Not everyone can learn calculus or even basic algebra. A person with a severe intellectual disability isn’t going to be able to learn these subjects. Between severely intellectually disabled and Einstein is a boundary between those who can and those who can’t. I don’t know where the boundary is but it exists.
It doesn’t just have to be severe disability. I’m pretty sure I’ve got what is now called NVLD (because my son is diagnosed with it and I share most of his symptoms). It doesn’t cause general intellectual issues, in fact I scored above 130 on the WISC IQ test as a kid. But one of the things it does cause is poor working memory. Trying to keep numbers in my head in order to calculate things is pretty much impossible. Give me calculator and I have no issue with the higher math concepts. But leave me on my own and I’ll give you the wrong answer due to bad calculation every time.
Anyway, this all led to me doing very poorly in math at school, and it’s entirely because my brain just isn’t good at it. Luckily these days the educators are smarter and my son has always been allowed a calculator for math class.
Students don’t fully grasp concepts at the early stages and they never will. Very few students who master the quadratic formula understand that this formula allows you to factor second degree polynomials. Many students will correctly solve x^2+x+1 = 0 while simultaneously believing that this polynomial doesn’t factor. Here’s a simple problem that tests understanding,
A second degree polynomial with leading coefficient 3 has zeros of -1 and 2. Find all the terms of the polynomial.
Most students can’t do this. Even most calculus students can’t do it.
We teach algorithms like long division and the quadratic formula because they are relatively easy computations to learn but they don’t in any way lead students to fully grasping a concept. It’s only with a certain level of mathematical maturity that one is able to understand the full import of even basic concepts.
I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example. They’ll play around with x’s in various complicated looking expressions. Even something as basic and fundamental as the meaning of an equation eludes people at this level even though they have been dealing with equations for years.
Well, such is my experience teaching math at a community college for over 20 years.
>I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example.
It's interesting that you picked 0=1 as your example, because I'd argue it stretches the definitions of "equation" and "solution" into semantic triviality. It's more of a falsehood than an "equation", since the two sides are trivially defined as not equal, and there's no variables to "solve". Using that as example exists somewhere between sophistry and pointing out the absurdity that mathematical definitions for terms technically hold even in trivially untrue situations. That's not how normal human communication works, and not recognizing that divide probably goes a long way in explaining the "inability" you see in students.
In other words, maybe you should have just used "0x=1" as your example :P
0x is the same thing as 0 so it appears my example is a good one in that you yourself don’t fully understand the concepts involved. This isn’t pejorative.
Suppose I said solve
x = x+1
You then subtract x from both sides and end up with
0 = 1
Then you conclude that the original equation has no solution. I’m guessing that you wouldn’t realize that the reason we conclude that the original equation has no solution is because the two equations
x = x + 1
and
0 = 1
have the same solution set since adding the opposite of x to both sides is a solution set preserving operation. It transforms a given equation into a new equation with the same solutions and clearly 0=1 has no solution. That is, 0=1 is a perfectly valid equation.
The larger point, that is missed by people, is that an equation in essence is asking for one to find the instances when two expressions are equal. To find an example of an equation with no solution just find two expressions that are never equal to each other.
>The larger point, that is missed by people, is that an equation in essence is asking for one to find the instances when two expressions are equal.
Respectfully, you've got this backwards. An equation, by definition, is an assertion that two expressions are equal. 0=1 is a logically consistent assertion, but it happens to be false. Most students will intuitively have trouble with the idea that you want them to make a false statement, even if they don't realize that, because their whole schooling has taught them the opposite.
The issue is precisely that we are teaching those students that that "an equation in essence is asking for one to find the instances when two expressions are equal". Mathematical statements don't "ask" anything, they simply are. That's a pedagogical definition, not a mathematical one, and by teaching students that, you're teaching them how to pass a math test rather than teaching them math. And there's no blame on you for that, since you're paid to teach students to pass math tests. But framing it that way doesn't teach them math, it teaches them how to guess the teacher's password[1]. It's a focus on getting an answer rather than understanding the actual axioms.
So of course students don't come up with an equation with nothing to solve, because you've taught them equations are things that only exist as things with unknowns to solve.
It might be obvious to someone who already is extremely well versed in mathematics that 0=1 is "an equation without a solution". But it's unfair to expect students who don't already have that answer to derive it, because they're working off of the wrong axioms. It's a communication failure, not a mathematical one.
