I know that many have had a hard time of learning math. Some of the horror stories that I have heard over the years would make your head spin. I do empathize a bit with math teachers however, because sometimes the words they use and procedures they understand to flow clearly from the concepts that they learned, perhaps many, many years ago, really "should just make sense to their students. In fact, none of us can truly take that next step, allowing higher concepts to flourish within us, until we truly understand the foundation upon which the mathematics are based.
I have heard many say that their misunderstanding, or difficulty with math, began with fractions. So, I was musing on the topic and came up with some simple steps to take into the realm of these quirky numbers, which in essence are operations within themselves. Perhaps starting even further back, let us compare whole numbers to fractions, so the idea of operations are more clear. Any number that can be expressed as a whole number seems pretty easy and straightforward. One apple, three apples five, etc. No mess, no smell, no persnickety knives to clean up after cleaving them into "equal parts". Division is the operation of fractions. One apple divided in two, or three, or five, etc.
To keep our examples easy to follow, let's use the time honored numbers that seem easiest, but we will look at them through different perspectives to see through whatever difficulties crept into how we understood/understand this quite simple operation, fractions. To keep this discussion simple, I'm going to attempt to write this whole post without numbers, but if it helps to understand, see them in your head as you read. For many, this will be unavoidable, yet not required. If all goes well, this will also address the many people who said that they first encountered trouble with math because of story problems. I prefer to not see them as "problems", just stories, so if any of my readers have anxiety, at this point, please, let it go. Increase your level of calm and go forth.
Let us say that four friends decide to buy something they cannot afford with each one purchasing it alone. Each only needs said item intermittently, so they can easily share. This could be a tool chest, sewing machine, car, boat or camper, a grocery store, bicycle or sacred land upon which to pray. It matters not what it is, the math works out the same. However, there is a problem. Two of the friends would use the item more, about twice as much and two would use it less, about half as much. One easy way to get to a fair "share" is divide the cost three ways and have two pay for a smaller part of the cost and two pay the majority of the cost. A simple solution is that one third of the price would get paid by each of the two investors that will use the item more and the other two would share the cost of the remaining third between them. One sixth of the cost each.
Co-ops usually have some sort of system to cover administrative costs, tracking future needs and sharing potential profits from future sales or "shares" in the group. however, for clarity, I'll keep things simple again, let us go into possible fractions that would occur under cooperative principles. Say there were a few more folks, say six who wanted the The item most of the time (two thirds) and four who would like to buy into the last third. All of a sudden there are ten shares to be paid, but because they are unequal shares, how would you satisfy the participants who still want to remain friends. One way is to continue speaking the language of "equality" a bit differently. The two thirds would still be shared by those who wanted to use the resource more and the remaining third would get split four ways. In certain cases there will be overhead or costs associated with ownership and to fund these things, several methods have been used. Perhaps the easiest would be to add ten percent service fee to each share, for those whose use characteristic is low and twenty percent for those whose use characteristic is high. I am stepping beyond the language of hard math to show solutions that speak a slightly different language. Since we are in story problem mode, trust that this will all make some sort of sense in the end.
Continuing with the same solution, say we pick a number like sixteen, because it gives us more "shares" to work with and in theory, smaller bits become easier to divide up. Also, you don't have to have a service fee, because extra shares cover the additional costs.You make the number based, not on the number of people involved, but the number of shares invested in. Ten people divided up sixteen ways may be just the right number because there is more variety in the use characteristic when there are more users. Six people paying the two thirds portion of the price would still pay about twice as much as the four who chip in to pay for the last third, but there is an overage because it is a way to avoid having fees for getting into the co-op relationship. Everyone pays slightly more to cover these associated costs.
Some groups and co-ops are formalized and have written agreements that address what happens if costs exceed investments, or vice versa, if there were to be too many funds for the actual costs, usually, this amounts to an end of the year rebate. In some cases it can be rolled into a general fund for future group investments. If there are outstanding costs, there could be fees levied on co-op members.
This is where we jump off into the math of our current system. It bears little resemblance to the earlier examples. Up until now, I have tried to keep things pretty easy to follow. Let us imagine instead fractions based on one hundred percent. Like with pennies to dollars, but we don't want to have fees or membership costs. We could work backward and determine total costs. Say they exceed the cost of the actual shared items by ten percent. This overage, in the cooperative schemes detailed above could be used for storage, maintenance, upkeep etc. Most of the time, everyone gets a better deal than having to foot the bill all by themselves. The kings of wall Street instead just absorb the overage.
