Is This the Key/Secret to Learning Math?

[Mister T]

I agree that teaching the "laws of math" is crucial but we need to distinguish teaching the concept and teaching its name. You don't really need to talk about associativity or distributivity formally before you've actually shown the student or pupil that there are several cases where it holds and cases where it doesn't.

First the thing, then the name of the thing. Famously said repeatedly by Arnold Arons.

 

[Greg Bernhardt]

That is certainly an important part of the effort. Unfortunately the way administrators, parents, and students treat teachers, and the way students are not held accountable for learning interfere with that effort. Most teachers have their spirits broken. Or never consider adopting teaching as a profession in the first place because of these issues and the low pay.

My wife is a first year elementary school math teacher. Parents disrespect her, administration disrespects her, she works 10-12 hour days and makes a bit more than a fast food manager. My eyes have never been wider on the primary education system in my life. The tragedy is that is is amazing with kids and is a great teacher. Give her the support, the respect, the tools and she becomes a life changer for these kids.

 

[stevendaryl]

I guess because it's a pet topic of mine, what struck me strongest about the original post wasn't the "laws of mathematics" business, but this line:

This month, she received a copyright for a diagnostic test that she says can assess specific gaps in students’ math knowledge in minutes

I don't have any idea about how to teach math, but I do know what drives kids out of math completely, and that's exactly "gaps in student's math knowledge". My experience with trying to help kids with math, at least, shows me that so often, kids have trouble with math at one level because they have never completely understood the foundations that it is supposed to be built on. If you don't know how to add, subtract and multiply, then you're going to have trouble doing fractions. If you don't really feel comfortable with fractions, you're going to have trouble in trigonometry and algebra. If you really don't understand algebra, you're going to have an enormous trouble in learning calculus. Math in particular is cumulative, so a gap in fundamentals can haunt a student for the rest of his academic career.

​

 

[Mister T]

If you don't know how to add, subtract and multiply, then you're going to have trouble doing fractions. If you don't really feel comfortable with fractions, you're going to have trouble in trigonometry and algebra. If you really don't understand algebra, you're going to have an enormous trouble in learning calculus. Math in particular is cumulative, so a gap in fundamentals can haunt a student for the rest of his academic career.

The other side of that coin is, because of the cumulative nature, students have repeated opportunities to pick up those gaps that slipped through in the past. Many times students won't really learn a topic until they find they need it to learn another topic.

 

[ZapperZ]

The real key to understanding math is to love it. If you love it, you are going to put in the time and effort to know more, to understand more. Also, everyone learns in different ways so anyone way of teaching math is not going to work for everyone.

I completely disagree with this.

I hate math, and had always hated learning math. However, I was quite good at it in college, so much so that a few instructors thought that I should pursue a theoretical physics career. When I told them that I can't stand math, they were surprised.

So no, it is not a necessary criteria to "love it" to be good at it.

Zz.

 

[mrnike992]

Just to throw in my 2¢ worth here, I think there's a pretty obvious answer to why her methods are so successful.. I doubt it has much at all to do with her teaching style, and more to do with the fact that she has dedicated, one-on-one access with the under-performing students. If all teachers could work after school one-on-one, or even with smaller class sizes, I believe most would be capable of getting those students up to speed.

 

[Jeff Rosenbury]

I remember being taught in the electronics in the AIr Force (1980s). I asked our instructor how he did when he took the class. He said he had never taken it. He simply followed the T.O. (technical order) and his training in training others. In 16 weeks I picked up more technical knowledge than in two years of E.T. coursework. (There were other things learned in college, but for pure technical training, the Air Force taught what it needed to quickly and efficiently.)

The Air Force didn't think teaching was some big mystery. They didn't even require the teacher know the subject, just follow a prearranged lesson plan.

 

[chiro]

I sense that high schoolers interpret mathematics in a very different way to what should exist.

Mathematics largely captures variation in an organized and consistent way and the study of mathematics is intended to lead to an understanding of said variation (again - in an organized and consistent way).

This is the real power of mathematics and I sense that the rules obfuscate this real understanding.