It is clear you are not a mathematician. When we write something like:
x^2 + x + 1 = 0
And say solve it we are definitely not asserting that the two expressions are the same. Indeed they are not the same polynomials and if your view were correct we wouldn’t spend time teaching how to solve the equation. There are values for which the two polynomials evaluate to the same number. Those are the solutions.
EDIT: In mathematical logic class one talks about predicates and you learn to think of equations as assertions that two expressions are the same. However, as people typically use and think about math they don’t think in these terms. Indeed, the graphical interpretation of an equation in one variable lends itself to the idea that solving an equation, in essence, is finding values of x that make two functions have the same value.
It is also equally clear that you haven’t taught basic mathematics to innumerate students. When students are taught to solve basic linear equations we include in our instruction that they can encounter situations like:
x+1 = x
And that they can see there is no solution because they reduce the equation to solving 0=1 and that equation has no solution.
You are in an absurd position when you think
0x = 1
is an equation but that
0=1
is not. I doubt that when you simplify:
x^2-2x - (x^2 -2x)
You write 0x^2 + 0x. What I wrote about solving equations has an important word in it. Namely “essence”. In essence…. I was not providing a mathematically rigorous definition. Indeed, the rigorous definition is far beyond the scope of students of basic mathematics. So we have to teach them the essence of things.
Given the first polynomial, when asked to solve it, there's an implied "for x" attached to the question. Even in higher level math you assume you're solving for a variable. When writing an equation without a solution, you don't naturally think about not including any variables. While 0 = 1 is an equation, it's not an equation you "solve". The meaning of equation is not in question, just the association of the terminology of equation to something without variables. Context is important, if the expression had a third order term and I had to use synthetic division, I would absolutely write include the zero terms.
I feel like you're talking past what I'm saying to continue teaching the same math lesson you've taught hundreds of times before, which is exactly the kind of discontinuity in communication that I'm trying to highlight (and, evidently, failing). It's difficult to articulate, and I already feel like this reply is rambling quite a bit, but hear goes:
> we are definitely not asserting that the two expressions are the same.
Correct. Not the same, equal. Because that's definitionally what the equals sign means. "A=B" is a symbolic representation of "'The expression A' equals 'The expression B'". I hope we can agree on that?
>if your view were correct we wouldn’t spend time teaching how to solve the equation.
What I actually said implies the exact opposite. You teach how to solve equations because that's the use case for equations as tools. That's not a bad thing, it's an extremely useful thing to teach.
But teaching how a tool is used is not the same as teaching the fundamentals of what a tool is. It can help in that goal, certainly, (and might even be required as a prerequisite) but it's not the same. It's exactly like you said:
>We teach algorithms like long division and the quadratic formula because they are relatively easy computations to learn but they don’t in any way lead students to fully grasping a concept.
It's not fair to blame students' "innumeracy" for not being able to derive "0=1" as "an equation without a solution", because they've successfully learned the thing that they were actually taught, that equations are "things with unknowns that we have to solve for". Of course generating a solution that has neither unknowns nor a solution is foreign, because everything they've learned about it as a tool goes against that.
(It's worth noting that there's another reason that the teachers teach this, one that's perhaps even more important for the school system; it's an easy thing to evaluate student understanding of. You can easily test whether a student can "solve" an equation, and return the correct answer. It's something you can get immediate, iterative feedback on. You can't really test if they actually grok a definition, because they can just parrot a definition with no understanding.)
Fundamentally, my argument is about language, not mathematics. You're saying that students aren't able to derive answers based on the definitions of terms, but not only are those definitions wildly divergent from their English meaning, they're divergent from the actual definitions the student are learning via practice.
Take, for example:
>There are values for which the two polynomials evaluate to the same number. Those are the solutions.
Values of x, you mean but didn't say. Because it's so heavily implied in the existence of an "equation to solve" that unknown quantities you are solving for are the ones written in the equation itself, that it's not even worth mentioning. But it's precisely this linguistic assumption that obscures what an equation actually is to students.
Actually the quadratic formula is a great example of exactly what I mean. I was taught it as a way to solve for roots of a second degree polynomial when "completing the square" didn't work. The terms in the quadratic formula are defined as the coefficients in the polynomial and the answer is written in the form "x = +-___". Looking back at that chapter in math, that along with the rational roots test for finding zeros in a higher order polynomial were genuine wastes of time. I don't mean the learning the intuition, I mean a question asking to list every single possible root given a second or higher order polynomial.
