Unpopular Opinion: Stuff taught at school

[Guest Mores]

36 minutes ago, anatess2 said:

Intro to Diff Eq is basically Alg2 on derivatives.  But we're not solving equations in the Intro class.  It's mainly understanding what it means and coming up with equations out of problems.

Yeah.  That's simply "pre-calculus" to us.  It is apparently a difference in terminology.  Compare --  Here's what I learned in "pre-calculus".

• What we had learned from Algebra as y' is now being called the "first derivative" now represented by dy/dx.
• Introduction to integrals.  We understand what a "differential element" is to determine the area under the graph as a function.
• We learned Newton's discovery that the Integral was actually the anti-derivative.
• We learned the simple means of manipulating the exponents and coefficients to go either direction and how best to determine the mysterious "+C" at the end of the integral.
• We learned the integral and derivatives of simple trigonometric functions.

Calculus was longer and deeper and used wilder and much more complex equations, and how to find very difficult integrals and derivatives.

Differential equations went further still.  The most clear cut things I can relate are (and to be honest, I can't remember anything else that was useful).

• Transformations (such as the LaPlace Transformation)
• The use of  ∂  as the partial derivative with respect to multi-variable calculus.  (multi-variable referring to the use of three or more variables in a single function).  And of course this would include the integral of multi-variable functions as well.

I believe the use of of the word "differential equations" is suffering from a translation error.  We did look at equations with regard to "differential" elements.  But those were certainly not what we refer to as "differential equations."

But everything you're describing as the actual definitions seems very much like what I had in high school.

Edited August 8, 2019 by Mores

[Guest Mores]

11 minutes ago, Fether said:

I did some more think and it was actually in the college calc class I took in high school that we did differentials.

Look at the definitions I gave @anatess2.  If you took it in high school, I doubt it was differential equations. To clarify, what was your degree in?

[Traveler]

If I were king and could determine the functions of public schools I would change a great deal:

#1. Children would be required to attend school - public, private or home for a minimum of 4 hours a day and be required to work in public service for a minimum of 2 hours a day - public service projects to be determined and provided by the community.

#2. Students would wear clean uniforms to class and there would be grooming standards (similar to the military) kids being home schooled would also be required to wear uniforms and meet grooming standards.  There would also be a working class uniform (similar to the military).

#3. Teachers would have full authority and power in the classroom.  Punishment however, would not be handled by the teacher except to expel a student from class.  Parents would be free to request their child in any class they can get their child to and by which the teacher would accept the student.  Teachers would have final say in what specific students could attend their class.  Teachers would be paid based on the number of students that passed the class curriculum (same for parents home schooling.  If a parent cannot find a class for their child or if a child cannot pass the curriculum tests then the state would take custody of the child's education/training.  A possible exception would be special needs children - but children with special needs  that cannot pass the curriculum for a class would not be allowed to enroll.  Teachers that cannot teach proficiencies to at least 2/3 of the class would lose their teaching credentials. 

#4. Students would be required to obtain levels of proficiency in basic fundamentals as listed:

    A. Math

    B. Reading and writing

    C. Computers

    D. Science

    E. Religion

    F. History

    G. Finance - general economics and personal finance

    H. Law and public behavior

    I. How to use firearms and other weapons to protect themselves and others

    J. Music and the arts

    K. language.  All students would be required to be proficient in a minimum of two languages. 

    L. Netritution.  Including how to plan a balanced and health diet

   M. Physical education - Students would be required to be physically fit.

   N. Basics of Trades and Professional careers.  Students would be required to be proficient in a trade.

#5. Students that complete required classes would be free to choose electives determined by the community or household in the case of homeschooling.  Electives could be taught in homes or churches or wherever = where the dress and grooming standards are maintained and 2/3 of the students continue to meet predetermined standards.

#6.  Starting in High School grade 10 through 12 students can choose exclusive trade education or if they meet proficiency standards they can choose trade and career education.

