closed under addition and multiplication; whenever you add or multiply two natural numbers together you get another natural number.
To continue building the hierarchy we notice that there are operations we would like to do but are not possible for every natural number (please note I am skipping over the formalisations from hereon and talking about the motivation for different types of numbers).
We notice that if we can add two numbers together we should be able to subtract them again. If a + b = c, then c - b = a. However (for example) 0 - 1 is not a natural number. So we extend the natural numbers to the integers, such that the integers are closed under subtraction.
If we can multiply it makes sense to try and divide, but 2 / 3 is not an integer so we extend integers to the rationals (ratios of integers) which are closed under division. We add an object called 2/3 so that now when we can say 2 / 3 = 2/3.
The next step in the hierarchy is a bit more complex. We notice that we can define a subset of rational numbers that all meet a certain criteria, for example all rational numbers that are less than 2. We call 2 an upper bound of that subset. Notice that 2 is a rational number, and that there are no rational numbers smaller than 2 that are also an upper bound of our subset - 2 is the least upper bound. Define a new subset, where we say a rational number x is in the subset if x * x < 2. We can easily see that 2 is an upper bound for this set, but so is the rational number 1.5, and 1.42, and 1.415. In fact, there is no least upper bound for this set that is a rational number. We extend the rational numbers to include a least upper bound for every subset of rationals, and we call this the real numbers. The real numbers have a lot of nice properties, most notably they are complete under the normal ordering, which essentially means that there are no gaps.
The reals don't have everything though! We notice that we can create polynomial equations, like x * x - 1 = 0 and that sometimes these can be solved (in this case x = 1 or x = -1 solves the equation) and in other cases they can't. For example, there are no real numbers that are the solution to the equation x * x + 1 = 0. We can extend the real numbers to the complex numbers by adding a new object called i, which has the property i * i = -1. A complex number has the form a + b * i, where a and b are real numbers. The complex numbers are called algebraically closed, and there is a really nice result that shows that all polynomials have solutions in the complex numbers.
The hierarchy actually keeps going, but hopefully you can see that numbers are just objects with properties that behave in useful and interesting ways under different operations. The formal definitions we have today have been refined over a long period of time to avoid contradictions and other issues, but there is nothing stopping you from making up your own numbers with their own properties. If they are useful or interesting other people will probably use them too!
I don't necessarily disagree with your point that for the given audience it's not appropriate to rigorously define the different sets of numbers.
However, I absolutely detest it when teachers just "sweep it under the rug", when they pretend that they just provided a definition when they evidently did not.
Like the commenter you replied to, this sort of stuff genuinely threw me off in high school and made me feel like I didn't understand mathematics.
You are right, I didn't teach low level math courses, but this brain washing is also precisely why I didn't understand math in high school. You cannot argue with this kind of definitions. Everything feels as if it was randomly defined by the teacher. This "intuition" simplifies teaching, but makes understanding harder.
It is like a game where you invent rules as you play. No student can win this game.
Here’s the definition of 2 using the standard construction with the Peano axioms. It’s the set containing 0 and 1. The number 1 is the set containing 0 and 0 exists by one of the axioms. It’s not something a person in intermediate algebra can understand. For one, the natural question then is, “what is a set?”. Whatever one does there has to be some brain washing in order to get started. This is unavoidable unless one thinks Principia Mathematica should be the starting point.
Well, the peano arithmetic can be described directly as first order logic without set theory ;)
I'm fine with having an intuition for sets, but I think reals really should be defined properly. At least, R should not be confused with the algebraic closure of Q.
The algebraic closure of the rationals is not the reals. The reals are the completion of the rationals using the standard metric.
The intention of my post was to point out the complexity of not brain washing students at a low level. Your comments have enhanced my point by bring up considerations I didn’t want to get into!
This volume does not confuse R as the algebraic completion of Q. It is completely reasonable for an Algebra 1/2 teacher to wait for a Calculus or Analysis teacher to discuss the metric completion of Q. Describing R as rational + irrational numbers is a completely solid description.
I think by the definition given at the beginning of the thread irrational numbers are numbers that can’t be expressed as a ratio of two natural numbers. i is irrational by that definition, but not a real.
Just to clarify: there are lots of numbers that aren't real numbers (for example, imaginary numbers). Intuitively the real numbers are all the points along the number line, including rationals and irrationals (such as root 2, pi, or e). There are lots of other comments in this thread that give a good explanation of how that works formally.
If the students haven't yet encountered complex numbers, infinitesimals, infinities etc. then it's perfectly reasonable to say that all numbers are assumed to be real (as follows strictly from the definition in the book).
i or some hyper real numbers are neither irrational nor rational.
Basically all crazy extensions of R used to solve even more crazy problems. They don't really exist naturally (and you can make up your own field extension of R as you wish), but they are handy if you want to compute stuff.
But by definition there are no real numbers that are neither rational or irrational.
You "win at the game" by learning what is taught, getting the "A", and then doing you own in-depth research about what interests you on your own time.
K-12 was, of course, invented by the Germans in order to create good little factory workers that would get up early and work all day and not complain too much. The fact we still use the word Kindergarten is a nod to this origin story.
They weren't at all interested in the students gaining any "understanding" and most certainly not in them "winning" in any sense of the word.
> Everything feels as if it was randomly defined by the teacher.
I suppose you prefer things randomly defined by Euclid? Just kidding... kinda. Seriously though, randomly defining things and then working through the consequences of that definition is a totally valid way to do math. Those random definitions are called postulates.
One of the standard construction of the real numbers is the set of equivalence classes of rational Cauchy sequences. This definition is equivalent of the handwavey definition above (irrationals are the Cauchy sequences that don’t converge to a rational number and the reals are the rationals plus the irrationals.)
However almost any construction of the real numbers is challenging to give a simple explanation for. Even leading 19th century mathematicians didn’t truly understand the real numbers until Cantor.
A problem is that lots of lower-education math instructors don't understand these concepts deeply themselves. I think it could be okay if these things were clearly framed as "true for the problems we're looking at, but not universal", but they were typically presented as universal by teachers who themselves don't know any better, and that really caught me up too.
Over the next year, a biosciences working group revised the program’s funding guidelines, stipulating in the final draft that it would not support any research into the first-order effects of genes on behavior or social outcomes.
The fact that this area is controversial suggests to me that it is worth exploration.
I’ve taught mathematics at community colleges for over 20 years and I’m absolutely convinced that not all people can learn algebra or calculus. To me it is obvious this is so since the mentally disabled can’t. There’s a level of “intelligence” that’s necessary to learn a given topic. Not everyone can learn all topics.
This belief of mine is considered heretical amongst leftist colleagues (I’m a liberal myself on almost every issue). As a college we act as if everyone can pass. Years of administration telling the math department that our passing rate is too low have led me to pretty much pass everyone who takes the final exam. Last semester 80% passed but only around 50% deserved to.
You're making a lot of leaps of logic here, which might be why your colleagues are disagreeing with you. You go from "not everyone can learn algebra or calculus" and "mental disabilities exist" to "therefore intelligence is a scale and controlled by genetics and unalterable," which doesn't really track.
Your previous education, your upbringing, your cultural values, these things all also have huge effects on your aptitudes, and you've just dismissed them out of hand, apparently in favour of pre-determined genetic intelligence. I mean, all I know of you is this comment, so I could easily be missing a lot more context about this argument you've had, but it sounds like you have an axe to grind, not a carefully-considered conclusion.
I think the more people can learn algebra than OP implies, but as a former math educator my experience also indicates some people are genetically limited in mathematics. For reference, my specialty was working with very remedial students and only ever had a few students not make progress (~2%) but I do think that small percent was genuinely hopeless and I don't say that lightly. I have a pet theory as to why those students could not do algebra that that you might find convincing:
To start, I believe that the idea of "working memory" is largely valid. Think of it as the number of distinct ideas you can hold in your head at once, sort of like trying to hold a phone number in your head when you've just heard it for the first time.
The general consensus in psych is that this number for the average person is in the single digits and is relatively static in adulthood till a decline in old age. It's been my observation that people with really incredibly small working memory cannot do algebra. The amount of numbers/ideas held in their head is too large, and multiple students in this group described the experience of attempting an algebra problem as feeling like sand constantly slipping through their fingers.
Many of these students grew up in rich neighborhoods with good parents. They had most advantages you can imagine, seem reasonably intelligent when you talk to them, but Algebra will always be beyond them.
My dad was a math and physics teacher and he had a very similar theory around the working memory. However he also thought that the problem for those students that had difficulties with math was that the working memory was often filled up with other stuff than the math that they were working on. It could be anything from difficulties to concentrate to problems with holding the pen correctly and therefore the working memory was overflowed.
This describes my experience with ADHD except your head is filled with impulsive thoughts (noise), so you struggle to string together a bunch of numbers in a logical forward progression through a formula, which also induces cognitive load when recalling the formula to your working memory when it is overwritten by the impulsive thoughts.
I can get around this by brute forcing the numbers into the formulas on paper and going through each step slowly but surely, even steps that people can do mentally. I just need significantly more time than my peers to finish.
Taking stimulants and increasing my dopamine levels across my synapses alleviates much of the working memory deficits I outlined.
I wonder how many of these students that struggle were just undiagnosed ADHD and not deficient in an low IQ mental handicap sense.
> I wonder how many of these students that struggle were just undiagnosed ADHD and not deficient in an low IQ mental handicap sense.
I've got ADHD that was undiagnosed until a decade after finishing my degree, and maths was my favourite - and best - subject all the way from age 4 through to doing solutions to Einstein's equations involving 10+ A4 pages (both sides) of tensor mechanics for my final year project. My results across that time showed the more maths in a subject the better I did, the more writing it had the worse I did.
Doing maths is like reading is for me - an external cognitive structure that I can follow to make my own brain calm down while in that process. While I don't do much maths nowadays, I literally read whenever I'm not actively doing something else - I read a page or two of my book in-between clicking reply to your comment and starting typing this reply.
Maths involves a lot of "muscle memory" once you get past the initial hump. But it's often poorly taught at an early age to the level of inducing near-phobic levels of discomfort with it which to me seems strange. But I don't think there's any extra issue with having ADHD and maths
ADHD is very comorbid with dyslexia, dyspraxia, dysgraphia and dyscalculia - around 20–60% of people with ADHD also have one or more learning difficulties. Dyscalculia affects as many people as dyslexia, but dyslexia is far more well-known and more likely to get diagnosed and helped with. There's a nice summary box of typical symptoms of dyscalculia in this paper:
> To start, I believe that the idea of "working memory" is largely valid. Think of it as the number of distinct ideas you can hold in your head at once, sort of like trying to hold a phone number in your head when you've just heard it for the first time.
Interestingly, testing myself with the Wechsler reverse-digit-span test, I found very large improvements in working memory from taking 20-minute afternoon naps and from modafinil. I'm not claiming everyone is alike, of course, but working memory is definitely not as fixed as height and eye color.
Specifically with regard to mental math, I find I can do a lot more when I'm lying in bed in a dark room than when I'm in an uncomfortable chair in a noisy cafe with people talking to me. Or, for that matter, classroom.
> Many of these students grew up in rich neighborhoods with good parents. They had most advantages you can imagine, seem reasonably intelligent when you talk to them, but Algebra will always be beyond them.
I tried marijuana once and found the opposite effect: when I was high, by the time I got to the end of saying a sentence, I couldn't remember how it had begun. But people report that I seemed like I was conversing normally; if they didn't know me, they wouldn't have realized anything was off. I wonder if these folks were experiencing something similar all their lives?
> I found very large improvements in working memory from taking 20-minute afternoon naps
Sure. My ability to do anything also drops to near zero if I don't sleep enough. Yet I have not found unlimited increases in capability if sleep more and more...
> modafinil
Yeah, drugs are a quite different beast. Doping happens when individual athlete realizes they have hit the limits what their "natural" biology can do and yet still are not going to win the competition.
Yeah, the drawback of modafinil is that it seems to reduce my ability to recall things from long-term memory. Also, it makes it hard to switch tasks when it's necessary.
I don't think I'm suffering from sleep deprivation. Certainly it is true that if you take enough stimulants your performance will worsen! In fact, if you take enough stimulants you'll die and your performance will be zilch!