This is particularly notable when you look at normal mathematics problems. In the context they are presented the understanding of variation is obfuscated by ridiculous problems wasting both the teachers and students time and presented in such a disorganized and unconnected way that many students forget everything a couple of weeks into their final break.

Focusing on the rules per se doesn't get to understanding the variation as well as understanding how more importantly to think about how this variation can - and does apply, to the real world.

They get so caught up in memorizing sine, cosine, tangent, quadratic formula, derivatives, different types of triangles and other stuff that the variation and its context is completely overlooked.

I did a couple of weeks doing student teaching in a very good school and unfortunately I saw first hand just how bad this can be.

Instead of having mathematics being a used to understand variation and consistency in many ways - which is also a survival attribute when you realize that people are constantly bombarded with information, claims and logic in which they need to be able to sort the BS from the non-BS, mathematics is instead a bunch of disconnected and seemingly random (and pointless) ideas shoveled down kids throats for which many of them will soon forget and far more will never end up appreciating it (mathematics) for what its value is - including the ability to make sense of the world and be able to mount some sort of critical defense to all of the BS information that people have to navigate through and fight against.

This is what mathematics is about and this is where it's value lies - it lies in being able to look at variation and uncertainty and navigate through it in the best possible way - something which most high school students never end up figuring out - and partly because of how the subject has been stripped of its meaning and been used to facilitate lots of garbage that does the opposite to what it should do in terms of facilitating the above.

 

[UncertaintyAjay]

I completely disagree with this.

I hate math, and had always hated learning math. However, I was quite good at it in college, so much so that a few instructors thought that I should pursue a theoretical physics career. When I told them that I can't stand math, they were surprised.

So no, it is not a necessary criteria to "love it" to be good at it.

I certainly didn't mean that it is a necessary criteria. You don't need to love something to be good at it ( biology, in my case) but if you love something, you will try to be good at it.

 

[gleem]

Learning math or lack of is further exacerbated in the home. Parents are continually requested to participate in the education of their children. But math teaching techniques have become so unfamiliar when a child asks for help the parent and the child become frustrated. The parent not understanding or appreciating the technique may refuse to help leaving the child in a quandary, " how can I learn it if my parents cannot or will not help" or if the parent tries to help ends up either confusing the child or causing him/her to just shut down.

This can be improved by maintaining a consistent teaching technique over a span of time that includes the educational experience of the parent and the child, about 30 years. How many time have math programs changed in the last 30 years?

 

[clope023]

The article doesn't really say much of what she does, so it's hard to have any thoughts on it. Mathematics doesn't have "laws".

Math follows the laws of logic last I checked.

 

[Jeff Rosenbury]

The article doesn't really say much of what she does, so it's hard to have any thoughts on it. Mathematics doesn't have "laws".

Not only does math have laws, it also has regulations, goals, and prizes. Here is an http://www2.ed.gov/programs/racetothetop/executive-summary.pdf.

Math follows the laws of logic last I checked.

Not as far as I can tell. But I admit I've never understood bureaucrats, so maybe I'm mistaken.

 

[clope023]

Not as far as I can tell. But I admit I've never understood bureaucrats, so maybe I'm mistaken.

Cute, but I'm not referring to how math education is managed.

 

[Jeff Rosenbury]

Cute, but I'm not referring to how math education is managed.

Sorry; I've been struggling with concrete thinking.

 

[Mark44]

Mathematics largely captures variation in an organized and consistent way and the study of mathematics is intended to lead to an understanding of said variation (again - in an organized and consistent way).

This description of what mathematics does and how it should be used is so high-level (a "50,000 foot view"), that is not very useful, IMO.

This is particularly notable when you look at normal mathematics problems. In the context they are presented the understanding of variation is obfuscated by ridiculous problems wasting both the teachers and students time and presented in such a disorganized and unconnected way that many students forget everything a couple of weeks into their final break.

I'm not convinced that an understanding of variation is important. Maybe you can give some examples of what you mean. I agree that concepts need to be organized, with connected themes running through the concepts, and that problems that waste time should be eliminated, but could you elaborate on the kinds of problems you're talking about?

Focusing on the rules per se doesn't get to understanding the variation as well as understanding how more importantly to think about how this variation can - and does apply, to the real world.