#7.  In order to be a citizen, serve in the military, own property (including a credit card or bank account), drive a powered vehicle or run for public office a student must complete High School.  Limited daytime driver permits (credit cards etc.) could be obtained when a student reaches grade 10 and they meet minimum requirements are under the supervision of a cognisant and responsible adult.  In other words if the student were to commit a crime (unpaid debt would be a crime) using a vehicle, credit card, bank account etc - the adult would be held accountable with the student and all temporary permits would be permanently suspended. 

 

The Traveler 

[Fether]

19 minutes ago, Mores said:

Look at the definitions I gave @anatess2.  If you took it in high school, I doubt it was differential equations. To clarify, what was your degree in?

The HS class was called Calculus AB. I took it so I didn’t have to take calculus in college. That little swirly symbol you used was definitely something we used often basis I that class. We used differentials to find velocity of a graph and things like that (honestly that was the only thing I remember. It was like 5 years ago and never took a calc class again)

[Fether]

25 minutes ago, Mores said:

Look at the definitions I gave @anatess2.  If you took it in high school, I doubt it was differential equations. To clarify, what was your degree in?

I also believe @anatess2 knows what she is saying, no translation error. Personally There were 3 Calc classes in my high school, I only took the first 2. I have not once in my time on this forum seen her be wrong (I mean that both in an honest sense and in a spiteful patronizing sense)

Edited August 8, 2019 by Fether

[Guest Mores]

5 minutes ago, Fether said:

I also believe @anatess2 knows what she is saying. I have not once in my time on this forum seen her be wrong (I mean that both in an honest sense and in a spiteful patronizing sense)

I don't believe she's wrong.  I believe it is a matter of terminology in differing cultures.  You, on the other hand...

8 minutes ago, Fether said:

The HS class was called Calculus AB. I took it so I didn’t have to take calculus in college. That little swirly symbol you used was definitely something we used often basis I that class. We used differentials to find velocity of a graph and things like that (honestly that was the only thing I remember. It was like 5 years ago and never took a calc class again)

That was not a differential equations class.  That was just calculus.  I don't know where you would have seen the del sign. But it's been longer for me than it was for you.  So, maybe there was a use of del in calculus that I'm forgetting. Or you could be mistaking it for the lower case delta which was certainly used in calculus.  I don't know.

What the heck do you mean by "find velocity of a graph"?  You can find the velocity fuction by taking the derivative of the acceleration function.  But that's just plain old calculus -- Intro to calculus to be honest.

Edited August 8, 2019 by Mores

[Fether]

8 minutes ago, Mores said:

I don't believe she's wrong.  I believe it is a matter of terminology in differing cultures.  You, on the other hand...

That was not a differential equations class.  That was just calculus.  I don't know where you would have seen the del sign. But it's been longer for me than it was for you.  So, maybe there was a use of del in calculus that I'm forgetting. Or you could be mistaking it for the lower case delta which was certainly used in calculus.  I don't know.

What the heck do you mean by "find velocity of a graph"?  You can find the velocity fuction by taking the derivative of the acceleration function.  But that's just plain old calculus -- Intro to calculus to be honest.

🤷🏻‍♂️  your probably right. Differential equation just sounds incredibly familiar from something that was at least discussed and taught  in my HS class.

And I just googled examples of differential equations and I’m more convinced I did it in HS

Edited August 8, 2019 by Fether

[anatess2]

Okay, I'm going by pure memory here...

3 hours ago, Mores said:

Yeah.  That's simply "pre-calculus" to us.  It is apparently a difference in terminology.  Compare --  Here's what I learned in "pre-calculus".

• What we had learned from Algebra 1 as y' is now being called the "first derivative" now represented by dy/dx. - Alg 2 touched on this
• Introduction to integrals.  We understand what a "differential element" is to determine the area under the graph as a function. - Alg2 and Trig introduced the concept (interestingly, only Trig is considered pre-calc)
• We learned Newton's discovery that the Integral was actually the anti-derivative. - Don't remember Newton's discovery but Integrals/anti-derivatives are in Trig and Intro to Diff Eq
• We learned the simple means of manipulating the exponents and coefficients to go either direction and how best to determine the mysterious "+C" at the end of the integral. - Intro to Diff Eq
• We learned the integral and derivatives of simple trigonometric functions. - Trig and Intro to Diff Eq

Calculus was longer and deeper and used wilder and much more complex equations, and how to find very difficult integrals and derivatives. - Calculus is now figuring out how to solve the equations we came up with in Intro to Diff Eq and then some.