> To start, I believe that the idea of "working memory" is largely valid.
Wait, are there mainstream schools of thought where working memory isn't considered valid? I'm a layman but I suffer from ADHD, and very much notice that my working memory fluctuates with my attention span (from lack of sleep/stress/etc).
Google's of no help to me, but I remember a story of some educators, looking at some kids who didnt go to school but worked selling concessions, but in turn, were actually quite good at math. They couldn't answer math questions when written out on a worksheet, but they could do the exact same questions when presented in the form of a complex order. (Double digit multiplication and summation isn't the same as algebra, but being able to do that implies a large working memory, which is claimed is the barrier to learning algebra.)
I do have poor working memory and I did struggle with algebra, especially with copying wrong sign from row to row and things like that.
But I did make paper sort of my working memory and when doing algebra I felt that I was just the very resource limited CPU that executed the instructions from paper-memory.
Algebra always made me feel like I was doing some mindfull exercise where I had to empty my mind, follow the paper script and hope I didn't mess anything while switching from row to row of calculations.
Even today, as a programmer, I struggle to remember class or function names, I just empty my mind and am really good at searching stuff in code.
I read this paragraph in a paper on ADHD and learning disabilities literally after replying to a comment above yours...
> There is also evidence of domain-specific cognitive deficits that contribute to specific learning-related disabilities. For example, phonological processing difficulties have been found in children with poor reading performance, whether or not they also exhibited problems with ADHD symptoms or math, but not in children with deficits in ADHD symptoms or math only. Similarly, both with and without a reading deficit, children with ADHD symptoms exhibit significantly impaired object naming and behavioral inhibition, and math-disabled groups demonstrate visuospatial and numerical processing deficits, while those with only reading problems sometimes do not.
There's significant co-morbidity between learning disorders and ADHD, and despite dyscalculia being as common as dyslexia (~3-7% of the population) it's a lot less well-known, isn't as frequently diagnosed and there are fewer tools to help people with it. It would be very possible to have both ADHA and dyscalculia, given you used symptoms listed for both that are almost word-for-word identical with those in the paper...
I had no problem with reading or abstract thinking or grasping mathematical concepts.
My problem was that oftentimes I would screw up calculations for a math problem and only get 7/10 score on it (the reasoning was mostly right, but the answer was not).
Well, I’m certain cockroaches can’t learn calculus. Neither can apes or monkeys. Some humans can. Therefore I think that brain composition has something to do with learning calculus. There is variation in said brain composition amongst humans. It seems reasonable to think that the set of all people is not a subset of all things that can learn calculus.
I haven’t performed any genetics tests but brain composition is partly determined by genetics so genetics might play role, right?
I always wondered if the teachers understood how much effort people put in. Especially in something like math, where you can't just waffle.
I got the feeling as a kid that a lot of my classmates would just give up. I mean they're sitting there acting like they're concentrating on math, but they aren't. In other classes intermittent attention is enough, you pick up social cues and repeated stories that you've heard, and poof you have a history essay. With math there's a need to have all the steps.
How do you control for that as a teacher observing the kids? How do you know whether a kid has actually tried to learn stuff at home?
I still remember my brother was baffled at how I got top grades in math when he never saw me studying. Of course I tended to do that when he wasn't around.
> How do you control for that as a teacher observing the kids?
The only solution I ever found was not having more than 3 students at a time, not an option for most teachers; I spent most of my career working with 1-3 students as a result. With that few students you can carefully observe the mistakes students are making and ask individual questions about their mental state. Experience will eventually tell you to differentiate students not putting in effort from students who are so lost that they're just flailing and hoping something sticks.
With more students, my experience was that both my attention became too split to give students the kind of careful diagnosis to control for effort.
I didn’t imply anything about the number of people who can’t learn algebra other than that I think the number of such people exceeds the number of intellectually disabled people. I used the word intelligence and put this in quotes because intelligence is not really definable but the word does convey a sense of what I mean. One must have a certain level of cognitive ability to learn something. Where that level is for a given topic I can’t say.
You've offered some evidence as to why you believe some individuals are incapable of learning algebra, but you haven't offered an explanation as to why what you observed is necessarily the result of genetics, there could be dozens of additional factors at play here which are not being considered.
HN crowd doesn't think twice about people who try to run DOOM on a stack of pennies, but are ready to give up before trying to get algebra to run on someone with less working memory. It's weird how different the attitude is.
Have you ever been a teacher? Even a teaching assistant?
I have. Some people are dumb.
You try teaching these people (anecdotally they're less than 1% of the population, with no obvious markers, so you have to actually find one first) and then come back and tell me how it's like "getting algebra to run on someone with less working memory". If I wanted to continue that analogy, I might say that the task is like getting algebra to run on someone whose brain has a power supply that randomly shorts out and spends half its time browned out.
Because some people out there are just not easy to teach.
I'm not sure I understand your comment, this is exactly what I spent hundreds of hours of my life trying to do. Was I unclear that my entire educational career was being the person who got through to students that traditional education was giving up on?
I disagree - we have excellent tooling avaliable to cram and compile code into smallest CPUs, but our teaching methods are still the same as they were in 1800's
I think this reveals a lot of ignorance on the huge amounts of different teaching methodologies and techniques that have risen and fallen over the last 100 years.
How do you know what effect upbringing and cultural values has on math performance? You know because there's research for that.
But you can't base any argument on that when research into alternative (or complementary), genetics-based explanations is being stiffled. Or well you can, but that's just society-scale version of googling for statements that you agree with instead of questions that they're supposed to answer.
When you stiffle research directions for political reasons, you're doing politics, not science, and the arguments based on lack of stiffled research don't hold any more water than arguments based on no research.
> society-scale version of googling for statements that you agree with instead of questions that they're supposed to answer
Unfortunately, in practice science often turns out to be almost as much of a vehicle for confirming answers one already has in mind, rather than open-ended investigation.
There were no leaps of logic made by me. I never indicated or said anything about genetics. I said that it’s clear some people can’t learn algebra and gave an extreme example of such a person by saying that an intellectually disabled person can’t learn algebra. I then said that I’m convinced that there is some level of “intelligence” required to learn algebra.
I said absolutely nothing about what percent of people this is true for and absolutely nothing about why this is true. I brought it up in the context of the article because saying that not everyone can learn algebra is as taboo in education as the thought that genetics plays a role in poverty and success in life in psychology. My reason for thinking this is responses like yours.
There are clearly people who can’t learn algebra: intellectually disable people are such an example. I believe some level of “intelligence” is require to learn algebra.
How can you conclude from what I’ve written that I haven’t thought much about my belief? How can it appear I have an axe to grind from what I’ve written? You have formed an image of me that is wildly incorrect. Do you initially assume that everyone who thinks not everyone can learn algebra has an axe to grind?
Other traits like height, health, beauty vary and are influenced by genetics. Intelligence varies between species, and is therefore, at least in part determined by genetics.
Presuming that there are only mentally disabled and normal people is the weird hypothesis. The baseline idea should be that it varies just like health and height. Affected by both genetics and outside factors such as malnutrition and injuries.
This is very nicely put. I wish I had thought of this phrasing when I posted my original comment. The point of my comment was to mention that there are heretical ideas that people won’t discuss in other areas of intellectual inquiry. That was the relation of my comment to the article.
"Nature vs nurture" in the hairless ape presupposes free will, which is a linguistic universal but a metaphysical unprovable.
Look closely enough and there is no essential difference between genetics and other causative factors. Other than maybe some people jumping to the conclusion that one has an axe to grind with minorities when one attempts to explain certain things with genetics. Which is just as much an arbitrary social taboo as the preceding taboos that constitute what we today call bigotry. (For the record, I'm a staunch opponent of all forms of violence and oppression.)
For me it makes exactly zero difference. Even if free will does exist in some essential sense, I do not believe that people generally choose what opinions to espouse. They simply acquire them through mimesis of their social environment. If that makes me a nihilist and a coward, then so be it.
Thought experiment: English Prime but also excluding any constructs expressing intentionality. I dream of a world where the concept of free will is considered just as poor taste as racial slurs. I think that, perhaps paradoxically, it will be a much more free and just world.
> Thought experiment: English Prime but also excluding any constructs expressing intentionality.
I like this idea of a "deterministic" language. In fact it reminds me of Nonviolent Communication, and is probably a good tactic for discussions that might otherwise devolve into personal attacks.
Saying "there is no essential difference between genetics and other causative factors" is arbitrarily different from saying "all is one, separateness is an illusion". That is, it differs in connotations and not in the essential content of the statement.
Which is exactly what you said, except that you chose to ignore that connotations conduct meaning, when you asked your rhetorical question. This is not e-prime, it is a plain old adverb answering the question "how?" like adverbs normally do.
No, I'm not. Can you be more precise in what you mean by "causative factors"? Without further context, and based on my understanding of the world, virtually anything internal or external to a person could cause them to be more or less skilled at something. It contains everything, and so seems that your statement could be interpreted that genetics and everything else in existence are one in the same.
You asked what was the difference between my statement (that genetics is not more special than other causes of being more or less skilled at something) and the statement that "all is one, separateness is an illusion". I answered that the difference between these two statements is arbitrary, which I still believe to be the case. Apologies if something else happened to you.
My original comment had the purpose of questioning the validity of the "nature vs nurture" distinction. It just seems like an unhelpful distinction, but then again I'm not a biologist, just a lay person who likes their concepts tidy.
Genetics is obviously not the same as everything else in existence; I'm not sure that even makes sense as a statement. You seem to have somehow derived that I am arguing against the concept of distinctions at all. I'm not sure if language would be feasible without distinctions.
I don't disagree with any of what you just said, but I fail to see what point you are trying to make or what you are arguing against. Without the (linguistic) act of making distinctions, everything is indeed one and the same, but that's... kind of pointless?
EDIT: Sibling poster also seems to fail to make the distinction whether (a) we're comparing genetics to other causative factors, or (b) we are comparing my statement about genetics to your "all is one" interpretation of it.
In case it's still unclear, (a) and (b) are two completely separate things and I'm not sure how this conversation got to the point of conflating them. It just serves to reinforce my belief that the ambiguity of our language's syntactic structures makes it inordinately difficult to reason about many things in everyday language. Or maybe I'm just a bad communicator. "Me bad", "you bad" that's supremely easy to express lol
EDIT2: Correction, TheSpiceIsLife does actually get it.
> My original comment had the purpose of questioning the validity of the "nature vs nurture" distinction. It just seems like an unhelpful distinction, but then again I'm not a biologist, just a lay person who likes their concepts tidy.
The confusion is in the difference between proximal and ultimate causes, the rest of the discussion is over the ultimate cause of certain phenotypic features being down to genetics, some other mechanism or not significant at all. You've then said "all these different things are just proximate causes and [because free will doesn't exist] the ultimate cause is the laws of physics" to which people have unsurprisingly gone "what the hell does that have to do with anything?" because, well, it doesn't.
The fact that the ultimate cause of me taking a dump is "the laws of physics" doesn't mean the proximal cause wasn't me 'deciding' to go to the loo, and the fact that you can always say "the laws of physics" (or some higher power) is the cause doesn't make talking about higher level causes any less useful.
I don't believe in free will but its such a good trick that you act as though it were true almost 100% of the time, and talking about my 'decisions' as causes is useful the same as talking about 'genetic' and 'environmental' is useful. We talk about the causes of the Big Bang usefully despite time only coming into existence when the Big Bang happened :)
What sprang to my mind when I read the arbitrarily response is that you have equally little control over the genes you’re born with as you do the place, time, family, society, economy, technology, and culture you’re born in to.
> How is this any different from saying "all is one, separateness is an illusion"?
> It's not arbitrary. "Other causative factors" is boundless.
These are distinctly different lines on inquiry. One is an inquiry to in the illusory nature of separateness, the other is an inquire in to the boundless nature of causative factors.
Fair enough. I still believe that tiptoeing around these issues gives power to those who consciously perpetrate and benefit from institutionalized violence.
Like another commenter said, operating with a comfortably skewed mental model doesn't help resolve the actual socioeconomic issues.
I was a little hyperbolic in my original answer. In all honesty, I think it's probably best to continue research in this area. However, in the current state of the world I don't see how that research is especially beneficial. Every finding would have to be taken with such a massive grain of salt that I have a hard time imagining we would find practical applications for it.