They get so caught up in memorizing sine, cosine, tangent, quadratic formula, derivatives, different types of triangles and other stuff that the variation and its context is completely overlooked.

Are you arguing against the memorization of these concepts? If so, I strongly disagree, as these are the fundamental concepts that need to be in a student's "toolbox" so that he/she can tackle applied problems that use these concepts.

Going back to my earlier analogies of music and sports, if a guitar player hasn't spent many hours learning how to shape (for example) a Bm chord followed quickly by D and A chords, the song being played won't sound good. And similarly, if each player in a football offensive team hasn't spent many hours committing each play to memory, the outcome for that team is not favorable. Why would things be different in the teaching of mathematics or any other academic study?

If a student in physics doesn't have the sine, cosine, and tangent functions and quadratic formula committed to memory, said student will not likely be able to even start applied problems involving multiple forces acting on an object, or involving an object that is thrown through the air.

You mentioned "understanding the variation" several times, so I gather that it is important to you. You didn't expand on what this means to you, but by itself, I don't see how this understanding is helpful to students of mathematics.

 

[Student100]

Math follows the laws of logic last I checked.

There's no such thing. This is a philosophy question, so I won't get into it. The word law shouldn't exist in the formal/natural sciences. Again, if you saw my earlier post, we could argue semantics all day long but I conceded it isn't useful to the thread.

Not only does math have laws, it also has regulations, goals, and prizes. Here is an http://www2.ed.gov/programs/racetothetop/executive-summary.pdf.

There are lots of those laws, I must agree.

 

[Mark44]

Math follows the laws of logic last I checked.

There's no such thing. This is a philosophy question, so I won't get into it.

Are you objecting to the word "laws"? Certainly proofs in mathematics follow the rules of logic

The word law shouldn't exist in the formal/natural sciences.

Why not? We already have the Law of Sines, Law of Cosines, and the Law of Pythagoras in mathematics, and Ohm's Law and Kirchhoff's Law in physics. I'm sure there are lots more.

 

[clope023]

There's no such thing. This is a philosophy question, so I won't get into it. The word law shouldn't exist in the formal/natural sciences. Again, if you saw my earlier post, we could argue semantics all day long but I conceded it isn't useful to the thread.

The word law might be a mis-nomer but they tend to be followed as such; 'Law's' of non-contradiction, causality, and such like are absolutely followed in math and physics.

 

[Student100]

Are you objecting to the word "laws"? Certainly proofs in mathematics follow the rules of logic

I am objecting to the word law, of course the foundations of mathematics follow logic.

Why not? We already have the Law of Sines, Law of Cosines, and the Law of Pythagoras in mathematics, and Ohm's Law and Kirchhoff's Law in physics. I'm sure there are lots more.

It does more harm than good. I understand, and you understand, what the context of the word is. Many students, especially in high school and introductory science courses, don't. It leads to confusion about what science is actually trying to do, what models actually say, and belief that scientific "laws" are somehow infallible simply due to a poor choice of words.

The word law might be a mis-nomer

That's all I'm arguing. If this teacher is drilling the "laws" of mathematics to students, these students will be in for a shock when these "laws" are no longer viable or true. I doubt she's accurately conveying what the word actually means in context, because so many others references and teachers also fail.

 

[Mister T]

Mathematics largely captures variation in an organized and consistent way and the study of mathematics is intended to lead to an understanding of said variation [...]
They get so caught up in memorizing sine, cosine, tangent, quadratic formula, derivatives, different types of triangles and other stuff that the variation and its context is completely overlooked.

I agree completely. This situation is a result of the way our American society has attempted to remedy the very problem we've created. We have failed to support and respect our teachers, and we have failed to hold our children accountable for learning. When confronted with the evidence of poor student performance in comparison to other countries, our response has been what I call educationism. We attempt to hold our teachers accountable by dissecting their subjects into pieces, followed by measurements of how well students perform each piece. So we end up with a lot of teachers who teach only the pieces. This seems to be especially true of mathematics.