I was taught that using the quadratic formula to find roots is "cheating" when I tried using it before we had covered it in class. What exactly does "completing the square" test for other than your skills at mental arithmetic?
The reason why many student believe that polynomial doesn't factor is because teachers do a lot of hand waving when it comes to explaining what it means to have no rational roots of a polynomial. Few teachers will take the time to teach the foundations of the cartesian coordinate system and how complex solutions don't map easily on the typical plane of rational numbers. All students learn is if there's an "i" the answer is "no solution".
Zeros being the solutions of functions is a question on the finding roots of a polynomial chapter in basically every single high school algebra 2 class. It's a prerequisite to learning how to graph second order polynomial functions. Many students learn and forget how to do it before reaching calculus, let alone college.
I genuinely haven't done long division in the last decade. I struggled to help my younger cousin with it recently and had to relearn myself because it's such a useless algorithm in the age of computers. Certain multiples and powers I remember, because of how often I come across the numbers, but in general I will choose a calculator every time. I would even choose a calculator to double check my own work with a paper and pencil. At this point what is the value in doing the work by hand? In many cases a decimal to the hundredths is required as well.
When I hear the question, write down an example of an equation with no solution, my intuition and experience doesn't lead me to writing an incorrect equation. It leads me to think about writing a polynomial function that I know will have complex roots because I was taught the answer to that is "no solution", or writing a system of equation where x is a specific number while at the same time having an equation where that specific number can't be a part of the domain. More fundamentally, a system of equation with no solution is one where the two lines that are graphed are parallel.
I have to admit, my experience is a little biased as I was placed in accelerated math since elementary school. It wasn't difficult for me or my peers. My math class senior year was fundamentals of multivariable calc and linear algebra as a senior in high school, having finished AP calc bc the year before. I was far from the only one in that situation, there were at least 60 of us that year, some seniors and some juniors. I can't say I have experience teaching a full class but I have been tutoring high schoolers in math for over 8 years. Many of my students have also been in accelerated math, but not all of them. I don't think anyone tested out of multivariable calculus, but I did have a friend who tested out of linear algebra at reputable universities.
I know that I have some time and experience left before I feel confident in my own mathematical maturity, but I'd like to imagine I'm somewhat good at math. At the very least I wouldn't consider myself bad at math, even though I still feel like I am at the early stages of learning in specific branches of math.
It leads me to think about writing a polynomial function that I know will have complex roots because I was taught the answer to that is "no solution",
By the time one reaches calculus they have been taught that complex solutions are valid solutions. They just aren’t real solutions. Therein lies one of the problems teachers of mathematics have. Conveying the concept of the answer depending on what the current algebraic object one is working on. We have to hand wave do some brain washing because the nuances involved are far too complicated for the students to understand at this level.
I think it’s clear what is meant by the title. In case it’s not to you then “aftermath” refers to: aftermath of the start of the war and the accompanying sanctions and ongoing global realignment.
I agree with your overall point but at this time there are, thankfully, few believers of Scientology. Hopefully at some point in the future encountering an anti-vaxer will be as rare as encountering a Scientologist.
Explain how the COVID vaccine which does not 1) Prevent becoming infected with COVID, 2) Does not prevent spreading COVID to others, 3) Does not prevent COVID symptoms is a vaccine again? Also using the CDC definition that was specifically changed because the COVID vaccine didn't meet the definition of vaccine doesn't count.
Additionally how about we don't just jump to lumping huge numbers of people into out groups to feel better about ourselves and create divisiveness? Or go to Reddit where you can cheer on people dying.
Vaccines have never been expected to have 100% efficacy except by the under-informed. Just off the top of my head I can list typhoid, dengue and malaria as diseases that have vaccines with fairly low efficacy rates.
I have no medical expertise but I do know that overwhelmingly those who do have expertise took the “vaccine” and the booster shots. It does not matter to me what word you use to denote the COVID vaccine. I do hope anti-vaxers become rare. They are a menace and are incredibly selfish and self-centered.
That is not a vaccine. No one calls corticosteroids or antibiotics vaccines yet both meet the definition of lowering the probability someone will dies from an infection. So does taking care of yourself, getting exercise, eating a healthy diet, not sitting still all day, and a large swath of other things none of which are called vaccines.
"a substance used to stimulate the production of antibodies and provide immunity against one or several diseases, prepared from the causative agent of a disease, its products, or a synthetic substitute, treated to act as an antigen without inducing the disease."