In the current state of the world, most scientific research will be co-opted by some violent apparatus or another. Does that mean we should lose hope and stop doing any research altogether?
By current state of the world, I mean that any research into this topic can't control for all the possible variables. That's why we'd have to take it with such a large grain of salt.
That's why I brought up metaphysics and e-prime actually. We can't resolve societal contradictions in a fundamental way if we do not have the tools to reason about them, and the main tool we have for that, human language, can at times be pitifully inconsistent and ambiguous - even if one does, in fact, control for people's automatic emotional reactions to controversial subjects.
How does this not track? It's clearly obvious that there are intellectual boundaries that exist (E.g. if you're not this smart you're going to struggle).
Previous education, upbringing, cultural values, etc are all separate effects that may influence your overall ability, but intelligence __definitely__ influences your ability.
Low intelligence + good education = poor overall ability
> Your previous education, your upbringing, your cultural values, these things all also have huge effects on your aptitudes, and you've just dismissed them out of hand, apparently in favour of pre-determined genetic intelligence.
They didn't seem to do this at all. It can simultaneously be true that there are some who, due to genetics, simply cannot complete a particular task, no matter how conducive an environment they are put in, while for others genetically do not have these hurdles but whose success is still dependent on their environment.
If anything, you are the one dismissing this possibility out of hand, and making leaps of logic.
The thing which I think gets missed out of these discussions is the notion of community. We're such horrific individualists in the West that we seem to see nature and nurture as distinct, which of course they aren't. The nurture of a child is a product of the combined abilities (from nature and nurture) of the people in their family and community. There is no "I"; our DNA and mores are all part of a greater whole.
That's because we haven't yet cornered good ol' "free will" yet like we have this subject, or if we have, it certainly isn't trotted out as regularly and forcefully by the Right, likely because the virtue of the concept is to support punitive discursive apparatuses.
My feeling with math is that moving on too soon is disastrous. If you move past a subject without having grasped it, you don't have a good base for the next subject. This compounds, and the explanations stop making sense.
At that point people are trying to help you but saying things to you as if you are stupid, but you still don't understand. That really sucks, so people get afraid of math. Avoiding it, and nodding when asked "do you understand" when really they do not.
This is hard to fix because you need to go back to the point they did not understand, but the fear and pain makes even teaching that a lot harder to fix as well.
My feeling is that most people should be able to understand algebra. However, I think that requires a very deliberate and personalized approach for some people. Certainly with collective classes, if you go at the speed of the slowest student there will be slow progress and a lot of people who are bored and mentally check-out.
If any approach is going to work for people who have real difficulties, it needs to be small-scale personal teaching, and it needs to come with trust. Someone needs to feel like they can keep saying "no I do not understand" without disapproval, disappointment, or frustration from the tutor.
This is so true. I'm Black and attended one of the worst performing elementary schools in my city for the first four years of school, which gave me an awful base for my math learning when I was finally transferred to a much higher performing school in a Jewish neighborhood on the other side of the city, not to mention high school and college. Only when I started working in programming did a lot of algebra and trig click (probably helps my first junior role threw me in the deep end working with linear algebra and trigonometry in animation and was lucky to have a senior around who chose to take on a mentor role). Math in public schools is basically magical spell incantation and it appears to most kids that you either have "it" or you don't. Math is a subject I believe requires long-term work that doesn't easily fit into the grade pass/fail structure of school, but then again a lot of aspects of mass education are fundamentally broken.
I like to feel the same way as your colleagues about "everyone can learn to code".
I can definitely say that there are many people for whom it is a struggle - even those on college courses.
But a hundred years ago we could easily have the same conversation about simply reading and writing - they (the poor, women, or "lower class" ) luke not be taught to read. But it turns out that if you start young enough, and put enough effort in, 99% of everyone can learn
So, perhaps society is not putting enough effort in, or asking enough effort from, our kids for them all to learn calculus.
As the Agile Manifesto says, the work delivered represents the effort input so far. If society wants all children to learn calculus, we need to pay that price.
Edit: I think that we do need a new conversation about education. We have My father left full time education at 14, at 18 most of my peers left school and only 25% of us went to college. Today it's 50% it the total spend has not gone up in line.
I honestly don't know what kind of world the "universal education" advocates of the 1870s were imagining - but I am damn sure it was not people doing college courses on their iPhones, but it is the world they ushered in.
We need to double down on what worked for the 20th Century (and avoid the, y'know wars and genocide and stuff).
"In the past, I have made no secret of my disdain for Chef Gusteau's famous motto: Anyone can cook. But I realize, only now do I truly understand what he meant. Not everyone can become a great artist, but a great artist can come from anywhere." [1]
> But a hundred years ago we could easily have the same conversation about simply reading and writing - they (the poor, women, or "lower class" ) luke not be taught to read. But it turns out that if you start young enough, and put enough effort in, 99% of everyone can learn
There is a thing called functional illiteracy [1], where people can write and read but mostly only their name and some very basic things like grocery lists. They also cannot comprehend texts even if they can read most words. It's more or less equivalent to being able to add numbers, maybe multiply numbers 1-10 but very far from algebra.
No first world country can claim a 99% literacy rate unless you count in these people, which would stretch things quite a bit.
I can accept your belief, in principle, about putting enough effort in early education and that lack of such efforts contributes to poor outcomes in college. I also think though that some people just can’t grok algebra. You would agree that a mentally disabled person can’t learn algebra, right? So where is the line below which a person can’t learn algebra. I suspect it is much higher than being mentally disabled.
I am not sure what "mentally disabled" means in this context.
I would be surprised that a psychiatrist would find it a useful term - in the same way that any doctor would not look at the paralympics and see only one illness / disability but dozens of conditions.
I am fairly certain there are Rain Man like people who can do algebra or calculus but would for most people sit under the term "disabled".
Perhaps it might be good for you to ask, "the people I see who can do algebra also can act in certain other ways, socially and mathematically. How much am I looking for people who can do maths, people who can do maths like I do, and people like me who can do maths"
Edit: yes I suspect I sit to the left of you, but the important takeaway here is not that some people are more "intelligent" than others (I agree for whatever definition we choose) but that raising the education level of all in society massively benefits us all. Oh and ranking peoples value in society based on intelligence is not a good idea. It's only just above ranking on physical disability, or your parents aristocratic connections.
I used the phrase in a colloquial sense. Replace mentally disabled with intellectually disabled. That’s the proper modern term for what was once called mental retardation.
I gave no indications that people ought to be ranked in terms of value to society based on intelligence or any other criteria.
I can’t prove this a priori, but I think like the line would be pretty low in a vacuum. It’s hard to tell in the real word though, since we have so many compounding factors in people’s educational careers that affect their abilities.
Mental disability and intelligence isn't one variable. And intelligence isn't one dimensional. Asking for a threshold makes about as much sense as asking how many pawns you need to win a chess game.
I started programming with a friend, I remember our first Js and PHP, he got stuck often, stopped for months while I was getting some first contracts, he was in awe about it and I thought he is hopeless, he always kept saying we will make it. He continued to go to local meetings where most people used phyton,c ,anything but what we used. Then I moved away, couple years later, he informed me he is some sort of full stack engineer at IBM, still is.
That was a big lesson to never judge one based on early results. This guy didn't understand function arguments and how return works for a long time.
I was bad at geometry initially as well, but with the help and encouragement of teachers, i managed to improve beyond class average.
Some other topics, there was no help nor will and I stayed well below average.
Motivation and dedication and external factors like help can neutralize intelligence adv/disavantages at times, seen it happening over and over.
To address your question about "universal education advocates", they were not the reason that we got it.
Their argument was purely an philosophical one, stating that a well informed society operates better, makes better decisions, ect. And a well educated person can make better decisions for themselves and have a higher quality of life.
The reason that universal education caught on was because large businesses realised that could have more efficient factories if they didn't have to teach everyone to read. Thus college and university because the place where the 'universal education' dream could be realised. Where every person could go and receive an education that would afford them a higher quality of life and allow them to better engage with the world.
Then, factories became more specialised and 90% of universities became slaves to some "industry"
Many universities "computer science" programs are hardly that. Rather they've turned into a coding bootcamp with slightly more math.
I think we can still have discussion about functional illiteracy. That is group who can technically read, but not at level needed for every day life. Very likely there is certain part of normal distribution who can't reach that level. But how large we can expect it to be is good question to consider.
Kids are savant at learning ”muscle memory skills”.
But there is a reason 2nd graders dont do book reviews of classic literature. Even though they can absolutely read the complex words out loud on autopilot any day of the week.
Yeah. it isn't that just saying there is a difference between a 10 year old and a 19 year old
Edit: Actually, I think your premise is wrong. Children don't read words they don't understand "like muscle memory". The differences in meaning between, idk, betrothed, engaged, wedded and living in sin are superficial but they carry with them implications of social value, history and more. Children are supposed to read books and pick up context and understand meaning as they go along. We don't expect a 10 year old to grok calculus any more than we expect them to grok social/sexual arrangements. But we have different standards for 19 year olds - in the case of sexual arrangements it's obvious why. Perhaps less so for calculus.
Reading and comprehension levels are hard to guage for this reason - at a certain point the words are placeholders for political positions. And it's hard to tell if someone does not understand the word or disagrees with the politics. Please see every election for past ten years for reference
Looking at your observation that "a hundred years ago we could easily have the same conversation about simply reading and writing - they (the poor, women, or "lower class" ) luke not be taught to read." I would not be so sure that the fact that nowadays 99% of everyone can learn proves that this historical observation was mistaken.
It's known that intelligence can be significantly negatively affected by environmental factors like malnutrition (both of child and mother during pregnancy), childhood disease, pollution and injury, and there is some evidence (also mentioned in the article we're discussing) that a century or two ago the average person was significantly dumber than now, presumably because of those factors - especially to the poor and women, which often experienced more severe childhood malnutrition than their male siblings.
So it would seem plausible that a hundred years ago a nontrivial percentage of people actually were too dumb to succeed in literacy and perhaps the fix for that wasn't just starting young enough and putting effort in, but rather that we started getting much fewer kids with severe environmental damage to their intelligence.
> if you start young enough, and put enough effort in, 99% of everyone can learn
The richest kids at my highschool were also the smartest. There were one or two rich kids who didn't make honor society. There were also only a 5-6 smart kids who were middle/lower-class. Graduating class of ~500 in 1985, about 50 honor society (top 10%).
> Years of administration telling the math department that our passing rate is too low have led me to pretty much pass everyone who takes the final exam. Last semester 80% passed but only around 50% deserved to
Doesn't this contribute to lower the reputation of the universities?
I mean, I did sociology in a small university in northern Spain and there was debate between the stats and methodology professors and pretty much everyone else about this.
In the end I'm grateful to my stats and methodology professors, because having to work very hard is the only thing that actually gave me something useful to go and fight with the world.
Many of my peers had to go to other unis because they couldn't pass there, so in the end they get a pass like me, the same in the eyes of employers, but they know it's bullshiting, and I know too.
IDK, It seems to me that there's a lot of lazyness in social sciences, and this kind of hard attitude to it is not only good for the students, but needed in the field.
IDK in the US, but this professors got a lot of heat for their stances on what a university is about, but they stood their ground, even when they've got hit by political bullshit, denied money for research, etc.
There's another thing (again, IDK if it's the same in the US), but why don't you see it in CS or Engineering faculties? I mean, I remember in my uni the students of this faculties had maybe a couple of professors that where ok to lower the bar a bit, but most of them just assumed that you gotta learn what you gotta learn. And many of them where pretty hostile to students, 180º stance from what I see in Stanford Moocs for example.
Yes, this has long term negative effects but I need a job and don’t need the stress of being hounded by administration. In the old days the state funded a much higher percent of a student’s higher education. In those days, society was the client. Today the student is the client and the student does not want to fail.
> Yes, this has long term negative effects but I need a job and don’t need the stress of being hounded by administration.
The way to make sure your students all pass is to do your job, those students need a teacher that cares and not one that 'needs a job', and preferably one that does not start off with the self-reinforcing viewpoint that a good fraction of them are too dumb to learn the material on offer.
Seriously, consider changing jobs to something with less of a negative effect on people's lives.