The latest version of educationism is placing an emphasis on student learning outcomes (SLO's). By listing the SLO's associated with the most populated college courses taught in the state of Texas, the Texas Higher Education Coordinating Board (THECB) has now assured us that each course is equivalent, regardless of the instructor or college where it's taught. Administrators at these colleges, in response, are now making instructors not only list these SLO's in their course syllabi, but show evidence, called assessment data, that each SLO is being addressed in the teaching of the course. And that each instructor is assessing each student's performance on each SLO. This will further assure us that these courses are equivalent. Of course, if the data show poor student performance, the instructor is supposed to make improvements to the curriculum, the teaching methods, or the assessment methods. If the data show good student performance the instructor is supposed to make improvements to the assessment methods because, presumably, the assessment instrument is not rigorous enough.

With my assessment data I have been able to show that everything is good and nothing is bad. As this process was becoming institutionalized during the last decade I warned anyone within earshot that the system would never measure anything of value. I stated obvious things that any instructor could do to produce good assessment data in the absence of good teaching and good learning. I was criticized as someone who is really good at coming up with ways to beat a system. So, I stopped.

 

[Rx7man]

I'm extremely surprised no one has brought this KEY failure in the educational system

Boredom!

This is especially true for brighter students.. They 'get' the concept the first time around... but they spend the entire rest of the year rehashing it.. by the end of the year, they're bored out of their minds, they've only learned half of what the could have learned, and have lost motivation.
For me, this was particularly true of English class... I think it was from about grade 2 to grade 10, I pretty much hashed and rehashed ad infinitum what verbs, nouns, and adjectives are.. 8 freaking years of this.. I always did poorly in English, it was an absolute bore.. Then I finally took English 101 in college, most of my classmates were Asian imports, and the teacher didn't really pay a whole lot of attention to sentence structure, etc.. he focused on reading comprehension, etc.. I actually got a B in that class!
I was homeschooled for grades 4-8, then I went to an unaccredited academy for 9-11... They had a terrible teaching program, and I ended up redoing grade 11 in public school. I was placed in Math 11, with about Math 8 skills, I had NO concept of algebra and my fractions were pretty shakey. I had a great teacher and she spend a little time with me to get my feet back under me.. We were on the quarter system, so each quarter was about 10 weeks of school with 2 classes per quarter.. It meant I studied each of the 2 subjects for 3 hours a day.. this was good, I was able to immerse myself into the subject and get to the bottom of it.. I think I had a high C.. I took Math 12 the next quarter and got an A, In the other science courses I took I got 98% in Physics 11, 90% in Calculus 12 (an elective), and 94% in Biology 12 (beating the teachers daughter).. I was interested and excited about the subjects and I applied myself
Then came university.. Like most people, I got distracted, class sizes were about 300 students, I was shy, so didn't really know anyone, which also meant I didn't have any 'competitors'... I was sedentary, I couldn't do any of the things I loved doing (fiddling with mechanical stuff, cattle, dirt bike rides,etc) and so I ended up just getting a string of dead-end jobs to pay bills.I think the educational system has to get it's priorities straight.. Give the kids who smoke weed behind the school enough skills to count out the dime bag, and leave it at that.. stop trying to make physicists out of them... The kids who want to learn, fast track them whenever you can.. don't bore them into smoking weed behind the school!

 

[Mark44]

Are you objecting to the word "laws"? Certainly proofs in mathematics follow the rules of logic

I am objecting to the word law, of course the foundations of mathematics follow logic.

Why not? We already have the Law of Sines, Law of Cosines, and the Law of Pythagoras in mathematics, and Ohm's Law and Kirchhoff's Law in physics. I'm sure there are lots more.

It does more harm than good. I understand, and you understand, what the context of the word is. Many students, especially in high school and introductory science courses, don't. It leads to confusion about what science is actually trying to do, what models actually say, and belief that scientific "laws" are somehow infallible simply due to a poor choice of words.

I don't believe that changing the word "law" to "property" or "principle" (or whatever) would decrease the confusion. The problem is, I believe, in not listing the conditions under which the law can be applied. Newton's Second Law of Motion is F = ma, which is not valid for an object whose mass is changing.

