Real Mathematics. (Not "Math")

[Alexander]

Here, I want to give some definitions of what makes mathematics, the art of doing it, how to teach it, and how it differs from "math", the subject familiar to us at school.

This is just the first post to get the subject line up on the board, while I think about how best to compose some examples.

a

[Wildflower]

Oh wonderful Alexander, I'm really excited about this! Especially in how this real math might figure into the lives of an unschooling family. Thanks for all the effort you're putting it, it is so appreciated!

[BusyMommy]

So, what do you think of Everyday Math?:

[SagMom]

I'm waiting impatiently here, Alexader. Do Tell.

Math is a mysterious thing. Growing up in American schools in the '60s and '70s, memorization of math "facts" was important. (To this day I haven't memorized them.) "Showing your work" was also important, (HOWEVER there was only one "right" way to solve each problem and I never seemed to take the correct path.) By the time I finished elementary school I hated math and was convinced that some people were good at it and others (like me) were simply not. I can function quite well in my world with my narrow understanding of math and my limited ability in calculations. However...

Imagine my surprise as an adult to find that there are people who ENJOY math. LOVE it. Think it FUN. (There are strange people in this world, eh? )

As it goes, I am the parent of one of these strange people. Her delight in space, puzzles, shapes and the games she finds in numbers makes me suspect I've missed out on something here--what IS this "math?" Where can I find out more about it? (And how can I keep up with my 7 year old?) :

[rsps]

Joan -- there's a couple of kids books you and your daughter might enjoy
"math for smarty pants" by Marilyn Burns
and "the I hate Mathematics book" by the same author.

[Alexander]

OK ok,

It's coming! Igot a life though, so, it's comin.

a

[Alexander]

OK. her we go. Just for starters.

Mathematics (Maths), not "Math".

I freely admit that I am no mathematician, but I do deal in the realm of the Information Era. The two intersect. That is why am am talking about it.

There are essentially three parts to what makes Maths.

1) The discovery of a new way to model functions, or ideas. These go on to become branches of Mathematics. Examples are: Trig. Euclidean Geom. Boolean Logic, and yes, Arithmetic.

2) working out the procedure that allows for solutions to be found.

3) A general proof that 2) works for all cases.

You might expect that 1) is the realm of pure mathematicians and professors, and to an extent, that would be right, but it boils down to "a new way of looking at things", and that means anyone can do it, even children. "How to draw a circle" is a good example, and we can come back to that later.

3) is what a a PhD is.

But 2) is what we are really interested in here, because this is where the confusion lies between "math" and Maths.

So let's take an example, that was described by none other than John Holt in his book "How children fail" (or was it the other book, "How children learn"?). Anyhow, the example is of when he was trying to teach a little girl how to add a single digit number to ten. AFAI can remember, she insisted on doing the exercises by counting out her fingers on her hands.

So he tried to "teach" her a technique for getting the answer quickly and reliably. He tried to show her that if she had to add ten to something, all she had to do was put a 1 in front of the single figure to get the right answer. For some reason she couldn't wouldn't follow method that she had been presented with. Perhaps she was not

So, John Holt gave up, and left her.

Much later, she came up to him in awe, with the revelation that when she put a 1 in front of the single figure. . . . . she got the right answer!!!!!!

What John Holt tried to do was teach her "math".

What the girl did was Mathematics.

This is because she noticed a pattern, and formulated a general method to solve problems of a particular type. That's mathematics (maths).

You will notice that she applied an idea that she got from recognising a pattern. We should not let the fact that it occurred in the branch of mathematics known as "Arithmetic", and involved moving a number around, from clouding the basic truth that the sums she she was solving was arithmetic, but the working out of a general method was mathematics.

If some-one had succeeded in showing her the method before she had worked it out, then she would never have encountered mathematics. She would have simply learned a method of moving digits around.

She would have learned a recipe.

In the Information Era, we don't want to train kids to remember other people's methods, we need them to come up with their own.

Hope this helps

a

PS More examples are available soon. How about one on subtraction?

[Alexander]

More clarity.

Mathematics (Maths), not "Math".

"Math" is teaching / learning the rules of a method to solve a problem, and solving the problems.