Differential equations went further still.  The most clear cut things I can relate are  - Okay, so what you have is Calculus is a different class than Differential Equations.  In my college Calculus is divided into 4 - Differential Calculus, Integral Calculus, Analytic Geometry, and Vector (This has an official name that I can’t remember.  I just know it’s a super tough class that all I can remember is Vectors).

• Transformations (such as the LaPlace Transformation) - taught in both Diff and Integ Calc
• The use of  ∂  as the partial derivative with respect to multi-variable calculus.  (multi-variable referring to the use of three or more variables in a single function).  And of course this would include the integral of a partial derivative as well. - also touched in both Diff and Integ Calc

I believe the use of of the word "differential equations" is suffering from a translation error.  We did look at equations with regard to "differential" elements.  But those were certainly not what we refer to as "differential equations." - We call anything that is a derivative of any order and its integral a differential equation.  Basically, dx/dy = f(x) is a differential equation and Calculus us a study of differential equations. That’s why @Vort saying differential equations is like saying thermodynamics to physics doesn’t make sense to me.

But everything you're describing as the actual definitions seems very much like what I had in high school. - I'm sure it is, because I didn't have to retake any high school courses or college math courses to get my transcripts US accredited. 

 

Edited August 8, 2019 by anatess2

[Guest Scott]

1 hour ago, anatess2 said:

Our Home Econ class did not cover credit cards.  I guess it just wasn’t common at that time in the Philippines.  They still don’t cover that in today’s Home Econ and credit card debt is skyrocketing in the Philippines.  They really should adjust that curriculum.

Home Econ is 2 years in my high school for the girls.  Boys only take it for 1 year because they get Shop for the next year (which should be called Home Maintenance if you ask me) - they learn plumbing carpentry and engine in Shop.

 

I didn't take home ec, but we learned about credit cards and interest in both accounting and math class.   We'd calculate interest in both.    In accounting we also learned to balance a checkbook, as well as the dangers of credit cards and unsecured debt.  

[anatess2]

19 minutes ago, Scott said:

I didn't take home ec, but we learned about credit cards and interest in both accounting and math class.   We'd calculate interest in both.    In accounting we also learned to balance a checkbook, as well as the dangers of credit cards and unsecured debt.  

I didn’t have accounting, I had business math.  It does touch on balancing checkbooks but is more focused on balancing ledgers.  It also didn’t touch on credit cards but it did teach credits and debits and interest rates including compounding ones but I remember this as focused on investment  rather than debt.

[Guest Mores]

54 minutes ago, anatess2 said:

Okay, I'm going by pure memory here...

Yeah, I guess we had different names for things because we covered analytic geometry in Algebra 2.  

As far as Newton:  People say he invented calculus.  That wasn't the whole story.  As I heard it:

The derivative (your differential equations) was already invented prior to Newton.  The use of integrals was also invented prior to Newton.  What he was responsible for was discovering the relationship between the two.  So, one could say that was when calculus was created.  But if the elements already existed and were used in the exact same manner as they are now, was it really "invented" by recognizing their relationship?

[anatess2]

1 minute ago, Mores said:

Yeah, I guess we had different names for things because we covered analytic geometry in Algebra 2.  

As far as Newton:  People say he invented calculus.  That wasn't the whole story.  As I heard it:

The derivative (your differential equations) was already invented prior to Newton.  The use of integrals was also invented prior to Newton.  What he was responsible for was discovering the relationship between the two.  So, one could say that was when calculus was created.  But if the elements already existed and were used in the exact same manner as they are now, was it really "invented" by recognizing their relationship?

Analytic Geometry in college was more Trig+Calc on Mountain Dew... or maybe I should just go ahead and say Meth.  This was one of those classes that ordinary mortals (non-engineers) that survive Integ Calc are found to be filtered out of the engineering program.