Steriods and antibodies don't meet this definition. No vaccine has a 100% success rate, so reducing symptoms doesn't make it not a vaccine.
Let’s say we are in base 3. Typically the symbols used to construct representations for numbers using the standard positional notation would be the numerals {0, 1, 2}. We could use any 3 symbols but these three work well since most people are comfortable with standard base 10 notation. When one says 0 is 0 in base 3 (or any other base) what the mean is precisely this:
To represent the number 0 in base 3 we just write: 0.
To represent the number 1 in base 3 we just write: 1.
To represent the number 3 in base 3 we must write: 10.
In general, for any positive number n the symbols used to construct representations for number will be {0, 1, 2,. . ., 9,. . ., n-1}. In base n the number 0 is represented as: 0. In base n the number 1 is represented as 1. The number n is represented as: 10.
Thus it makes sense to always use the symbol 0 and 1 as part of the set of numerals in whatever base you are in and if one does this then 0 is 0 and 1 is 1.
Don’t have to take off your shoes with pre-check and I’ve found that the probability of being groped during a secondary screening is far lower with pre-check.
It is meaningful to point out. As the poverty rate declines worldwide more and more people will start consuming like Americans do. There aren’t enough resources in the world to sustain that long term unless we start mining asteroids and find a cheap way to send our garbage to the sun.
Russia’s economy is contracting so in what sense are sanctions not working? They aren’t 100% effective but they do appear to be having an effect and long term the effects, if sanctions remain, will force Russia into a pseudo vassal client of China.
Most economies worldwide have been contracting this year, along with price inflation on commodities including food.
If the intent of Russian sanctions was to increase prices of food and fuel in USA to harm the working poor in USA, then Russian sanctions are going great.
Russia’s economy, from what I’ve read, is contracting much more with sanctions than without them. I think it’s clear and obvious that the EU did not impose sanctions on Russia for the purpose of increasing prices of food and fuel for poor people in the U.S. The EU and the U.S. have announced the reason for sanctions. There are secondary effects of the sanctions and I believe that food price increases are more a result of global supply chain issues and Russia preventing Ukraine from fully exporting its products.
The problem is that a contracting economy doesn’t faze ordinary Russians outside of Moscow and St Petersburg all that much.
They were already poor. They’ll get a little poorer. But they’ll have enough to eat and enough to heat their homes. And they’ll still believe that Russian pride is intact, and that’s all that matters.
The entire sanctions approach fails so spectacularly because the capitalist Western ideology sees everything in terms of money and wealth. Outside of the west, most people will happily suffer if they believe that their pride and identity is being protected. This is something Americans just don’t understand at all.
The European gas sanctions specifically aren't working:
1. The gas is just being arbitraged via China. It's a commodity, it makes no sense to stop buying from source A and start buying from source B if B is willing to buy from A.
2. Gazprom is now making record-breaking profits.
Europe is now facing an unprecedented and mostly self inflicted crisis of the type that didn't occur even during the cold war. The Russians claim they are willing to fill up to half of Nord Stream 2 the moment Germany decides to accept it. If that's a lie then the energy crisis is real. If it's not a lie then European leaders are manipulating the public - sanctioning Russia so they feel good and virtuous whilst simultaneously claiming the sanctions are the other way around and there's nothing they can do to get more gas.
Sanctions on non-commodities like chips, specialist tools, infrastructure access? That can work and is probably having an impact. Sanctions on a commodity that can be moved around as a liquid? That can only work in theory if everyone does it despite the huge incentives to defect.
This economic war is being fought differently from the two sides: Russia is trying to squeeze as much money as possible out of Europe for gas and oil, Europe is trying to starve Russia of every thing else.
Of course Gazprom is making record profit it is currently an unregulated adversarial monopoly. The sanction effect are thing like how the other week Russia was unable to operate some of its warship for lack of replacement parts.
Economic war? This is a new cold war with countries* being compelled to choose sides and split banking channels. The war material manufacturers are creating career paths for their grand children.
* Turkey is attempting to make money from everyone, while remaining in NATO.
Which other Russian companies supply Russian gas to Europe in significant quantities? I thought Gazprom was pretty much the only company doing this but I have only a cursory knowledge of this matter.
The point of sanctions is a combination of two things. One is to punish a country and the other is to make a moral stance. If in the course of punishing the country a particular company makes more money that does not necessarily negate the efficacy of the sanctions or necessarily negate the morality of the sanctions.