Mathematics is:

Noticing that a problem exists, and inventing the method to solve the problems

Hope this helps

a

[SagMom]

Alexander--gotcha. This is a surprisingly simple distinction. What you explain about mathematics is how I feel about learning in general. However, I always treated maths as something separate from all other learning ...probably because I grew to believe that that was no discovery to be made in math...all it ever was, was the act of applying recipes or formulas that someone had given to me in order to get an answer. (Further, the answer was simply a number--it had absolutely no meaning.) I can't recall a single "discovery" I ever made for myself involving math.

rsps: Thanks for the book ideas--how co-incidental is this--I just requested one of those titles from inter-library loan last week.

[Alexander]

Originally posted by BusyMommy
So, what do you think of Everyday Math?:

In what context?

If you mean should we be teaching "School Math" to children, then N O, not unless they ask for it.

a

[lilyka]

Ok I am stil a little confused (maybe but maybe not)

So when my teachers taught me to add and subtract from right to left that was "math" because I was just following the rules.

When I learned to add form left to right (much eiser for me and works every time thank you very much) that was mathmatics because I figured it out by using what I knew about general math principals and discover for myself how to make it work all the time even though it still just arithmatic.

edited because although I can add, I still don't know my right from left

[sleepies]

if you don't teach "School Math"
then what will your kids do when they need to take an ACT or SAT?

you have to know math to get a good score on there.

and if you get a bad score, college can be hard to get into

if at all.

[SagMom]

"if you don't teach "School Math"
then what will your kids do when they need to take an ACT or SAT? "

ASSUMING they want to go to college, and ASSUMING they will be required to take the SAT for admission, and ASSUMING they need to score "well" (whatever the definition of that is) in the math portion... ...I'll take a shot at answering that question:

Not teaching math the way schools do (I think that's what you mean by "school math,") does not mean that a child won't have an understanding of math. I think that, if a child is allowed the time and space to explore and discover concepts/ideas on their own and in their own way, they gain a better understanding than if they are "taught" the information. Memorizing bits of information is very different from understanding.

Aside from that, standardized tests are simply hoops to jump through. A quick look in any bookstore will show shelves of books on how to prepare for the ACT/SAT etc. If all one needed to know to get a good score on the test was taught in school, then why are there so many books on test taking? Why are there SAT "prep" classes?

(Incidentally, I scored a horrid 300 on the math portion of the SAT. It didn't hurt my college career in the least. )

[Britishmum]

I'm not familiar with maths in American schools, but I gather from this that there is an assumption there are few, if any, teachers out there in schools encouraging children to do real maths, to find their own ways, to investigate, and to find their own solutions to problems? Are all teachers really just teaching systems?

Are we talking reality here, or just drawing upon our own (often bad) memories of maths being taught to us at school? What is it really like in schools? Are they all really just teaching by rote, or are there some good maths teachers out there? What do people find when their children are in school?

What exactly do you mean by 'school math'?

Just curious..............

[SagMom]

Britishmum, from what I've seen, methods of teaching math vary from classroom to classroom. I commented on my own personal experience. (My "reality") Certainly some teachers are better (however we define that) than others. However, in every school there are grade levels and "scope and sequence" outlines detailing what a child will learn and when they will learn it. To me, this doesn't leave a lot of room for individual discovery. My children aren't in school, but I hope that their experience with mathematics will be very different from my own...I figured a discussion like this one may help me to sort it all out.

[Britishmum]

Joan, thanks for the input - along with schools having outlines for what will be learned and when, is there any outline about how? What happens if a child doesn't learn what is proscribed at the right time?

A bad teacher can stifle any creative thought on the part of his or her students, which is why I believe so many people say that they 'hate' maths or are 'useless' at it. A few bad experiences, or careless comments, or ridicule, sadly, can do that. But of course, some good experiences with a great teacher can fire enthusiasm for maths, as with any learning experience.

Maths seems to be a subject that people get turned off from very easily. I wonder why confidence and enthusiasm for numbers can be destroyed more easily than that with the spoken or written word? Mind you, I heard once of someone who never learned to swim because a PE teacher told her that she was so brainy that her head would be too heavy to float. Incredible, but true.

[SagMom]

"What happens if a child doesn't learn what is proscribed at the right time?"

No simple answer there. It depends on the school, on the teacher, on the subject. A child who doesn't learn to read by the X grade, for instance, could be removed from regular classes and given extra "help," or ... not. Generally, there are remedial classes available for the "basic skills" (meaning, reading, writing and arithmatic) but not for the sciences or any other subjects. The quality of the remedial classes fluctuates as much as the quality in the regular classes. Alternately, a child may repeat a grade if they haven't learned what they were "supposed to" learn. Increasingly, children are sent for learning evaluations.

