How to teach mathematics

[lpetrich]

Richard Feynman on Teaching Math to Kids and the Lessons of Knowledge

He got into considering lower-level education when he thought about what sort of education his children was to receive. He rejected a first-grade physics textbook that attributed several things' motions to "energy", because he thought that that was not saying very much. Attributing falling to "gravity" or wear to "friction" he thought equally vacuous.

Shoe leather wears out because it rubs against the sidewalk and the little notches and bumps on the sidewalk grab pieces and pull them off.” That is knowledge. “To simply say, ‘It is because of friction,’ is sad, because it’s not science.”

He may have been too picky there, especially about energy and gravity.

In 1964, he joined a commission in California for assessing mathematics textbooks. Those were the days of the "New Math", introducing lots of abstractions and advanced concepts at an early age.

Feynman was skeptical of this approach but rather than simply let it go, he popped the balloon.

He argued to his fellow commissioners that sets, as presented in the reformers’ textbooks, were an example of the most insidious pedantry: new definitions for the sake of definition, a perfect case of introducing words without introducing ideas.

A proposed primer instructed first-graders: “Find out if the set of the lollipops is equal in number to the set of the girls.”

To Feynman this was a disease. It confused without adding precision to the normal sentence: “Find out if there are just enough lollipops for the girls.”

According to Feynman, specialized language should wait until it is needed.

He argued that this was knowledge of words, not actual knowledge -- and convincingly so.

Biographer James Gleick noted: "Feynman proposed that first-graders learn to add and subtract more or less the way he worked out complicated integrals— free to select any method that seems suitable for the problem at hand."

Like adding 29 and 3 by finding the next number for 29 three times: 29, 30, 31, 32.

He proposed that kids be given simple algebra problems (2 times what plus 3 is 7) and be encouraged to solve them through the scientific method, which is tantamount to trial and error. This, he argued, is what real scientists do.

...
It was better in the end to have a bag of tricks at your disposal that could be used to solve problems than one orthodox method. Indeed, part of Feynman's genius was his ability to solve problems that were baffling others because they were using the standard method to try and solve them. He would come along and approach the problem with a different tool, which often led to simple and beautiful solutions.

I couldn't find much more than that in this blog entry, however.

 

[steve_bank]

My view us simple. The basic rules are taught. You then learn math by doing progressively harder problems.

There was a book on Quantum Mechanics written as a project by Japanese students who had no prior knowledge of sconce and math.

The approach they ended up with is treating it like learning a language. It is not enough to just learn facts, you have to speak it in a math-science environment.

I heard a math teacher say on a show when learning math getting the right answer is not important. I've heard other claims like that from so called education experts/


Back in the 70s in Hartford Ct Pratt And Whitney found kids coming out of high school did not have the basic skills needed for skilled appreticips to replace retiring workers.

Algebra, arithmetic, geometry, trig. They had to create remedial programs.

Math skills stared dropping when education embarked on a liberal course abandoning a lot of torsional approaches.

The great education experiment has failed.

 

[Malintent]

The old fashioned way of learning math, that Steve_bank describes as "torsional approaches"... which I describe as "wrote repetition" creates people with equivalent mathematical ability as their memory.

Common Core was developed in response to the understanding that numbers are not finite, static "things" like the quantification of objects (there are 3 apples and johnny eats 1, how many apples remain). Numbers are fluid, dynamic properties... not things in themselves that can be memorized (12 x 2 is 24, 12 x 3 is 36...), but rather (15 is 3 groups of 5, or 5 groups of 3, or "one of ten and half of another ten".. etc..)

Procedurally, one can do "long math" to derive that 29 * 5 = 145. OR, more intelligently and usefully, one can logically produce the correct answer by drawing on experience that 30 * 5 is 3 * 5 * 10 which is easily seen as 150... but there is 5 times 1 less, so 150 - 5 = 145. That is easy to do in your head, whereas "9 * 5, carry the 4, plus 10, plus the original 0 from the 40 that was carried, equals 145, is far more difficulat to do in your head.

Children are no longer taught strict "procedure" for finding answers to problems... instead, a "method" of estimating and correcting to find the answer is now used... and hopefully, this will improve forming brains' ability to make flexible and sound decisions.

 

[bilby]

The old fashioned way of learning math, that Steve_bank describes as "torsional approaches"... which I describe as "wrote repetition" creates people with equivalent mathematical ability as their memory.