Europe doesn't only use Russian gas, large parts of it hardly rely on Russian gas at all. For example the UK only gets ~5% of gas from Russia. Europe is dotted with LNG import terminals that can import gas via ship from anywhere.
The reason there's a price spike across Europe is that Germany is suddenly trying to buy the supplies it needs from elsewhere, and gas is a (mostly) fungible commodity. So there's effectively less supply now in Europe, but with same or greater demand (to fill storage), which means the global LNG price goes up for everyone.
"The point of sanctions is a combination of two things. One is to punish a country and the other is to make a moral stance."
For as long as any country is willing to arbitrage gas it simply punishes the buyer, not the seller. From this whole thread it seems the basic economics of this seem to have got lost somewhere so let's do a worked example:
1. Russia sells $1000 worth of gas per day to Europe and $500 worth to China (or India, or wherever).
2. Europe says to Russia we don't want to buy your gas anymore. Russia is now $1000/day poorer, but has lots of spare gas. Now Europe needs to buy $1000 worth from somewhere else like China.
3. China looks at its natural reserves and thinks, well, we don't really have much more to sell from local production. But hey, we have pipelines and LNG terminals and gas is the same no matter where it comes from. Russia, would you be willing to sell us all your spare gas?
4. Russia says sure, that'll be $1000/day. China says OK, we'll take it.
5. Now China turns around to Europe and says sure, we can sell you gas. That'll be $1200/day. Europe is desperate for gas and so says yes; they don't have many other options because most sellers can't meet such a huge block of demand.
Result: Russia makes the same money as before and doesn't care, China takes a profit off the top, Europe ends up shooting itself in the foot. But at least its politicians can feel virtuous.
What's happening isn't as pure cut as that example. Obviously, Europe is trying to get gas from lots of alternative locations, only some is being arbitraged via third party countries. And in reality the arbitrageurs would pay less because there's greater supply, pushing down the price. But in any situation where you're trying to buy something that is a fungible commodity sanctions like that only make sense if there are no groups that break them. The moment someone does they make huge profits. So it doesn't make sense in an international market, it'd be like trying to sanction oil or grain. Someone will just buy it at the new lower price, relabel it and sell it back to you.
As for the moral stance argument, German politicians have no moral authority to bankrupt the poor throughout Europe or break down the electricity grid. They were not elected to do that, and it is deeply immoral to do that even if it worked which it does not. The working classes of Europe are not cannon fodder to freeze to death so Scholz can take a "moral stance".
So Gazprom is a monopoly within Russia. Germany is not solely the reason for the EU sanctions. Other countries were involved in the sanctions decision. The Baltics and Poland, etc. have been far more vocal about wanting sanctions.
The fact is Russia hasn’t been stably non expansionist for centuries and it is they who have been aggressors in Moldova, Ukraine, Belarus, and Georgia. At some point it’s a reasonable response by those threatened by Russia (the Baltics) to respond by demanding action by their allies. Russia alone is at fault and it alone deserves ire.
The moral imperative clearly lies with those desiring sanctions. Europe is going through the painful process to divest itself now of Russian resources. This will pay dividends in the future.
But it would change the equation a lot. Russia in very reliant on its fossil fuel exports both to finance its imports and its internal economy; the EU would like sanctions to affect both of those fields, but one is better than zero.
If China pays less it just means the above situation works out to China's advantage even more than with my toy numbers, and even less to Europe's. Regardless of what numbers you use, Europe loses more than Russia. Sanctions only "work" (by the very low standards of workingness they're held to) if the sanctioned party needs to sell a lot more than the sanctioner needs to buy. With Europe that's clearly not the case - Russia can easily sell to others but Europe faces more or less immediate collapse. The advantage here is Russia's and the inability of the EU to recognize that they miscalculated threatens its population with dire levels of deprivation of economic destruction.
I don’t understand your point. The person you responded to isn’t generalizing to all people based off of school age kids. They noticed something about attitudes and speech in their kids versus when they were growing up and this observation comports with well a known fact. Namely, that regulation of speech can frame peoples’ thinking.
None of us are experts in everything and we need to try to make sense of things as we go about life. We all use examples/experiences in our lives to make general conclusions about the efficacy of policies.
If they're arguing against a general statement (by Fry) with a specific, anecdotal example, it's fair to question whether that specific example is generalizable.
I’m interested in the reasons for downvoting this comment. What is written is thought out and done in good faith. I don’t like regulation of word usage but it is true that propagandists can be effective at changing thinking patterns by changing speech patterns.