I've been on the receiving end of this attitude and it harmed me quite a bit, later on, with better teachers some of the damage got fixed but I most certainly would have traded two of my math teachers that 'needed a job' for some of the (much) better ones. Thanks to Fred Pach and one Mr. Groot I'm not a total loss at math and it ended up not being my skills but those of the teachers that were deficient.
You know nothing about my teaching or what I do for my students. Did I say a good fraction of the students can’t learn algebra? I never said or implied anything of the sort. All I said is that some people can’t learn it. I gave an extreme example of such a person. I didn’t say anything regarding how large the set of people who can’t learn algebra is other than that this set is larger than the set of intellectually disabled people.
It appears you are letting your past personal experiences distort your judgments about me. It’s ok. I understand this. It’s natural but it makes it hard to have a discussion about whether or not it’s the case that everyone can learn algebra.
You made a statement that the way to pass everyone is to do my job. Clearly you have not taught much in a classroom with a wide variety of students. Consider the possibility that you simply don’t know anything about the craft of teaching other than your very limited experience of being a student and that as such you should be wary of making negative conclusions about me.
Here’s one reason I know you’ve don’t have much experience teaching. You’ve mentioned that the way to pass all of my students is to do my job. No one who has extensive experience in classrooms with a wide variety of student backgrounds can possibly make such a statement. Even the most ardent, hardcore teachers at my college who advocate that everyone can learn any topic don’t think this.
When I mention that I just mostly pass my students now they understand that grades are an administrative aspect to the job and know that this is detached from my desire to get as many students as possible to learn as much as possible. People with extensive experience don’t harp on my statements about grading. At most they’ll claim they don’t lower their standards but they don’t make any wild conclusions about me like you did.
But you, you’ve made wholly unjustified conclusions about me and that says you don’t know the nuances involved. You don’t immediately understand that one can think not everyone can learn a topic and still be passionate about learning.
You don’t because I’ve never said anything about my teaching. Your conclusions are not supported by the available evidence.
You’ve obviously not taught much in the classroom. Why is it so obvious? Consider the possibility that your thinking on this issue is clouded by your past.
I have always found the university ranking game to be screwed up.
Are the best Universities good because they have the best teaching, or are they just selecting the best people, who would be successfull no matter what?
The universities are penalised in rankings when a student fails to pass, but they are not penalised if they are so selective that the student never got a chance to study in the first place. This breeds an elitist system that does not let students rise to the challenge. Thats also one of the reasons why european universities do worse in rabking than UK ones - it's common there to start with a huge class, but only small part will co plete the course. In UK its harder to get in, but vast majorty seems to pass.
If universities worked like a normal business does, you'd expect that the best universities would grow, expand, and eventually there would an Oxford in every major city, with oxford teaching methods. But that's not how it works. But people still keep trying to appply 'free market theory' to education, when it obviously works very differently.
Lastly there is perpetual conflict between research and teaching - many professors want to do research and don't like teaching, many post graduate teaching assistants are folks that, no matter how briliant, were afraid of the job market, etc.
Maybe this is a stupid question, but is there any indication that most US universities cared about their academic reputation in the first place? I don't see any.
Well in Spain everybody says to care, yet I see what I see. Not to mention that everyone complains about the low rankings of spanish unis, but I'd say that's mostly a consquence of money and some biases at play.
> I’ve taught mathematics at community colleges for over 20 years and I’m absolutely convinced that not all people can learn algebra or calculus. To me it is obvious this is so since the mentally disabled can’t. There’s a level of “intelligence” that’s necessary to learn a given topic. Not everyone can learn all topics.
I think most people agree that there are extremes. There probably exists a small number of people who literally cannot (just like how 1 in a million could probably teach themselves calculus at age 10, also exists). However it's a big jump to conclude that even a small percentage of the people at your college cannot in the literal sense - i.e. if their life depended on it, they had all the resources they could want, they had no other distractions.
> i.e. if their life depended on it, they had all the resources they could want, they had no other distractions.
You just gave the definition of "cannot". Those circumstances will never happen, they're pure hypotheticals. Sure, hypothetically I might be able to play competitive tennis against Nadal. Practically, though, that'll never happen, I cannot and will never be able to do that - pretending otherwise is just naive and purely theoretical discussion.
I'll go one further and say that very few people in the entire human population would be competitive with Usain Bolt in the 100 meter when he was setting records, no matter what kind of training they had. There would be a few more beyond the world class sprinters Bolt was demolishing. Similarly, there would be very few people who could beat Nadal in 5 sets on a clay surface. Djokovic happens to be one of those few. But, those are extreme outliers.
Still though, we could ask how many people in the entire human population could run under a 4 minute mile with any amount of training, time and resources? Obviously more than have done it, but how many? Certainly not remotely close to everyone.
If the original claim was merely that a certain percentage of the population are not going to learn algebra - it would be uncontroversial and i would agree with it. After all, we all know that some people fail classes.
If the claim was merely that math comes easier to some than others - again it would be an uncontroversial claim and i would agree with it.
However, my reading of the original comment was that a good portion of the population literally are physically incapable under any circumstance, including the silly hypothetical circumstances i laid out, to learn algebra. I think that is false. It sounds like you do too. So i guess we're in agreement.
Not exactly. I read the original claim that they are "incapable to learn algebra even in very good/way above average (but still realistic) circumstances".
* How do you differentiate between someone who doesn't get it because it's beyond their mental skill level versus someone who doesn't get it because they'd rather be doing literally anything else?
* When you say "mentally disabled", what exactly does that mean in the context of a spectrum of ability levels across the species?
* There are a lot of reasons people might fail your class. What are they? What's the percentage breakdown?
* What are the dangers of thinking that not all people can learn math? What are the dangers of not thinking that?
Oh, I’ve thought a great about these things. None of your questions are new to me or are ones I’ve not considered. I did not come to my conclusion quickly.
Here are some more thoughts to ponder:
Children are easier to brain wash than adults and teaching basic math involves a great deal of brain washing. To what extent is it the case that a person with no exposure to algebraic concepts growing up is at a disadvantage to learning algebra as an adult? There is a theory that if one doesn’t learn a language in childhood then it is impossible for them to learn one as an adult. Is something similar applicable to Mathematics?
I do think something similar is applicable - for any subject. I was/am a math teacher and what I've found is those who couldn't learn algebra often didn't understand the 'language' or have a good grasp of what math means (like what is a fraction, what does it represent in real life, etc). Taking time to go over these basic concepts with them vastly improved their mathematical abilities, and eventually most got to the point where they could easily do algebra.
I don't think the issue is necessarily that they're incapable, it's more that they don't have the necessary background intuition or knowledge. At least in the state i taught in, this is not surprising as elementary school teachers often dread math and fail their standardized teaching test math portion several times sometimes. It's the blind leading the blind.
I've also noticed it in reading, when they have teachers who aren't readers themselves trying to teach it to them.
Though I don't know of any studies about algebra in particular, kids show greater neuroplasticity in general than adults. So learning anything as an adult tends to be more difficult.
But I live with an American who reasonably fluent in Norwegian after 2 years of study in their 30s. It's certainly not impossible.
I have no doubt that math, like virtually everything, is more difficult for adults to pick up from scratch.
What is theorized to be impossible is learning a language as an adult if you did not learn a language as a child. That is, if at age 20 you know 0 languages then you’ll never be able really learn a language. For obvious reasons experiments in this regard are few.
My conclusion is that some people simply can’t learn algebra. Be it genetic, environmental, lack of intellectual fortitude, lack of “intelligence”, or any combination thereof it is simply the case that some people can’t learn algebra.
I wonder if you are equating "can't" and "lack of intellectual fortitude" with "aren't truly motivated to do so."
Considering that a small percentage of careers require more than basic algebra/geometry/statistics, it's not hard to see that math has a "why should I learn this?" problem more than a "am I able to learn this?" problem.
> I’m absolutely convinced that not all people can learn algebra or calculus. To me it is obvious this is so since the mentally disabled can’t.
Not all people can read, it's obvious since the mentally disabled can't. Yet with proper education, all non-disabled people can be taught reading.
But that doesn't mean everyone can read, if you're illiterate by your 20s, you're gonna have a hard time catching up. Same for mathematics: most people reaching even high school are too mathematically illiterate to catch up[1]. Is genetics a factor: definitely, but it's among many others.
The reason why it's a partisan issue is the following: if I say genetics is a decisive factor, then I can say «it's natural, there's nothing we can do so we don't need to spend all that government money trying to help those people». The left-sided point of view goes as «There's nothing we can do about genetics, but we can change everything else. Then we need to find what are all the other factors, because those are the actionable ones». The conservative focus on genetics is mainly a justification for doing nothing.
> There’s a level of “intelligence” that’s necessary to learn a given topic. Not everyone can learn all topics.
“intelligence” is conveniently pretty ill-defined, but I don't think I'm more intelligent than my doctor friends, yet they struggled a lot to grasp even the most basic concepts of algebra when I tried to help them during our studies. “Not everyone can learn all topic” but I have yet to find evidences that your ability to learn a random topic you're not interested into is correlated with the common acceptance of the word “intelligence”.
[1]: that doesn't mean it's impossible, just likely well beyond the amount of effort they can (or want to) afford.
IMO, biological differences and barriers certainly exist and are non-trivial, but the differences between private and public schools demonstrates the degree to which institutional finesse can accommodate student strengths and shortcomings, including differences in intelligence or disability. In other words, we have a long way to go.
In some schools around the Bay Area, we have half the students on the fastest track, and about half the math faculty as calculus teachers. An immodest minority of students complete Calculus BC early and go on to the local college to continue their math education for their remaining years in HS. And these aren't even the top 50 schools in the nation.
I do wonder what’s gained by this. There are plenty of physics professors (perhaps even most) that didn’t do calculus until college or at least senior year of high school, what’s the advantage?
I'm not sure it's easy to discuss the high school transcripts of people with university posts. How did you get in touch with this information?
The gains become clear depending on the degree to which you know what you want, and especially if you're thinking about graduate programs.
1. Calculus in some sense is somewhat disorganized, and it sucks that it's a gateway to more math in the US. By finishing Calculus early you can move on to the math you wanted to learn, such as Linear Algebra.
2. University students may have to spend about a year learning Calculus. That's a long time, and to some people they'd rather spend their money better by learning something else or graduating early.
3. If you're thinking about graduate studies in something technical such as Econ or Stats, then you'll probably want at least Analysis. The problem is that in the grand scheme of things, even Analysis does not leave you very prepared to do things, it just makes you prepared to learn more, so you may want to get ahead of this problem.
> I'm not sure it's easy to discuss the high school transcripts of people with university posts. How did you get in touch with this information?
Because 95% of the time high school work doesn't transfer to the college level even if you do semesters at De Anza. You're back to scratch unless you go to flagship in-state, and even then you can only transfer up to a certain amount (at my undergrad it was up to Calc 2 - several people I know took Calc 3 in the same institution in high school but had to take it again).
Every university has a different agreement on how skipping reqs and credits work, but I'm really surprised that you're saying you need to go to a flagship school for Calc 2, because I believe that's normally covered just by the Calc BC AP exam.
Even if a college didn't want to transfer the credits for Calc 3, I'm surprised that they wouldn't allow you to skip the course. Also, for your friends, if they took a class at the same institution... doesn't that mean they got a repeat? Strange.
I also wouldn't generalize these things to people who occupy university posts, as they probably had interesting trajectories.
As a scientist who thinks professionally about things like culture, behavioral genetics, and psychology, I can say: the idea that there /would not be/ humans who cannot learn calculus is the one that is hard to believe.
Some years ago, California tried to make it so that you had to pass Algebra II to graduate high school. I think the state backed down from that, because it meant only 25% of students could pass. Or maybe they just bowdlerized the definition of algebra II.
It’s too bad, because I don’t think anyone should be able to leave high school without understanding compound interest-sorta vital for participating in a modern economy. Also, it’s not like we can stop loaning money to the innumerate, even if that might be the ethical thing to do.
Do you really need to know Algebra II to understand exponential growth? Couldn't you just get by with multiplication and seeing a pattern and learning the FV function in excel?
Could you link to an article about this? I couldn’t find anything substantial.