Common Core was developed in response to the understanding that numbers are not finite, static "things" like the quantification of objects (there are 3 apples and johnny eats 1, how many apples remain). Numbers are fluid, dynamic properties... not things in themselves that can be memorized (12 x 2 is 24, 12 x 3 is 36...), but rather (15 is 3 groups of 5, or 5 groups of 3, or "one of ten and half of another ten".. etc..)

Procedurally, one can do "long math" to derive that 29 * 5 = 145. OR, more intelligently and usefully, one can logically produce the correct answer by drawing on experience that 30 * 5 is 3 * 5 * 10 which is easily seen as 150... but there is 5 times 1 less, so 150 - 5 = 145. That is easy to do in your head, whereas "9 * 5, carry the 4, plus 10, plus the original 0 from the 40 that was carried, equals 145, is far more difficulat to do in your head.

Children are no longer taught strict "procedure" for finding answers to problems... instead, a "method" of estimating and correcting to find the answer is now used... and hopefully, this will improve forming brains' ability to make flexible and sound decisions.

Ah, yes; But you forget that the way I learned to do something is the right way, and that therefore any other approach is wrong, wrong, wrong, awful, bad, stupid, nonsense and must die in a fire.

 

[rousseau]

One way or another people with an aptitude for math will learn it, and people without an aptitude for math probably won't, with a bit of grey area in between. My memory of high school math class isn't gone yet, and IIRC most people either could, or could not, and were divided into post-secondary (or lack thereof) disciplines accordingly.

In practice public school is more of a sorting hat than an educator. Those who know how to listen and figure things out succeed, and those who can't... don't. The room for pedagogy to completely transform a person is probably over-stated, imo. At graduation people usually fall within the frame of their natural ability in reference to the bell curve.

Really, if you want kids to reach their potential.. make sure they're well fed and that their teachers aren't completely useless.

 

[steve_bank]

Math education has failed. Educators are looking for a pie in the sky solution that will entice kids to learn.

Bill Gates said in an interview in his opinion computers should not be used below a certain grade level.

It stunts the 'learning how to learn' process.

Engineers in my generation I have talked to have seen a change in new college grads. If they can't find an existing solution, find it on the net, or find an app yhey are at a loss.

In an intrerview on a newd segment somone from busness said new high school grads compared to privious generations are less able to self organize and solve problems without guidance and supervision. He attributed in part to too much structure in primary education.

That idea that there is a process that will generate a homogeneous math capacityn across the entire population is fantasy.

Language,science, math, history and the rest all require wrote leaning and the ability to assimilate and organize facts. Whatever you learn requires repitition to create long term memory. The principle of neural plasticity says the brain responds to stimulus, even in old people. Stimulation creates brain cell growth. Raise kids with processes that do not require work and practice will wire the brains and condition thinking accordingly. Raise kids on learning computer apps and that is what you get.

 

[Jimmy Higgins]

With a young daughter, I have slowly been incorporating math and science. The key is about knowing what you are talking about and then being able to reach the student. What is an odd or even number? How can you break it down to a relatable level? That is teaching.

Math education has failed. Educators are looking for a pie in the sky solution that will entice kids to learn.

Bill Gates said in an interview in his opinion computers should not be used below a certain grade level.

It stunts the 'learning how to learn' process.

Engineers in my generation I have talked to have seen a change in new college grads. If they can't find an existing solution, find it on the net, or find an app yhey are at a loss.

I think one major thing these days, as I am on the cusp of both generations, is that the huge difference is computers. The ease at computing and recomputing has changed how problems are solved. Making errors at first isn't a big deal. 40 years ago, it was.

 

[bilby]

With a young daughter, I have slowly been incorporating math and science. The key is about knowing what you are talking about and then being able to reach the student. What is an odd or even number? How can you break it down to a relatable level? That is teaching.

Math education has failed. Educators are looking for a pie in the sky solution that will entice kids to learn.

Bill Gates said in an interview in his opinion computers should not be used below a certain grade level.

It stunts the 'learning how to learn' process.

Engineers in my generation I have talked to have seen a change in new college grads. If they can't find an existing solution, find it on the net, or find an app yhey are at a loss.