In any case, there are many laws that are named after the persons who discovered them. See the wiki article for a long list of such laws: https://en.wikipedia.org/wiki/List_of_scientific_laws_named_after_people. It makes no sense to me to revise history by renaming, say, Kirchhoff's Law or Boyle's Law.

The word law might be a mis-nomer

That's all I'm arguing. If this teacher is drilling the "laws" of mathematics to students, these students will be in for a shock when these "laws" are no longer viable or true. I doubt she's accurately conveying what the word actually means in context, because so many others references and teachers also fail.

It seems to me that your disagreement is due to the way these "laws" are presented (or misrepresented) rather than with the use of the word "law" per se; i.e., without any "fine print" giving the limitations. One example is saying that ##\sqrt{ab} = \sqrt{a}\sqrt{b}## without also listing the restrictions on a and b.

Also, do you have any evidence that she is inaccurately conveying these laws? Just because some teachers and some references are sloppy doesn't mean you can extrapolate this sloppiness to every teacher.

 

[MidgetDwarf]

I think the secret is having knowledgeable teachers in the classroom and changing the culture between: parents, students, and administration. Of major in importance is access to textbooks for students. Many children still do not have textbooks for classroom use and take home purposes. I know atleast 20 people who struggled with math, statistics were there highest math class taken, majored in the arts, and who are now teaching mathematics to young students.

 

[Mister T]

I'm extremely surprised no one has brought this KEY failure in the educational system

Boredom!

That's not a failure of the educational system. It's a failure of our society. Students who are bored with learning could be removed from those classes in which they're bored and put to work doing something productive. But our society will not allow that. Instead we as parents and educators coddle these "bored" students and pass them along to the next grade, where of course they continue to be bored.

If you're bored, guess what? It's not anyone else's fault. No one else is responsible for your failures or your emotional state. Find something interesting to do, a way to do it, and stop whining.

 

[clope023]

That's all I'm arguing. If this teacher is drilling the "laws" of mathematics to students, these students will be in for a shock when these "laws" are no longer viable or true. I doubt she's accurately conveying what the word actually means in context, because so many others references and teachers also fail.

People can't read and look up the nuance when they get more mathematically mature?

 

[gleem]

That's all I'm arguing. If this teacher is drilling the "laws" of mathematics to students, these students will be in for a shock when these "laws" are no longer viable or true. I doubt she's accurately conveying what the word actually means in context, because so many others references and teachers also fail.

I'm perplex about what "laws" will change or not be true.

 

[Rx7man]

That's not a failure of the educational system. It's a failure of our society. Students who are bored with learning could be removed from those classes in which they're bored and put to work doing something productive. But our society will not allow that. Instead we as parents and educators coddle these "bored" students and pass them along to the next grade, where of course they continue to be bored.

If you're bored, guess what? It's not anyone elses fault. No one else is responsible for your failures or your emotional state. Find something interesting to do, a way to do it, and stop whining.

Perhaps there should be a distinction between easily distracted students and interested students who lose interest. This is of particular importance in the younger grades.

By college, you're right, its the student's responsibility to keep interest in something and they have the freedom to chose what they want to pursue, and that was MY failure, I accept that.
In earlier grades, where the student really has no choice, it is absolutely the fault of the educational system... I don't think 8 years of picking out nouns and verbs did anything for my english writing skills... Just reading books for 7 of those 8 years would have expanded my horizons far more.

Perhaps slightly tangential to this, here's a video I stumbled on... I can't agree with everything, but they make some good points

 

[Mister T]

In earlier grades, where the student really has no choice, it is absolutely the fault of the educational system...

Our society will not tolerate giving those students a choice. So instead we blame the problem on the education system. But we are the ones who govern that system! So hopefully you see the problem.

I don't think 8 years of picking out nouns and verbs did anything for my english writing skills... Just reading books for 7 of those 8 years would have expanded my horizons far more.

Then you could have read. But instead you grew up to blame the education system. And you will likely do the same as a parent.

The solution is to realize that it's the fault of our society. From there we can perhaps begin to proceed towards improvements. Meanwhile, blaming it on boredom, or anything else, is a distraction that allows us to continue to avoid facing the reality.