I have heard many say that their misunderstanding, or difficulty with math, began with fractions. So, I was musing on the topic and came up with some simple steps to take into the realm of these quirky numbers, which in essence are operations within themselves. Perhaps starting even further back, let us compare whole numbers to fractions, so the idea of operations are more clear. Any number that can be expressed as a whole number seems pretty easy and straightforward. One apple, three apples five, etc. No mess, no smell, no persnickety knives to clean up after cleaving them into "equal parts". Division is the operation of fractions. One apple divided in two, or three, or five, etc.
 |
| Where the sidewalk ends... |
To keep our examples easy to follow, let's use the time honored numbers that seem easiest, but we will look at them through different perspectives to see through whatever difficulties crept into how we understood/understand this quite simple operation, fractions. To keep this discussion simple, I'm going to attempt to write this whole post without numbers, but if it helps to understand, see them in your head as you read. For many, this will be unavoidable, yet not required. If all goes well, this will also address the many people who said that they first encountered trouble with math because of story problems. I prefer to not see them as "problems", just stories, so if any of my readers have anxiety, at this point, please, let it go. Increase your level of calm and go forth.
Let us say that four friends decide to buy something they cannot afford with each one purchasing it alone. Each only needs said item intermittently, so they can easily share. This could be a tool chest, sewing machine, car, boat or camper, a grocery store, bicycle or sacred land upon which to pray. It matters not what it is, the math works out the same. However, there is a problem. Two of the friends would use the item more, about twice as much and two would use it less, about half as much. One easy way to get to a fair "share" is divide the cost three ways and have two pay for a smaller part of the cost and two pay the majority of the cost. A simple solution is that one third of the price would get paid by each of the two investors that will use the item more and the other two would share the cost of the remaining third between them. One sixth of the cost each.
Co-ops usually have some sort of system to cover administrative costs, tracking future needs and sharing potential profits from future sales or "shares" in the group. however, for clarity, I'll keep things simple again, let us go into possible fractions that would occur under cooperative principles. Say there were a few more folks, say six who wanted the The item most of the time (two thirds) and four who would like to buy into the last third. All of a sudden there are ten shares to be paid, but because they are unequal shares, how would you satisfy the participants who still want to remain friends. One way is to continue speaking the language of "equality" a bit differently. The two thirds would still be shared by those who wanted to use the resource more and the remaining third would get split four ways. In certain cases there will be overhead or costs associated with ownership and to fund these things, several methods have been used. Perhaps the easiest would be to add ten percent service fee to each share, for those whose use characteristic is low and twenty percent for those whose use characteristic is high. I am stepping beyond the language of hard math to show solutions that speak a slightly different language. Since we are in story problem mode, trust that this will all make some sort of sense in the end.
Continuing with the same solution, say we pick a number like sixteen, because it gives us more "shares" to work with and in theory, smaller bits become easier to divide up. Also, you don't have to have a service fee, because extra shares cover the additional costs.You make the number based, not on the number of people involved, but the number of shares invested in. Ten people divided up sixteen ways may be just the right number because there is more variety in the use characteristic when there are more users. Six people paying the two thirds portion of the price would still pay about twice as much as the four who chip in to pay for the last third, but there is an overage because it is a way to avoid having fees for getting into the co-op relationship. Everyone pays slightly more to cover these associated costs.
Some groups and co-ops are formalized and have written agreements that address what happens if costs exceed investments, or vice versa, if there were to be too many funds for the actual costs, usually, this amounts to an end of the year rebate. In some cases it can be rolled into a general fund for future group investments. If there are outstanding costs, there could be fees levied on co-op members.
This is where we jump off into the math of our current system. It bears little resemblance to the earlier examples. Up until now, I have tried to keep things pretty easy to follow. Let us imagine instead fractions based on one hundred percent. Like with pennies to dollars, but we don't want to have fees or membership costs. We could work backward and determine total costs. Say they exceed the cost of the actual shared items by ten percent. This overage, in the cooperative schemes detailed above could be used for storage, maintenance, upkeep etc. Most of the time, everyone gets a better deal than having to foot the bill all by themselves. The kings of wall Street instead just absorb the overage.
No comments:
Post a Comment