I think one major thing these days, as I am on the cusp of both generations, is that the huge difference is computers. The ease at computing and recomputing has changed how problems are solved. Making errors at first isn't a big deal. 40 years ago, it was.

People have been complaining that college graduates are incompetent (and not up to the standards that were expected as the bare minimum back when they graduated) for as long as there have been college graduates to whinge about.

People are VERY bad at recalling just how incapable they used to be before they actually got some real world experience under their belts.

And of course, that goes double when the new guys are using techniques and/or technologies that the old guys never learned in the first place and don't really understand.

 

[Malintent]

The old fashioned way of learning math, that Steve_bank describes as "torsional approaches"... which I describe as "wrote repetition" creates people with equivalent mathematical ability as their memory.

Common Core was developed in response to the understanding that numbers are not finite, static "things" like the quantification of objects (there are 3 apples and johnny eats 1, how many apples remain). Numbers are fluid, dynamic properties... not things in themselves that can be memorized (12 x 2 is 24, 12 x 3 is 36...), but rather (15 is 3 groups of 5, or 5 groups of 3, or "one of ten and half of another ten".. etc..)

Procedurally, one can do "long math" to derive that 29 * 5 = 145. OR, more intelligently and usefully, one can logically produce the correct answer by drawing on experience that 30 * 5 is 3 * 5 * 10 which is easily seen as 150... but there is 5 times 1 less, so 150 - 5 = 145. That is easy to do in your head, whereas "9 * 5, carry the 4, plus 10, plus the original 0 from the 40 that was carried, equals 145, is far more difficulat to do in your head.

Children are no longer taught strict "procedure" for finding answers to problems... instead, a "method" of estimating and correcting to find the answer is now used... and hopefully, this will improve forming brains' ability to make flexible and sound decisions.

Ah, yes; But you forget that the way I learned to do something is the right way, and that therefore any other approach is wrong, wrong, wrong, awful, bad, stupid, nonsense and must die in a fire.

That is how I felt about common core when I first heard about it from an older teacher friend. And then I learned more about it from a younger teacher friend... that grew up learning both the old and the new ways.

 

[Jimmy Higgins]

People have been complaining that college graduates are incompetent (and not up to the standards that were expected as the bare minimum back when they graduated) for as long as there have been college graduates to whinge about.

I'm not complaining, clearly smart kids are graduating college these days. My comment was more to the thoroughness of calculations before hand via computers. Granted, I remember doing Geotech homework in college, by hand, because that is how you are taught (then and likely now), and a single problem could take a couple hours to finish.

The oddity is that spreadsheets can actually help to teach a subject because the hardest thing to program are differing conditions, ie for it to work, you must know all of the variables. So who knows, it probably works out in the end.

People are VERY bad at recalling just how incapable they used to be before they actually got some real world experience under their belts.

Oh goodness, how I try to remind myself before ever criticizing something.

That is how I felt about common core when I first heard about it from an older teacher friend. And then I learned more about it from a younger teacher friend... that grew up learning both the old and the new ways.

There was an article about number lines or something and people were criticizing it because it was different than what they were taught, but it seemed apparent to me that the new method was meant to capable of handling in the brain and not needing a calculator. As you noted, us intelligent but not computer like people don't calculate a product straight up. There is almost always a way to reorganize the numbers in a way that becomes easy to calculate (like 16x14 -> 16 x 10 + 16 + 4). memorization is good, but unless your brain can memorize everything, it is best off to learn how to do stuff in a manageable way.

 

[Malintent]

Math education has failed.

That sounds like political rhetoric more than anything.. except despair.

Educators are looking for a pie in the sky solution that will entice kids to learn.

More rhetoric... pie in the sky? silver bullet? magic carpet to success? Educators want to educate with increasing effectiveness in whatever ways our knowledge of learning provide.

Bill Gates said in an interview in his opinion computers should not be used below a certain grade level.

was that the same interview where he said no one would ever need more than 640K of memory?

It stunts the 'learning how to learn' process.

Calculators can stunt the learning process if used in an Arithmetic class. It sounds like another nearsighted Bill Gates statement regarding the utility of computers prior to the world wide web.

Engineers in my generation I have talked to have seen a change in new college grads. If they can't find an existing solution, find it on the net, or find an app yhey are at a loss.

In an intrerview on a newd segment somone from busness said new high school grads compared to privious generations are less able to self-organize and solve problems without guidance and supervision. He attributed in part to too much structure in primary education.