Context: Recently there was some controversy about Oregon graduation requirements [0] and social media represented the issue much differently than primary/secondary sources.
I wonder how much of the difficulty is due to large class size. If the student to teacher ratio were smaller, would teachers be better able to tailor instruction to struggling students?
My high school had one teacher capable of teaching calculus. His seniority allowed him to fob it off on someone who could not. She left the profession entirely a few years after I graduated.
Of course, one may learn calculus from a book, which was what many of us did in that class.
This is a big problem imo. In my experience teaching high school, a lot of elementary school teachers struggle with basic math... And that's where the problems arise that we have to try to fix later on.
Likewise, it's my my less experienced impression that some people just can't get (or at least it's overwhelmingly more difficult) basic programming concepts with even a visual building block system
I do have an anecdotal conter-example to your theory.
In my country, highschool last three years. The first year, i was living in the dorms with some of my classmates. One, let's call him M, was fairly dissipated and, while everyone would rather play and discuss than study, and had shitty grades in everything but biology. Math was his weakest, with grade ranging from E- to D+ (i try to convert grades here, my grades wer in the A range and the average was B-), and English/spanish were not a lot better.
At the time, at the end of the first year, we had to choose specialities and which kind of diploma we could do. the "general" kind, with to specialities, economics/humanities or science, or the "technological" one, with three (electric, mechanic, civil). He realized at the end of the year that the only thing he wanted to do was biology, but as this subject was only available in the science speciality, he was fucked. Impossible to get to the "general"branch with his grade. He struck a deal with the school administration: 4 hour a week added to his curriculum, one hour every night after eating and before going in our dorms to make sure he worked. The year, in math, did not start well. He had a F/F- as his first grade (the average was D+, top of the class was B), but our professor was exceptionnal. we had 3 math illiterate in our class, and while the notation was pretty harsh on them, he took time to help everyone understand with interesting examples, stories and exercises. He even took hours of his own time for supplementary lessons. Our class relation was special too, we were only 20, and spent a lot of time together, even after college. Last time i talked to M he was doing a Master in marine biology, and aimed for a doctorate
End of the 3rd year, we have to get the national test. Our class ended up with an average grade of A-, M had a B+ and wasn't the lowest scorer in math (of our class). Miles ahead of the nationnal average (C+/B- that year). It was a public technological highscool in a rural area. The other close HS was one dedicated to farming. Culturally, we did not start at an advantage compared to the average kid of our country, i'd say we were at a disadvantage. But great teachers and small, but close class can help you emancipate from some of your determinations.
It seems like you're essentially implying the mean value theorem here. Because there exist some people who cannot do algebra (folks with mental disabilities), and there exist some people who can, there likely must be some sort of transition point where someone doesn't have mental disabilities but can't do algebra. In that graph, you're implying that "mental intelligence" is the independent variable(s) and algebraic ability is the dependent variable.
Another assumption is that the function is continuous. That's one I'm not sure about. I don't think there's a continuum between folks with mental disabilities and folks without. I think there's a discontinuity -- or perhaps overlapping spectrums on different dimensions -- but not a continuum. That's why many mental disabilities are the result of very concrete genetic variations. You can't have 80% of a generic variation. You either have it or you don't, which implies some sort of inherent discontinuity.
My guess would be algebraic ability is a normal distribution. But that's not a death sentence. People who might, because of genetics or more likely nurture/circumstance be unlikely to learn a lot in algebra, are not doomed to that fate. It just means it might require a lot more time and hard work for them and their teachers -- to the point where they might not find it feasible, and instead choose to do something else.
I believe the universe is discrete and that everything in it is discrete. I believe that “intellignece” is not really measurable. A person might have an intelligence as it pertains to art or critiquing art and be utterly clueless in another area. Indeed, I believe all people have a lack of ability in some area. For all of us there are things we just aren’t ever going to be able to learn and can’t learn. There are lots of reasons for this. I don’t know but can believe that genetics plays a role. There is variation on how our brains work so it seems reasonable that genetics plays some role.
I hope you don't think this article in any way supports your belief that some people can't learn algebra/calculus due to any kind of genetic traits, because I don't think the complexity of either rises to the level of "difficulty" that only gifted minds can comprehend. Very, very little in this world does.
Your belief should probably be considered heretical, even among your conservative colleagues, because your colleagues should recognize how deficiencies in one area can be made up for in other areas, sometimes (but not always) at a higher efficiency cost.
Where genetic gifts are lacking, determination and perseverance can almost always make up the difference, especially at a basic algebra/calculus level.
While I'm sure almost everyone can learn the basics of algebra or calculus, no amount of wishful thinking will give everyone the brain needed to master them or even be proficient. I'm not sure where your belief that perseverance can overcome all even comes from.
I have been trying to get better at chess lately. I grew up playing it so I'm already fairly proficient, and I've been putting a decent amount of free time into studying openings and playing puzzles. Yesterday, I played my friend who hasn't played in literally years and never really played seriously anyway, and he beat me handily 3/3 games. I know that I do not have the brain to be a chess master, no matter how much time I put in, because I don't have enough working memory to keep the board in my head.
I disagree, as it's not a sufficiently difficult topic to warrant some kind of genetic stratification.
Life circumstance stratification, certainly, and you will get no disagreement from me that some people don't have the time or energy necessary to devote to something as not-immediately-useful as calculus, but from an intellect perspective, no.
Everyone who is not experiencing some kind of mental illness has the intellectual capacity to learn calculus, though I will capitulate it is fortunate for me and this statement that we do classify a low enough IQ as a mental illness!
It’s not a sufficiently difficult topic for you. Have you ever taught algebra to a classroom of people from a wide range of intellectual capabilities and experiences? Are you even aware of some of the pain points to learning algebra?
Do I need experience teaching calculus in order to suggest that it's got more to do with environment and situation than it does intellectual capability? No, I don't think that's true.
Are you seriously suggesting trying to raise multiple children while simultaneously learning calculus is approximately as hard as living alone and trying to learn calculus?
Honestly, I think you're trying to find a neat solution where none exists. You haven't stumbled on anything here, you're more likely ignoring the real-life circumstances your students find themselves in, and your colleagues are not.
> Do I need experience teaching calculus in order to suggest that it's got more to do with environment and situation than it does intellectual capability?
Yes, because I'd expect HN readers to understand an ad hominem when they see one.
I could be living in a ditch down by the river, or I could be the leading researcher into how people learn calculus and it wouldn't make a difference with regard to my argument (though, practically speaking if I were the lead researcher on how people learn calculus, I'd be more likely to back my argument up with objective research, and that would strengthen my argument).
Ad hominem occurs when you use someone's reputation in another area to discredit someone on an unrelated matter. For example, you're the worst golfer on the planet so I say you don't know how to cook. Golf expertise has no bearing on your cooking skills. But if you consistently burned toast or you mainly use your oven to heat up frozen pizza, I'm going to take anything you say about baking a cake with a giant grain of salt.
No, ad hominem occurs when you use who a person is rather than their argument to claim they are incorrect. Here's a definition from Google:
> (of an argument or reaction) directed against a person rather than the position they are maintaining.
Nothing about a reputation or using it to discredit them.
For example, I'm not a calculus teacher, therefore I couldn't possibly form a valid argument about how teaching works. That would be an ad hominem, because it focuses on who I am (not a teacher), rather than what I've said (the teacher I was replying to hasn't eliminated any variables at all before drawing their conclusion). (It's also nonsensical, considering how many other things people teach, and how small a percentage of all teachers calculus teaching ends up being, and how unrelated-to-the-science-of-education calculus is).
It's actually funny, because the example you've given isn't an ad hominem, since you have evidence to support the idea that I can't cook (the burnt toast). You're equating an absence of information about me with specific data, which is different.
You don’t know what an ad hominem is. You are incorrectly applying the definition. It is not an argumentative fallacy to ask the basis by which a person’s assertions have been formed. You’ve made lots of claims about teaching but clearly you have no experience to back it up and (this is important) you have not provided any citations to back up your assertions.
People have to make judgments with imperfect knowledge. It’s reasonable to discount the unsubstantiated opinions of someone with no experience with the topic at hand.
It should at least be interesting to you that it was obvious from your comments that you don’t have experience teaching mathematics in the classroom. Why was that so obvious to those of us with that experience? The previous question is rhetorical.
I didn't say it was an argumentative fallacy to ask the basis by which a person's assertions have been formed, I said it was a fallacy to say a person's assertions are wrong because of some aspect of themselves, which is what's taken place here by insisting I must be a calculus teacher in order to challenge your assertion that IQ is the primary source of the problems your students have with learning in your classrooms.
What is interesting to me is the fact that you retreated to this ad hominem the moment you were challenged, because it tells me you don't have any real explanation for how you eliminated other possible causes for your students sometimes performing poorly.
You'd prefer to live in a world where your experience has meaning than to live in a world where your experience is not valuable when faced with this question, which is completely human of you, but ultimately not useful in this discussion, due to its anecdotal and un-rigorously collected nature.
I can't stop you from throwing this New Yorker article, and the other works of Dr. Harden, in the face of your colleagues, but I can hopefully dissuade others from making the same logical mistakes you're making. I believe I've succeeded at that, by clearly highlighting the carelessness of what you've said here.
Ultimately, what I find most fascinating is, in real time, you've demonstrated how right Dr. Turkheimer ultimately is and how dangerous this research can be when put in the hands of folks who don't understand its delicacy or even the basic facts surrounding these arguments.
I'm grateful for your engagement with me, it's been helpful to work through this with someone like you, but I'll commit to the thing you tried and failed to do; I'm no longer going to reply to your comments in this thread. You're clearly (and I mean clearly) wrapped up in a need to think of some of your students as too dumb to learn calculus, and there's literally nothing I or anyone else here can say that would convince you otherwise, and at this point I've done my part in preventing others from thinking that your insight is useful or helpful in this conversation.
I committed to not responding to you in a different thread. I did not commit to not responding to you in all threads. The teacher in me forced me to try to explain to you why your use of ad hominem was incorrect. My previous response had nothing to do with what was being discussed as such. Your conclusions have not been logically valid.
Here’s an example:
… I must be a calculus teacher in order to challenge your assertion that IQ is the primary source of the problems your students have with learning in your classrooms..
No one has said any of these things and no one has implied any of these things. I never said or implied that IQ is the primary source of anything. No one believes that you must be a teacher of calculus to be right. What people have wondered is if you are a teacher because some of your statements seem to the the type of statements only a non teacher would make.
Oh, well, if you're a leading researcher in math pedagogy, you should just say so and save time. Such credibility would offer serious firepower to the debate.
> I have been trying to get better at chess lately. I grew up playing it so I'm already fairly proficient, and I've been putting a decent amount of free time into studying openings and playing puzzles. Yesterday, I played my friend who hasn't played in literally years and never really played seriously anyway, and he beat me handily 3/3 games. I know that I do not have the brain to be a chess master, no matter how much time I put in, because I don't have enough working memory to keep the board in my head.
Given the immense complexity involved with learning, there's essentially no way for you to describe the situation with enough detail for anyone to be able to make an informed judgement. It's extremely common for bad players to describe themselves as 'proficient', spending a 'decent amount of time' can mean anything, studying openings and playing puzzles is likely the wrong way to practice, the skill level of your friend is impossible to ascertain, etc.
The way you actually get better at anything is by:
1. Dedicating enough time to it, consistently (~20h per week or so maybe if you want any sort of quick results). The incredibly common trap here is that people think that spending time on a game equals getting better at it. There's no way to improve without spending a certain amount of time playing, but spending time playing does not make you better on its own.
2. Doing whatever it is you want to get good at (doing anything else does not count - if you want to learn how to play regular chess, play regular chess; puzzles or anything else does not count).
3. Reviewing your games to find mistakes you've made - this part is crucial, if you can't see any mistakes you made, you cannot improve. If that's the case, get someone better than you to review your game(s).
4. Playing while trying to work on addressing one mistake at a time, until you don't make it anymore.
Chess is actually really easy to improve in - fast, trivially repeatable games, chess engines, lots of learning resources, objective rating system.