 

[Jeff Rosenbury]

That's not a failure of the educational system. It's a failure of our society. Students who are bored with learning could be removed from those classes in which they're bored and put to work doing something productive. But our society will not allow that. Instead we as parents and educators coddle these "bored" students and pass them along to the next grade, where of course they continue to be bored.

If you're bored, guess what? It's not anyone else's fault. No one else is responsible for your failures or your emotional state. Find something interesting to do, a way to do it, and stop whining.

When I was a child, I was forced to sit in a seat and endure boring lectures all day. Fortunately some compassionate teacher noticed when I started pounding my head against the desk. Unfortunately the response by the bureaucracy was to label me and have me detained for incorrigibility.

So while I agree with your sentiment, I'm unclear how much of the blame lies with the student and how much with the system.

 

[Mister T]

All of the blame lies with our society. We created this problem, we cause it to persist, and we delay its remedy by avoiding its cause.

Yes, the poor child is bored. It's not his fault. The fault lies with the way our society has decided to respond to the issue. Instead of addressing the problem we blame the system. Or the teacher. The real fault lies with us.

 

[Rx7man]

Our society will not tolerate giving those students a choice. So instead we blame the problem on the education system. But we are the ones who govern that system! So hopefully you see the problem.
Then you could have read. But instead you grew up to blame the education system. And you will likely do the same as a parent.

The solution is to realize that it's the fault of our society. From there we can perhaps begin to proceed towards improvements. Meanwhile, blaming it on boredom, or anything else, is a distraction that allows us to continue to avoid facing the reality.

I'm well aware there's more than one cause to this problem... How much of a say in the educational system do the parents have these days? Mister T, how much of a say do YOU have in how you teach (if you're a public school teacher). I'm not a parent (unlikely that will ever happen), and I'm not even in the same country.. But from what I see, it looks like there's a bunch of people who really don't care too much about their kids education, as long as they're in school.. and the rest that are concerned about how things happen are frustrated by the system and can't or don't know how to effect any change.. there's always a bureaucrat at the top that makes the calls but is quite disconnected from the classroom.. and they make all the decisions.

With the exception of my english classes, and partly because I was homeschooled for some time, I didn't get bored, but I know a lot of my public schooled friends suffered from it... One in particular was pretty much at the bottom of his class in everything, they had him do IQ tests and everything, and he wasn't a dumb kid by any means.. If you gave him something he was interested in, he excelled at it, and luckily for him he was given the opportunity to do that when he wasn't in school.. many kids don't.

Now if you publicly come out and say it, it would raise a huge PR nightmare, but if kids that are interested in trades learned more to do with trades in general they would try harder... They don't need to be able to quote Shakespeare, they need to have descent fundamental math, it's good if they can spell (a lost art?), but knowing the intricate details of sentence structure is a waste... a carpenter needs to be able to determine slope and pitch of a roof, square footage and all that... And english major doesn't. The long and the short of what I'm saying is that it would be called a 'two tier' system, and the usual outrage of kids being deprived of a 'full' education would certainly ensue.. On the other hand, they'd probably be better at what they WANT to do.

In my school there were some kids with Down's syndrome, and many with Fetal Alcohol Syndrome... the resources they required were ENORMOUS (averaging nearly 1 dedicated staff member to each)... I think it's great they were IN SCHOOL, but not because they had to learn intellectual things, it's because they need the socialization, and it's good for other kids to learn to accept them. Teach them basic life skills and call it a day, but the budget for the 10 dedicated special needs educators could have made a much greater impact elsewhere.

The current system desperately tries to make one size fits all, but it just isn't that simple. I'm not claiming I have the magic fix to it either, these are just observations.
Mister T, you seem to be taking this far more personally than intended.. I really don't believe 'the teachers' are the problem (exceptions to every rule applies).

 

[Mister T]

How much of a say in the educational system do the parents have these days?

Everything. They elect the members of the school board. The school board is in entrusted by the taxpayers to run the school.

As far as speaking out goes, one person's complaints to a school board, or even praise for that matter, have little effect. It takes a majority to win an election.