Alas - in all my years of engineering I've never heard of Newton inventing calculus.  I suck.  So yes, I'd say Newton invented calculus if he's the guy that realized integral is to differential in the same manner that division is to multiplication... only because you kinda need that concept to make sense out of the whole thing.  

 

[Vort]

8 hours ago, Mores said:

Here's the problem with it being taught in schools.

In general, a public school teacher should have a good fundamental understanding of a course BEFORE s/he teaches it. Most Americans, including public school teachers, don't have a good handle on personal finance. It's like having public school teachers teach sex education. Do you really trust that they (a) know what they're talking about and (b) can present that information within a moral framework that puts things in a proper perspective? I certainly do not.

[Vort]

6 hours ago, Scott said:

Even the Church doesn't seem to hammer on avoiding debt as much as they used to.  When I was younger, Church leaders would constantly say that the only reasons you should ever go into debt were for a modest house and for an education.  That was it.   It was mentioned all the time in meetings, conferences, etc.  Although avoiding debt is still occasionally mentioned, it doesn't seem to be mentioned as much anymore and when it is, it doesn't seem to be mentioned as much that the only reasons to go into debt should be for a modest house and education.

To some extent, I think this is an "ears to hear" problem. Every General Conference includes a statement like the following:

The Church follows the practices taught to its members of living within a budget, avoiding debt, and saving against a time of need.

Anyone who actually pays attention gets this at least one time every six months. If it's not stressed more often than that (and I'm pretty sure it is), perhaps that's because our leaders have decided that if people don't listen, eventually it's useless to keep hounding them. You can't teach calculus to a child who doesn't yet know arithmetic.

[Vort]

5 hours ago, Mores said:

6 hours ago, anatess2 said:

Ok, in my High School, Pre-Calc spans 2 classes - Trig and Intro to Diff Eq.  All students have to take Trig.  Students applying for Engineering college take Intro to Diff Eq.  Students enrolling in architecture take Drafting.  Students enrolling in other college courses or trade school or no idea what they’re gonna do with the rest of their lives take Numerical Methods.

You're not really giving definitions.  What do you mean when you say "differential equations"?

6 hours ago, Fether said:

I took pre-calc in HS and we covered differentials.

I'd ask you the same question.

Good point. Differentials are basic "math of the infinitesimal", absolutely foundational for understanding any kind of calculus. Differential equations are a whole 'nother animal.

[Guest Mores]

6 minutes ago, Vort said:

Differential equations are a whole 'nother animal.

Forgive me as I recover from the fingernails on the chalkboard.   

[Vort]

5 hours ago, Mores said:

Differential equations went further still.  The most clear cut things I can relate are (and to be honest, I can't remember anything else that was useful).

• Transformations (such as the LaPlace Transformation)
• The use of  ∂  as the partial derivative with respect to multi-variable calculus.  (multi-variable referring to the use of three or more variables in a single function).  And of course this would include the integral of multi-variable functions as well.

Differential equations are awesome (and I say that as someone who received a D—my only college D—in the class; what can I say? I was an undisciplined undergraduate student). They are self-referential equations where you have an equation that uses a function and also its derivatives. For example, assume we have a function F(x) where:

F + dF/dx + d²F/dx² = C

Wha.....? That's mind-bending! How do you even begin to go about solving such a problem? Then make F a function of two variables (e.g. F(x,y)) and try that. My brain fogs up just thinking about it (and I ostensibly know differential equations, since after my D I actually studied them closely to figure them out, at least at a Khan Academy level).

Do you want to launch rockets into outer space? Better brush up on your differential equations.

I very strongly suspect that neither @Fether nor @anatess2 took a high school class that broached differential equations in any sense. I could be wrong...but I would bet a substantial amount of money that I'm right.

[Guest Mores]

2 minutes ago, Vort said:

Then make F a function of two variables (e.g. F(x,y)) and try that.

That just reminded me of composed functions f o g (x).  I don't know if I've ever used that in professional life.  Was that in diff. eq.?  or was that Algebra?  I can't remember now.

