Math using objects instead of numbers.

[stephenbosley]

I know this is a little off base but I thought I would bring up a strange question. We have always been using math by using the symbols of numbers. Is there a math using objects instead and how would one create this type of math? I believe it is what we need in order for people having a hard time with math. Not to go into a long drawn out explanation about human thought processes, but I think thoses that have holistic thinking can benifit from this type of math. If fact help those with creative side of the brain to understand math instead of being so frustrated from it and giving up. Thank you for your thoughts.

 

[Jack21222]

That depends on your definition of math.

The trivial case is teaching a 3 year old that 1 apple plus another 1 apples means you have 2 apples. In this way, it's absolutely clear that you can learn to count and add with objects.

Another case is trying to prove some obscure result in linear algebra using objects. What sort of object would you use to represent an arbitrary n-dimensional vector space?

 

[stephenbosley]

Good question. This is the reason for the question I am asking. Yes you are correct in you example but the question still remain. Not easy to create when all we have ever use are symbols. I was thinking of a way that would create an object that would connect to another object that would create an answer. Yes, hard to conceive but not impossible. Of course rules must be in place in order to create this kind of math. I know you have heard that a picture speaks a thousand words. By creating this kind of math we can find answers more quickly. Take the english language it is also created up of symbols. What if we had a language that was created up of objects. Do you think it would be easier to understand the language? I know the human brain can understand information easier in the form of an object instead of a series of symbols. So why does man not explore this area and see if it really true or not?

 

[Office_Shredder]

What is your definition of object, and how does it exclude numbers from being objects? I think you really need to give an example here

 

[1MileCrash]

Objects work really well in set theory. If A is the set of sports cars and B is the set of red vehicles, the intersection of A and B contains only red sports cars.

That's why I'm starting to like dealing with sets. In equations, you can replace arbitrary variables with numbers; in set theory, you can replace arbitrary sets with a set of pudding, oreos, milk, rabbits, traffic lights, horse shoes.. and it of course all still works.

 

[micromass]

I know this is a little off base but I thought I would bring up a strange question. We have always been using math by using the symbols of numbers. Is there a math using objects instead and how would one create this type of math? I believe it is what we need in order for people having a hard time with math. Not to go into a long drawn out explanation about human thought processes, but I think thoses that have holistic thinking can benifit from this type of math. If fact help those with creative side of the brain to understand math instead of being so frustrated from it and giving up. Thank you for your thoughts.

Ever heard of geometry??

 

[zoobyshoe]

Ever heard of geometry??

Yeah, I think old fashioned Greek Geometry best represents what he's asking for. You can visually represent all kinds of things with Geometry. That, though, doesn't necessarily end up making them easier to grasp in many cases.

 

[Moonbear]

It's also a great way to teach fractions. Pizza slices or pie slices make it much easier for kids to understand the concept by seeing it, especially when you start multiplying and dividing fractions.

 

[stephenbosley]

I think you are being to understand what I am trying to say. I am not trying to be difficult in the explanation of what I am saying and yes Geometry has its place but it is still not quite right. Like a puzzle each piece has its place when put together in the correct fashion it comes out with the solution which is very simple. I hope that clears thing a little, and no I can’t find anything like this anywhere. I feel those who are of holistic thought are left out of the beauty of math do to frustration of the linear concept. By creating such math we can get everyone involved in solving some of our most difficult questions. I am trying to find someone that is open minded enough to work on such a concept in order for mankind to take a large leap forward. I know this is possible; it is being able to see math at a completely different perspective that is not normally seen at all.

 

[Hobin]

It's also a great way to teach fractions. Pizza slices or pie slices make it much easier for kids to understand the concept by seeing it, especially when you start multiplying and dividing fractions.

I suppose that works when multiplying or dividing by integers ("divide x/y pizza by z persons", or "multiply x/y pizza by z"), but doesn't that become somewhat problematic when you start multiplying and dividing fractions by other fractions? "Now, let's multiply one-third of a pizza by two-fifth of a pizza." (Of course, you could just tell them to multiply by two, then divide by five persons, but that's, well, cheating.)

I've always preferred to try to teach people the underlying concepts; to really get them to understand why these things work. I have yet to find someone who couldn't understand fractions when you go into more detail each time they don't 'get' a step. (In the past, I gave some extra lessons to people in high school who were particularly bad at math.)

Then again, I would make a terrible teacher at an elementary school. I kind of assume the people I teach can think rationally; I'm not an expert, but I suppose you can't expect that from an 8-year old.

 

[DaveC426913]

I suppose that works when multiplying or dividing by integers ("divide x/y pizza by z persons", or "multiply x/y pizza by z"), but doesn't that become somewhat problematic when you start multiplying and dividing fractions by other fractions?

Of course it can be done.

"Now, let's multiply one-third of a pizza by two-fifth of a pizza."

This is not a valid formula, since you'd end up with a unit of pizza².

But multiplying by fractions using real objects is certainly possible.