As far as course content, I pretty much control that myself. I also set the standards. But in doing both those things I have to work within the constraints set by the society in which I live. And I'm happy to do that. But I teach physics at a public two-year college, so that's a situation unlike many others for many different reasons.

 

[Rx7man]

The school board apparently doesn't set the curriculum though.. that's the Ministry of Education (at least here) that does that, and they aren't elected

About as much authority as the school board has around here is to shuffle funds from one place to another.. Repair the leaking roof and lay off 2 teachers, or cancel field trips to resurface gym floor...

 

[SredniVashtar]

I remember being taught in the electronics in the AIr Force (1980s). I asked our instructor how he did when he took the class. He said he had never taken it. He simply followed the T.O. (technical order) and his training in training others. In 16 weeks I picked up more technical knowledge than in two years of E.T. coursework. (There were other things learned in college, but for pure technical training, the Air Force taught what it needed to quickly and efficiently.)

The Air Force didn't think teaching was some big mystery. They didn't even require the teacher know the subject, just follow a prearranged lesson plan.

And who wrote the plan, and according to what ideas?
My guess is that you might have been exposed to Bob Widlar's wake.

https://books.google.it/books?id=2c...ved=0ahUKEwj09fGjxuLKAhWH1BoKHfsrCfwQ6AEIKTAA

"Young Bob Widlar joined the USAF in February 1958. One of his duties was teaching classes on electronic equipment and devices. The very first Widlar publication was a crispy clear textbook "Introduction to Semiconductor Devices". When I was reading this text, I realized why Bob Widlar was so successful in his future work. He had an extraordinary capability to simplify complex problems"

Bo Jolek, "History of Semiconductor Engineering", p.254​

 

[chiro]

For the reply to Mark44.

Capturing variation is literally that - capturing complexity. This is what mathematics does - it captures complexity and tries to organize it in the best way possible to make sense of it. You can call it variation, complexity, abstractness - even entropy - they all mean the same thing.

Mathematics has variation through variables primarily. The whole point of mathematics is allow one to take this complexity and do two things - organize it effectively and make it consistent. This is all mathematics is - an attempt to take higher and higher levels of complexity and make it consistent.

The way it's done depends on the field. Probabilities do it for probability - calculus does it for derivative and integrals, geometry does it for distance and angle, topology does it for continuity, and other fields do it in their respective way.

It's the ability to be able to translate between different situations and relate that information to something consistent and organized that facilitates understanding and critical thinking.

Knowing sines, cosines and tangents is useful for geometry and I would expect physicists and engineers to be able to recite results and apply it.

But knowledge by itself and the recall of facts means nothing with the relations that come with it.

A student in high school is not going to see that mathematics helps with understanding complexity. They are going to see a bunch of random examples from two dimensional geometry, some algebra, polynomials and they are going to not appreciate that mathematics is used to make sense of complexity - something everyone faces in their daily lives.

They don't see that complexity entails all sorts of information whether that information be about what home loan to get, how to make sense of lies and evaluate them for consistency, or being able to accurately read statistics, fractions, and other claims via information.

The ability to extract the useful attributes of complexity and make sense of the consistency is what real mathematics is about.

Any bozo can memorize things and learn to recite them - it's the ability to deal with uncertainty that is more impressive.

It is a survival skill to be able to face uncertainty and deal with all of the BS that is faced in the world for some reason and another.

The best thing that anyone can do for any set of logical relations when it comes to evaluating them is to evaluate them for consistency - and mathematics is the primary area of knowledge that evaluates consistency.

You may think I'm talking about the axiomatic pure mathematic logic stuff but I'm not.

The ability to show inconsistency whether it's via predicate logic, statistics, sets of linear equations or inequalities, optimization or any other thing is something that allows a person to deal with uncertainty in the best way possible.

The arguments of science use it and certainly engineers use it for the same purposes - but this is not the only forte of using mathematics and consistency to evaluate things.

People are bombarded by claims, information, results, and they often have no real way to deal with it. If they understood what mathematics actually is as opposed to what they think it is they would probably be a bit more interested because they would realize how useful it is to making big life decisions and being able to find a way to apply consistency to an unfamiliar situation - something I think many parents would want their children to be able to do.