[Vort]

3 hours ago, Mores said:

As far as Newton:  People say he invented calculus.  That wasn't the whole story.  As I heard it:

The derivative (your differential equations) was already invented prior to Newton.  The use of integrals was also invented prior to Newton.  What he was responsible for was discovering the relationship between the two.  So, one could say that was when calculus was created.  But if the elements already existed and were used in the exact same manner as they are now, was it really "invented" by recognizing their relationship?

My understanding (so don't believe it) is that Leibniz developed his very elegant calculus notation around the same time as Newton, possibly before or possibly after, but when Newton found out about it, he rushed to publish HIS calculus first so that he would get credit for inventing it, then spent the rest of his life taking a spiteful attitude toward Leibniz (who by all accounts was a very decent fellow). It seems that Newton, despite his fervid religiosity and unparalleled intelligence, wasn't a very nice fellow in many cases.

Leibniz may have lost that battle, but he won the war. Calculus taught today features Leibniz notation as a central feature. Anyone who wants to do calculus by Newton's methods has to go research Newton's notes and then try to figure out his arcane methodology.

[Guest Mores]

11 minutes ago, Vort said:

My understanding (so don't believe it) is that Leibniz developed his very elegant calculus notation around the same time as Newton, possibly before or possibly after, but when Newton found out about it, he rushed to publish HIS calculus first so that he would get credit for inventing it, then spent the rest of his life taking a spiteful attitude toward Leibniz (who by all accounts was a very decent fellow). It seems that Newton, despite his fervid religiosity and unparalleled intelligence, wasn't a very nice fellow in many cases.

Leibniz may have lost that battle, but he won the war. Calculus taught today features Leibniz notation as a central feature. Anyone who wants to do calculus by Newton's methods has to go research Newton's notes and then try to figure out his arcane methodology.

Now you've got my curiosity up.  I've never verified the story I heard.  I gotta go look it up.  Sounds like a hoot.

 

[anatess2]

1 hour ago, Vort said:

Differential equations are awesome (and I say that as someone who received a D—my only college D—in the class; what can I say? I was an undisciplined undergraduate student). They are self-referential equations where you have an equation that uses a function and also its derivatives. For example, assume we have a function F(x) where:

F + dF/dx + d²F/dx² = C

Wha.....? That's mind-bending! How do you even begin to go about solving such a problem? Then make F a function of two variables (e.g. F(x,y)) and try that. My brain fogs up just thinking about it (and I ostensibly know differential equations, since after my D I actually studied them closely to figure them out, at least at a Khan Academy level).

Do you want to launch rockets into outer space? Better brush up on your differential equations.

I very strongly suspect that neither @Fether nor @anatess2 took a high school class that broached differential equations in any sense. I could be wrong...but I would bet a substantial amount of money that I'm right.

Okay, my understanding of what you call "differential equations"...

So, I'm trying to think of an analogy...

okay, so it's like x = 4 * 1 as multiplication and x = 4 * 4 is still multiplication but x = (4 * 4)^10/(3 * 4) is not multiplication anymore, it's called multiplicative equations such that x = 4 * 1 is not a multiplicative equation (doesn't make sense but I'm trying to find an analogy).

So, this is where I'm confused.  I'm wondering how complex does a differential have to be for it to qualify for what you'd call "differential equations".... because in  my neck of the woods, if it's a differential and it has an = sign in it, it's a differential equation.  So dx/dy = f(x) is a differential equation.

And I'm confused why you have a class called "Differential Equations" class which is after "Calculus" class signifying Calculus doesn't have differential equations... I don't understand how you can have a Calculus class without differential equations... 

Edited August 8, 2019 by anatess2

[Texan]

1 hour ago, Mores said:

Now you've got my curiosity up.  I've never verified the story I heard.  I gotta go look it up.  Sounds like a hoot.

 

These multiple discoveries and simultaneous inventions happen sometimes.  The classic example is Darwin and Wallace, of course, but a few years ago I had to do some reading on the discovery of Neptune.  