> I know that I do not have the brain to be a chess master
I'm not sure what exactly you mean by 'chess master', but for example getting into the top 10% of players is pretty easy. You are indeed incredibly unlikely to be one of the best, because for that you will have to dedicate your entire life to playing chess. But unless you literally want to be one of the best (top 1%+ of players), genetics will not limit you. You may take more time to get to any given level than someone gifted with more working memory or whatnot, but it's still doable.
> because I don't have enough working memory to keep the board in my head.
Because of humans' limited memory and processing power, we play games not by exhaustively analyzing but via building simplified models and heuristics so I don't see why you'd need to 'keep the board in your head'.
My belief comes my experience teaching math the past 20 years. I think we both agree that mentally disabled people can’t learn algebra. Do you have any evidence that these are the only people incapable of learning algebra? I have lots of anecdotal evidence that the set of people who can’t learn algebra is much larger than the set of mentally disabled people.
Do I require evidence to not believe something you've stated? Or do you require evidence to assert something you've stated?
I don't think the onus is on me to prove you wrong, so much as the onus is on you to prove that your experience is scientifically rigorous and representative of more than just your personal experience.
And to be clear, I don't doubt that many people "can't" learn calculus in the same way I "can't" run a marathon; we don't have the desire, discipline, and free time necessary to do the work required. This does not speak in any way towards our capacities to do so, only our desire.
So I suspect the vast majority of your students didn't have the sufficient desire to learn a lot more than they were incapable, and I think it's a critical distinction, because you can change people's motivations, but you cannot change their capacity.
There’s no onus on either of us to do anything. We are just strangers posting on the internet without pay or other compensation. Your experience and knowledge about teaching help me to decide the merits of your beliefs. It appears from my perspective that you know quite little about what you are posting. I know slightly more by having a lot of anecdotal evidence but this certainly doesn’t imply that I’m more likely to be right.
As stated in my original comment I’m quite liberal. In education we talk a lot about social conditions and their effects on education. It’s why teachers’ unions strongly support universal healthcare, school lunch programs, etc.
After 20 years on the job I’ve come to the conclusion that some can’t learn it. This isn’t controversial in some sense since we know retarded people can’t learn algebra. So what level of capability is necessary to learn it? I don’t know but I suspect it’s well above being mentally retarded.
I could be completely wrong but given your lack of experience in the topic my experience ought to at least make you pause a bit. In another comment you wrote about me:
… you're more likely ignoring the real-life circumstances your students find themselves in, and your colleagues are not.
Talk about making an assertion with no evidence! At least I waited 20 years before stating my absurd assertion.
It's clear to me that this belief has wormed its way into your identity somehow, and you're unable to discuss it in a detached and curious way. I'm sad for you that such is the case, but I am now more empathetic towards your colleagues who have to deal with your false assertion that some people are too stupid to learn the math you teach.
Also not for nothing, but "retarded" hasn't been a medical term for some time now, with institutions such as the AMA and SSA both replacing the term with "intellectual disability". The only remaining reasons you'd use it are either because you are thus uninformed, or you're signaling something...
It’s easier to type retarded than intellectually disabled on an iPad. I wrote the modern version in my original comments. Clearly I have been discussing this topic in a detached way. You’ve formed an image in your mind about me that isn’t supported by the evidence.
I’ll read whatever response you have but won’t comment further. I wish you well. Keep up the good fight!
You can use "cognitively impaired" as a more neutral and factual descriptor than either (and "cognitive impairment" as the generic term). It's also more broadly applicable. (I.e. If you got whacked in the head once too many while playing rugby, that might not count as "intellectual disability" according to some since it's not developmental. But it might interfere with learning math, so it's pretty indistinguishable in a practical sense!)
Thanks, I appreciate you taking the time to talk about a view you've held that's been unpopular with your colleagues.
I think you, and everyone who thinks like you, need to take a very hard look at what you've done to eliminate the environmental factors related to your viewpoint that some people aren't smart enough to learn something, because its extremely easy to trick yourself into thinking someone is incapable when in reality other factors abound.
As the article explains, if society refused to educate/feed/raise/nourish red-headed children, there would be a genetic correlation between red-headedness and intelligence.
How absolutely confident are you that you've accounted for every explanation besides the genetic one in determining why some people can't learn algebra?
Its kinda crazy how casual people are with their constraints on explantion when the topic is intellectual disparity. Person you're replying to claimed they weren't arguing for a gentic component to another reply but also doesn't do anything to filter any other factor than a congential threshold for learning math. This is absolutely why the Left is generally leary of this type of discussion because it is so easy to use the information for all sorts of ill-informed or cross purposes.
It's pretty clear you've never instructed a remedial math class.
My university would conditionally admit students who had a math score below a certain threshold on their ACT, I think it was 19 or something. Anyway, as part of their admittance criteria, they had to attend an after class lab for an additional hour an a half for a total of three hours per week dedicated to learning pre-algrebra. There were four modules where they would do some reading, work through some example problems through interactive software, and then have some homework to work through that was almost identical to the examples. Students would try and try and try and try to learn the material, and they would take days to work through the problems on their own (often with my guidance, giving pointers on how to think about the problems) to finish the module so they could take the quiz and pass the module. Near the end of the semester many students made appreciable progress, but for others the inability to retain and apply what they've spent so much time on results in tears, especially because they don't know if this requirement will keep them from being able to graduate.
Given infinite time, could these guys all have figured out pre-algrebra enough to pass? Maybe. But the amount of time it takes them to learn math concepts that are very easy for us means that it's entirely impractical to expect them to ever achieve proficiency in advanced mathematics.
> There were four modules where they would do some reading, work through some example problems through interactive software, and then have some homework to work through that was almost identical to the examples.
This just does not look like a universally effective way of teaching to me, irrespective of the topic. It's hardly any wonder that some people fell through the cracks if they were unfamiliar with the subject in the first place. What about leveraging stuff that's actually been tried and tested, like the Khan Academy videos and their automated interactive, school-like environment?
> And to be clear, I don't doubt that many people "can't" learn calculus in the same way I "can't" run a marathon; we don't have the desire, discipline, and free time necessary to do the work required. This does not speak in any way towards our capacities to do so, only our desire.
Your VO2max, max heart rate, and other factors appear to be significantly determined by genetics, and will absolutely contribute to your capacity to run a marathon. If it takes you a year of hard training, but it takes me a few practice runs a few weeks before, it's not fair to say we required the same amount of desire, discipline, or free time to succeed. I would instead say we had a very different capacity to run a marathon.
Are you saying that similar things could not possibly be true for learning math? Or really anything else that humans do?
I don't think any of those things contribute to whether or not I could complete a marathon, though they do contribute to how pleasant the experience would be. The same holds for learning calculus, or doing anything intellectually taxing. It's differing in difficulty for people, however it's not unachievable for anyone who isn't experiencing some kind of mental (or physical, in this marathon example) impairment.
All healthy people can learn calculus and run marathons, with varying degrees of success and effort, due to genetics and environmental factors.
I don't see how if you can agree that it requires varying degrees of effort for different people to run a marathon or learn calculus, that you then think it is impossible that some people won't be able to do those things. Just like the amount of effort required will be small for some, it will be impractically large for others. Even in the theoretical sense, there is only so much time in a day.
I also don't think "healthy" and (presumably) "not healthy" are useful categorizations. There are many people in that fuzzy in between area between "healthy" and "not healthy", for both physical and mental health.
Is it really worth so vigorously arguing the semantics of "some people are incapable of learning algebra" and "it requires an impractically large amount of effort for some people to learn algebra"?"
Yes, I think it's absolutely worth arguing the difference between "it's hard" and "it's impossible", because those are two fundamentally different things.
You can overcome difficulty, you cannot overcome (by definition) impossibility.
What you keep ignoring in what I'm saying however, is that I do think certain things are impossible for some people. I will never play in the NBA, for example, but that's a far cry from being "very good" at basketball.
Learning calculus and completing a marathon are not the point at which "healthy" (and yes, it's fuzzy, but precision is impossible on this topic) people are sometimes unable to do things. Winning a marathon and getting a Ph.D. in mathematics, I would acquiesce to your argument.
In other words, I think anyone can dabble in anything, but you do need an alignment of genetic and environmental circumstances to be in the top 1% of something. I could be argued into top 10%, but below that, it appears the data supports almost anyone being able to do almost anything, or at least well beyond whatever artificial lines we might draw to discourage people from achieving.
> I don't think the complexity of either rises to the level of "difficulty" that only gifted minds can comprehend
This is a strawman. They never said 'gifted', they just said that some people have it and some don't.
> deficiencies in one area can be made up for in other areas
Not what we're talking about. GP is probably great at things that aren't chess. That doesn't mean they're good at chess.
> Where genetic gifts are lacking, determination and perseverance can almost always make up the difference
Determination and perseverance aren't (at least partially) genetic gifts?
Honestly I think 'genetics' are a bit of a red herring here anyway because the real underlying assertion being challenged is that all humans have equal potential in all things (which anyone who has ever taught any mildly difficult topic will know is trivially disprovable). The precise mechanism whereby innate human potential differs isn't important if you can't even agree that it differs.
There are some questions science is not allowed to ask now. This is, of course, because they fear the possible findings. And I laugh every time some progressive I know shits on the Republicans for being "anti-science." (not that they are any better)
My english teacher/class master till 8th grade insisted I follow a profession that doesn't involve math -- she even recommend me to be a radio host for a children program -- she thought highly of me, but she thought math was not my thing.
She was half right -- I have ADHD and even though my blood relatives are highly successful (doctors, lawyers, judges) NOBODY was good at math.
Anyway, my parents thought otherwise and promised to buy me a PC if I get admitted to a math/programming class in highschool and paid for math tutoring. My math skills completely turned around in a year and I ultimately loved math, especially calculus.
If you want everyone to pass, lower the standards. The end result will be that certificates from schools without high standards will be worth less in the eyes of employers (while still costing the same to the student). As they progress through such a system, some students will be placed in advance of their abilities and fail out at a higher rate than they might otherwise when they reach an institution with non-negotiable standards, like the hard sciences, demoralizing those who might otherwise have found their place of achievement, self-respect, and independence. Or, even those institutions will cave to such pressures and simply "mark 'em up, ship 'em on", as you have described. Caveat emptor.
First, there are clearly a large number of people out there who have no business trying to pass algebra 101. Even if they theoretically could with sufficient effort and tutoring, it is not worth their time, nor is it going to lead to that much that is beneficial.
Second, teaching is already hard. Teaching math also requires an understanding of math that is rare. Hence finding people who are good math teachers is hard. Finding people who are good math teachers and willing to do it for a math teachers salary is even harder. If it were trivial to improve our teachers we would off course do it. But it is not at all trivial.
I think this matters less in a liberal arts college, but especially if you are going to ask students to build on their algebra 101 with other courses. If passing algebra 101 took a lot of effort, chances are that the other courses are going to take the same amount of effort, if not more.
Why, indeed. Maybe our standards for teachers are too lenient. In any case, it is possible that there is no quality of teaching that will change a student's natural aptitude. It's a disservice to huge swaths of the population to funnel them, ill-prepared, into colleges. Maybe a network of high-quality trade schools would be a better fit for some.
The obvious answer is you lower the standards that year because somewhere the system broke down. (It being much more likely that there is an issue with the system than some kind of statistical anomaly with the students)
Then you evaluate and attempt to improve techniques as is done every year.
I come from a third world country where to graduate high school, you HAVE to take Calculus I, II and little III. Planar Geometry, linear algebra etc. And thats only if you through the Life Sciences track (Biology and Organic Chemistry)
I truly don’t believe my people are genetically smarter because we have a truly awful abysmal track history in governing.
My guess as an immigrant who went to community college here and then to University here is that students just have too much choices/freedom. They are told they should strive to whatever they want. I and my friends grew up told you need to choose a major that will bring bread on the table, that should be the only priority.
> I’ve taught mathematics at community colleges for over 20 years and I’m absolutely convinced that not all people can learn algebra or calculus.
This is the "As a mom"-argument applied to iq. When it comes to child-rearing mothers will not infrequently claim to know what works best in general because they have had children of their own. I mean an equally valid interpretation of your anecdata is that not all people can learn how to teach mathematics at community colleges.
Certainly not all people can learn how to teach mathematics. That is not at all disputable in my mind. I may be in the set of people who can’t learn to teach math. I’ve tried for over 20 years and appear to be just average at it. Maybe I’m below average.