A 1/2 of a quarter of a pizza is easy enough to grasp. It's going to take 8 of those to make a whole pizza.

(Of course, you could just tell them to multiply by two, then divide by five persons, but that's, well, cheating.)

Why is that cheating? That's what you're supposed to do. (though it's not the best example)

I've always preferred to try to teach people the underlying concepts; to really get them to understand why these things work.

That;s what you're doing when you're using objects.

I have yet to find someone who couldn't understand fractions when you go into more detail each time they don't 'get' a step.

Clearly the OP is one. Some people need physical objects to grasp concepts. Abstract numerals floating in the air don't cut it.

Then again, I would make a terrible teacher at an elementary school. I kind of assume the people I teach can think rationally; I'm not an expert, but I suppose you can't expect that from an 8-year old.

People are thinking rationally! Why would you claim this isn't rational?

 

[Hobin]

Of course it can be done.

Sure, it just seems much more complicated to me than doing the same calculations without the pizza-metaphor.

Why is that cheating? That's what you're supposed to do. (though it's not the best example)

Because, if you know to do that, you might as well simply explain how those rules work without using pizzas, as pizzas will only add additional complexity from that moment on. Don't get me wrong, if the pizza example teaches people to do that, I'm fine with it, but I also know a few people who can't do such calculations without resorting to pizza metaphors. If you think of all fractions as pizzas, a fraction times a fraction makes very little sense. (It's really not that uncommon, I think. My parents, for example, simply don't think of the fact that 3/5 = 3*(1/5) when trying to calculate something. Of course, I might simply be from a weird family.)

Clearly the OP is one.

I didn't immediately assume that, actually. I was under the impression that the OP was trying to find a method to teach *other* people who were having difficulty with math. Although now that I reread it, both assumptions seem somewhat unfounded.

People are thinking rationally! Why would you claim this isn't rational?

Let me rephrase: when I gave extra lessons to people in high school, I sort of relied on the fact that they would be able to follow the logical processes used in math. Younger children still have to learn those processes, something I wouldn't like to teach. That's why I would make a bad teacher at an elementary school.

 

[DaveC426913]

Sure, it just seems much more complicated to me than doing the same calculations without the pizza-metaphor.

But the pizza is not a metaphor.

Pizza is more concrete than numeral-based math. The numerals are actually the metaphor, since they only represent things being divided and multiplied.

If you think of all fractions as pizzas, a fraction times a fraction makes very little sense.

That's right it doesn't. Because you've done it wrong. You don't think of all fractions as pizza slices.

Reexamine my pizza example.

If you have a quarter of a pizza, and you cut it in half, it makes sense that you now have a slice of pizza - of which it will take eight to make a whole pizza.

 

[Hobin]

But the pizza is not a metaphor.

Pizza is more concrete than numeral-based math. The numerals are actually the metaphor, since they only represent things being divided and multiplied.

In the case of teaching a math, it's a metaphor. You're not trying to teach people how to divide a pizza, but how to use math. Thus, the former is a metaphor to teach the latter. Whether dividing pizzas seem more real to us, or whether math in itself is ultimately a tool used to quantify physical objects is irrelevant.

That's right it doesn't. Because you've done it wrong. You don't think of all fractions as pizza slices.

Reexamine my pizza example.

I know all that, and that wasn't my point. The point I was trying to make is that some people I know who've only learned the pizza method, can *only* think of fractions in the sense of pizzas (although we used pies). In the sense of 'half of x/y pizza', these people don't really think of 'half' as a fraction, they simply see half as slicing what they have in two. However, when they really have to calculate a fraction of a fraction (more precisely, when none of these fractions is of the form 1/c), they don't get it, because they're stuck with two strange pieces of pizza.

Perhaps I should make it clear that I'm not saying we shouldn't use the pizza example. I think it's a great example. I simply think it's not very useful when trrying to teach people how to calculate more complicated combinations of fractions.

 

[MarcoD]

Is there a math using objects instead and how would one create this type of math?

I think the closest you can get to that is category theory which tries to abstract away from more concrete mathematical notions. It has been booming in math and CS for the last decades.

Personally, I loath it, since I am from CS and don't really think it 'adds value' to my profession. But I am rather alone in that.

(The reason I loath it is that I believe it actually distracts from the concrete problems at hand.)

 

[Ygggdrasil]

Steven Strogatz wrote a nice series of articles on math in the NY Times, many of which explain some fundamental (and even some fairly advanced) math topics using objects. I'll link to two below one on group theory (which lends itself very well to performing mathematics on objects because group theory is the language of symmetry) and a piece on thinking about math visually by performing arithmetic with objects:

http://opinionator.blogs.nytimes.com/2010/05/02/group-think/
http://opinionator.blogs.nytimes.com/2010/02/07/rock-groups/

 

[achyut.karve]

This is the first step in teaching maths. It is the abacus with nine beads in each line. Before learning computing one should learn counting. One could see fractions as numbers by just changing the base that is changing the number of beeds in each column.

 

[mathwonk]

take a look at euclid, archimedes, and galileo.