Is all of this that different from your country?

[larsy]

I notice that the consequences of a child not learning what the school has proscribed to be learned by a certain grade/age, are all contrived by the school- to be taken out and put in remedial class or be held back a grade- which then has social implications. There is no real reason why a child must read by any certain age, or learn the times tables at a certain age, is there, other than someone deciding it must be so?

Why do so many people get a bad attitude about math? Because it is forced upon them when they are not interested/ready to learn about it.

The difference between learning math and learning mathematics seems like a distillation of the problem of what good education for the information era vs what constituted a good education for the industrial age. The industrial age required people who could perform certain tasks and were willing to be controlled by the industrial complex, and schools were good institutions to burp out good citizens with those characteristics. Learning the right answer to the math problem worked for that society.

The information era will need people who can be self-directed and who can think beyond what was the right answer for the previous generation(s)- and maybe this has been true of every generation that has ever come along, needing to find different and better solutions than the previous generation because humand are able to create new knowledge, and do! Mathematics- as opposed to math- reflects this problem-solving attitude, to engage creativity and look for better solutions.

Thanks, Alexander!

[Britishmum]

Joan, I think the US system does differ in a lot of ways. In the UK, it is extremely rare, almost unheard of, for a child to drop back a year. There is no concept of 'failing' a grade or year.

There are National Curriculum target levels which children are supposed to reach, but not all. It is a percentage target, which the government keep altering, but it's somewhere in the region of 85% of children should reach National Curriculum Level 4 by the end of Year 6 (fifth grade). Some reach Level 5, some only Level 3, some not Level 3.

If a child has significant difficulties they have to be put on the Special Needs register at the school, which means that the school has to work with the parents to provide an individual programme to help the child. The word 'remedial' hasn't been used in years. Part of inspections by OFSTED, the national inspectors, is to study the provision for these children. (Ofsted has many faults and is not liked by most of the profession, but it does ensure that there is some sort of accountability for these children getting provision). Furthermore, schools get some funding according to the number of children with these needs, although of course it is never enough.

It is hard now for teachers to be creative in the way that they teach, whether maths or any other subject, but it is not impossible.

Larsy, I think you are right that people are put off maths by it being forced on them, but I think that the issue is not what is 'taught' (I know you dislike that word) but how it is taught. I think that a child's natural curiosity leads him or her to explore 'real maths' from birth. If a teacher is creative, she can work most aspects of the curriculum to be relevant and creative. A workshop approach to maths can deliver the same curriculum as rote learning, but is meaningful and can lead to real learning, not just of maths facts, but also of problem solving, which in my opinion is of greatest importance.

I think it is simplistic to always blame everything on children being forced to do things that are against their will. I'm still interested in thinking about what exactly it is about maths that leads people to be turned off so easily - is it perhaps the idea that there are right and wrong answers, and nothing in between?

Of course, you can leave children to discover things for themselves, for as long as it takes. I've been through that school of thought in education in the 80s, but never fully bought into it. Or you can facilitate learning by enriching their world with suggestions and ideas, and be a partner in their learning. I think that a 'teacher's' role, or a facilitator, or whatever you want to call a teacher, is to help lead the child on to further discovery.

[SagMom]

"is it perhaps the idea that there are right and wrong answers, and nothing in between?"
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This is interesting...that IS what turned me off of math when I was in school. Maybe some math-savy person here can expand on this---is math that rigid? Are there really only right and wrong answers and nothing in between? I remember, as a child, being instructed to set up subtraction problems with the larger number on top because, of course, you can't subtract 10 from 8. I was very annoyed when I later learned about negative numbers.

One of the reasons I loved English/literature was that there was room for debate and discussion and interpretation. I didn't think that was possible with math. There was a certain fear involved in my math education because of this. I knew that in other areas, as long as I could defend my position, my answer was okay. Not so with math. I felt I had one chance to be right and infinite chances of being laughed at.
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"Of course, you can leave children to discover things for themselves, for as long as it takes. I've been through that school of thought in education in the 80s, but never fully bought into it. Or you can facilitate learning by enriching their world with suggestions and ideas, and be a partner in their learning. I think that a 'teacher's' role, or a facilitator, or whatever you want to call a teacher, is to help lead the child on to further discovery."
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You and I are close in our thoughts here. I would rather the child do the leading, but I do see the adult as a resource.