My collection of anecdotes that not everyone can learn math works as an anecdote that not everyone can teach math.
I think in the paedagogy community genetics is often an excuse for the unwillingness to develop right teaching concepts. It's then the "fault" of the pupils.
Polya did good works, especial regarding Mathematics.
Of course there are genetic influences and not everybody can be an Einstein or rocket scientist. But below that level there is a lot possible that has nothing to do with genetics.
Most things are more influenced by motivation than the ability. And on that the real cause does not matter.
If the people would spend only a fraction of the energy, they waste with questioning if something is genetic or not, in the development of skills by learning - they would improve beyond the proclaimed "genetic" level.
Reduce media consume. Reduce politics. Almost the whole discussion is toxic.
The right proclaims everything is genetic but on the other side punish people not having the "good genetics".
"The right proclaims everything is genetic". I don't know what kind of "right" are you talking about. I don't believe it is mainstream conservative thought to claim family/school training/displine/education does not matter but only genetics matters.
Some people can certainly reach an intuitive understanding of mathematics that will never come naturally to "normals".
It's worth noting however that we often consider these people "disabled" (or euphemistically "differently abled"). The point is that there's no genetic dial you can turn from "worse" to "better", it's more like a massive board of switches that feed into each other.
The problem with learning advanced math is that math is isolated.
Like if I learn accounting, I know why I'm learning it; to learn how to create, audit, or read the financial statements of a business. If I learn math, it comes across more as "here are these arcane puzzles you need to solve." Like no one does double entry bookkeeping in the abstract; you learn the history, the methods, etc in the confines of an express purpose.
But trying to learn precalc as an adult (which i do get), what's striking is how little purpose or context there is to it. Why do I need to know how to factor imaginary numbers, or know the slope of something?
I think math when isolated is a big reason why its so hard to learn. They teach the toolset but people don't have the need to use the tools unless they go into a separate subject.
Here's a thought I've been tossing around: It's a community college, everyone is supposed to pass. At some point at the bottom of intelligence but with motivation to go to school and try and work and do the assignments, meritocracy literally doesn't matter. Because a diploma is needed even for minimum wage jobs.
Why penalize people who are trying by failing them out of the simplest classes?
Rich idiots fail up, poor idiots end up homeless. No reason to penalize the poor who try when the entire game is arbitrary. This position tends to make certain groups bristle, especially those with a classist sense of "fairness".
It does matter in this sense. Students who are poor end up taking on debt for college and this debt is currently (mostly) unforgivable. If we had universal higher education this objection of mine goes by the wayside. But then a new objection rises up.
perhaps some people cannot learn algebra / calculus the way it is traditionally taught in the classroom. perhaps the students who are failing just need a different environment, more time, more patience, different resources?
Obviously there are people who couldn't learn a given topic under one type of educational regime but who could under a different regime. But that doesn't eliminate the obvious: Some people can't learn a given topic at all, under any circumstances. OP's example of the mentally retarded (which I learned recently is a valid medical descriptor) is just an extreme example.
It's so funny watching people scramble to avoid admitting that genetics has a huge impact on humans and their potentialities.
Granted, as a species, we are the closest to blank-slate out of any species ("niche-switching is our niche"), but reality doesn't go away just 'cause we don't like it.
A good deal of the folks enmeshed in various delusions related to their belief that reality is socially constructed, I've found, are folks that have little concrete experience with reality. Academic types, those who've exclusively worked in knowledge-production or in offices. Rock climbers and farmers are very much not prone to these delusions, for a couple of examples.
Try to convince a dog breeder that dopey English Mastiffs are just an environmental change away from gaining the intelligence of the German Short-Haired Pointer, which can practically solve Sudokus.
Look, there's a "valid medical descriptor" for grandpa who is in a nursing home with Alzheimer's disease, but this kind of thing is totally immaterial to people who are in school today. There's no way that they'd have that level of cognitive impairment. Saying that "some people just can't learn" so-called "advanced" math such as college algebra and calculus, or programming for that matter, is just pointless speculation with zero evidence to back it. Most likely they can, we just can't be assed to teach them effectively.
> Saying that "some people just can't learn" so-called "advanced" math such as college algebra and calculus, or programming for that matter, is just pointless speculation with zero evidence to back it.
What about the anecdotes of millions of people who self-profess that despite very much effort, they just can't wrap their head around some advanced math concepts? That doesn't count as evidence?
There are also plenty of anecdotes of people who self-profess that for years or decades they couldn't wrap their head around some math concepts, and then one day they met a teacher who explained it in a different way than any teacher before had done, and it "clicked" for them as adults.
I don't know how to weigh these anecdotes, but I think that's suggestive that the methods of teaching might be relevant even to people who struggle with math for decades.
Honestly no. Most people say i can't do X when really they mean, i've decided that its not worth the effort/i dont want to.
If you were arguing that math comes easier for some people than others, sure that's strong evidence. If you're arguing that they are literally incapable, and no set of curcumstances would allow them to learn - that is a very different claim and needs very different evidence.
I'm a farmer, not a teacher, so I can't answer this question as asked.
But I would speculate that of the set of people who bombed the class, they could fall into a number of buckets. E.g., one bucket is people who were mentally capable of learning the concepts, but were to lazy to put in the effort (then we can quibble about whether inherent laziness puts people into the "not capable" bucket). Another bucket is people for whom alternative learning environments might have brought them to understanding and a passing grade. Another bucket is people who just lacked the preliminary background and with a couple years of effort could be made to pass the class as it exists. And finally, another bucket is people who are genuinely incapable, regardless of environment, of understanding the concepts.
This shouldn't be surprising. I have tried to deeply understand quantum mechanics, and while I can parrot some of the most well-known and more simple concepts, I truly believe that I lack the capability of grasping the very core, deep insights in an intuitive way. I might pass undergraduate level classes in the topic, but I am fairly certain I couldn't achieve a PhD. I'm not the dumbest bulb in the shed, but I can see that there are people much, much brighter than I, and it is obvious that their ability to understand more advanced and deep concepts is greater than mine; This leads to the observation that of the set of understandable knowledge in the universe, some of it is available to some people but not available to me, no matter how hard I try. (I take solace in Feynman's quote, "If you think you understand quantum mechanics, you don't understand quantum mechanics.")
Then look at every human capability and its distribution across the universe of humans, and it's pretty clear that we all can't do everything, every one person has some cutoff beyond which they aren't capable of understanding in any given topic. For some people (and hey, maybe it's a really small slice of the population), that cutoff is somewhere before Calc II.
I think we're getting closer to the actual situation with this description. There are a lot of buckets of people who don't do well in a particular class.
And we both admittedly don't know the size of the bucket of people who cannot, given a lifetime of 80 years continuous study and tutoring, understand a topic.
But in my teaching experience, it's dwarfed by the group of people who doesn't care about the topic and bombs because they don't put in the effort.
From my point of view it seems that the US educational system clutters their curriculum with too much unnecessary cruft.
Students need to focus on the basics and no where is this more true than in mathematics. Too many students muddle through middle and high school mathematics without gaining mastery or with multiple gaps in their knowledge. By the time they get to calculus they're simply unprepared to put all their previous knowledge to use. "Calc 101" is the first time that many students are required to apply theorems and then use algebra, trigonometry and arithmetic to arrive at results to problems. If there's any weaknesses in their fundamentals it's going to make the problems intractable (and, yes, they are for many students).
It's better to track students and only advance them to the next level in math when they've demonstrated mastery of previous topics. That would mean, of course, that a good fraction of students would never "reach" calculus (or even algebra)-- but that's OK if it means they have enough numeracy to balance a ledger or learn avoid blowing money on the lotto. At least their time would not be wasted on trying to do "Business Calculus" in college.
That's not really rebutting the original claim, is it? If, due to genetics, they need a different environment, resources, more time and patience.. how is that not agreeing with the premise that genetics matter?
Yes, different environments exclude and include some people, those who may be on the border of capability seem likeliest to be impacted here. That still doesn't mean that ALL people can learn calculus, given enough attempts to find the right environment for each of them.
Shows, also, the silliness of this argument. Some people can pick up calc at age 10 no sweat whatsoever. Others struggle mightily with the basics in their 30s. Should we as a society invest 1000x the resources in the strugglers to ensure they can achieve the same understanding?
>Should we as a society invest 1000x the resources in the strugglers to ensure they can achieve the same understanding?
There's no need for everybody to reach the same level of understanding, but I think the pandemic has shown the importance of teaching as many as possible the basic concepts of calculus. "Flatten the curve" doesn't mean much when you've never heard of integration. The same applies to climate change. People will have more faith in mathematical models if they think it's something they could have done themselves if they really wanted to (overly optimistic judgement or not) instead of some bullshit the so-called experts made up to bamboozle them.
Recently I read things about our school system in tabloid. Wholly unscientific, but popular anyway. The result is that trying to teach these students in regular environments might negatively effect everyone and specially those who are borderline. That is there is a group who need extra support and inside regular lecture could learn, but can't as groups needing even more support take resources.
This doesn’t necessarily have to do with genetics but with the environment you live in when going to school. My parents and grand parents were both born in the countryside and couldn’t get algebra or calculus (whereas myself and my siblings didn’t have issues learning) because they were lacking the basis that you now more easily get in primary and secondary school if you have supportive parents and teachers.
By the time you’re in college, it’s already over. It’s why good daycare / preschool / school for everyone matters, otherwise you’re just missing out on a lot of potential talent.
> I’ve taught mathematics at community colleges for over 20 years and I’m absolutely convinced that not all people can learn algebra or calculus. To me it is obvious this is so since the mentally disabled can’t. There’s a level of “intelligence” that’s necessary to learn a given topic. Not everyone can learn all topics.
You know, if you were trying to teach Chinese to 18-year-old English-speakers at community colleges with three hours a week of lecture, you might come to the conclusion that only a few rare geniuses had the ability to learn Chinese at all, and none would ever learn more than a few hundred words. But of course over 98% of people born and raised in China learn to speak Chinese fluently by the age of 5, and that's not because of genetics; it includes almost 98% of white people born and raised there too. There are several factors that I think of as key to this difference:
1. Plausibly there is a critical period for language acquisition, and if so, it very likely ends before age 18. Looking at an 18-year-old you can't tell whether their deficits in Chinese-speaking ability are due to genetics or environmental effects in the previous 18 years. There probably isn't an early-childhood critical period for calculus (I think you need formal operational reasoning for that) but maybe there are other things you need to develop early on for calculus to be easy for you when the time comes. Like, fluency in reading, for example.
2. Three hours a week, 36 weeks a year, for two years, is a total of 216 hours. Native language acquisition more typically involves over 20,000 hours of language exposure by age 5. Sometimes it's surprising how much more you can achieve when you apply literally over a hundred times more effort. Or, to look at it a different way, how little you will achieve when you're applying less than 1% of the effort necessary to get good results.
3. Schooling is a really terrible way to learn things. Extrinsic motivation displaces intrinsic motivation, massed practice displaces spaced practice because it gets better exam scores (especially if you know when midterms are coming up), the feedback necessary for improving is delayed by hours or days by the nature of homework grading, and the pacing is inevitably far too fast for some students and far too slow for others. It's well established that individual tutoring produces results two standard deviations better than ordinary classes (Bloom's two sigma problem). That's 30 IQ points. And Bloom wasn't even spacing out practice over decades the way you ought to; he was working within the semester structure of traditional schooling, extrinsic motivation and all.
Is there a possible human culture where 98% of everybody routinely learns to do symbolic integration? Or does human nature render that impossible? Maybe. (Hell, when I have an integral to do in my head, I myself invariably settle for just approximating it unless it's a fucking monomial. Maybe I should spend a few semesters doing them on the blackboard in front of a class, I bet that would help.) But the meat grinder of community college doesn't give us much evidence about that one way or the other, except to know that we don't live in that culture today. It doesn't help us at all with the question of what to attribute to environmental effects and what to attribute to genetics.
(One indication that such radical changes may be possible is the gradual transformation of literacy; hieroglyphics were the province of the priesthood and the quasi-priestly scribes, and even after the invention of alphabets, Charlemagne and Genghis Khan were illiterate. Can you imagine trying to teach a classroom full of 18-year-olds hieroglyphics in three hours a week, if they had no previous experience with reading and writing? Yet today literacy rates are over 95% in most countries, though countries with non-phonetic orthographies like Chinese and English lag a bit behind.)