That idea that there is a process that will generate a homogeneous math capacityn across the entire population is fantasy.

.. or... not enough structure?
It is true that different people learn different ways. That is one of the key elements driving innovation in Education Sciences.

Language,science, math, history and the rest all require wrote leaning and the ability to assimilate and organize facts. Whatever you learn requires repitition to create long term memory. The principle of neural plasticity says the brain responds to stimulus, even in old people. Stimulation creates brain cell growth. Raise kids with processes that do not require work and practice will wire the brains and condition thinking accordingly. Raise kids on learning computer apps and that is what you get.

"Practice makes permanent". (not "perfect") as the newer saying goes. Repetition is excellent for kinematic learning (muscle memory), important for musicians and athletes. Memorizing wrote facts require the same for permanence.
Even processes require repetition to run smoothly.
However, deriving problem-solving processes.. innovating.... being creative... are not things that are learned through repetition... but "practice" of the methods learned about how to learn.

 

[Wiploc]

Ah, yes; But you forget that the way I learned to do something is the right way, and that therefore any other approach is wrong, wrong, wrong, awful, bad, stupid, nonsense and must die in a fire.

I gave bilby a thumbs up for this post. But now I see that he left out the Oxford comma, so, I dunno.

 

[Wiploc]

Engineers in my generation I have talked to have seen a change in new college grads. If they can't find an existing solution, find it on the net, or find an app yhey are at a loss.

Quora is now flooded with people asking, "What's wrong with this sentence?" Crowd-sourcing your homework must now be standard.

 

[bilby]

Engineers in my generation I have talked to have seen a change in new college grads. If they can't find an existing solution, find it on the net, or find an app yhey are at a loss.

Quora is now flooded with people asking, "What's wrong with this sentence?" Crowd-sourcing your homework must now be standard.

It always was. Students have always asked other students, parents, friends and acquaintances to do their (home)work for them. The only thing that's changed is that it is now a lot easier to ask a lot of people, which somewhat increases your chances of being spoon-fed the right answer, rather than being assisted to find it your self, or told to do your own work.

Perhaps the real problem is that the system has always given students the impression that what matters is giving the right answer, so you don't feel stupid or get into trouble.

If students (and teachers) were aware that the objective is to learn how to solve problems, rather than being able to provide the correct solution to just those examples that are in front of them right now, that sort of thing would go away. But while the aim of education remains 'passing tests' rather than 'learning stuff', it will persist.

 

[steve_bank]

Perhaps the question should be reframed. How do we learn anything? Other than complexity and amount of information is there a fundamental difference in learning to be an electrician and a mathematician?

Best way? Correct me if I am wrong. There is short term and long term memory. It takes time and repetition to transfer to long term memory.

Learning is a genetic survival trait. Chimps learned to quarry and work rocks into tools to open nuts. I've scene video. They select rocks in a quarry area and carry them to a work area, Young chimps learn monkey see monkey do. It was developed and passed on without writing and articulate speech.

Motivation is nature vs nurture. Looking back at 1-12n school and my work experience I'd say environment and expectaions are a major component of any academic learning.

In Europe traditionally the well rounded man was both physically and academically fit. I've heard it said in Europe a PHD commands a lot more social respect over here.

Over here the word nerd applied to kids pursuing science pains me. Look at TV and movie depictions of academic culture. Mostly negative. With some exceptions. Contrast how sports are presented.

For myself I have not played a video game since around 1983. I'd rather open a book and work problems, makes me feel good.

 

[Malintent]

Well that certainly is a far broader question... how do we learn (at all)?

As you pointed out, learning is pretty fundamental, as even monkeys can do it... what they lack that we have, though, is language... the ability to archive knowledge to more efficiently pass it on to the next generation for the next generation to build even more upon. "Standing on the shoulders of giants", is an expression for that. Monkeys can only learn what is demonstrated to them... they can't read a book about "how to fashion tools" (or look it up on ee-ee-ee-tube.com), they can only copy what they are directly exposed to. Without language, perhaps we wouldn't be much more advanced than the monkeys.

For myself I have not played a video game since around 1983. I'd rather open a book and work problems, makes me feel good.

playing video games, nowadays, involves a good deal of problem solving. I find the games that I enjoy are centered around the kind of problem solving that my engineering mind likes to wrap itself around.