The existence of Neptune was visually confirmed by telescope after its position had been predicted by mathematics, which I find astounding.  But two people (a Brit and a Frenchman) were working out the math at the same time, and there was some dispute over who should get the credit.  The French guy won, and then I guess a hundred years after the discovery one of the Brit's letters was found in South America and it became clear that he had come late to the party after all.  Or something along those lines.  The story is a lot juicier then the Leibnitz/Newton thing.  Check out the "Discovery of Neptune" article in Wikipedia.  It's not a nail-biter, but it kept me awake one night.

[Vort]

22 minutes ago, anatess2 said:

So, this is where I'm confused.  I'm wondering how complex does a differential have to be for it to qualify for what you'd call "differential equations".... because in  my neck of the woods, if it's a differential and it has an = sign in it, it's a differential equation.  So dx/dy = f(x) is a differential equation.

For a given function F, if you have an equation that uses F and its differential (e.g. F + F' = 0, where F' represents the differential of F), THAT is a differential equation.

Now, if you want to be hyperliteral, then any equation involving F is a differential equation if F is a constant. You can find a great many trivial cases of simple function-based equations that might qualify as differential equations in some limited sense. But that's not what I'm talking about when I say "differential equation".

An extremely simple example is the equation F = F', or in other words, F(x) = d[F(x)]/dx. What's the solution to that differential equation? Obviously, F(x) = e^x is a solution. That's a differential equation that can be identified merely by inspection; a reasonably clever student who's mid-semester in introductory calculus could figure that one out. Other differential equations are far more complex. A differential equations class at a US university presupposes a solid base in introductory calculus, and in effect teaches you a bag of tricks for dealing with various differential equation types.

22 minutes ago, anatess2 said:

And I'm confused why you have a class called "Differential Equations" class which is after "Calculus" class signifying Calculus doesn't have differential equations... I don't understand how you can have a Calculus class without differential equations... 

When I talk about a "calculus class", I mean a first-year introductory calculus class, one of the first two or three semesters of engineering-level calculus. Sure, "calculus" includes differential equations, just like "algebra" includes calculus. But when I say "algebra", I'm not talking about calculus. I'm talking about manipulation of equations with an unknown, maybe even using trig. That's about it. In the same sense, I understand "calculus" (in this sense) to mean the general idea of understanding limits, differentials, and integral sums, and not necessarily higher-order stuff like F + aF' + bF'' = c.

[Vort]

@anatess2, Wikipedia offers a pretty solid explanation of what is meant by a Differential equation.

[anatess2]

33 minutes ago, Vort said:

For a given function F, if you have an equation that uses F and its differential (e.g. F + F' = 0, where F' represents the differential of F), THAT is a differential equation.

Now, if you want to be hyperliteral, then any equation involving F is a differential equation if F is a constant. You can find a great many trivial cases of simple function-based equations that might qualify as differential equations in some limited sense. But that's not what I'm talking about when I say "differential equation".

An extremely simple example is the equation F = F', or in other words, F(x) = d[F(x)]/dx. What's the solution to that differential equation? Obviously, F(x) = e^x is a solution. That's a differential equation that can be identified merely by inspection; a reasonably clever student who's mid-semester in introductory calculus could figure that one out. Other differential equations are far more complex. A differential equations class at a US university presupposes a solid base in introductory calculus, and in effect teaches you a bag of tricks for dealing with various differential equation types.

When I talk about a "calculus class", I mean a first-year introductory calculus class, one of the first two or three semesters of engineering-level calculus. Sure, "calculus" includes differential equations, just like "algebra" includes calculus. But when I say "algebra", I'm not talking about calculus. I'm talking about manipulation of equations with an unknown, maybe even using trig. That's about it. In the same sense, I understand "calculus" (in this sense) to mean the general idea of understanding limits, differentials, and integral sums, and not necessarily higher-order stuff like F + aF' + bF'' = c.

Well, there's the difference.  Because we don't have "Calculus" class or "Differential Equations" class in college.  We have Differential Calculus, Integral Calculus, Analytic Geometry, and one other class I can't remember the name of that talks a lot about vectors.  All these classes deal with differential equations of any order including the "introductory" first order.

Edited August 8, 2019 by anatess2