Math notation is very consistent and it’s optimized for convenience. It’s been constantly refined over last centuries. People who complain about notation usually actually have problem with the substance, and the complaints about notation is just a coping mechanism, to deny hurtful reality that one can’t understand something hard, blaming external factors instead. Ask yourself: if notation was genuinely confusing, why would mathematicians make themselves suffer needlessly?
Now, to be sure, some people and some books are better at teaching than others, but it typically has nothing to do with notation used, and everything to do with the order of introduction of concepts, level of detail of explanation (which can be both too high and too low), amount and quality of examples, etc. However, the core issue here is that some things are actually genuinely hard, and people of average intelligence simply cannot grasp them without expending ludicrous amounts of effort.
If you have some concrete suggestions about mathematical notation, ways it could be improved in more than superficial manner, I (and the rest of mathematical community) is very much open to hear them. Improvements in notation do happen regularly, and when they are valuable, they reach wide acceptance. For example, in the second half of 20th century, the notation of commutative diagrams have been invented, and it spread like a wildfire, because it genuinely facilitates understanding.
>Ask yourself: if notation was genuinely confusing, why would mathematicians make themselves suffer needlessly?
I think its pretty clear that momentum causes a lot of nomenclature pain. You can't just redesign entire fields of understanding every couple decades. For instance, what other fields use single greek characters to label concepts in a seemingly unpredictable arrangement? Often, local maxima are found because concepts are added in the context of the field as it already existed.
And its not just math. Most sciences have this problem. It is what it is but its silly to say we're in the best of all possible worlds just because math has been around a long time.
> For instance, what other fields use single greek characters to label concepts in a seemingly unpredictable arrangement?
This is a perfect example of what I was talking about. Greek letters are typically used in opposition to Latin ones in order to distinguish what programmers can think of as “type” of concepts. For example, when doing geometry, you might want to designate angles with Greek letters, and points or line segments with Latin ones. This makes it much easier to mentally keep track of what name correspond to which object.
Yes, learning Greek letters for the first time is some amount of initial overhead (not much, as typically used ones are similar to Latin anyway, nobody starts with psi or xi). However, crucially, this overhead is paid once, and pales compared to the difficulty of learning the concepts being represented in the first place. It is never the case that replacing Greek letters with Latin makes students go “oh thank you, now I understand everything!”, instead, things are typically just as hard as before. However, replacing Latin with Greek might actually do that, by reducing the mental overhead through introduction of categories (types) of objects.
>Math notation is very consistent and it’s optimized for convenience.
Hard disagree. The most important thing I learned is that it is all made up on the spot to the point that the lecture material explicitly says that books have used 6 different forms of notation for the exact same concept. When you understand that you drop any pretense of "design" in the notation. That helps you abandon foolish ideas that it is "consistent" and that the only thing it is optimized for is the author. When you understand that then it's just a meaningless barrier to overcome but it also becomes easy to overcome precisely because it is that trivial. You just get used to it and e.g. learn the alphabets of the dozen languages (including klingon because the lecturer had to make that joke) from which the variable names where sourced from. Once you did the meaningless grind the barriers are gone.
> People who complain about notation usually actually have problem with the substance, and the complaints about notation is just a coping mechanism, to deny hurtful reality that one can’t understand something hard, blaming external factors instead.
No it is quite simple. You can't understand an easy or hard concept if you can't read it. I still remember how I understood nothing in the first semester. Then when I was preparing for the exam everything was extremely easy because the notation was understood by that point.
>If you have some concrete suggestions about mathematical notation, ways it could be improved in more than superficial manner, I (and the rest of mathematical community) is very much open to hear them.
As I already said that is meaningless because there is no universal notation. "The mathematical community" will adopt a fraction of proposals and further splinter into separate "factions".
> if notation was genuinely confusing, why would mathematicians make themselves suffer needlessly?
Because they are unable to change it. Just like with any thing evolved over a long time, like music notation, languages, even, to some extent, programming languages. Every change brings a lot of pushback, it's a monumental task to create a new one and even more so is getting any traction with it.
That doesn’t square well with the fact that notation keeps getting refined and improved. There is no pushback for genuine improvements. Biggest problem here is that there rarely are changes that clearly and meaningfully improve situation over status quo. I gave one example above, but overall, I am not going to take complaints about notation being obstacle to understanding seriously without concrete ways how to meaningfully improve it. You can of course keep complaining that it’s confusing, but without proposals for improvements, you’re actually complaining about the difficulty of substance, not the notation, and it says more about you than about notation.
That's a bad example: first, it's a question of definition, not of notation. Second, it's defined pretty consistently within physics, and pretty consistently within electrical engineering (none of which is mathematics).
> Ask yourself: if notation was genuinely confusing, why would mathematicians make themselves suffer needlessly?
Sometimes notation elides 'obvious details'. The details are obvious to those that have already gone through the learning curve, which is easier if one is strongly connected to other working mathematicians than for outsiders. Mathematicians barely notice inadequate or unusual notation. Outsiders struggle and spend significant energy just deciphering the notation.
Anecdote 1: Sometimes notation is too terse: single letters. Granted, efficient for whiteboard scribbling. Would be really nice to standardize an appendix for notation. E[X] = <expr>. Hmmm, what could E be? By the fifth paper, somebody bothers to write 'expected value' in plain English and the mystery in unambiguously clarified. In a voice-based interaction this is a non-issue, not so for those that only have text to deal with. This compounds as a novice has to juggle a set of mysterious symbols with tens of elements.
Anecdote 2: Long long time ago (before ubiquitous email or http:// took off) a young student spent some time working through a handful of type theory academic papers that somehow trickled into his corner of the Universe. The type inference rule notation (https://en.wikipedia.org/wiki/Type_rule), which is rather straightforward in retrospect, ate up more time that he's willing to admit. Γ is just a set of judgments and Γ |- expr is just a notation for 'Γ includes the judgement expr'? Then why don't they use the standard expr ∈ Γ, this is confusing? Are there some examples? Is there some code that one could possibly run through a debugger? The questions remained unanswered, as there was noone he knew in the same linguistic sphere interested in the topic. And the papers themselves never detailed such obvious notation details.
PS. I agree that the bigger obstacle is the lack of proper big picture 'why do we even bother with these concepts / theorems'. At an extreme, there is (used to be?) a certain style of math books consisting exclusively of a dry litany of 'Definition 1.2.3', 'Theorem 3.2.4', 'Corollary 2.3.1'. Very rigorous and very difficult to ascertain what problems they were trying to solve.
> Anecdote 1: Sometimes notation is too terse: single letters. Granted, efficient for whiteboard scribbling. Would be really nice to standardize an appendix for notation. E[X] = <expr>. Hmmm, what could E be? By the fifth paper, somebody bothers to write 'expected value' in plain English and the mystery in unambiguously clarified. In a voice-based interaction this is a non-issue, not so for those that only have text to deal with. This compounds as a novice has to juggle a set of mysterious symbols with tens of elements.
I really cannot conceive how one can learn what the concept of “expected value” of a “random variable” means, without encountering E[X] notation. This is a technical concept having a technical meaning, and any place that actually defines this meaning will teach you this notation. If you see this notation for the first time in some academic paper, but haven’t ever read any probability textbook, it means that you almost never actually learned the concept, which is my entire point. You might have some intuitive understanding derived purely from the literal meaning of the words “expected value”, but without actually getting technical, this intuitive understanding is mostly superficial, and, as such, not very useful. You won’t be, for example, be able to answer such fundamentally important questions like “is expected value of sum of random variables a sum of expected values of each? Is expected value of product a product of expectations?”. You can’t know answers to these questions without having ever seen E[X] notation, and if you don’t know the answers, your problem is with the actual concept, not notation.
> PS. I agree that the bigger obstacle is the lack of proper big picture 'why do we even bother with these concepts / theorems'. At an extreme, there is (used to be?) a certain style of math books consisting exclusively of a dry litany of 'Definition 1.2.3', 'Theorem 3.2.4', 'Corollary 2.3.1'. Very rigorous and very difficult to ascertain what problems they were trying to solve.
I agree that it very much often is a problem. It’s not a problem of notation, though.
Technically speaking, there are other cultural spaces than US/English. In one such space E[X] is written M(X) and called 'average value'. But that's quibbling. 'Expected value' is simply 'weighted sum' over possible values with respective probabilities as weights: weighted average value. Not exactly rocket science if one groks what a probability distribution is. But even that is quibbling. The more interesting point is that some would rather starting learning from concrete applications instead of pacing through a seemingly endless dry litany of definitions. Cryptic notation is unhelpful for this style of learning.
Ken Iverson thought mathematical notation was inconsistent, so he wanted to invent a better notation for thinking, which became the programming language APL. But APL syntax never captured the mainstream. Neither did Lisp for that matter. It was C-style syntax that ended up dominating.
Do you believe everyone can learn algebra? What evidence do you have for this belief? Have you taught in the classroom much to students with a wide range of educational backgrounds?
I don’t believe everyone has equal intellectual talent in all areas. My talent for math far exceeds my talent for physics. I’ve tried to learn physics but I just can’t. I have no intuition for it. There is variation amongst our brains and how the connections in our brains formed in childhood. In some people the wiring is such that learning a given topic is not feasible. Such is my belief.
Do I believe that everyone should attend college? I do not. Of the people who should attend college, is it frequent that people can't grok single-variable algebra?
I'm not disputing your superior experience on this topic (hence "well-informed position"); I'm just saying that it's fairly depressing.
In case you don’t know this, community colleges are generally required to accept everyone who applies that has a high school diploma or a GED. We are open enrollment institutions.
I understand now what you meant by fairly depressing. Thank you for the clarification.
Hello, fellow mathematician. The first step in your proof which doesn't convince me is the claim that, in general, "levels of intelligence" exist, admit a partial order, and can reliably predict whether certain people can learn certain topics.
In general, you'll find that "leftists" believe that psychometrics is bogus in terms of science. Instead, it masquerades as science in order to fool the public into believing that biased public policy is neutral. There's no denial that some folks have brain damage, developmental disabilities, etc. but a denial that standardized tests are an appropriate proxy for genuine medical diagnoses.
Community colleges should present themselves as a public benefit. We shouldn't ask that everybody pass any class, simply that everybody has the opportunity to attend/audit any class. It is unfortunately true that the typical university administration is clueless and money-grubbing, preferring graduation rates to other metrics.
I don’t claim that there is a partial order to intelligence. Indeed, I think it’s clear that no such can exist. Let me put it this way. The set of people who can’t learn algebra (using any reasonable standard for this) is nonempty as it includes the set of mentally retarded people. I further suspect that this set is considerably larger than the set of mentally retarded people.
I further suspect but didn’t state this that if one doesn’t have exposure to algebra growing up then it is extremely difficult to learn as an adult. Children are easier to brain wash and in low level Mathematics we do a great deal of brain washing.
Correlation doesn't always imply causation, right? For example, there's a correlation between IQ and income. There are several causative possibilities:
* IQ causes income
* Income causes IQ
* Some unknown thing causes both income and IQ
* The correlation is spurious; IQ and income are unrelated
Evidence strongly suggests that the third bullet point is true; socioeconomic class causes both income and IQ. Richer people living in nicer neighborhoods both have better opportunities for income, and also better opportunities for education; education causes IQ. This is why redlining is brought up so often as a root cause of so many of the disparities in quality of life; redlining deepened socioeconomic divides.
Is a mental disability not caused by genetic disorder, injury or illness. This would invalidate your claim. You wouldn't use a contergan victim as a example for the thesis that not all can learn shot put.
Isn't the whole point that you can't simply look at the gene of a person and know what his mind is capable of except for genetic defects and even then it's hard to estimate.
The problem with never consenting to a search or never talking to police is the ease with which cops can ruin your weekend (and even life) if they get pissed off at you. I may be able afford a lawyer but is the night in jail worth the effort to assert my rights?
This is why cops on Interstates will target out of state plates. People just want to get on to their destination. And not being a state resident, it makes it harder for them to fight after the fact. [citation needed]
